ALBERT EINSTEIN REFERENCE ARCHIVE
RELATIVITY: THE SPECIAL AND GENERAL THEORY
BY ALBERT EINSTEIN
Written: 1916 (this revised edition: 1924)
Source: Relativity: The Special and General Theory (1920)
Publisher: Methuen & Co Ltd
First Published: December, 1916
Translated: Robert W. Lawson (Authorised translation)
Transcription/Markup: Brian Basgen <brian@marxists.org>
Transcription to text: Gregory B. Newby <gbnewby@petascale.org>
Thanks to: Einstein Reference Archive (marxists.org)
The Einstein Reference Archive is online at:
http://www.marxists.org/reference/archive/einstein/index.htm
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CONTENTS
Preface
Part I: The Special Theory of Relativity
01. Physical Meaning of Geometrical Propositions
02. The System of Co-ordinates
03. Space and Time in Classical Mechanics
04. The Galileian System of Co-ordinates
05. The Principle of Relativity (in the Restricted Sense)
06. The Theorem of the Addition of Velocities employed in
Classical Mechanics
07. The Apparent Incompatability of the Law of Propagation of
Light with the Principle of Relativity
08. On the Idea of Time in Physics
09. The Relativity of Simultaneity
10. On the Relativity of the Conception of Distance
11. The Lorentz Transformation
12. The Behaviour of Measuring-Rods and Clocks in Motion
13. Theorem of the Addition of Velocities. The Experiment of Fizeau
14. The Hueristic Value of the Theory of Relativity
15. General Results of the Theory
16. Expereince and the Special Theory of Relativity
17. Minkowski's Four-dimensial Space
Part II: The General Theory of Relativity
18. Special and General Principle of Relativity
19. The Gravitational Field
20. The Equality of Inertial and Gravitational Mass as an Argument
for the General Postulate of Relativity
21. In What Respects are the Foundations of Classical Mechanics
and of the Special Theory of Relativity Unsatisfactory?
22. A Few Inferences from the General Principle of Relativity
23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of
Reference
24. Euclidean and non-Euclidean Continuum
25. Gaussian Co-ordinates
26. The Space-Time Continuum of the Speical Theory of Relativity
Considered as a Euclidean Continuum
27. The Space-Time Continuum of the General Theory of Relativity
is Not a Eculidean Continuum
28. Exact Formulation of the General Principle of Relativity
29. The Solution of the Problem of Gravitation on the Basis of the
General Principle of Relativity
Part III: Considerations on the Universe as a Whole
30. Cosmological Difficulties of Netwon's Theory
31. The Possibility of a "Finite" and yet "Unbounded" Universe
32. The Structure of Space According to the General Theory of
Relativity
Appendices:
01. Simple Derivation of the Lorentz Transformation (sup. ch. 11)
02. Minkowski's Four-Dimensional Space ("World") (sup. ch 17)
03. The Experimental Confirmation of the General Theory of Relativity
04. The Structure of Space According to the General Theory of
Relativity (sup. ch 32)
05. Relativity and the Problem of Space
Note: The fifth Appendix was added by Einstein at the time of the
fifteenth re-printing of this book; and as a result is still under
copyright restrictions so cannot be added without the permission of
the publisher.
PREFACE
(December, 1916)
The present book is intended, as far as possible, to give an exact
insight into the theory of Relativity to those readers who, from a
general scientific and philosophical point of view, are interested in
the theory, but who are not conversant with the mathematical apparatus
of theoretical physics. The work presumes a standard of education
corresponding to that of a university matriculation examination, and,
despite the shortness of the book, a fair amount of patience and force
of will on the part of the reader. The author has spared himself no
pains in his endeavour to present the main ideas in the simplest and
most intelligible form, and on the whole, in the sequence and
connection in which they actually originated. In the interest of
clearness, it appeared to me inevitable that I should repeat myself
frequently, without paying the slightest attention to the elegance of
the presentation. I adhered scrupulously to the precept of that
brilliant theoretical physicist L. Boltzmann, according to whom
matters of elegance ought to be left to the tailor and to the cobbler.
I make no pretence of having withheld from the reader difficulties
which are inherent to the subject. On the other hand, I have purposely
treated the empirical physical foundations of the theory in a
"step-motherly" fashion, so that readers unfamiliar with physics may
not feel like the wanderer who was unable to see the forest for the
trees. May the book bring some one a few happy hours of suggestive
thought!
December, 1916
A. EINSTEIN
PART I
THE SPECIAL THEORY OF RELATIVITY
PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS
In your schooldays most of you who read this book made acquaintance
with the noble building of Euclid's geometry, and you remember --
perhaps with more respect than love -- the magnificent structure, on
the lofty staircase of which you were chased about for uncounted hours
by conscientious teachers. By reason of our past experience, you would
certainly regard everyone with disdain who should pronounce even the
most out-of-the-way proposition of this science to be untrue. But
perhaps this feeling of proud certainty would leave you immediately if
some one were to ask you: "What, then, do you mean by the assertion
that these propositions are true?" Let us proceed to give this
question a little consideration.
Geometry sets out form certain conceptions such as "plane," "point,"
and "straight line," with which we are able to associate more or less
definite ideas, and from certain simple propositions (axioms) which,
in virtue of these ideas, we are inclined to accept as "true." Then,
on the basis of a logical process, the justification of which we feel
ourselves compelled to admit, all remaining propositions are shown to
follow from those axioms, i.e. they are proven. A proposition is then
correct ("true") when it has been derived in the recognised manner
from the axioms. The question of "truth" of the individual geometrical
propositions is thus reduced to one of the "truth" of the axioms. Now
it has long been known that the last question is not only unanswerable
by the methods of geometry, but that it is in itself entirely without
meaning. We cannot ask whether it is true that only one straight line
goes through two points. We can only say that Euclidean geometry deals
with things called "straight lines," to each of which is ascribed the
property of being uniquely determined by two points situated on it.
The concept "true" does not tally with the assertions of pure
geometry, because by the word "true" we are eventually in the habit of
designating always the correspondence with a "real" object; geometry,
however, is not concerned with the relation of the ideas involved in
it to objects of experience, but only with the logical connection of
these ideas among themselves.
It is not difficult to understand why, in spite of this, we feel
constrained to call the propositions of geometry "true." Geometrical
ideas correspond to more or less exact objects in nature, and these
last are undoubtedly the exclusive cause of the genesis of those
ideas. Geometry ought to refrain from such a course, in order to give
to its structure the largest possible logical unity. The practice, for
example, of seeing in a "distance" two marked positions on a
practically rigid body is something which is lodged deeply in our
habit of thought. We are accustomed further to regard three points as
being situated on a straight line, if their apparent positions can be
made to coincide for observation with one eye, under suitable choice
of our place of observation.
If, in pursuance of our habit of thought, we now supplement the
propositions of Euclidean geometry by the single proposition that two
points on a practically rigid body always correspond to the same
distance (line-interval), independently of any changes in position to
which we may subject the body, the propositions of Euclidean geometry
then resolve themselves into propositions on the possible relative
position of practically rigid bodies.* Geometry which has been
supplemented in this way is then to be treated as a branch of physics.
We can now legitimately ask as to the "truth" of geometrical
propositions interpreted in this way, since we are justified in asking
whether these propositions are satisfied for those real things we have
associated with the geometrical ideas. In less exact terms we can
express this by saying that by the "truth" of a geometrical
proposition in this sense we understand its validity for a
construction with rule and compasses.
Of course the conviction of the "truth" of geometrical propositions in
this sense is founded exclusively on rather incomplete experience. For
the present we shall assume the "truth" of the geometrical
propositions, then at a later stage (in the general theory of
relativity) we shall see that this "truth" is limited, and we shall
consider the extent of its limitation.
Notes
*) It follows that a natural object is associated also with a
straight line. Three points A, B and C on a rigid body thus lie in a
straight line when the points A and C being given, B is chosen such
that the sum of the distances AB and BC is as short as possible. This
incomplete suggestion will suffice for the present purpose.
THE SYSTEM OF CO-ORDINATES
On the basis of the physical interpretation of distance which has been
indicated, we are also in a position to establish the distance between
two points on a rigid body by means of measurements. For this purpose
we require a " distance " (rod S) which is to be used once and for
all, and which we employ as a standard measure. If, now, A and B are
two points on a rigid body, we can construct the line joining them
according to the rules of geometry ; then, starting from A, we can
mark off the distance S time after time until we reach B. The number
of these operations required is the numerical measure of the distance
AB. This is the basis of all measurement of length. *
Every description of the scene of an event or of the position of an
object in space is based on the specification of the point on a rigid
body (body of reference) with which that event or object coincides.
This applies not only to scientific description, but also to everyday
life. If I analyse the place specification " Times Square, New York,"
**A I arrive at the following result. The earth is the rigid body
to which the specification of place refers; " Times Square, New York,"
is a well-defined point, to which a name has been assigned, and with
which the event coincides in space.**B
This primitive method of place specification deals only with places on
the surface of rigid bodies, and is dependent on the existence of
points on this surface which are distinguishable from each other. But
we can free ourselves from both of these limitations without altering
the nature of our specification of position. If, for instance, a cloud
is hovering over Times Square, then we can determine its position
relative to the surface of the earth by erecting a pole
perpendicularly on the Square, so that it reaches the cloud. The
length of the pole measured with the standard measuring-rod, combined
with the specification of the position of the foot of the pole,
supplies us with a complete place specification. On the basis of this
illustration, we are able to see the manner in which a refinement of
the conception of position has been developed.
(a) We imagine the rigid body, to which the place specification is
referred, supplemented in such a manner that the object whose position
we require is reached by. the completed rigid body.
(b) In locating the position of the object, we make use of a number
(here the length of the pole measured with the measuring-rod) instead
of designated points of reference.
(c) We speak of the height of the cloud even when the pole which
reaches the cloud has not been erected. By means of optical
observations of the cloud from different positions on the ground, and
taking into account the properties of the propagation of light, we
determine the length of the pole we should have required in order to
reach the cloud.
From this consideration we see that it will be advantageous if, in the
description of position, it should be possible by means of numerical
measures to make ourselves independent of the existence of marked
positions (possessing names) on the rigid body of reference. In the
physics of measurement this is attained by the application of the
Cartesian system of co-ordinates.
This consists of three plane surfaces perpendicular to each other and
rigidly attached to a rigid body. Referred to a system of
co-ordinates, the scene of any event will be determined (for the main
part) by the specification of the lengths of the three perpendiculars
or co-ordinates (x, y, z) which can be dropped from the scene of the
event to those three plane surfaces. The lengths of these three
perpendiculars can be determined by a series of manipulations with
rigid measuring-rods performed according to the rules and methods laid
down by Euclidean geometry.
In practice, the rigid surfaces which constitute the system of
co-ordinates are generally not available ; furthermore, the magnitudes
of the co-ordinates are not actually determined by constructions with
rigid rods, but by indirect means. If the results of physics and
astronomy are to maintain their clearness, the physical meaning of
specifications of position must always be sought in accordance with
the above considerations. ***
We thus obtain the following result: Every description of events in
space involves the use of a rigid body to which such events have to be
referred. The resulting relationship takes for granted that the laws
of Euclidean geometry hold for "distances;" the "distance" being
represented physically by means of the convention of two marks on a
rigid body.
Notes
* Here we have assumed that there is nothing left over i.e. that
the measurement gives a whole number. This difficulty is got over by
the use of divided measuring-rods, the introduction of which does not
demand any fundamentally new method.
**A Einstein used "Potsdamer Platz, Berlin" in the original text.
In the authorised translation this was supplemented with "Tranfalgar
Square, London". We have changed this to "Times Square, New York", as
this is the most well known/identifiable location to English speakers
in the present day. [Note by the janitor.]
**B It is not necessary here to investigate further the significance
of the expression "coincidence in space." This conception is
sufficiently obvious to ensure that differences of opinion are
scarcely likely to arise as to its applicability in practice.
*** A refinement and modification of these views does not become
necessary until we come to deal with the general theory of relativity,
treated in the second part of this book.
SPACE AND TIME IN CLASSICAL MECHANICS
The purpose of mechanics is to describe how bodies change their
position in space with "time." I should load my conscience with grave
sins against the sacred spirit of lucidity were I to formulate the
aims of mechanics in this way, without serious reflection and detailed
explanations. Let us proceed to disclose these sins.
It is not clear what is to be understood here by "position" and
"space." I stand at the window of a railway carriage which is
travelling uniformly, and drop a stone on the embankment, without
throwing it. Then, disregarding the influence of the air resistance, I
see the stone descend in a straight line. A pedestrian who observes
the misdeed from the footpath notices that the stone falls to earth in
a parabolic curve. I now ask: Do the "positions" traversed by the
stone lie "in reality" on a straight line or on a parabola? Moreover,
what is meant here by motion "in space" ? From the considerations of
the previous section the answer is self-evident. In the first place we
entirely shun the vague word "space," of which, we must honestly
acknowledge, we cannot form the slightest conception, and we replace
it by "motion relative to a practically rigid body of reference." The
positions relative to the body of reference (railway carriage or
embankment) have already been defined in detail in the preceding
section. If instead of " body of reference " we insert " system of
co-ordinates," which is a useful idea for mathematical description, we
are in a position to say : The stone traverses a straight line
relative to a system of co-ordinates rigidly attached to the carriage,
but relative to a system of co-ordinates rigidly attached to the
ground (embankment) it describes a parabola. With the aid of this
example it is clearly seen that there is no such thing as an
independently existing trajectory (lit. "path-curve"*), but only
a trajectory relative to a particular body of reference.
In order to have a complete description of the motion, we must specify
how the body alters its position with time ; i.e. for every point on
the trajectory it must be stated at what time the body is situated
there. These data must be supplemented by such a definition of time
that, in virtue of this definition, these time-values can be regarded
essentially as magnitudes (results of measurements) capable of
observation. If we take our stand on the ground of classical
mechanics, we can satisfy this requirement for our illustration in the
following manner. We imagine two clocks of identical construction ;
the man at the railway-carriage window is holding one of them, and the
man on the footpath the other. Each of the observers determines the
position on his own reference-body occupied by the stone at each tick
of the clock he is holding in his hand. In this connection we have not
taken account of the inaccuracy involved by the finiteness of the
velocity of propagation of light. With this and with a second
difficulty prevailing here we shall have to deal in detail later.
Notes
*) That is, a curve along which the body moves.
THE GALILEIAN SYSTEM OF CO-ORDINATES
As is well known, the fundamental law of the mechanics of
Galilei-Newton, which is known as the law of inertia, can be stated
thus: A body removed sufficiently far from other bodies continues in a
state of rest or of uniform motion in a straight line. This law not
only says something about the motion of the bodies, but it also
indicates the reference-bodies or systems of coordinates, permissible
in mechanics, which can be used in mechanical description. The visible
fixed stars are bodies for which the law of inertia certainly holds to
a high degree of approximation. Now if we use a system of co-ordinates
which is rigidly attached to the earth, then, relative to this system,
every fixed star describes a circle of immense radius in the course of
an astronomical day, a result which is opposed to the statement of the
law of inertia. So that if we adhere to this law we must refer these
motions only to systems of coordinates relative to which the fixed
stars do not move in a circle. A system of co-ordinates of which the
state of motion is such that the law of inertia holds relative to it
is called a " Galileian system of co-ordinates." The laws of the
mechanics of Galflei-Newton can be regarded as valid only for a
Galileian system of co-ordinates.
THE PRINCIPLE OF RELATIVITY
(IN THE RESTRICTED SENSE)
In order to attain the greatest possible clearness, let us return to
our example of the railway carriage supposed to be travelling
uniformly. We call its motion a uniform translation ("uniform" because
it is of constant velocity and direction, " translation " because
although the carriage changes its position relative to the embankment
yet it does not rotate in so doing). Let us imagine a raven flying
through the air in such a manner that its motion, as observed from the
embankment, is uniform and in a straight line. If we were to observe
the flying raven from the moving railway carriage. we should find that
the motion of the raven would be one of different velocity and
direction, but that it would still be uniform and in a straight line.
Expressed in an abstract manner we may say : If a mass m is moving
uniformly in a straight line with respect to a co-ordinate system K,
then it will also be moving uniformly and in a straight line relative
to a second co-ordinate system K1 provided that the latter is
executing a uniform translatory motion with respect to K. In
accordance with the discussion contained in the preceding section, it
follows that:
If K is a Galileian co-ordinate system. then every other co-ordinate
system K' is a Galileian one, when, in relation to K, it is in a
condition of uniform motion of translation. Relative to K1 the
mechanical laws of Galilei-Newton hold good exactly as they do with
respect to K.
We advance a step farther in our generalisation when we express the
tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate
system devoid of rotation, then natural phenomena run their course
with respect to K1 according to exactly the same general laws as with
respect to K. This statement is called the principle of relativity (in
the restricted sense).
As long as one was convinced that all natural phenomena were capable
of representation with the help of classical mechanics, there was no
need to doubt the validity of this principle of relativity. But in
view of the more recent development of electrodynamics and optics it
became more and more evident that classical mechanics affords an
insufficient foundation for the physical description of all natural
phenomena. At this juncture the question of the validity of the
principle of relativity became ripe for discussion, and it did not
appear impossible that the answer to this question might be in the
negative.
Nevertheless, there are two general facts which at the outset speak
very much in favour of the validity of the principle of relativity.
Even though classical mechanics does not supply us with a sufficiently
broad basis for the theoretical presentation of all physical
phenomena, still we must grant it a considerable measure of " truth,"
since it supplies us with the actual motions of the heavenly bodies
with a delicacy of detail little short of wonderful. The principle of
relativity must therefore apply with great accuracy in the domain of
mechanics. But that a principle of such broad generality should hold
with such exactness in one domain of phenomena, and yet should be
invalid for another, is a priori not very probable.
We now proceed to the second argument, to which, moreover, we shall
return later. If the principle of relativity (in the restricted sense)
does not hold, then the Galileian co-ordinate systems K, K1, K2, etc.,
which are moving uniformly relative to each other, will not be
equivalent for the description of natural phenomena. In this case we
should be constrained to believe that natural laws are capable of
being formulated in a particularly simple manner, and of course only
on condition that, from amongst all possible Galileian co-ordinate
systems, we should have chosen one (K[0]) of a particular state of
motion as our body of reference. We should then be justified (because
of its merits for the description of natural phenomena) in calling
this system " absolutely at rest," and all other Galileian systems K "
in motion." If, for instance, our embankment were the system K[0] then
our railway carriage would be a system K, relative to which less
simple laws would hold than with respect to K[0]. This diminished
simplicity would be due to the fact that the carriage K would be in
motion (i.e."really")with respect to K[0]. In the general laws of
nature which have been formulated with reference to K, the magnitude
and direction of the velocity of the carriage would necessarily play a
part. We should expect, for instance, that the note emitted by an
organpipe placed with its axis parallel to the direction of travel
would be different from that emitted if the axis of the pipe were
placed perpendicular to this direction.
Now in virtue of its motion in an orbit round the sun, our earth is
comparable with a railway carriage travelling with a velocity of about
30 kilometres per second. If the principle of relativity were not
valid we should therefore expect that the direction of motion of the
earth at any moment would enter into the laws of nature, and also that
physical systems in their behaviour would be dependent on the
orientation in space with respect to the earth. For owing to the
alteration in direction of the velocity of revolution of the earth in
the course of a year, the earth cannot be at rest relative to the
hypothetical system K[0] throughout the whole year. However, the most
careful observations have never revealed such anisotropic properties
in terrestrial physical space, i.e. a physical non-equivalence of
different directions. This is very powerful argument in favour of the
principle of relativity.
THE THEOREM OF THE
ADDITION OF VELOCITIES
EMPLOYED IN CLASSICAL MECHANICS
Let us suppose our old friend the railway carriage to be travelling
along the rails with a constant velocity v, and that a man traverses
the length of the carriage in the direction of travel with a velocity
w. How quickly or, in other words, with what velocity W does the man
advance relative to the embankment during the process ? The only
possible answer seems to result from the following consideration: If
the man were to stand still for a second, he would advance relative to
the embankment through a distance v equal numerically to the velocity
of the carriage. As a consequence of his walking, however, he
traverses an additional distance w relative to the carriage, and hence
also relative to the embankment, in this second, the distance w being
numerically equal to the velocity with which he is walking. Thus in
total be covers the distance W=v+w relative to the embankment in the
second considered. We shall see later that this result, which
expresses the theorem of the addition of velocities employed in
classical mechanics, cannot be maintained ; in other words, the law
that we have just written down does not hold in reality. For the time
being, however, we shall assume its correctness.
THE APPARENT INCOMPATIBILITY OF THE
LAW OF PROPAGATION OF LIGHT WITH THE
PRINCIPLE OF RELATIVITY
There is hardly a simpler law in physics than that according to which
light is propagated in empty space. Every child at school knows, or
believes he knows, that this propagation takes place in straight lines
with a velocity c= 300,000 km./sec. At all events we know with great
exactness that this velocity is the same for all colours, because if
this were not the case, the minimum of emission would not be observed
simultaneously for different colours during the eclipse of a fixed
star by its dark neighbour. By means of similar considerations based
on observa- tions of double stars, the Dutch astronomer De Sitter was
also able to show that the velocity of propagation of light cannot
depend on the velocity of motion of the body emitting the light. The
assumption that this velocity of propagation is dependent on the
direction "in space" is in itself improbable.
In short, let us assume that the simple law of the constancy of the
velocity of light c (in vacuum) is justifiably believed by the child
at school. Who would imagine that this simple law has plunged the
conscientiously thoughtful physicist into the greatest intellectual
difficulties? Let us consider how these difficulties arise.
Of course we must refer the process of the propagation of light (and
indeed every other process) to a rigid reference-body (co-ordinate
system). As such a system let us again choose our embankment. We shall
imagine the air above it to have been removed. If a ray of light be
sent along the embankment, we see from the above that the tip of the
ray will be transmitted with the velocity c relative to the
embankment. Now let us suppose that our railway carriage is again
travelling along the railway lines with the velocity v, and that its
direction is the same as that of the ray of light, but its velocity of
course much less. Let us inquire about the velocity of propagation of
the ray of light relative to the carriage. It is obvious that we can
here apply the consideration of the previous section, since the ray of
light plays the part of the man walking along relatively to the
carriage. The velocity w of the man relative to the embankment is here
replaced by the velocity of light relative to the embankment. w is the
required velocity of light with respect to the carriage, and we have
w = c-v.
The velocity of propagation ot a ray of light relative to the carriage
thus comes cut smaller than c.
But this result comes into conflict with the principle of relativity
set forth in Section V. For, like every other general law of
nature, the law of the transmission of light in vacuo [in vacuum]
must, according to the principle of relativity, be the same for the
railway carriage as reference-body as when the rails are the body of
reference. But, from our above consideration, this would appear to be
impossible. If every ray of light is propagated relative to the
embankment with the velocity c, then for this reason it would appear
that another law of propagation of light must necessarily hold with
respect to the carriage -- a result contradictory to the principle of
relativity.
In view of this dilemma there appears to be nothing else for it than
to abandon either the principle of relativity or the simple law of the
propagation of light in vacuo. Those of you who have carefully
followed the preceding discussion are almost sure to expect that we
should retain the principle of relativity, which appeals so
convincingly to the intellect because it is so natural and simple. The
law of the propagation of light in vacuo would then have to be
replaced by a more complicated law conformable to the principle of
relativity. The development of theoretical physics shows, however,
that we cannot pursue this course. The epoch-making theoretical
investigations of H. A. Lorentz on the electrodynamical and optical
phenomena connected with moving bodies show that experience in this
domain leads conclusively to a theory of electromagnetic phenomena, of
which the law of the constancy of the velocity of light in vacuo is a
necessary consequence. Prominent theoretical physicists were theref
ore more inclined to reject the principle of relativity, in spite of
the fact that no empirical data had been found which were
contradictory to this principle.
At this juncture the theory of relativity entered the arena. As a
result of an analysis of the physical conceptions of time and space,
it became evident that in realily there is not the least
incompatibilitiy between the principle of relativity and the law of
propagation of light, and that by systematically holding fast to both
these laws a logically rigid theory could be arrived at. This theory
has been called the special theory of relativity to distinguish it
from the extended theory, with which we shall deal later. In the
following pages we shall present the fundamental ideas of the special
theory of relativity.
ON THE IDEA OF TIME IN PHYSICS
Lightning has struck the rails on our railway embankment at two places
A and B far distant from each other. I make the additional assertion
that these two lightning flashes occurred simultaneously. If I ask you
whether there is sense in this statement, you will answer my question
with a decided "Yes." But if I now approach you with the request to
explain to me the sense of the statement more precisely, you find
after some consideration that the answer to this question is not so
easy as it appears at first sight.
After some time perhaps the following answer would occur to you: "The
significance of the statement is clear in itself and needs no further
explanation; of course it would require some consideration if I were
to be commissioned to determine by observations whether in the actual
case the two events took place simultaneously or not." I cannot be
satisfied with this answer for the following reason. Supposing that as
a result of ingenious considerations an able meteorologist were to
discover that the lightning must always strike the places A and B
simultaneously, then we should be faced with the task of testing
whether or not this theoretical result is in accordance with the
reality. We encounter the same difficulty with all physical statements
in which the conception " simultaneous " plays a part. The concept
does not exist for the physicist until he has the possibility of
discovering whether or not it is fulfilled in an actual case. We thus
require a definition of simultaneity such that this definition
supplies us with the method by means of which, in the present case, he
can decide by experiment whether or not both the lightning strokes
occurred simultaneously. As long as this requirement is not satisfied,
I allow myself to be deceived as a physicist (and of course the same
applies if I am not a physicist), when I imagine that I am able to
attach a meaning to the statement of simultaneity. (I would ask the
reader not to proceed farther until he is fully convinced on this
point.)
After thinking the matter over for some time you then offer the
following suggestion with which to test simultaneity. By measuring
along the rails, the connecting line AB should be measured up and an
observer placed at the mid-point M of the distance AB. This observer
should be supplied with an arrangement (e.g. two mirrors inclined at
90^0) which allows him visually to observe both places A and B at the
same time. If the observer perceives the two flashes of lightning at
the same time, then they are simultaneous.
I am very pleased with this suggestion, but for all that I cannot
regard the matter as quite settled, because I feel constrained to
raise the following objection:
"Your definition would certainly be right, if only I knew that the
light by means of which the observer at M perceives the lightning
flashes travels along the length A arrow M with the same velocity as
along the length B arrow M. But an examination of this supposition
would only be possible if we already had at our disposal the means of
measuring time. It would thus appear as though we were moving here in
a logical circle."
After further consideration you cast a somewhat disdainful glance at
me -- and rightly so -- and you declare:
"I maintain my previous definition nevertheless, because in reality it
assumes absolutely nothing about light. There is only one demand to be
made of the definition of simultaneity, namely, that in every real
case it must supply us with an empirical decision as to whether or not
the conception that has to be defined is fulfilled. That my definition
satisfies this demand is indisputable. That light requires the same
time to traverse the path A arrow M as for the path B arrow M is in
reality neither a supposition nor a hypothesis about the physical
nature of light, but a stipulation which I can make of my own freewill
in order to arrive at a definition of simultaneity."
It is clear that this definition can be used to give an exact meaning
not only to two events, but to as many events as we care to choose,
and independently of the positions of the scenes of the events with
respect to the body of reference * (here the railway embankment).
We are thus led also to a definition of " time " in physics. For this
purpose we suppose that clocks of identical construction are placed at
the points A, B and C of the railway line (co-ordinate system) and
that they are set in such a manner that the positions of their
pointers are simultaneously (in the above sense) the same. Under these
conditions we understand by the " time " of an event the reading
(position of the hands) of that one of these clocks which is in the
immediate vicinity (in space) of the event. In this manner a
time-value is associated with every event which is essentially capable
of observation.
This stipulation contains a further physical hypothesis, the validity
of which will hardly be doubted without empirical evidence to the
contrary. It has been assumed that all these clocks go at the same
rate if they are of identical construction. Stated more exactly: When
two clocks arranged at rest in different places of a reference-body
are set in such a manner that a particular position of the pointers of
the one clock is simultaneous (in the above sense) with the same
position, of the pointers of the other clock, then identical "
settings " are always simultaneous (in the sense of the above
definition).
Notes
*) We suppose further, that, when three events A, B and C occur in
different places in such a manner that A is simultaneous with B and B
is simultaneous with C (simultaneous in the sense of the above
definition), then the criterion for the simultaneity of the pair of
events A, C is also satisfied. This assumption is a physical
hypothesis about the the of propagation of light: it must certainly be
fulfilled if we are to maintain the law of the constancy of the
velocity of light in vacuo.
THE RELATIVITY OF SIMULATNEITY
Up to now our considerations have been referred to a particular body
of reference, which we have styled a " railway embankment." We suppose
a very long train travelling along the rails with the constant
velocity v and in the direction indicated in Fig 1. People travelling
in this train will with a vantage view the train as a rigid
reference-body (co-ordinate system); they regard all events in
Fig. 01: file fig01.gif
reference to the train. Then every event which takes place along the
line also takes place at a particular point of the train. Also the
definition of simultaneity can be given relative to the train in
exactly the same way as with respect to the embankment. As a natural
consequence, however, the following question arises :
Are two events (e.g. the two strokes of lightning A and B) which are
simultaneous with reference to the railway embankment also
simultaneous relatively to the train? We shall show directly that the
answer must be in the negative.
When we say that the lightning strokes A and B are simultaneous with
respect to be embankment, we mean: the rays of light emitted at the
places A and B, where the lightning occurs, meet each other at the
mid-point M of the length A arrow B of the embankment. But the events
A and B also correspond to positions A and B on the train. Let M1 be
the mid-point of the distance A arrow B on the travelling train. Just
when the flashes (as judged from the embankment) of lightning occur,
this point M1 naturally coincides with the point M but it moves
towards the right in the diagram with the velocity v of the train. If
an observer sitting in the position M1 in the train did not possess
this velocity, then he would remain permanently at M, and the light
rays emitted by the flashes of lightning A and B would reach him
simultaneously, i.e. they would meet just where he is situated. Now in
reality (considered with reference to the railway embankment) he is
hastening towards the beam of light coming from B, whilst he is riding
on ahead of the beam of light coming from A. Hence the observer will
see the beam of light emitted from B earlier than he will see that
emitted from A. Observers who take the railway train as their
reference-body must therefore come to the conclusion that the
lightning flash B took place earlier than the lightning flash A. We
thus arrive at the important result:
Events which are simultaneous with reference to the embankment are not
simultaneous with respect to the train, and vice versa (relativity of
simultaneity). Every reference-body (co-ordinate system) has its own
particular time ; unless we are told the reference-body to which the
statement of time refers, there is no meaning in a statement of the
time of an event.
Now before the advent of the theory of relativity it had always
tacitly been assumed in physics that the statement of time had an
absolute significance, i.e. that it is independent of the state of
motion of the body of reference. But we have just seen that this
assumption is incompatible with the most natural definition of
simultaneity; if we discard this assumption, then the conflict between
the law of the propagation of light in vacuo and the principle of
relativity (developed in Section 7) disappears.
We were led to that conflict by the considerations of Section 6,
which are now no longer tenable. In that section we concluded that the
man in the carriage, who traverses the distance w per second relative
to the carriage, traverses the same distance also with respect to the
embankment in each second of time. But, according to the foregoing
considerations, the time required by a particular occurrence with
respect to the carriage must not be considered equal to the duration
of the same occurrence as judged from the embankment (as
reference-body). Hence it cannot be contended that the man in walking
travels the distance w relative to the railway line in a time which is
equal to one second as judged from the embankment.
Moreover, the considerations of Section 6 are based on yet a second
assumption, which, in the light of a strict consideration, appears to
be arbitrary, although it was always tacitly made even before the
introduction of the theory of relativity.
ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE
Let us consider two particular points on the train * travelling
along the embankment with the velocity v, and inquire as to their
distance apart. We already know that it is necessary to have a body of
reference for the measurement of a distance, with respect to which
body the distance can be measured up. It is the simplest plan to use
the train itself as reference-body (co-ordinate system). An observer
in the train measures the interval by marking off his measuring-rod in
a straight line (e.g. along the floor of the carriage) as many times
as is necessary to take him from the one marked point to the other.
Then the number which tells us how often the rod has to be laid down
is the required distance.
It is a different matter when the distance has to be judged from the
railway line. Here the following method suggests itself. If we call
A^1 and B^1 the two points on the train whose distance apart is
required, then both of these points are moving with the velocity v
along the embankment. In the first place we require to determine the
points A and B of the embankment which are just being passed by the
two points A^1 and B^1 at a particular time t -- judged from the
embankment. These points A and B of the embankment can be determined
by applying the definition of time given in Section 8. The distance
between these points A and B is then measured by repeated application
of thee measuring-rod along the embankment.
A priori it is by no means certain that this last measurement will
supply us with the same result as the first. Thus the length of the
train as measured from the embankment may be different from that
obtained by measuring in the train itself. This circumstance leads us
to a second objection which must be raised against the apparently
obvious consideration of Section 6. Namely, if the man in the
carriage covers the distance w in a unit of time -- measured from the
train, -- then this distance -- as measured from the embankment -- is
not necessarily also equal to w.
Notes
*) e.g. the middle of the first and of the hundredth carriage.
THE LORENTZ TRANSFORMATION
The results of the last three sections show that the apparent
incompatibility of the law of propagation of light with the principle
of relativity (Section 7) has been derived by means of a
consideration which borrowed two unjustifiable hypotheses from
classical mechanics; these are as follows:
(1) The time-interval (time) between two events is independent of the
condition of motion of the body of reference.
(2) The space-interval (distance) between two points of a rigid body
is independent of the condition of motion of the body of reference.
If we drop these hypotheses, then the dilemma of Section 7
disappears, because the theorem of the addition of velocities derived
in Section 6 becomes invalid. The possibility presents itself that
the law of the propagation of light in vacuo may be compatible with
the principle of relativity, and the question arises: How have we to
modify the considerations of Section 6 in order to remove the
apparent disagreement between these two fundamental results of
experience? This question leads to a general one. In the discussion of
Section 6 we have to do with places and times relative both to the
train and to the embankment. How are we to find the place and time of
an event in relation to the train, when we know the place and time of
the event with respect to the railway embankment ? Is there a
thinkable answer to this question of such a nature that the law of
transmission of light in vacuo does not contradict the principle of
relativity ? In other words : Can we conceive of a relation between
place and time of the individual events relative to both
reference-bodies, such that every ray of light possesses the velocity
of transmission c relative to the embankment and relative to the train
? This question leads to a quite definite positive answer, and to a
perfectly definite transformation law for the space-time magnitudes of
an event when changing over from one body of reference to another.
Before we deal with this, we shall introduce the following incidental
consideration. Up to the present we have only considered events taking
place along the embankment, which had mathematically to assume the
function of a straight line. In the manner indicated in Section 2
we can imagine this reference-body supplemented laterally and in a
vertical direction by means of a framework of rods, so that an event
which takes place anywhere can be localised with reference to this
framework. Fig. 2 Similarly, we can imagine the train travelling with
the velocity v to be continued across the whole of space, so that
every event, no matter how far off it may be, could also be localised
with respect to the second framework. Without committing any
fundamental error, we can disregard the fact that in reality these
frameworks would continually interfere with each other, owing to the
impenetrability of solid bodies. In every such framework we imagine
three surfaces perpendicular to each other marked out, and designated
as " co-ordinate planes " (" co-ordinate system "). A co-ordinate
system K then corresponds to the embankment, and a co-ordinate system
K' to the train. An event, wherever it may have taken place, would be
fixed in space with respect to K by the three perpendiculars x, y, z
on the co-ordinate planes, and with regard to time by a time value t.
Relative to K1, the same event would be fixed in respect of space and
time by corresponding values x1, y1, z1, t1, which of course are not
identical with x, y, z, t. It has already been set forth in detail how
these magnitudes are to be regarded as results of physical
measurements.
Obviously our problem can be exactly formulated in the following
manner. What are the values x1, y1, z1, t1, of an event with respect
to K1, when the magnitudes x, y, z, t, of the same event with respect
to K are given ? The relations must be so chosen that the law of the
transmission of light in vacuo is satisfied for one and the same ray
of light (and of course for every ray) with respect to K and K1. For
the relative orientation in space of the co-ordinate systems indicated
in the diagram ([7]Fig. 2), this problem is solved by means of the
equations :
eq. 1: file eq01.gif
y1 = y
z1 = z
eq. 2: file eq02.gif
This system of equations is known as the " Lorentz transformation." *
If in place of the law of transmission of light we had taken as our
basis the tacit assumptions of the older mechanics as to the absolute
character of times and lengths, then instead of the above we should
have obtained the following equations:
x1 = x - vt
y1 = y
z1 = z
t1 = t
This system of equations is often termed the " Galilei
transformation." The Galilei transformation can be obtained from the
Lorentz transformation by substituting an infinitely large value for
the velocity of light c in the latter transformation.
Aided by the following illustration, we can readily see that, in
accordance with the Lorentz transformation, the law of the
transmission of light in vacuo is satisfied both for the
reference-body K and for the reference-body K1. A light-signal is sent
along the positive x-axis, and this light-stimulus advances in
accordance with the equation
x = ct,
i.e. with the velocity c. According to the equations of the Lorentz
transformation, this simple relation between x and t involves a
relation between x1 and t1. In point of fact, if we substitute for x
the value ct in the first and fourth equations of the Lorentz
transformation, we obtain:
eq. 3: file eq03.gif
eq. 4: file eq04.gif
from which, by division, the expression
x1 = ct1
immediately follows. If referred to the system K1, the propagation of
light takes place according to this equation. We thus see that the
velocity of transmission relative to the reference-body K1 is also
equal to c. The same result is obtained for rays of light advancing in
any other direction whatsoever. Of cause this is not surprising, since
the equations of the Lorentz transformation were derived conformably
to this point of view.
Notes
*) A simple derivation of the Lorentz transformation is given in
Appendix I.
THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION
Place a metre-rod in the x1-axis of K1 in such a manner that one end
(the beginning) coincides with the point x1=0 whilst the other end
(the end of the rod) coincides with the point x1=I. What is the length
of the metre-rod relatively to the system K? In order to learn this,
we need only ask where the beginning of the rod and the end of the rod
lie with respect to K at a particular time t of the system K. By means
of the first equation of the Lorentz transformation the values of
these two points at the time t = 0 can be shown to be
eq. 05a: file eq05a.gif
eq. 05b: file eq05b.gif
the distance between the points being eq. 06 .
But the metre-rod is moving with the velocity v relative to K. It
therefore follows that the length of a rigid metre-rod moving in the
direction of its length with a velocity v is eq. 06 of a metre.
The rigid rod is thus shorter when in motion than when at rest, and
the more quickly it is moving, the shorter is the rod. For the
velocity v=c we should have eq. 06a ,
and for stiII greater velocities the square-root becomes imaginary.
From this we conclude that in the theory of relativity the velocity c
plays the part of a limiting velocity, which can neither be reached
nor exceeded by any real body.
Of course this feature of the velocity c as a limiting velocity also
clearly follows from the equations of the Lorentz transformation, for
these became meaningless if we choose values of v greater than c.
If, on the contrary, we had considered a metre-rod at rest in the
x-axis with respect to K, then we should have found that the length of
the rod as judged from K1 would have been eq. 06 ;
this is quite in accordance with the principle of relativity which
forms the basis of our considerations.
A Priori it is quite clear that we must be able to learn something
about the physical behaviour of measuring-rods and clocks from the
equations of transformation, for the magnitudes z, y, x, t, are
nothing more nor less than the results of measurements obtainable by
means of measuring-rods and clocks. If we had based our considerations
on the Galileian transformation we should not have obtained a
contraction of the rod as a consequence of its motion.
Let us now consider a seconds-clock which is permanently situated at
the origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of
this clock. The first and fourth equations of the Lorentz
transformation give for these two ticks :
t = 0
and
eq. 07: file eq07.gif
As judged from K, the clock is moving with the velocity v; as judged
from this reference-body, the time which elapses between two strokes
of the clock is not one second, but
eq. 08: file eq08.gif
seconds, i.e. a somewhat larger time. As a consequence of its motion
the clock goes more slowly than when at rest. Here also the velocity c
plays the part of an unattainable limiting velocity.
THEOREM OF THE ADDITION OF VELOCITIES.
THE EXPERIMENT OF FIZEAU
Now in practice we can move clocks and measuring-rods only with
velocities that are small compared with the velocity of light; hence
we shall hardly be able to compare the results of the previous section
directly with the reality. But, on the other hand, these results must
strike you as being very singular, and for that reason I shall now
draw another conclusion from the theory, one which can easily be
derived from the foregoing considerations, and which has been most
elegantly confirmed by experiment.
In Section 6 we derived the theorem of the addition of velocities
in one direction in the form which also results from the hypotheses of
classical mechanics- This theorem can also be deduced readily horn the
Galilei transformation (Section 11). In place of the man walking
inside the carriage, we introduce a point moving relatively to the
co-ordinate system K1 in accordance with the equation
x1 = wt1
By means of the first and fourth equations of the Galilei
transformation we can express x1 and t1 in terms of x and t, and we
then obtain
x = (v + w)t
This equation expresses nothing else than the law of motion of the
point with reference to the system K (of the man with reference to the
embankment). We denote this velocity by the symbol W, and we then
obtain, as in Section 6,
W=v+w         A)
But we can carry out this consideration just as well on the basis of
the theory of relativity. In the equation
x1 = wt1         B)
we must then express x1and t1 in terms of x and t, making use of the
first and fourth equations of the Lorentz transformation. Instead of
the equation (A) we then obtain the equation
eq. 09: file eq09.gif
which corresponds to the theorem of addition for velocities in one
direction according to the theory of relativity. The question now
arises as to which of these two theorems is the better in accord with
experience. On this point we axe enlightened by a most important
experiment which the brilliant physicist Fizeau performed more than
half a century ago, and which has been repeated since then by some of
the best experimental physicists, so that there can be no doubt about
its result. The experiment is concerned with the following question.
Light travels in a motionless liquid with a particular velocity w. How
quickly does it travel in the direction of the arrow in the tube T
(see the accompanying diagram, Fig. 3) when the liquid above
mentioned is flowing through the tube with a velocity v ?
In accordance with the principle of relativity we shall certainly have
to take for granted that the propagation of light always takes place
with the same velocity w with respect to the liquid, whether the
latter is in motion with reference to other bodies or not. The
velocity of light relative to the liquid and the velocity of the
latter relative to the tube are thus known, and we require the
velocity of light relative to the tube.
It is clear that we have the problem of Section 6 again before us. The
tube plays the part of the railway embankment or of the co-ordinate
system K, the liquid plays the part of the carriage or of the
co-ordinate system K1, and finally, the light plays the part of the
Figure 03: file fig03.gif
man walking along the carriage, or of the moving point in the present
section. If we denote the velocity of the light relative to the tube
by W, then this is given by the equation (A) or (B), according as the
Galilei transformation or the Lorentz transformation corresponds to
the facts. Experiment * decides in favour of equation (B) derived
from the theory of relativity, and the agreement is, indeed, very
exact. According to recent and most excellent measurements by Zeeman,
the influence of the velocity of flow v on the propagation of light is
represented by formula (B) to within one per cent.
Nevertheless we must now draw attention to the fact that a theory of
this phenomenon was given by H. A. Lorentz long before the statement
of the theory of relativity. This theory was of a purely
electrodynamical nature, and was obtained by the use of particular
hypotheses as to the electromagnetic structure of matter. This
circumstance, however, does not in the least diminish the
conclusiveness of the experiment as a crucial test in favour of the
theory of relativity, for the electrodynamics of Maxwell-Lorentz, on
which the original theory was based, in no way opposes the theory of
relativity. Rather has the latter been developed trom electrodynamics
as an astoundingly simple combination and generalisation of the
hypotheses, formerly independent of each other, on which
electrodynamics was built.
Notes
*) Fizeau found eq. 10 , where eq. 11
is the index of refraction of the liquid. On the other hand, owing to
the smallness of eq. 12 as compared with I,
we can replace (B) in the first place by eq. 13 , or to the same order
of approximation by
eq. 14 , which agrees with Fizeau's result.
THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY
Our train of thought in the foregoing pages can be epitomised in the
following manner. Experience has led to the conviction that, on the
one hand, the principle of relativity holds true and that on the other
hand the velocity of transmission of light in vacuo has to be
considered equal to a constant c. By uniting these two postulates we
obtained the law of transformation for the rectangular co-ordinates x,
y, z and the time t of the events which constitute the processes of
nature. In this connection we did not obtain the Galilei
transformation, but, differing from classical mechanics, the Lorentz
transformation.
The law of transmission of light, the acceptance of which is justified
by our actual knowledge, played an important part in this process of
thought. Once in possession of the Lorentz transformation, however, we
can combine this with the principle of relativity, and sum up the
theory thus:
Every general law of nature must be so constituted that it is
transformed into a law of exactly the same form when, instead of the
space-time variables x, y, z, t of the original coordinate system K,
we introduce new space-time variables x1, y1, z1, t1 of a co-ordinate
system K1. In this connection the relation between the ordinary and
the accented magnitudes is given by the Lorentz transformation. Or in
brief : General laws of nature are co-variant with respect to Lorentz
transformations.
This is a definite mathematical condition that the theory of
relativity demands of a natural law, and in virtue of this, the theory
becomes a valuable heuristic aid in the search for general laws of
nature. If a general law of nature were to be found which did not
satisfy this condition, then at least one of the two fundamental
assumptions of the theory would have been disproved. Let us now
examine what general results the latter theory has hitherto evinced.
GENERAL RESULTS OF THE THEORY
It is clear from our previous considerations that the (special) theory
of relativity has grown out of electrodynamics and optics. In these
fields it has not appreciably altered the predictions of theory, but
it has considerably simplified the theoretical structure, i.e. the
derivation of laws, and -- what is incomparably more important -- it
has considerably reduced the number of independent hypothese forming
the basis of theory. The special theory of relativity has rendered the
Maxwell-Lorentz theory so plausible, that the latter would have been
generally accepted by physicists even if experiment had decided less
unequivocally in its favour.
Classical mechanics required to be modified before it could come into
line with the demands of the special theory of relativity. For the
main part, however, this modification affects only the laws for rapid
motions, in which the velocities of matter v are not very small as
compared with the velocity of light. We have experience of such rapid
motions only in the case of electrons and ions; for other motions the
variations from the laws of classical mechanics are too small to make
themselves evident in practice. We shall not consider the motion of
stars until we come to speak of the general theory of relativity. In
accordance with the theory of relativity the kinetic energy of a
material point of mass m is no longer given by the well-known
expression
eq. 15: file eq15.gif
but by the expression
eq. 16: file eq16.gif
This expression approaches infinity as the velocity v approaches the
velocity of light c. The velocity must therefore always remain less
than c, however great may be the energies used to produce the
acceleration. If we develop the expression for the kinetic energy in
the form of a series, we obtain
eq. 17: file eq17.gif
When eq. 18 is small compared with unity, the third of these terms is
always small in comparison with the second,
which last is alone considered in classical mechanics. The first term
mc^2 does not contain the velocity, and requires no consideration if
we are only dealing with the question as to how the energy of a
point-mass; depends on the velocity. We shall speak of its essential
significance later.
The most important result of a general character to which the special
theory of relativity has led is concerned with the conception of mass.
Before the advent of relativity, physics recognised two conservation
laws of fundamental importance, namely, the law of the canservation of
energy and the law of the conservation of mass these two fundamental
laws appeared to be quite independent of each other. By means of the
theory of relativity they have been united into one law. We shall now
briefly consider how this unification came about, and what meaning is
to be attached to it.
The principle of relativity requires that the law of the concervation
of energy should hold not only with reference to a co-ordinate system
K, but also with respect to every co-ordinate system K1 which is in a
state of uniform motion of translation relative to K, or, briefly,
relative to every " Galileian " system of co-ordinates. In contrast to
classical mechanics; the Lorentz transformation is the deciding factor
in the transition from one such system to another.
By means of comparatively simple considerations we are led to draw the
following conclusion from these premises, in conjunction with the
fundamental equations of the electrodynamics of Maxwell: A body moving
with the velocity v, which absorbs * an amount of energy E[0] in
the form of radiation without suffering an alteration in velocity in
the process, has, as a consequence, its energy increased by an amount
eq. 19: file eq19.gif
In consideration of the expression given above for the kinetic energy
of the body, the required energy of the body comes out to be
eq. 20: file eq20.gif
Thus the body has the same energy as a body of mass
eq.21: file eq21.gif
moving with the velocity v. Hence we can say: If a body takes up an
amount of energy E[0], then its inertial mass increases by an amount
eq. 22: file eq22.gif
the inertial mass of a body is not a constant but varies according to
the change in the energy of the body. The inertial mass of a system of
bodies can even be regarded as a measure of its energy. The law of the
conservation of the mass of a system becomes identical with the law of
the conservation of energy, and is only valid provided that the system
neither takes up nor sends out energy. Writing the expression for the
energy in the form
eq. 23: file eq23.gif
we see that the term mc^2, which has hitherto attracted our attention,
is nothing else than the energy possessed by the body ** before it
absorbed the energy E[0].
A direct comparison of this relation with experiment is not possible
at the present time (1920; see *** Note, p. 48), owing to the fact that
the changes in energy E[0] to which we can Subject a system are not
large enough to make themselves perceptible as a change in the
inertial mass of the system.
eq. 22: file eq22.gif
is too small in comparison with the mass m, which was present before
the alteration of the energy. It is owing to this circumstance that
classical mechanics was able to establish successfully the
conservation of mass as a law of independent validity.
Let me add a final remark of a fundamental nature. The success of the
Faraday-Maxwell interpretation of electromagnetic action at a distance
resulted in physicists becoming convinced that there are no such
things as instantaneous actions at a distance (not involving an
intermediary medium) of the type of Newton's law of gravitation.
According to the theory of relativity, action at a distance with the
velocity of light always takes the place of instantaneous action at a
distance or of action at a distance with an infinite velocity of
transmission. This is connected with the fact that the velocity c
plays a fundamental role in this theory. In Part II we shall see in
what way this result becomes modified in the general theory of
relativity.
Notes
*) E[0] is the energy taken up, as judged from a co-ordinate system
moving with the body.
**) As judged from a co-ordinate system moving with the body.
***[Note] The equation E = mc^2 has been thoroughly proved time and
again since this time.
EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY
To what extent is the special theory of relativity supported by
experience?  This question is not easily answered for the reason
already mentioned in connection with the fundamental experiment of
Fizeau. The special theory of relativity has crystallised out from the
Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of
experience which support the electromagnetic theory also support the
theory of relativity. As being of particular importance, I mention
here the fact that the theory of relativity enables us to predict the
effects produced on the light reaching us from the fixed stars. These
results are obtained in an exceedingly simple manner, and the effects
indicated, which are due to the relative motion of the earth with
reference to those fixed stars are found to be in accord with
experience. We refer to the yearly movement of the apparent position
of the fixed stars resulting from the motion of the earth round the
sun (aberration), and to the influence of the radial components of the
relative motions of the fixed stars with respect to the earth on the
colour of the light reaching us from them. The latter effect manifests
itself in a slight displacement of the spectral lines of the light
transmitted to us from a fixed star, as compared with the position of
the same spectral lines when they are produced by a terrestrial source
of light (Doppler principle). The experimental arguments in favour of
the Maxwell-Lorentz theory, which are at the same time arguments in
favour of the theory of relativity, are too numerous to be set forth
here. In reality they limit the theoretical possibilities to such an
extent, that no other theory than that of Maxwell and Lorentz has been
able to hold its own when tested by experience.
But there are two classes of experimental facts hitherto obtained
which can be represented in the Maxwell-Lorentz theory only by the
introduction of an auxiliary hypothesis, which in itself -- i.e.
without making use of the theory of relativity -- appears extraneous.
It is known that cathode rays and the so-called b-rays emitted by
radioactive substances consist of negatively electrified particles
(electrons) of very small inertia and large velocity. By examining the
deflection of these rays under the influence of electric and magnetic
fields, we can study the law of motion of these particles very
exactly.
In the theoretical treatment of these electrons, we are faced with the
difficulty that electrodynamic theory of itself is unable to give an
account of their nature. For since electrical masses of one sign repel
each other, the negative electrical masses constituting the electron
would necessarily be scattered under the influence of their mutual
repulsions, unless there are forces of another kind operating between
them, the nature of which has hitherto remained obscure to us.*   If
we now assume that the relative distances between the electrical
masses constituting the electron remain unchanged during the motion of
the electron (rigid connection in the sense of classical mechanics),
we arrive at a law of motion of the electron which does not agree with
experience. Guided by purely formal points of view, H. A. Lorentz was
the first to introduce the hypothesis that the form of the electron
experiences a contraction in the direction of motion in consequence of
that motion. the contracted length being proportional to the
expression
eq. 05: file eq05.gif
This, hypothesis, which is not justifiable by any electrodynamical
facts, supplies us then with that particular law of motion which has
been confirmed with great precision in recent years.
The theory of relativity leads to the same law of motion, without
requiring any special hypothesis whatsoever as to the structure and
the behaviour of the electron. We arrived at a similar conclusion in
Section 13 in connection with the experiment of Fizeau, the result
of which is foretold by the theory of relativity without the necessity
of drawing on hypotheses as to the physical nature of the liquid.
The second class of facts to which we have alluded has reference to
the question whether or not the motion of the earth in space can be
made perceptible in terrestrial experiments. We have already remarked
in Section 5 that all attempts of this nature led to a negative
result. Before the theory of relativity was put forward, it was
difficult to become reconciled to this negative result, for reasons
now to be discussed. The inherited prejudices about time and space did
not allow any doubt to arise as to the prime importance of the
Galileian transformation for changing over from one body of reference
to another. Now assuming that the Maxwell-Lorentz equations hold for a
reference-body K, we then find that they do not hold for a
reference-body K1 moving uniformly with respect to K, if we assume
that the relations of the Galileian transformstion exist between the
co-ordinates of K and K1. It thus appears that, of all Galileian
co-ordinate systems, one (K) corresponding to a particular state of
motion is physically unique. This result was interpreted physically by
regarding K as at rest with respect to a hypothetical �ther of space.
On the other hand, all coordinate systems K1 moving relatively to K
were to be regarded as in motion with respect to the �ther. To this
motion of K1 against the �ther ("�ther-drift " relative to K1) were
attributed the more complicated laws which were supposed to hold
relative to K1. Strictly speaking, such an �ther-drift ought also to
be assumed relative to the earth, and for a long time the efforts of
physicists were devoted to attempts to detect the existence of an
�ther-drift at the earth's surface.
In one of the most notable of these attempts Michelson devised a
method which appears as though it must be decisive. Imagine two
mirrors so arranged on a rigid body that the reflecting surfaces face
each other. A ray of light requires a perfectly definite time T to
pass from one mirror to the other and back again, if the whole system
be at rest with respect to the �ther. It is found by calculation,
however, that a slightly different time T1 is required for this
process, if the body, together with the mirrors, be moving relatively
to the �ther. And yet another point: it is shown by calculation that
for a given velocity v with reference to the �ther, this time T1 is
different when the body is moving perpendicularly to the planes of the
mirrors from that resulting when the motion is parallel to these
planes. Although the estimated difference between these two times is
exceedingly small, Michelson and Morley performed an experiment
involving interference in which this difference should have been
clearly detectable. But the experiment gave a negative result -- a
fact very perplexing to physicists. Lorentz and FitzGerald rescued the
theory from this difficulty by assuming that the motion of the body
relative to the �ther produces a contraction of the body in the
direction of motion, the amount of contraction being just sufficient
to compensate for the differeace in time mentioned above. Comparison
with the discussion in Section 11 shows that also from the
standpoint of the theory of relativity this solution of the difficulty
was the right one. But on the basis of the theory of relativity the
method of interpretation is incomparably more satisfactory. According
to this theory there is no such thing as a " specially favoured "
(unique) co-ordinate system to occasion the introduction of the
�ther-idea, and hence there can be no �ther-drift, nor any experiment
with which to demonstrate it. Here the contraction of moving bodies
follows from the two fundamental principles of the theory, without the
introduction of particular hypotheses ; and as the prime factor
involved in this contraction we find, not the motion in itself, to
which we cannot attach any meaning, but the motion with respect to the
body of reference chosen in the particular case in point. Thus for a
co-ordinate system moving with the earth the mirror system of
Michelson and Morley is not shortened, but it is shortened for a
co-ordinate system which is at rest relatively to the sun.
Notes
*) The general theory of relativity renders it likely that the
electrical masses of an electron are held together by gravitational
forces.
MINKOWSKI'S FOUR-DIMENSIONAL SPACE
The non-mathematician is seized by a mysterious shuddering when he
hears of "four-dimensional" things, by a feeling not unlike that
awakened by thoughts of the occult. And yet there is no more
common-place statement than that the world in which we live is a
four-dimensional space-time continuum.
Space is a three-dimensional continuum. By this we mean that it is
possible to describe the position of a point (at rest) by means of
three numbers (co-ordinales) x, y, z, and that there is an indefinite
number of points in the neighbourhood of this one, the position of
which can be described by co-ordinates such as x[1], y[1], z[1], which
may be as near as we choose to the respective values of the
co-ordinates x, y, z, of the first point. In virtue of the latter
property we speak of a " continuum," and owing to the fact that there
are three co-ordinates we speak of it as being " three-dimensional."
Similarly, the world of physical phenomena which was briefly called "
world " by Minkowski is naturally four dimensional in the space-time
sense. For it is composed of individual events, each of which is
described by four numbers, namely, three space co-ordinates x, y, z,
and a time co-ordinate, the time value t. The" world" is in this sense
also a continuum; for to every event there are as many "neighbouring"
events (realised or at least thinkable) as we care to choose, the
co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely
small amount from those of the event x, y, z, t originally considered.
That we have not been accustomed to regard the world in this sense as
a four-dimensional continuum is due to the fact that in physics,
before the advent of the theory of relativity, time played a different
and more independent role, as compared with the space coordinates. It
is for this reason that we have been in the habit of treating time as
an independent continuum. As a matter of fact, according to classical
mechanics, time is absolute, i.e. it is independent of the position
and the condition of motion of the system of co-ordinates. We see this
expressed in the last equation of the Galileian transformation (t1 =
t)
The four-dimensional mode of consideration of the "world" is natural
on the theory of relativity, since according to this theory time is
robbed of its independence. This is shown by the fourth equation of
the Lorentz transformation:
eq. 24: file eq24.gif
Moreover, according to this equation the time difference Dt1 of two
events with respect to K1 does not in general vanish, even when the
time difference Dt1 of the same events with reference to K vanishes.
Pure " space-distance " of two events with respect to K results in "
time-distance " of the same events with respect to K. But the
discovery of Minkowski, which was of importance for the formal
development of the theory of relativity, does not lie here. It is to
be found rather in the fact of his recognition that the
four-dimensional space-time continuum of the theory of relativity, in
its most essential formal properties, shows a pronounced relationship
to the three-dimensional continuum of Euclidean geometrical
space.*  In order to give due prominence to this relationship,
however, we must replace the usual time co-ordinate t by an imaginary
magnitude eq. 25 proportional to it. Under these conditions, the
natural laws satisfying the demands of the (special) theory of
relativity assume mathematical forms, in which the time co-ordinate
plays exactly the same role as the three space co-ordinates. Formally,
these four co-ordinates correspond exactly to the three space
co-ordinates in Euclidean geometry. It must be clear even to the
non-mathematician that, as a consequence of this purely formal
addition to our knowledge, the theory perforce gained clearness in no
mean measure.
These inadequate remarks can give the reader only a vague notion of
the important idea contributed by Minkowski. Without it the general
theory of relativity, of which the fundamental ideas are developed in
the following pages, would perhaps have got no farther than its long
clothes. Minkowski's work is doubtless difficult of access to anyone
inexperienced in mathematics, but since it is not necessary to have a
very exact grasp of this work in order to understand the fundamental
ideas of either the special or the general theory of relativity, I
shall leave it here at present, and revert to it only towards the end
of Part 2.
Notes
*) Cf. the somewhat more detailed discussion in Appendix II.
PART II
THE GENERAL THEORY OF RELATIVITY
SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY
The basal principle, which was the pivot of all our previous
considerations, was the special principle of relativity, i.e. the
principle of the physical relativity of all uniform motion. Let as
once more analyse its meaning carefully.
It was at all times clear that, from the point of view of the idea it
conveys to us, every motion must be considered only as a relative
motion. Returning to the illustration we have frequently used of the
embankment and the railway carriage, we can express the fact of the
motion here taking place in the following two forms, both of which are
equally justifiable :
(a) The carriage is in motion relative to the embankment,
(b) The embankment is in motion relative to the carriage.
In (a) the embankment, in (b) the carriage, serves as the body of
reference in our statement of the motion taking place. If it is simply
a question of detecting or of describing the motion involved, it is in
principle immaterial to what reference-body we refer the motion. As
already mentioned, this is self-evident, but it must not be confused
with the much more comprehensive statement called "the principle of
relativity," which we have taken as the basis of our investigations.
The principle we have made use of not only maintains that we may
equally well choose the carriage or the embankment as our
reference-body for the description of any event (for this, too, is
self-evident). Our principle rather asserts what follows : If we
formulate the general laws of nature as they are obtained from
experience, by making use of
(a) the embankment as reference-body,
(b) the railway carriage as reference-body,
then these general laws of nature (e.g. the laws of mechanics or the
law of the propagation of light in vacuo) have exactly the same form
in both cases. This can also be expressed as follows : For the
physical description of natural processes, neither of the reference
bodies K, K1 is unique (lit. " specially marked out ") as compared
with the other. Unlike the first, this latter statement need not of
necessity hold a priori; it is not contained in the conceptions of "
motion" and " reference-body " and derivable from them; only
experience can decide as to its correctness or incorrectness.
Up to the present, however, we have by no means maintained the
equivalence of all bodies of reference K in connection with the
formulation of natural laws. Our course was more on the following
Iines. In the first place, we started out from the assumption that
there exists a reference-body K, whose condition of motion is such
that the Galileian law holds with respect to it : A particle left to
itself and sufficiently far removed from all other particles moves
uniformly in a straight line. With reference to K (Galileian
reference-body) the laws of nature were to be as simple as possible.
But in addition to K, all bodies of reference K1 should be given
preference in this sense, and they should be exactly equivalent to K
for the formulation of natural laws, provided that they are in a state
of uniform rectilinear and non-rotary motion with respect to K ; all
these bodies of reference are to be regarded as Galileian
reference-bodies. The validity of the principle of relativity was
assumed only for these reference-bodies, but not for others (e.g.
those possessing motion of a different kind). In this sense we speak
of the special principle of relativity, or special theory of
relativity.
In contrast to this we wish to understand by the "general principle of
relativity" the following statement : All bodies of reference K, K1,
etc., are equivalent for the description of natural phenomena
(formulation of the general laws of nature), whatever may be their
state of motion. But before proceeding farther, it ought to be pointed
out that this formulation must be replaced later by a more abstract
one, for reasons which will become evident at a later stage.
Since the introduction of the special principle of relativity has been
justified, every intellect which strives after generalisation must
feel the temptation to venture the step towards the general principle
of relativity. But a simple and apparently quite reliable
consideration seems to suggest that, for the present at any rate,
there is little hope of success in such an attempt; Let us imagine
ourselves transferred to our old friend the railway carriage, which is
travelling at a uniform rate. As long as it is moving unifromly, the
occupant of the carriage is not sensible of its motion, and it is for
this reason that he can without reluctance interpret the facts of the
case as indicating that the carriage is at rest, but the embankment in
motion. Moreover, according to the special principle of relativity,
this interpretation is quite justified also from a physical point of
view.
If the motion of the carriage is now changed into a non-uniform
motion, as for instance by a powerful application of the brakes, then
the occupant of the carriage experiences a correspondingly powerful
jerk forwards. The retarded motion is manifested in the mechanical
behaviour of bodies relative to the person in the railway carriage.
The mechanical behaviour is different from that of the case previously
considered, and for this reason it would appear to be impossible that
the same mechanical laws hold relatively to the non-uniformly moving
carriage, as hold with reference to the carriage when at rest or in
uniform motion. At all events it is clear that the Galileian law does
not hold with respect to the non-uniformly moving carriage. Because of
this, we feel compelled at the present juncture to grant a kind of
absolute physical reality to non-uniform motion, in opposition to the
general principle of relatvity. But in what follows we shall soon see
that this conclusion cannot be maintained.
THE GRAVITATIONAL FIELD
"If we pick up a stone and then let it go, why does it fall to the
ground ?" The usual answer to this question is: "Because it is
attracted by the earth." Modern physics formulates the answer rather
differently for the following reason. As a result of the more careful
study of electromagnetic phenomena, we have come to regard action at a
distance as a process impossible without the intervention of some
intermediary medium. If, for instance, a magnet attracts a piece of
iron, we cannot be content to regard this as meaning that the magnet
acts directly on the iron through the intermediate empty space, but we
are constrained to imagine -- after the manner of Faraday -- that the
magnet always calls into being something physically real in the space
around it, that something being what we call a "magnetic field." In
its turn this magnetic field operates on the piece of iron, so that
the latter strives to move towards the magnet. We shall not discuss
here the justification for this incidental conception, which is indeed
a somewhat arbitrary one. We shall only mention that with its aid
electromagnetic phenomena can be theoretically represented much more
satisfactorily than without it, and this applies particularly to the
transmission of electromagnetic waves. The effects of gravitation also
are regarded in an analogous manner.
The action of the earth on the stone takes place indirectly. The earth
produces in its surrounding a gravitational field, which acts on the
stone and produces its motion of fall. As we know from experience, the
intensity of the action on a body dimishes according to a quite
definite law, as we proceed farther and farther away from the earth.
From our point of view this means : The law governing the properties
of the gravitational field in space must be a perfectly definite one,
in order correctly to represent the diminution of gravitational action
with the distance from operative bodies. It is something like this:
The body (e.g. the earth) produces a field in its immediate
neighbourhood directly; the intensity and direction of the field at
points farther removed from the body are thence determined by the law
which governs the properties in space of the gravitational fields
themselves.
In contrast to electric and magnetic fields, the gravitational field
exhibits a most remarkable property, which is of fundamental
importance for what follows. Bodies which are moving under the sole
influence of a gravitational field receive an acceleration, which does
not in the least depend either on the material or on the physical
state of the body. For instance, a piece of lead and a piece of wood
fall in exactly the same manner in a gravitational field (in vacuo),
when they start off from rest or with the same initial velocity. This
law, which holds most accurately, can be expressed in a different form
in the light of the following consideration.
According to Newton's law of motion, we have
(Force) = (inertial mass) x (acceleration),
where the "inertial mass" is a characteristic constant of the
accelerated body. If now gravitation is the cause of the acceleration,
we then have
(Force) = (gravitational mass) x (intensity of the gravitational
field),
where the "gravitational mass" is likewise a characteristic constant
for the body. From these two relations follows:
eq. 26: file eq26.gif
If now, as we find from experience, the acceleration is to be
independent of the nature and the condition of the body and always the
same for a given gravitational field, then the ratio of the
gravitational to the inertial mass must likewise be the same for all
bodies. By a suitable choice of units we can thus make this ratio
equal to unity. We then have the following law: The gravitational mass
of a body is equal to its inertial law.
It is true that this important law had hitherto been recorded in
mechanics, but it had not been interpreted. A satisfactory
interpretation can be obtained only if we recognise the following fact
: The same quality of a body manifests itself according to
circumstances as " inertia " or as " weight " (lit. " heaviness '). In
the following section we shall show to what extent this is actually
the case, and how this question is connected with the general
postulate of relativity.
THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS
AS AN ARGUMENT FOR THE GENERAL POSTULE OF RELATIVITY
We imagine a large portion of empty space, so far removed from stars
and other appreciable masses, that we have before us approximately the
conditions required by the fundamental law of Galilei. It is then
possible to choose a Galileian reference-body for this part of space
(world), relative to which points at rest remain at rest and points in
motion continue permanently in uniform rectilinear motion. As
reference-body let us imagine a spacious chest resembling a room with
an observer inside who is equipped with apparatus. Gravitation
naturally does not exist for this observer. He must fasten himself
with strings to the floor, otherwise the slightest impact against the
floor will cause him to rise slowly towards the ceiling of the room.
To the middle of the lid of the chest is fixed externally a hook with
rope attached, and now a " being " (what kind of a being is immaterial
to us) begins pulling at this with a constant force. The chest
together with the observer then begin to move "upwards" with a
uniformly accelerated motion. In course of time their velocity will
reach unheard-of values -- provided that we are viewing all this from
another reference-body which is not being pulled with a rope.
But how does the man in the chest regard the Process ? The
acceleration of the chest will be transmitted to him by the reaction
of the floor of the chest. He must therefore take up this pressure by
means of his legs if he does not wish to be laid out full length on
the floor. He is then standing in the chest in exactly the same way as
anyone stands in a room of a home on our earth. If he releases a body
which he previously had in his land, the accelertion of the chest will
no longer be transmitted to this body, and for this reason the body
will approach the floor of the chest with an accelerated relative
motion. The observer will further convince himself that the
acceleration of the body towards the floor of the chest is always of
the same magnitude, whatever kind of body he may happen to use for the
experiment.
Relying on his knowledge of the gravitational field (as it was
discussed in the preceding section), the man in the chest will thus
come to the conclusion that he and the chest are in a gravitational
field which is constant with regard to time. Of course he will be
puzzled for a moment as to why the chest does not fall in this
gravitational field. just then, however, he discovers the hook in the
middle of the lid of the chest and the rope which is attached to it,
and he consequently comes to the conclusion that the chest is
suspended at rest in the gravitational field.
Ought we to smile at the man and say that he errs in his conclusion ?
I do not believe we ought to if we wish to remain consistent ; we must
rather admit that his mode of grasping the situation violates neither
reason nor known mechanical laws. Even though it is being accelerated
with respect to the "Galileian space" first considered, we can
nevertheless regard the chest as being at rest. We have thus good
grounds for extending the principle of relativity to include bodies of
reference which are accelerated with respect to each other, and as a
result we have gained a powerful argument for a generalised postulate
of relativity.
We must note carefully that the possibility of this mode of
interpretation rests on the fundamental property of the gravitational
field of giving all bodies the same acceleration, or, what comes to
the same thing, on the law of the equality of inertial and
gravitational mass. If this natural law did not exist, the man in the
accelerated chest would not be able to interpret the behaviour of the
bodies around him on the supposition of a gravitational field, and he
would not be justified on the grounds of experience in supposing his
reference-body to be " at rest."
Suppose that the man in the chest fixes a rope to the inner side of
the lid, and that he attaches a body to the free end of the rope. The
result of this will be to strech the rope so that it will hang "
vertically " downwards. If we ask for an opinion of the cause of
tension in the rope, the man in the chest will say: "The suspended
body experiences a downward force in the gravitational field, and this
is neutralised by the tension of the rope ; what determines the
magnitude of the tension of the rope is the gravitational mass of the
suspended body." On the other hand, an observer who is poised freely
in space will interpret the condition of things thus : " The rope must
perforce take part in the accelerated motion of the chest, and it
transmits this motion to the body attached to it. The tension of the
rope is just large enough to effect the acceleration of the body. That
which determines the magnitude of the tension of the rope is the
inertial mass of the body." Guided by this example, we see that our
extension of the principle of relativity implies the necessity of the
law of the equality of inertial and gravitational mass. Thus we have
obtained a physical interpretation of this law.
From our consideration of the accelerated chest we see that a general
theory of relativity must yield important results on the laws of
gravitation. In point of fact, the systematic pursuit of the general
idea of relativity has supplied the laws satisfied by the
gravitational field. Before proceeding farther, however, I must warn
the reader against a misconception suggested by these considerations.
A gravitational field exists for the man in the chest, despite the
fact that there was no such field for the co-ordinate system first
chosen. Now we might easily suppose that the existence of a
gravitational field is always only an apparent one. We might also
think that, regardless of the kind of gravitational field which may be
present, we could always choose another reference-body such that no
gravitational field exists with reference to it. This is by no means
true for all gravitational fields, but only for those of quite special
form. It is, for instance, impossible to choose a body of reference
such that, as judged from it, the gravitational field of the earth (in
its entirety) vanishes.
We can now appreciate why that argument is not convincing, which we
brought forward against the general principle of relativity at theend
of Section 18. It is certainly true that the observer in the
railway carriage experiences a jerk forwards as a result of the
application of the brake, and that he recognises, in this the
non-uniformity of motion (retardation) of the carriage. But he is
compelled by nobody to refer this jerk to a " real " acceleration
(retardation) of the carriage. He might also interpret his experience
thus: " My body of reference (the carriage) remains permanently at
rest. With reference to it, however, there exists (during the period
of application of the brakes) a gravitational field which is directed
forwards and which is variable with respect to time. Under the
influence of this field, the embankment together with the earth moves
non-uniformly in such a manner that their original velocity in the
backwards direction is continuously reduced."
IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE
SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?
We have already stated several times that classical mechanics starts
out from the following law: Material particles sufficiently far
removed from other material particles continue to move uniformly in a
straight line or continue in a state of rest. We have also repeatedly
emphasised that this fundamental law can only be valid for bodies of
reference K which possess certain unique states of motion, and which
are in uniform translational motion relative to each other. Relative
to other reference-bodies K the law is not valid. Both in classical
mechanics and in the special theory of relativity we therefore
differentiate between reference-bodies K relative to which the
recognised " laws of nature " can be said to hold, and
reference-bodies K relative to which these laws do not hold.
But no person whose mode of thought is logical can rest satisfied with
this condition of things. He asks : " How does it come that certain
reference-bodies (or their states of motion) are given priority over
other reference-bodies (or their states of motion) ? What is the
reason for this Preference? In order to show clearly what I mean by
this question, I shall make use of a comparison.
I am standing in front of a gas range. Standing alongside of each
other on the range are two pans so much alike that one may be mistaken
for the other. Both are half full of water. I notice that steam is
being emitted continuously from the one pan, but not from the other. I
am surprised at this, even if I have never seen either a gas range or
a pan before. But if I now notice a luminous something of bluish
colour under the first pan but not under the other, I cease to be
astonished, even if I have never before seen a gas flame. For I can
only say that this bluish something will cause the emission of the
steam, or at least possibly it may do so. If, however, I notice the
bluish something in neither case, and if I observe that the one
continuously emits steam whilst the other does not, then I shall
remain astonished and dissatisfied until I have discovered some
circumstance to which I can attribute the different behaviour of the
two pans.
Analogously, I seek in vain for a real something in classical
mechanics (or in the special theory of relativity) to which I can
attribute the different behaviour of bodies considered with respect to
the reference systems K and K1.*  Newton saw this objection and
attempted to invalidate it, but without success. But E. Mach recognsed
it most clearly of all, and because of this objection he claimed that
mechanics must be placed on a new basis. It can only be got rid of by
means of a physics which is conformable to the general principle of
relativity, since the equations of such a theory hold for every body
of reference, whatever may be its state of motion.
Notes
*) The objection is of importance more especially when the state of
motion of the reference-body is of such a nature that it does not
require any external agency for its maintenance, e.g. in the case when
the reference-body is rotating uniformly.
A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY
The considerations of Section 20 show that the general principle of
relativity puts us in a position to derive properties of the
gravitational field in a purely theoretical manner. Let us suppose,
for instance, that we know the space-time " course " for any natural
process whatsoever, as regards the manner in which it takes place in
the Galileian domain relative to a Galileian body of reference K. By
means of purely theoretical operations (i.e. simply by calculation) we
are then able to find how this known natural process appears, as seen
from a reference-body K1 which is accelerated relatively to K. But
since a gravitational field exists with respect to this new body of
reference K1, our consideration also teaches us how the gravitational
field influences the process studied.
For example, we learn that a body which is in a state of uniform
rectilinear motion with respect to K (in accordance with the law of
Galilei) is executing an accelerated and in general curvilinear motion
with respect to the accelerated reference-body K1 (chest). This
acceleration or curvature corresponds to the influence on the moving
body of the gravitational field prevailing relatively to K. It is
known that a gravitational field influences the movement of bodies in
this way, so that our consideration supplies us with nothing
essentially new.
However, we obtain a new result of fundamental importance when we
carry out the analogous consideration for a ray of light. With respect
to the Galileian reference-body K, such a ray of light is transmitted
rectilinearly with the velocity c. It can easily be shown that the
path of the same ray of light is no longer a straight line when we
consider it with reference to the accelerated chest (reference-body
K1). From this we conclude, that, in general, rays of light are
propagated curvilinearly in gravitational fields. In two respects this
result is of great importance.
In the first place, it can be compared with the reality. Although a
detailed examination of the question shows that the curvature of light
rays required by the general theory of relativity is only exceedingly
small for the gravitational fields at our disposal in practice, its
estimated magnitude for light rays passing the sun at grazing
incidence is nevertheless 1.7 seconds of arc. This ought to manifest
itself in the following way. As seen from the earth, certain fixed
stars appear to be in the neighbourhood of the sun, and are thus
capable of observation during a total eclipse of the sun. At such
times, these stars ought to appear to be displaced outwards from the
sun by an amount indicated above, as compared with their apparent
position in the sky when the sun is situated at another part of the
heavens. The examination of the correctness or otherwise of this
deduction is a problem of the greatest importance, the early solution
of which is to be expected of astronomers.[2]*
In the second place our result shows that, according to the general
theory of relativity, the law of the constancy of the velocity of
light in vacuo, which constitutes one of the two fundamental
assumptions in the special theory of relativity and to which we have
already frequently referred, cannot claim any unlimited validity. A
curvature of rays of light can only take place when the velocity of
propagation of light varies with position. Now we might think that as
a consequence of this, the special theory of relativity and with it
the whole theory of relativity would be laid in the dust. But in
reality this is not the case. We can only conclude that the special
theory of relativity cannot claim an unlinlited domain of validity ;
its results hold only so long as we are able to disregard the
influences of gravitational fields on the phenomena (e.g. of light).
Since it has often been contended by opponents of the theory of
relativity that the special theory of relativity is overthrown by the
general theory of relativity, it is perhaps advisable to make the
facts of the case clearer by means of an appropriate comparison.
Before the development of electrodynamics the laws of electrostatics
were looked upon as the laws of electricity. At the present time we
know that electric fields can be derived correctly from electrostatic
considerations only for the case, which is never strictly realised, in
which the electrical masses are quite at rest relatively to each
other, and to the co-ordinate system. Should we be justified in saying
that for this reason electrostatics is overthrown by the
field-equations of Maxwell in electrodynamics ? Not in the least.
Electrostatics is contained in electrodynamics as a limiting case ;
the laws of the latter lead directly to those of the former for the
case in which the fields are invariable with regard to time. No fairer
destiny could be allotted to any physical theory, than that it should
of itself point out the way to the introduction of a more
comprehensive theory, in which it lives on as a limiting case.
In the example of the transmission of light just dealt with, we have
seen that the general theory of relativity enables us to derive
theoretically the influence of a gravitational field on the course of
natural processes, the Iaws of which are already known when a
gravitational field is absent. But the most attractive problem, to the
solution of which the general theory of relativity supplies the key,
concerns the investigation of the laws satisfied by the gravitational
field itself. Let us consider this for a moment.
We are acquainted with space-time domains which behave (approximately)
in a " Galileian " fashion under suitable choice of reference-body,
i.e. domains in which gravitational fields are absent. If we now refer
such a domain to a reference-body K1 possessing any kind of motion,
then relative to K1 there exists a gravitational field which is
variable with respect to space and time.[3]**  The character of this
field will of course depend on the motion chosen for K1. According to
the general theory of relativity, the general law of the gravitational
field must be satisfied for all gravitational fields obtainable in
this way. Even though by no means all gravitationial fields can be
produced in this way, yet we may entertain the hope that the general
law of gravitation will be derivable from such gravitational fields of
a special kind. This hope has been realised in the most beautiful
manner. But between the clear vision of this goal and its actual
realisation it was necessary to surmount a serious difficulty, and as
this lies deep at the root of things, I dare not withhold it from the
reader. We require to extend our ideas of the space-time continuum
still farther.
Notes
*) By means of the star photographs of two expeditions equipped by
a Joint Committee of the Royal and Royal Astronomical Societies, the
existence of the deflection of light demanded by theory was first
confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
III.)
**) This follows from a generalisation of the discussion in
Section 20
BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE
Hitherto I have purposely refrained from speaking about the physical
interpretation of space- and time-data in the case of the general
theory of relativity. As a consequence, I am guilty of a certain
slovenliness of treatment, which, as we know from the special theory
of relativity, is far from being unimportant and pardonable. It is now
high time that we remedy this defect; but I would mention at the
outset, that this matter lays no small claims on the patience and on
the power of abstraction of the reader.
We start off again from quite special cases, which we have frequently
used before. Let us consider a space time domain in which no
gravitational field exists relative to a reference-body K whose state
of motion has been suitably chosen. K is then a Galileian
reference-body as regards the domain considered, and the results of
the special theory of relativity hold relative to K. Let us supposse
the same domain referred to a second body of reference K1, which is
rotating uniformly with respect to K. In order to fix our ideas, we
shall imagine K1 to be in the form of a plane circular disc, which
rotates uniformly in its own plane about its centre. An observer who
is sitting eccentrically on the disc K1 is sensible of a force which
acts outwards in a radial direction, and which would be interpreted as
an effect of inertia (centrifugal force) by an observer who was at
rest with respect to the original reference-body K. But the observer
on the disc may regard his disc as a reference-body which is " at rest
" ; on the basis of the general principle of relativity he is
justified in doing this. The force acting on himself, and in fact on
all other bodies which are at rest relative to the disc, he regards as
the effect of a gravitational field. Nevertheless, the
space-distribution of this gravitational field is of a kind that would
not be possible on Newton's theory of gravitation.* But since the
observer believes in the general theory of relativity, this does not
disturb him; he is quite in the right when he believes that a general
law of gravitation can be formulated- a law which not only explains
the motion of the stars correctly, but also the field of force
experienced by himself.
The observer performs experiments on his circular disc with clocks and
measuring-rods. In doing so, it is his intention to arrive at exact
definitions for the signification of time- and space-data with
reference to the circular disc K1, these definitions being based on
his observations. What will be his experience in this enterprise ?
To start with, he places one of two identically constructed clocks at
the centre of the circular disc, and the other on the edge of the
disc, so that they are at rest relative to it. We now ask ourselves
whether both clocks go at the same rate from the standpoint of the
non-rotating Galileian reference-body K. As judged from this body, the
clock at the centre of the disc has no velocity, whereas the clock at
the edge of the disc is in motion relative to K in consequence of the
rotation. According to a result obtained in Section 12, it follows
that the latter clock goes at a rate permanently slower than that of
the clock at the centre of the circular disc, i.e. as observed from K.
It is obvious that the same effect would be noted by an observer whom
we will imagine sitting alongside his clock at the centre of the
circular disc. Thus on our circular disc, or, to make the case more
general, in every gravitational field, a clock will go more quickly or
less quickly, according to the position in which the clock is situated
(at rest). For this reason it is not possible to obtain a reasonable
definition of time with the aid of clocks which are arranged at rest
with respect to the body of reference. A similar difficulty presents
itself when we attempt to apply our earlier definition of simultaneity
in such a case, but I do not wish to go any farther into this
question.
Moreover, at this stage the definition of the space co-ordinates also
presents insurmountable difficulties. If the observer applies his
standard measuring-rod (a rod which is short as compared with the
radius of the disc) tangentially to the edge of the disc, then, as
judged from the Galileian system, the length of this rod will be less
than I, since, according to Section 12, moving bodies suffer a
shortening in the direction of the motion. On the other hand, the
measaring-rod will not experience a shortening in length, as judged
from K, if it is applied to the disc in the direction of the radius.
If, then, the observer first measures the circumference of the disc
with his measuring-rod and then the diameter of the disc, on dividing
the one by the other, he will not obtain as quotient the familiar
number p = 3.14 . . ., but a larger number,[4]** whereas of course,
for a disc which is at rest with respect to K, this operation would
yield p exactly. This proves that the propositions of Euclidean
geometry cannot hold exactly on the rotating disc, nor in general in a
gravitational field, at least if we attribute the length I to the rod
in all positions and in every orientation. Hence the idea of a
straight line also loses its meaning. We are therefore not in a
position to define exactly the co-ordinates x, y, z relative to the
disc by means of the method used in discussing the special theory, and
as long as the co- ordinates and times of events have not been
defined, we cannot assign an exact meaning to the natural laws in
which these occur.
Thus all our previous conclusions based on general relativity would
appear to be called in question. In reality we must make a subtle
detour in order to be able to apply the postulate of general
relativity exactly. I shall prepare the reader for this in the
following paragraphs.
Notes
*) The field disappears at the centre of the disc and increases
proportionally to the distance from the centre as we proceed outwards.
**) Throughout this consideration we have to use the Galileian
(non-rotating) system K as reference-body, since we may only assume
the validity of the results of the special theory of relativity
relative to K (relative to K1 a gravitational field prevails).
EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM
The surface of a marble table is spread out in front of me. I can get
from any one point on this table to any other point by passing
continuously from one point to a " neighbouring " one, and repeating
this process a (large) number of times, or, in other words, by going
from point to point without executing "jumps." I am sure the reader
will appreciate with sufficient clearness what I mean here by "
neighbouring " and by " jumps " (if he is not too pedantic). We
express this property of the surface by describing the latter as a
continuum.
Let us now imagine that a large number of little rods of equal length
have been made, their lengths being small compared with the dimensions
of the marble slab. When I say they are of equal length, I mean that
one can be laid on any other without the ends overlapping. We next lay
four of these little rods on the marble slab so that they constitute a
quadrilateral figure (a square), the diagonals of which are equally
long. To ensure the equality of the diagonals, we make use of a little
testing-rod. To this square we add similar ones, each of which has one
rod in common with the first. We proceed in like manner with each of
these squares until finally the whole marble slab is laid out with
squares. The arrangement is such, that each side of a square belongs
to two squares and each corner to four squares.
It is a veritable wonder that we can carry out this business without
getting into the greatest difficulties. We only need to think of the
following. If at any moment three squares meet at a corner, then two
sides of the fourth square are already laid, and, as a consequence,
the arrangement of the remaining two sides of the square is already
completely determined. But I am now no longer able to adjust the
quadrilateral so that its diagonals may be equal. If they are equal of
their own accord, then this is an especial favour of the marble slab
and of the little rods, about which I can only be thankfully
surprised. We must experience many such surprises if the construction
is to be successful.
If everything has really gone smoothly, then I say that the points of
the marble slab constitute a Euclidean continuum with respect to the
little rod, which has been used as a " distance " (line-interval). By
choosing one corner of a square as " origin" I can characterise every
other corner of a square with reference to this origin by means of two
numbers. I only need state how many rods I must pass over when,
starting from the origin, I proceed towards the " right " and then "
upwards," in order to arrive at the corner of the square under
consideration. These two numbers are then the " Cartesian co-ordinates
" of this corner with reference to the " Cartesian co-ordinate system"
which is determined by the arrangement of little rods.
By making use of the following modification of this abstract
experiment, we recognise that there must also be cases in which the
experiment would be unsuccessful. We shall suppose that the rods "
expand " by in amount proportional to the increase of temperature. We
heat the central part of the marble slab, but not the periphery, in
which case two of our little rods can still be brought into
coincidence at every position on the table. But our construction of
squares must necessarily come into disorder during the heating,
because the little rods on the central region of the table expand,
whereas those on the outer part do not.
With reference to our little rods -- defined as unit lengths -- the
marble slab is no longer a Euclidean continuum, and we are also no
longer in the position of defining Cartesian co-ordinates directly
with their aid, since the above construction can no longer be carried
out. But since there are other things which are not influenced in a
similar manner to the little rods (or perhaps not at all) by the
temperature of the table, it is possible quite naturally to maintain
the point of view that the marble slab is a " Euclidean continuum."
This can be done in a satisfactory manner by making a more subtle
stipulation about the measurement or the comparison of lengths.
But if rods of every kind (i.e. of every material) were to behave in
the same way as regards the influence of temperature when they are on
the variably heated marble slab, and if we had no other means of
detecting the effect of temperature than the geometrical behaviour of
our rods in experiments analogous to the one described above, then our
best plan would be to assign the distance one to two points on the
slab, provided that the ends of one of our rods could be made to
coincide with these two points ; for how else should we define the
distance without our proceeding being in the highest measure grossly
arbitrary ? The method of Cartesian coordinates must then be
discarded, and replaced by another which does not assume the validity
of Euclidean geometry for rigid bodies.*  The reader will notice
that the situation depicted here corresponds to the one brought about
by the general postitlate of relativity (Section 23).
Notes
*) Mathematicians have been confronted with our problem in the
following form. If we are given a surface (e.g. an ellipsoid) in
Euclidean three-dimensional space, then there exists for this surface
a two-dimensional geometry, just as much as for a plane surface. Gauss
undertook the task of treating this two-dimensional geometry from
first principles, without making use of the fact that the surface
belongs to a Euclidean continuum of three dimensions. If we imagine
constructions to be made with rigid rods in the surface (similar to
that above with the marble slab), we should find that different laws
hold for these from those resulting on the basis of Euclidean plane
geometry. The surface is not a Euclidean continuum with respect to the
rods, and we cannot define Cartesian co-ordinates in the surface.
Gauss indicated the principles according to which we can treat the
geometrical relationships in the surface, and thus pointed out the way
to the method of Riemman of treating multi-dimensional, non-Euclidean
continuum. Thus it is that mathematicians long ago solved the formal
problems to which we are led by the general postulate of relativity.
GAUSSIAN CO-ORDINATES
According to Gauss, this combined analytical and geometrical mode of
handling the problem can be arrived at in the following way. We
imagine a system of arbitrary curves (see Fig. 4) drawn on the surface
of the table. These we designate as u-curves, and we indicate each of
them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in
the diagram. Between the curves u= 1 and u= 2 we must imagine an
infinitely large number to be drawn, all of which correspond to real
numbers lying between 1 and 2. fig. 04 We have then a system of
u-curves, and this "infinitely dense" system covers the whole surface
of the table. These u-curves must not intersect each other, and
through each point of the surface one and only one curve must pass.
Thus a perfectly definite value of u belongs to every point on the
surface of the marble slab. In like manner we imagine a system of
v-curves drawn on the surface. These satisfy the same conditions as
the u-curves, they are provided with numbers in a corresponding
manner, and they may likewise be of arbitrary shape. It follows that a
value of u and a value of v belong to every point on the surface of
the table. We call these two numbers the co-ordinates of the surface
of the table (Gaussian co-ordinates). For example, the point P in the
diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring
points P and P1 on the surface then correspond to the co-ordinates
P:       u,v
P1:     u + du, v + dv,
where du and dv signify very small numbers. In a similar manner we may
indicate the distance (line-interval) between P and P1, as measured
with a little rod, by means of the very small number ds. Then
according to Gauss we have
ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2
where g[11], g[12], g[22], are magnitudes which depend in a perfectly
definite way on u and v. The magnitudes g[11], g[12] and g[22],
determine the behaviour of the rods relative to the u-curves and
v-curves, and thus also relative to the surface of the table. For the
case in which the points of the surface considered form a Euclidean
continuum with reference to the measuring-rods, but only in this case,
it is possible to draw the u-curves and v-curves and to attach numbers
to them, in such a manner, that we simply have :
ds2 = du2 + dv2
Under these conditions, the u-curves and v-curves are straight lines
in the sense of Euclidean geometry, and they are perpendicular to each
other. Here the Gaussian coordinates are samply Cartesian ones. It is
clear that Gauss co-ordinates are nothing more than an association of
two sets of numbers with the points of the surface considered, of such
a nature that numerical values differing very slightly from each other
are associated with neighbouring points " in space."
So far, these considerations hold for a continuum of two dimensions.
But the Gaussian method can be applied also to a continuum of three,
four or more dimensions. If, for instance, a continuum of four
dimensions be supposed available, we may represent it in the following
way. With every point of the continuum, we associate arbitrarily four
numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates."
Adjacent points correspond to adjacent values of the coordinates. If a
distance ds is associated with the adjacent points P and P1, this
distance being measurable and well defined from a physical point of
view, then the following formula holds:
ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2,
where the magnitudes g[11], etc., have values which vary with the
position in the continuum. Only when the continuum is a Euclidean one
is it possible to associate the co-ordinates x[1] . . x[4]. with the
points of the continuum so that we have simply
ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.
In this case relations hold in the four-dimensional continuum which
are analogous to those holding in our three-dimensional measurements.
However, the Gauss treatment for ds2 which we have given above is not
always possible. It is only possible when sufficiently small regions
of the continuum under consideration may be regarded as Euclidean
continua. For example, this obviously holds in the case of the marble
slab of the table and local variation of temperature. The temperature
is practically constant for a small part of the slab, and thus the
geometrical behaviour of the rods is almost as it ought to be
according to the rules of Euclidean geometry. Hence the imperfections
of the construction of squares in the previous section do not show
themselves clearly until this construction is extended over a
considerable portion of the surface of the table.
We can sum this up as follows: Gauss invented a method for the
mathematical treatment of continua in general, in which "
size-relations " (" distances " between neighbouring points) are
defined. To every point of a continuum are assigned as many numbers
(Gaussian coordinates) as the continuum has dimensions. This is done
in such a way, that only one meaning can be attached to the
assignment, and that numbers (Gaussian coordinates) which differ by an
indefinitely small amount are assigned to adjacent points. The
Gaussian coordinate system is a logical generalisation of the
Cartesian co-ordinate system. It is also applicable to non-Euclidean
continua, but only when, with respect to the defined "size" or
"distance," small parts of the continuum under consideration behave
more nearly like a Euclidean system, the smaller the part of the
continuum under our notice.
THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A
EUCLIDEAN CONTINUUM
We are now in a position to formulate more exactly the idea of
Minkowski, which was only vaguely indicated in Section 17. In
accordance with the special theory of relativity, certain co-ordinate
systems are given preference for the description of the
four-dimensional, space-time continuum. We called these " Galileian
co-ordinate systems." For these systems, the four co-ordinates x, y,
z, t, which determine an event or -- in other words, a point of the
four-dimensional continuum -- are defined physically in a simple
manner, as set forth in detail in the first part of this book. For the
transition from one Galileian system to another, which is moving
uniformly with reference to the first, the equations of the Lorentz
transformation are valid. These last form the basis for the derivation
of deductions from the special theory of relativity, and in themselves
they are nothing more than the expression of the universal validity of
the law of transmission of light for all Galileian systems of
reference.
Minkowski found that the Lorentz transformations satisfy the following
simple conditions. Let us consider two neighbouring events, the
relative position of which in the four-dimensional continuum is given
with respect to a Galileian reference-body K by the space co-ordinate
differences dx, dy, dz and the time-difference dt. With reference to a
second Galileian system we shall suppose that the corresponding
differences for these two events are dx1, dy1, dz1, dt1. Then these
magnitudes always fulfil the condition*
dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2.
The validity of the Lorentz transformation follows from this
condition. We can express this as follows: The magnitude
ds2 = dx2 + dy2 + dz2 - c^2dt2,
which belongs to two adjacent points of the four-dimensional
space-time continuum, has the same value for all selected (Galileian)
reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1],
x[2], x[3], x[4], we also obtaill the result that
ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.
is independent of the choice of the body of reference. We call the
magnitude ds the " distance " apart of the two events or
four-dimensional points.
Thus, if we choose as time-variable the imaginary variable sq. rt. -I
. ct instead of the real quantity t, we can regard the space-time
contintium -- accordance with the special theory of relativity -- as a
", Euclidean " four-dimensional continuum, a result which follows from
the considerations of the preceding section.
Notes
*) Cf. Appendixes I and 2. The relations which are derived
there for the co-ordlnates themselves are valid also for co-ordinate
differences, and thus also for co-ordinate differentials (indefinitely
small differences).
THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A
ECULIDEAN CONTINUUM
In the first part of this book we were able to make use of space-time
co-ordinates which allowed of a simple and direct physical
interpretation, and which, according to Section 26, can be regarded
as four-dimensional Cartesian co-ordinates. This was possible on the
basis of the law of the constancy of the velocity of tight. But
according to Section 21 the general theory of relativity cannot
retain this law. On the contrary, we arrived at the result that
according to this latter theory the velocity of light must always
depend on the co-ordinates when a gravitational field is present. In
connection with a specific illustration in Section 23, we found
that the presence of a gravitational field invalidates the definition
of the coordinates and the ifine, which led us to our objective in the
special theory of relativity.
In view of the resuIts of these considerations we are led to the
conviction that, according to the general principle of relativity, the
space-time continuum cannot be regarded as a Euclidean one, but that
here we have the general case, corresponding to the marble slab with
local variations of temperature, and with which we made acquaintance
as an example of a two-dimensional continuum. Just as it was there
impossible to construct a Cartesian co-ordinate system from equal
rods, so here it is impossible to build up a system (reference-body)
from rigid bodies and clocks, which shall be of such a nature that
measuring-rods and clocks, arranged rigidly with respect to one
another, shaIll indicate position and time directly. Such was the
essence of the difficulty with which we were confronted in Section
23.
But the considerations of Sections 25 and 26 show us the way to
surmount this difficulty. We refer the fourdimensional space-time
continuum in an arbitrary manner to Gauss co-ordinates. We assign to
every point of the continuum (event) four numbers, x[1], x[2], x[3],
x[4] (co-ordinates), which have not the least direct physical
significance, but only serve the purpose of numbering the points of
the continuum in a definite but arbitrary manner. This arrangement
does not even need to be of such a kind that we must regard x[1],
x[2], x[3], as "space" co-ordinates and x[4], as a " time "
co-ordinate.
The reader may think that such a description of the world would be
quite inadequate. What does it mean to assign to an event the
particular co-ordinates x[1], x[2], x[3], x[4], if in themselves these
co-ordinates have no significance ? More careful consideration shows,
however, that this anxiety is unfounded. Let us consider, for
instance, a material point with any kind of motion. If this point had
only a momentary existence without duration, then it would to
described in space-time by a single system of values x[1], x[2], x[3],
x[4]. Thus its permanent existence must be characterised by an
infinitely large number of such systems of values, the co-ordinate
values of which are so close together as to give continuity;
corresponding to the material point, we thus have a (uni-dimensional)
line in the four-dimensional continuum. In the same way, any such
lines in our continuum correspond to many points in motion. The only
statements having regard to these points which can claim a physical
existence are in reality the statements about their encounters. In our
mathematical treatment, such an encounter is expressed in the fact
that the two lines which represent the motions of the points in
question have a particular system of co-ordinate values, x[1], x[2],
x[3], x[4], in common. After mature consideration the reader will
doubtless admit that in reality such encounters constitute the only
actual evidence of a time-space nature with which we meet in physical
statements.
When we were describing the motion of a material point relative to a
body of reference, we stated nothing more than the encounters of this
point with particular points of the reference-body. We can also
determine the corresponding values of the time by the observation of
encounters of the body with clocks, in conjunction with the
observation of the encounter of the hands of clocks with particular
points on the dials. It is just the same in the case of
space-measurements by means of measuring-rods, as a litttle
consideration will show.
The following statements hold generally : Every physical description
resolves itself into a number of statements, each of which refers to
the space-time coincidence of two events A and B. In terms of Gaussian
co-ordinates, every such statement is expressed by the agreement of
their four co-ordinates x[1], x[2], x[3], x[4]. Thus in reality, the
description of the time-space continuum by means of Gauss co-ordinates
completely replaces the description with the aid of a body of
reference, without suffering from the defects of the latter mode of
description; it is not tied down to the Euclidean character of the
continuum which has to be represented.
EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
We are now in a position to replace the pro. visional formulation of
the general principle of relativity given in Section 18 by an exact
formulation. The form there used, "All bodies of reference K, K1,
etc., are equivalent for the description of natural phenomena
(formulation of the general laws of nature), whatever may be their
state of motion," cannot be maintained, because the use of rigid
reference-bodies, in the sense of the method followed in the special
theory of relativity, is in general not possible in space-time
description. The Gauss co-ordinate system has to take the place of the
body of reference. The following statement corresponds to the
fundamental idea of the general principle of relativity: "All Gaussian
co-ordinate systems are essentially equivalent for the formulation of
the general laws of nature."
We can state this general principle of relativity in still another
form, which renders it yet more clearly intelligible than it is when
in the form of the natural extension of the special principle of
relativity. According to the special theory of relativity, the
equations which express the general laws of nature pass over into
equations of the same form when, by making use of the Lorentz
transformation, we replace the space-time variables x, y, z, t, of a
(Galileian) reference-body K by the space-time variables x1, y1, z1,
t1, of a new reference-body K1. According to the general theory of
relativity, on the other hand, by application of arbitrary
substitutions of the Gauss variables x[1], x[2], x[3], x[4], the
equations must pass over into equations of the same form; for every
transformation (not only the Lorentz transformation) corresponds to
the transition of one Gauss co-ordinate system into another.
If we desire to adhere to our "old-time" three-dimensional view of
things, then we can characterise the development which is being
undergone by the fundamental idea of the general theory of relativity
as follows : The special theory of relativity has reference to
Galileian domains, i.e. to those in which no gravitational field
exists. In this connection a Galileian reference-body serves as body
of reference, i.e. a rigid body the state of motion of which is so
chosen that the Galileian law of the uniform rectilinear motion of
"isolated" material points holds relatively to it.
Certain considerations suggest that we should refer the same Galileian
domains to non-Galileian reference-bodies also. A gravitational field
of a special kind is then present with respect to these bodies (cf.
Sections 20 and 23).
In gravitational fields there are no such things as rigid bodies with
Euclidean properties; thus the fictitious rigid body of reference is
of no avail in the general theory of relativity. The motion of clocks
is also influenced by gravitational fields, and in such a way that a
physical definition of time which is made directly with the aid of
clocks has by no means the same degree of plausibility as in the
special theory of relativity.
For this reason non-rigid reference-bodies are used, which are as a
whole not only moving in any way whatsoever, but which also suffer
alterations in form ad lib. during their motion. Clocks, for which the
law of motion is of any kind, however irregular, serve for the
definition of time. We have to imagine each of these clocks fixed at a
point on the non-rigid reference-body. These clocks satisfy only the
one condition, that the "readings" which are observed simultaneously
on adjacent clocks (in space) differ from each other by an
indefinitely small amount. This non-rigid reference-body, which might
appropriately be termed a "reference-mollusc", is in the main
equivalent to a Gaussian four-dimensional co-ordinate system chosen
arbitrarily. That which gives the "mollusc" a certain
comprehensibility as compared with the Gauss co-ordinate system is the
(really unjustified) formal retention of the separate existence of the
space co-ordinates as opposed to the time co-ordinate. Every point on
the mollusc is treated as a space-point, and every material point
which is at rest relatively to it as at rest, so long as the mollusc
is considered as reference-body. The general principle of relativity
requires that all these molluscs can be used as reference-bodies with
equal right and equal success in the formulation of the general laws
of nature; the laws themselves must be quite independent of the choice
of mollusc.
The great power possessed by the general principle of relativity lies
in the comprehensive limitation which is imposed on the laws of nature
in consequence of what we have seen above.
THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
PRINCIPLE OF RELATIVITY
If the reader has followed all our previous considerations, he will
have no further difficulty in understanding the methods leading to the
solution of the problem of gravitation.
We start off on a consideration of a Galileian domain, i.e. a domain
in which there is no gravitational field relative to the Galileian
reference-body K. The behaviour of measuring-rods and clocks with
reference to K is known from the special theory of relativity,
likewise the behaviour of "isolated" material points; the latter move
uniformly and in straight lines.
Now let us refer this domain to a random Gauss coordinate system or to
a "mollusc" as reference-body K1. Then with respect to K1 there is a
gravitational field G (of a particular kind). We learn the behaviour
of measuring-rods and clocks and also of freely-moving material points
with reference to K1 simply by mathematical transformation. We
interpret this behaviour as the behaviour of measuring-rods, docks and
material points tinder the influence of the gravitational field G.
Hereupon we introduce a hypothesis: that the influence of the
gravitational field on measuringrods, clocks and freely-moving
material points continues to take place according to the same laws,
even in the case where the prevailing gravitational field is not
derivable from the Galfleian special care, simply by means of a
transformation of co-ordinates.
The next step is to investigate the space-time behaviour of the
gravitational field G, which was derived from the Galileian special
case simply by transformation of the coordinates. This behaviour is
formulated in a law, which is always valid, no matter how the
reference-body (mollusc) used in the description may be chosen.
This law is not yet the general law of the gravitational field, since
the gravitational field under consideration is of a special kind. In
order to find out the general law-of-field of gravitation we still
require to obtain a generalisation of the law as found above. This can
be obtained without caprice, however, by taking into consideration the
following demands:
(a) The required generalisation must likewise satisfy the general
postulate of relativity.
(b) If there is any matter in the domain under consideration, only its
inertial mass, and thus according to Section 15 only its energy is
of importance for its etfect in exciting a field.
(c) Gravitational field and matter together must satisfy the law of
the conservation of energy (and of impulse).
Finally, the general principle of relativity permits us to determine
the influence of the gravitational field on the course of all those
processes which take place according to known laws when a
gravitational field is absent i.e. which have already been fitted into
the frame of the special theory of relativity. In this connection we
proceed in principle according to the method which has already been
explained for measuring-rods, clocks and freely moving material
points.
The theory of gravitation derived in this way from the general
postulate of relativity excels not only in its beauty ; nor in
removing the defect attaching to classical mechanics which was brought
to light in Section 21; nor in interpreting the empirical law of
the equality of inertial and gravitational mass ; but it has also
already explained a result of observation in astronomy, against which
classical mechanics is powerless.
If we confine the application of the theory to the case where the
gravitational fields can be regarded as being weak, and in which all
masses move with respect to the coordinate system with velocities
which are small compared with the velocity of light, we then obtain as
a first approximation the Newtonian theory. Thus the latter theory is
obtained here without any particular assumption, whereas Newton had to
introduce the hypothesis that the force of attraction between mutually
attracting material points is inversely proportional to the square of
the distance between them. If we increase the accuracy of the
calculation, deviations from the theory of Newton make their
appearance, practically all of which must nevertheless escape the test
of observation owing to their smallness.
We must draw attention here to one of these deviations. According to
Newton's theory, a planet moves round the sun in an ellipse, which
would permanently maintain its position with respect to the fixed
stars, if we could disregard the motion of the fixed stars themselves
and the action of the other planets under consideration. Thus, if we
correct the observed motion of the planets for these two influences,
and if Newton's theory be strictly correct, we ought to obtain for the
orbit of the planet an ellipse, which is fixed with reference to the
fixed stars. This deduction, which can be tested with great accuracy,
has been confirmed for all the planets save one, with the precision
that is capable of being obtained by the delicacy of observation
attainable at the present time. The sole exception is Mercury, the
planet which lies nearest the sun. Since the time of Leverrier, it has
been known that the ellipse corresponding to the orbit of Mercury,
after it has been corrected for the influences mentioned above, is not
stationary with respect to the fixed stars, but that it rotates
exceedingly slowly in the plane of the orbit and in the sense of the
orbital motion. The value obtained for this rotary movement of the
orbital ellipse was 43 seconds of arc per century, an amount ensured
to be correct to within a few seconds of arc. This effect can be
explained by means of classical mechanics only on the assumption of
hypotheses which have little probability, and which were devised
solely for this purponse.
On the basis of the general theory of relativity, it is found that the
ellipse of every planet round the sun must necessarily rotate in the
manner indicated above ; that for all the planets, with the exception
of Mercury, this rotation is too small to be detected with the
delicacy of observation possible at the present time ; but that in the
case of Mercury it must amount to 43 seconds of arc per century, a
result which is strictly in agreement with observation.
Apart from this one, it has hitherto been possible to make only two
deductions from the theory which admit of being tested by observation,
to wit, the curvature of light rays by the gravitational field of the
sun,*x and a displacement of the spectral lines of light reaching
us from large stars, as compared with the corresponding lines for
light produced in an analogous manner terrestrially (i.e. by the same
kind of atom).**  These two deductions from the theory have both
been confirmed.
Notes
*) First observed by Eddington and others in 1919. (Cf. Appendix
III, pp. 126-129).
**) Established by Adams in 1924. (Cf. p. 132)
PART III
CONSIDERATIONS ON THE UNIVERSE AS A WHOLE
COSMOLOGICAL DIFFICULTIES OF NEWTON'S THEORY
Part from the difficulty discussed in Section 21, there is a second
fundamental difficulty attending classical celestial mechanics, which,
to the best of my knowledge, was first discussed in detail by the
astronomer Seeliger. If we ponder over the question as to how the
universe, considered as a whole, is to be regarded, the first answer
that suggests itself to us is surely this: As regards space (and time)
the universe is infinite. There are stars everywhere, so that the
density of matter, although very variable in detail, is nevertheless
on the average everywhere the same. In other words: However far we
might travel through space, we should find everywhere an attenuated
swarm of fixed stars of approrimately the same kind and density.
This view is not in harmony with the theory of Newton. The latter
theory rather requires that the universe should have a kind of centre
in which the density of the stars is a maximum, and that as we proceed
outwards from this centre the group-density of the stars should
diminish, until finally, at great distances, it is succeeded by an
infinite region of emptiness. The stellar universe ought to be a
finite island in the infinite ocean of space.*
This conception is in itself not very satisfactory. It is still less
satisfactory because it leads to the result that the light emitted by
the stars and also individual stars of the stellar system are
perpetually passing out into infinite space, never to return, and
without ever again coming into interaction with other objects of
nature. Such a finite material universe would be destined to become
gradually but systematically impoverished.
In order to escape this dilemma, Seeliger suggested a modification of
Newton's law, in which he assumes that for great distances the force
of attraction between two masses diminishes more rapidly than would
result from the inverse square law. In this way it is possible for the
mean density of matter to be constant everywhere, even to infinity,
without infinitely large gravitational fields being produced. We thus
free ourselves from the distasteful conception that the material
universe ought to possess something of the nature of a centre. Of
course we purchase our emancipation from the fundamental difficulties
mentioned, at the cost of a modification and complication of Newton's
law which has neither empirical nor theoretical foundation. We can
imagine innumerable laws which would serve the same purpose, without
our being able to state a reason why one of them is to be preferred to
the others ; for any one of these laws would be founded just as little
on more general theoretical principles as is the law of Newton.
Notes
*) Proof -- According to the theory of Newton, the number of "lines
of force" which come from infinity and terminate in a mass m is
proportional to the mass m. If, on the average, the Mass density p[0]
is constant throughout tithe universe, then a sphere of volume V will
enclose the average man p[0]V. Thus the number of lines of force
passing through the surface F of the sphere into its interior is
proportional to p[0] V. For unit area of the surface of the sphere the
number of lines of force which enters the sphere is thus proportional
to p[0] V/F or to p[0]R. Hence the intensity of the field at the
surface would ultimately become infinite with increasing radius R of
the sphere, which is impossible.
THE POSSIBILITY OF A "FINITE" AND YET "UNBOUNDED" UNIVERSE
But speculations on the structure of the universe also move in quite
another direction. The development of non-Euclidean geometry led to
the recognition of the fact, that we can cast doubt on the
infiniteness of our space without coming into conflict with the laws
of thought or with experience (Riemann, Helmholtz). These questions
have already been treated in detail and with unsurpassable lucidity by
Helmholtz and Poincar�, whereas I can only touch on them briefly here.
In the first place, we imagine an existence in two dimensional space.
Flat beings with flat implements, and in particular flat rigid
measuring-rods, are free to move in a plane. For them nothing exists
outside of this plane: that which they observe to happen to themselves
and to their flat " things " is the all-inclusive reality of their
plane. In particular, the constructions of plane Euclidean geometry
can be carried out by means of the rods e.g. the lattice construction,
considered in Section 24. In contrast to ours, the universe of
these beings is two-dimensional; but, like ours, it extends to
infinity. In their universe there is room for an infinite number of
identical squares made up of rods, i.e. its volume (surface) is
infinite. If these beings say their universe is " plane," there is
sense in the statement, because they mean that they can perform the
constructions of plane Euclidean geometry with their rods. In this
connection the individual rods always represent the same distance,
independently of their position.
Let us consider now a second two-dimensional existence, but this time
on a spherical surface instead of on a plane. The flat beings with
their measuring-rods and other objects fit exactly on this surface and
they are unable to leave it. Their whole universe of observation
extends exclusively over the surface of the sphere. Are these beings
able to regard the geometry of their universe as being plane geometry
and their rods withal as the realisation of " distance " ? They cannot
do this. For if they attempt to realise a straight line, they will
obtain a curve, which we " three-dimensional beings " designate as a
great circle, i.e. a self-contained line of definite finite length,
which can be measured up by means of a measuring-rod. Similarly, this
universe has a finite area that can be compared with the area, of a
square constructed with rods. The great charm resulting from this
consideration lies in the recognition of the fact that the universe of
these beings is finite and yet has no limits.
But the spherical-surface beings do not need to go on a world-tour in
order to perceive that they are not living in a Euclidean universe.
They can convince themselves of this on every part of their " world,"
provided they do not use too small a piece of it. Starting from a
point, they draw " straight lines " (arcs of circles as judged in
three dimensional space) of equal length in all directions. They will
call the line joining the free ends of these lines a " circle." For a
plane surface, the ratio of the circumference of a circle to its
diameter, both lengths being measured with the same rod, is, according
to Euclidean geometry of the plane, equal to a constant value p, which
is independent of the diameter of the circle. On their spherical
surface our flat beings would find for this ratio the value
eq. 27: file eq27.gif
i.e. a smaller value than p, the difference being the more
considerable, the greater is the radius of the circle in comparison
with the radius R of the " world-sphere." By means of this relation
the spherical beings can determine the radius of their universe ("
world "), even when only a relatively small part of their worldsphere
is available for their measurements. But if this part is very small
indeed, they will no longer be able to demonstrate that they are on a
spherical " world " and not on a Euclidean plane, for a small part of
a spherical surface differs only slightly from a piece of a plane of
the same size.
Thus if the spherical surface beings are living on a planet of which
the solar system occupies only a negligibly small part of the
spherical universe, they have no means of determining whether they are
living in a finite or in an infinite universe, because the " piece of
universe " to which they have access is in both cases practically
plane, or Euclidean. It follows directly from this discussion, that
for our sphere-beings the circumference of a circle first increases
with the radius until the " circumference of the universe " is
reached, and that it thenceforward gradually decreases to zero for
still further increasing values of the radius. During this process the
area of the circle continues to increase more and more, until finally
it becomes equal to the total area of the whole " world-sphere."
Perhaps the reader will wonder why we have placed our " beings " on a
sphere rather than on another closed surface. But this choice has its
justification in the fact that, of all closed surfaces, the sphere is
unique in possessing the property that all points on it are
equivalent. I admit that the ratio of the circumference c of a circle
to its radius r depends on r, but for a given value of r it is the
same for all points of the " worldsphere "; in other words, the "
world-sphere " is a " surface of constant curvature."
To this two-dimensional sphere-universe there is a three-dimensional
analogy, namely, the three-dimensional spherical space which was
discovered by Riemann. its points are likewise all equivalent. It
possesses a finite volume, which is determined by its "radius"
(2p2R3). Is it possible to imagine a spherical space? To imagine a
space means nothing else than that we imagine an epitome of our "
space " experience, i.e. of experience that we can have in the
movement of " rigid " bodies. In this sense we can imagine a spherical
space.
Suppose we draw lines or stretch strings in all directions from a
point, and mark off from each of these the distance r with a
measuring-rod. All the free end-points of these lengths lie on a
spherical surface. We can specially measure up the area (F) of this
surface by means of a square made up of measuring-rods. If the
universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is
always less than 4pR2. With increasing values of r, F increases from
zero up to a maximum value which is determined by the " world-radius,"
but for still further increasing values of r, the area gradually
diminishes to zero. At first, the straight lines which radiate from
the starting point diverge farther and farther from one another, but
later they approach each other, and finally they run together again at
a "counter-point" to the starting point. Under such conditions they
have traversed the whole spherical space. It is easily seen that the
three-dimensional spherical space is quite analogous to the
two-dimensional spherical surface. It is finite (i.e. of finite
volume), and has no bounds.
It may be mentioned that there is yet another kind of curved space: "
elliptical space." It can be regarded as a curved space in which the
two " counter-points " are identical (indistinguishable from each
other). An elliptical universe can thus be considered to some extent
as a curved universe possessing central symmetry.
It follows from what has been said, that closed spaces without limits
are conceivable. From amongst these, the spherical space (and the
elliptical) excels in its simplicity, since all points on it are
equivalent. As a result of this discussion, a most interesting
question arises for astronomers and physicists, and that is whether
the universe in which we live is infinite, or whether it is finite in
the manner of the spherical universe. Our experience is far from being
sufficient to enable us to answer this question. But the general
theory of relativity permits of our answering it with a moduate degree
of certainty, and in this connection the difficulty mentioned in
Section 30 finds its solution.
THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
According to the general theory of relativity, the geometrical
properties of space are not independent, but they are determined by
matter. Thus we can draw conclusions about the geometrical structure
of the universe only if we base our considerations on the state of the
matter as being something that is known. We know from experience that,
for a suitably chosen co-ordinate system, the velocities of the stars
are small as compared with the velocity of transmission of light. We
can thus as a rough approximation arrive at a conclusion as to the
nature of the universe as a whole, if we treat the matter as being at
rest.
We already know from our previous discussion that the behaviour of
measuring-rods and clocks is influenced by gravitational fields, i.e.
by the distribution of matter. This in itself is sufficient to exclude
the possibility of the exact validity of Euclidean geometry in our
universe. But it is conceivable that our universe differs only
slightly from a Euclidean one, and this notion seems all the more
probable, since calculations show that the metrics of surrounding
space is influenced only to an exceedingly small extent by masses even
of the magnitude of our sun. We might imagine that, as regards
geometry, our universe behaves analogously to a surface which is
irregularly curved in its individual parts, but which nowhere departs
appreciably from a plane: something like the rippled surface of a
lake. Such a universe might fittingly be called a quasi-Euclidean
universe. As regards its space it would be infinite. But calculation
shows that in a quasi-Euclidean universe the average density of matter
would necessarily be nil. Thus such a universe could not be inhabited
by matter everywhere ; it would present to us that unsatisfactory
picture which we portrayed in Section 30.
If we are to have in the universe an average density of matter which
differs from zero, however small may be that difference, then the
universe cannot be quasi-Euclidean. On the contrary, the results of
calculation indicate that if matter be distributed uniformly, the
universe would necessarily be spherical (or elliptical). Since in
reality the detailed distribution of matter is not uniform, the real
universe will deviate in individual parts from the spherical, i.e. the
universe will be quasi-spherical. But it will be necessarily finite.
In fact, the theory supplies us with a simple connection *  between
the space-expanse of the universe and the average density of matter in
it.
Notes
*) For the radius R of the universe we obtain the equation
eq. 28: file eq28.gif
The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27;
p is the average density of the matter and k is a constant connected
with the Newtonian constant of gravitation.
APPENDIX I
SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION
(SUPPLEMENTARY TO SECTION 11)
For the relative orientation of the co-ordinate systems indicated in
Fig. 2, the x-axes of both systems pernumently coincide. In the
present case we can divide the problem into parts by considering first
only events which are localised on the x-axis. Any such event is
represented with respect to the co-ordinate system K by the abscissa x
and the time t, and with respect to the system K1 by the abscissa x'
and the time t'. We require to find x' and t' when x and t are given.
A light-signal, which is proceeding along the positive axis of x, is
transmitted according to the equation
x = ct
or
x - ct = 0     .     .     .    (1).
Since the same light-signal has to be transmitted relative to K1 with
the velocity c, the propagation relative to the system K1 will be
represented by the analogous formula
x' - ct' = O     .     .     .    (2)
Those space-time points (events) which satisfy (x) must also satisfy
(2). Obviously this will be the case when the relation
(x' - ct') = l (x - ct)     .     .     .    (3).
is fulfilled in general, where l indicates a constant ; for, according
to (3), the disappearance of (x - ct) involves the disappearance of
(x' - ct').
If we apply quite similar considerations to light rays which are being
transmitted along the negative x-axis, we obtain the condition
(x' + ct') = �(x + ct)    .     .     .    (4).
By adding (or subtracting) equations (3) and (4), and introducing for
convenience the constants a and b in place of the constants l and �,
where
eq. 29: file eq29.gif
and
eq. 30: file eq30.gif
we obtain the equations
eq. 31: file eq31.gif
We should thus have the solution of our problem, if the constants a
and b were known. These result from the following discussion.
For the origin of K1 we have permanently x' = 0, and hence according
to the first of the equations (5)
eq. 32: file eq32.gif
If we call v the velocity with which the origin of K1 is moving
relative to K, we then have
eq. 33: file eq33.gif
The same value v can be obtained from equations (5), if we calculate
the velocity of another point of K1 relative to K, or the velocity
(directed towards the negative x-axis) of a point of K with respect to
K'. In short, we can designate v as the relative velocity of the two
systems.
Furthermore, the principle of relativity teaches us that, as judged
from K, the length of a unit measuring-rod which is at rest with
reference to K1 must be exactly the same as the length, as judged from
K', of a unit measuring-rod which is at rest relative to K. In order
to see how the points of the x-axis appear as viewed from K, we only
require to take a " snapshot " of K1 from K; this means that we have
to insert a particular value of t (time of K), e.g. t = 0. For this
value of t we then obtain from the first of the equations (5)
x' = ax
Two points of the x'-axis which are separated by the distance Dx' = I
when measured in the K1 system are thus separated in our instantaneous
photograph by the distance
eq. 34: file eq34.gif
But if the snapshot be taken from K'(t' = 0), and if we eliminate t
from the equations (5), taking into account the expression (6), we
obtain
eq. 35: file eq35.gif
From this we conclude that two points on the x-axis separated by the
distance I (relative to K) will be represented on our snapshot by the
distance
eq. 36: file eq36.gif
But from what has been said, the two snapshots must be identical;
hence Dx in (7) must be equal to Dx' in (7a), so that we obtain
eq. 37: file eq37.gif
The equations (6) and (7b) determine the constants a and b. By
inserting the values of these constants in (5), we obtain the first
and the fourth of the equations given in Section 11.
eq. 38: file eq38.gif
Thus we have obtained the Lorentz transformation for events on the
x-axis. It satisfies the condition
x'2 - c^2t'2 = x2 - c^2t2    .     .     .    (8a).
The extension of this result, to include events which take place
outside the x-axis, is obtained by retaining equations (8) and
supplementing them by the relations
eq. 39: file eq39.gif
In this way we satisfy the postulate of the constancy of the velocity
of light in vacuo for rays of light of arbitrary direction, both for
the system K and for the system K'. This may be shown in the following
manner.
We suppose a light-signal sent out from the origin of K at the time t
= 0. It will be propagated according to the equation
eq. 40: file eq40.gif
or, if we square this equation, according to the equation
x2 + y2 + z2 = c^2t2 = 0    .     .     .    (10).
It is required by the law of propagation of light, in conjunction with
the postulate of relativity, that the transmission of the signal in
question should take place -- as judged from K1 -- in accordance with
the corresponding formula
r' = ct'
or,
x'2 + y'2 + z'2 - c^2t'2 = 0    .     .     .    (10a).
In order that equation (10a) may be a consequence of equation (10), we
must have
x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2)       (11).
Since equation (8a) must hold for points on the x-axis, we thus have s
= I. It is easily seen that the Lorentz transformation really
satisfies equation (11) for s = I; for (11) is a consequence of (8a)
and (9), and hence also of (8) and (9). We have thus derived the
Lorentz transformation.
The Lorentz transformation represented by (8) and (9) still requires
to be generalised. Obviously it is immaterial whether the axes of K1
be chosen so that they are spatially parallel to those of K. It is
also not essential that the velocity of translation of K1 with respect
to K should be in the direction of the x-axis. A simple consideration
shows that we are able to construct the Lorentz transformation in this
general sense from two kinds of transformations, viz. from Lorentz
transformations in the special sense and from purely spatial
transformations. which corresponds to the replacement of the
rectangular co-ordinate system by a new system with its axes pointing
in other directions.
Mathematically, we can characterise the generalised Lorentz
transformation thus :
It expresses x', y', x', t', in terms of linear homogeneous functions
of x, y, x, t, of such a kind that the relation
x'2 + y'2 + z'2 - c^2t'2 = x2 + y2 + z2 - c^2t2       (11a).
is satisficd identically. That is to say: If we substitute their
expressions in x, y, x, t, in place of x', y', x', t', on the
left-hand side, then the left-hand side of (11a) agrees with the
right-hand side.
APPENDIX II
MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD")
(SUPPLEMENTARY TO SECTION 17)
We can characterise the Lorentz transformation still more simply if we
introduce the imaginary eq. 25 in place of t, as time-variable. If, in
accordance with this, we insert
x[1] = x
x[2] = y
x[3] = z
x[4] = eq. 25
and similarly for the accented system K1, then the condition which is
identically satisfied by the transformation can be expressed thus :
x[1]'2 + x[2]'2 + x[3]'2 + x[4]'2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2
(12).
That is, by the afore-mentioned choice of " coordinates," (11a) [see
the end of Appendix II] is transformed into this equation.
We see from (12) that the imaginary time co-ordinate x[4], enters into
the condition of transformation in exactly the same way as the space
co-ordinates x[1], x[2], x[3]. It is due to this fact that, according
to the theory of relativity, the " time "x[4], enters into natural
laws in the same form as the space co ordinates x[1], x[2], x[3].
A four-dimensional continuum described by the "co-ordinates" x[1],
x[2], x[3], x[4], was called "world" by Minkowski, who also termed a
point-event a " world-point." From a "happening" in three-dimensional
space, physics becomes, as it were, an " existence " in the
four-dimensional " world."
This four-dimensional " world " bears a close similarity to the
three-dimensional " space " of (Euclidean) analytical geometry. If we
introduce into the latter a new Cartesian co-ordinate system (x'[1],
x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are
linear homogeneous functions of x[1], x[2], x[3] which identically
satisfy the equation
x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2
The analogy with (12) is a complete one. We can regard Minkowski's "
world " in a formal manner as a four-dimensional Euclidean space (with
an imaginary time coordinate) ; the Lorentz transformation corresponds
to a " rotation " of the co-ordinate system in the fourdimensional "
world."
APPENDIX III
THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY
From a systematic theoretical point of view, we may imagine the
process of evolution of an empirical science to be a continuous
process of induction. Theories are evolved and are expressed in short
compass as statements of a large number of individual observations in
the form of empirical laws, from which the general laws can be
ascertained by comparison. Regarded in this way, the development of a
science bears some resemblance to the compilation of a classified
catalogue. It is, as it were, a purely empirical enterprise.
But this point of view by no means embraces the whole of the actual
process ; for it slurs over the important part played by intuition and
deductive thought in the development of an exact science. As soon as a
science has emerged from its initial stages, theoretical advances are
no longer achieved merely by a process of arrangement. Guided by
empirical data, the investigator rather develops a system of thought
which, in general, is built up logically from a small number of
fundamental assumptions, the so-called axioms. We call such a system
of thought a theory. The theory finds the justification for its
existence in the fact that it correlates a large number of single
observations, and it is just here that the " truth " of the theory
lies.
Corresponding to the same complex of empirical data, there may be
several theories, which differ from one another to a considerable
extent. But as regards the deductions from the theories which are
capable of being tested, the agreement between the theories may be so
complete that it becomes difficult to find any deductions in which the
two theories differ from each other. As an example, a case of general
interest is available in the province of biology, in the Darwinian
theory of the development of species by selection in the struggle for
existence, and in the theory of development which is based on the
hypothesis of the hereditary transmission of acquired characters.
We have another instance of far-reaching agreement between the
deductions from two theories in Newtonian mechanics on the one hand,
and the general theory of relativity on the other. This agreement goes
so far, that up to the preseat we have been able to find only a few
deductions from the general theory of relativity which are capable of
investigation, and to which the physics of pre-relativity days does
not also lead, and this despite the profound difference in the
fundamental assumptions of the two theories. In what follows, we shall
again consider these important deductions, and we shall also discuss
the empirical evidence appertaining to them which has hitherto been
obtained.
(a) Motion of the Perihelion of Mercury
According to Newtonian mechanics and Newton's law of gravitation, a
planet which is revolving round the sun would describe an ellipse
round the latter, or, more correctly, round the common centre of
gravity of the sun and the planet. In such a system, the sun, or the
common centre of gravity, lies in one of the foci of the orbital
ellipse in such a manner that, in the course of a planet-year, the
distance sun-planet grows from a minimum to a maximum, and then
decreases again to a minimum. If instead of Newton's law we insert a
somewhat different law of attraction into the calculation, we find
that, according to this new law, the motion would still take place in
such a manner that the distance sun-planet exhibits periodic
variations; but in this case the angle described by the line joining
sun and planet during such a period (from perihelion--closest
proximity to the sun--to perihelion) would differ from 360^0. The line
of the orbit would not then be a closed one but in the course of time
it would fill up an annular part of the orbital plane, viz. between
the circle of least and the circle of greatest distance of the planet
from the sun.
According also to the general theory of relativity, which differs of
course from the theory of Newton, a small variation from the
Newton-Kepler motion of a planet in its orbit should take place, and
in such away, that the angle described by the radius sun-planet
between one perhelion and the next should exceed that corresponding to
one complete revolution by an amount given by
eq. 41: file eq41.gif
(N.B. -- One complete revolution corresponds to the angle 2p in the
absolute angular measure customary in physics, and the above
expression giver the amount by which the radius sun-planet exceeds
this angle during the interval between one perihelion and the next.)
In this expression a represents the major semi-axis of the ellipse, e
its eccentricity, c the velocity of light, and T the period of
revolution of the planet. Our result may also be stated as follows :
According to the general theory of relativity, the major axis of the
ellipse rotates round the sun in the same sense as the orbital motion
of the planet. Theory requires that this rotation should amount to 43
seconds of arc per century for the planet Mercury, but for the other
Planets of our solar system its magnitude should be so small that it
would necessarily escape detection. *
In point of fact, astronomers have found that the theory of Newton
does not suffice to calculate the observed motion of Mercury with an
exactness corresponding to that of the delicacy of observation
attainable at the present time. After taking account of all the
disturbing influences exerted on Mercury by the remaining planets, it
was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
perihelial movement of the orbit of Mercury remained over, the amount
of which does not differ sensibly from the above mentioned +43 seconds
of arc per century. The uncertainty of the empirical result amounts to
a few seconds only.
(b) Deflection of Light by a Gravitational Field
In Section 22 it has been already mentioned that according to the
general theory of relativity, a ray of light will experience a
curvature of its path when passing through a gravitational field, this
curvature being similar to that experienced by the path of a body
which is projected through a gravitational field. As a result of this
theory, we should expect that a ray of light which is passing close to
a heavenly body would be deviated towards the latter. For a ray of
light which passes the sun at a distance of D sun-radii from its
centre, the angle of deflection (a) should amount to
eq. 42: file eq42.gif
It may be added that, according to the theory, half of Figure 05 this
deflection is produced by the Newtonian field of attraction of the
sun, and the other half by the geometrical modification (" curvature
") of space caused by the sun.
This result admits of an experimental test by means of the
photographic registration of stars during a total eclipse of the sun.
The only reason why we must wait for a total eclipse is because at
every other time the atmosphere is so strongly illuminated by the
light from the sun that the stars situated near the sun's disc are
invisible. The predicted effect can be seen clearly from the
accompanying diagram. If the sun (S) were not present, a star which is
practically infinitely distant would be seen in the direction D[1], as
observed front the earth. But as a consequence of the deflection of
light from the star by the sun, the star will be seen in the direction
D[2], i.e. at a somewhat greater distance from the centre of the sun
than corresponds to its real position.
In practice, the question is tested in the following way. The stars in
the neighbourhood of the sun are photographed during a solar eclipse.
In addition, a second photograph of the same stars is taken when the
sun is situated at another position in the sky, i.e. a few months
earlier or later. As compared whh the standard photograph, the
positions of the stars on the eclipse-photograph ought to appear
displaced radially outwards (away from the centre of the sun) by an
amount corresponding to the angle a.
We are indebted to the [British] Royal Society and to the Royal
Astronomical Society for the investigation of this important
deduction. Undaunted by the [first world] war and by difficulties of
both a material and a psychological nature aroused by the war, these
societies equipped two expeditions -- to Sobral (Brazil), and to the
island of Principe (West Africa) -- and sent several of Britain's most
celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson),
in order to obtain photographs of the solar eclipse of 29th May, 1919.
The relative discrepancies to be expected between the stellar
photographs obtained during the eclipse and the comparison photographs
amounted to a few hundredths of a millimetre only. Thus great accuracy
was necessary in making the adjustments required for the taking of the
photographs, and in their subsequent measurement.
The results of the measurements confirmed the theory in a thoroughly
satisfactory manner. The rectangular components of the observed and of
the calculated deviations of the stars (in seconds of arc) are set
forth in the following table of results :
Table 01: file table01.gif
(c) Displacement of Spectral Lines Towards the Red
In Section 23 it has been shown that in a system K1 which is in
rotation with regard to a Galileian system K, clocks of identical
construction, and which are considered at rest with respect to the
rotating reference-body, go at rates which are dependent on the
positions of the clocks. We shall now examine this dependence
quantitatively. A clock, which is situated at a distance r from the
centre of the disc, has a velocity relative to K which is given by
V = wr
where w represents the angular velocity of rotation of the disc K1
with respect to K. If v[0], represents the number of ticks of the
clock per unit time (" rate " of the clock) relative to K when the
clock is at rest, then the " rate " of the clock (v) when it is moving
relative to K with a velocity V, but at rest with respect to the disc,
will, in accordance with Section 12, be given by
eq. 43: file eq43.gif
or with sufficient accuracy by
eq. 44: file eq44.gif
This expression may also be stated in the following form:
eq. 45: file eq45.gif
If we represent the difference of potential of the centrifugal force
between the position of the clock and the centre of the disc by f,
i.e. the work, considered negatively, which must be performed on the
unit of mass against the centrifugal force in order to transport it
from the position of the clock on the rotating disc to the centre of
the disc, then we have
eq. 46: file eq46.gif
From this it follows that
eq. 47: file eq47.gif
In the first place, we see from this expression that two clocks of
identical construction will go at different rates when situated at
different distances from the centre of the disc. This result is aiso
valid from the standpoint of an observer who is rotating with the
disc.
Now, as judged from the disc, the latter is in a gravititional field
of potential f, hence the result we have obtained will hold quite
generally for gravitational fields. Furthermore, we can regard an atom
which is emitting spectral lines as a clock, so that the following
statement will hold:
An atom absorbs or emits light of a frequency which is dependent on
the potential of the gravitational field in which it is situated.
The frequency of an atom situated on the surface of a heavenly body
will be somewhat less than the frequency of an atom of the same
element which is situated in free space (or on the surface of a
smaller celestial body).
Now f = - K (M/r), where K is Newton's constant of gravitation, and M
is the mass of the heavenly body. Thus a displacement towards the red
ought to take place for spectral lines produced at the surface of
stars as compared with the spectral lines of the same element produced
at the surface of the earth, the amount of this displacement being
eq. 48: file eq48.gif
For the sun, the displacement towards the red predicted by theory
amounts to about two millionths of the wave-length. A trustworthy
calculation is not possible in the case of the stars, because in
general neither the mass M nor the radius r are known.
It is an open question whether or not this effect exists, and at the
present time (1920) astronomers are working with great zeal towards
the solution. Owing to the smallness of the effect in the case of the
sun, it is difficult to form an opinion as to its existence. Whereas
Grebe and Bachem (Bonn), as a result of their own measurements and
those of Evershed and Schwarzschild on the cyanogen bands, have placed
the existence of the effect almost beyond doubt, while other
investigators, particularly St. John, have been led to the opposite
opinion in consequence of their measurements.
Mean displacements of lines towards the less refrangible end of the
spectrum are certainly revealed by statistical investigations of the
fixed stars ; but up to the present the examination of the available
data does not allow of any definite decision being arrived at, as to
whether or not these displacements are to be referred in reality to
the effect of gravitation. The results of observation have been
collected together, and discussed in detail from the standpoint of the
question which has been engaging our attention here, in a paper by E.
Freundlich entitled "Zur Pr�fung der allgemeinen
Relativit&umlaut;ts-Theorie" (Die Naturwissenschaften, 1919, No. 35,
p. 520: Julius Springer, Berlin).
At all events, a definite decision will be reached during the next few
years. If the displacement of spectral lines towards the red by the
gravitational potential does not exist, then the general theory of
relativity will be untenable. On the other hand, if the cause of the
displacement of spectral lines be definitely traced to the
gravitational potential, then the study of this displacement will
furnish us with important information as to the mass of the heavenly
bodies. [5][A]
Notes
*) Especially since the next planet Venus has an orbit that is
almost an exact circle, which makes it more difficult to locate the
perihelion with precision.
The displacentent of spectral lines towards the red end of the
spectrum was definitely established by Adams in 1924, by observations
on the dense companion of Sirius, for which the effect is about thirty
times greater than for the Sun. R.W.L. -- translator
APPENDIX IV
THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
(SUPPLEMENTARY TO SECTION 32)
Since the publication of the first edition of this little book, our
knowledge about the structure of space in the large (" cosmological
problem ") has had an important development, which ought to be
mentioned even in a popular presentation of the subject.
My original considerations on the subject were based on two
hypotheses:
(1) There exists an average density of matter in the whole of space
which is everywhere the same and different from zero.
(2) The magnitude (" radius ") of space is independent of time.
Both these hypotheses proved to be consistent, according to the
general theory of relativity, but only after a hypothetical term was
added to the field equations, a term which was not required by the
theory as such nor did it seem natural from a theoretical point of
view (" cosmological term of the field equations ").
Hypothesis (2) appeared unavoidable to me at the time, since I thought
that one would get into bottomless speculations if one departed from
it.
However, already in the 'twenties, the Russian mathematician Friedman
showed that a different hypothesis was natural from a purely
theoretical point of view. He realized that it was possible to
preserve hypothesis (1) without introducing the less natural
cosmological term into the field equations of gravitation, if one was
ready to drop hypothesis (2). Namely, the original field equations
admit a solution in which the " world radius " depends on time
(expanding space). In that sense one can say, according to Friedman,
that the theory demands an expansion of space.
A few years later Hubble showed, by a special investigation of the
extra-galactic nebulae (" milky ways "), that the spectral lines
emitted showed a red shift which increased regularly with the distance
of the nebulae. This can be interpreted in regard to our present
knowledge only in the sense of Doppler's principle, as an expansive
motion of the system of stars in the large -- as required, according
to Friedman, by the field equations of gravitation. Hubble's discovery
can, therefore, be considered to some extent as a confirmation of the
theory.
There does arise, however, a strange difficulty. The interpretation of
the galactic line-shift discovered by Hubble as an expansion (which
can hardly be doubted from a theoretical point of view), leads to an
origin of this expansion which lies " only " about 10^9 years ago,
while physical astronomy makes it appear likely that the development
of individual stars and systems of stars takes considerably longer. It
is in no way known how this incongruity is to be overcome.
I further want to rernark that the theory of expanding space, together
with the empirical data of astronomy, permit no decision to be reached
about the finite or infinite character of (three-dimensional) space,
while the original " static " hypothesis of space yielded the closure
(finiteness) of space.
K = co-ordinate system
x, y = two-dimensional co-ordinates
x, y, z = three-dimensional co-ordinates
x, y, z, t = four-dimensional co-ordinates
t = time
I = distance
v = velocity
F = force
G = gravitational field
EXPERIMENTS WITH ALTERNATE CURRENTS OF HIGH POTENTIAL AND HIGH FREQUENCY
A Lecture Delivered before the Institution of Electrical Engineers, London
by
NIKOLA TESLA
With a Portrait and Biographical Sketch of the Author
NEW YORK
1892
Biographical Sketch of Nikola Tesla.
While a large portion of the European family has been surging westward
during the last three or four hundred years, settling the vast
continents of America, another, but smaller, portion has been doing
frontier work in the Old World, protecting the rear by beating back
the "unspeakable Turk" and reclaiming gradually the fair lands that
endure the curse of Mohammedan rule. For a long time the Slav
people--who, after the battle of Kosovopjolje, in which the Turks
defeated the Servians, retired to the confines of the present
Montenegro, Dalmatia, Herzegovina and Bosnia, and "Borderland" of
Austria--knew what it was to deal, as our Western pioneers did, with
foes ceaselessly fretting against their frontier; and the races of
these countries, through their strenuous struggle against the armies
of the Crescent, have developed notable qualities of bravery and
sagacity, while maintaining a patriotism and independence unsurpassed
in any other nation.
It was in this interesting border region, and from among these valiant
Eastern folk, that Nikola Tesla was born in the year 1857, and the
fact that he, to-day, finds himself in America and one of our foremost
electricians, is striking evidence of the extraordinary attractiveness
alike of electrical pursuits and of the country where electricity
enjoys its widest application. Mr. Tesla's native place was Smiljan,
Lika, where his father was an eloquent clergyman of the Greek Church,
in which, by the way, his family is still prominently represented. His
mother enjoyed great fame throughout the countryside for her skill and
originality in needlework, and doubtless transmitted her ingenuity to
Nikola; though it naturally took another and more masculine direction.
The boy was early put to his books, and upon his father's removal to
Gospic he spent four years in the public school, and later, three
years in the Real School, as it is called. His escapades were such as
most quick witted boys go through, although he varied the programme on
one occasion by getting imprisoned in a remote mountain chapel rarely
visited for service; and on another occasion by falling headlong into
a huge kettle of boiling milk, just drawn from the paternal herds. A
third curious episode was that connected with his efforts to fly when,
attempting to navigate the air with the aid of an old umbrella, he
had, as might be expected, a very bad fall, and was laid up for six
weeks.
About this period he began to take delight in arithmetic and physics.
One queer notion he had was to work out everything by three or the
power of three. He was now sent to an aunt at Cartstatt, Croatia, to
finish his studies in what is known as the Higher Real School. It was
there that, coming from the rural fastnesses, he saw a steam engine
for the first time with a pleasure that he remembers to this day. At
Cartstatt he was so diligent as to compress the four years' course
into three, and graduated in 1873. Returning home during an epidemic
of cholera, he was stricken down by the disease and suffered so
seriously from the consequences that his studies were interrupted for
fully two years. But the time was not wasted, for he had become
passionately fond of experimenting, and as much as his means and
leisure permitted devoted his energies to electrical study and
investigation. Up to this period it had been his father's intention to
make a priest of him, and the idea hung over the young physicist like
a very sword of Damocles. Finally he prevailed upon his worthy but
reluctant sire to send him to Gratz in Austria to finish his studies
at the Polytechnic School, and to prepare for work as professor of
mathematics and physics. At Gratz he saw and operated a Gramme machine
for the first time, and was so struck with the objections to the use
of commutators and brushes that he made up his mind there and then to
remedy that defect in dynamo-electric machines. In the second year of
his course he abandoned the intention of becoming a teacher and took
up the engineering curriculum. After three years of absence he
returned home, sadly, to see his father die; but, having resolved to
settle down in Austria, and recognizing the value of linguistic
acquirements, he went to Prague and then to Buda-Pesth with the view
of mastering the languages he deemed necessary. Up to this time he had
never realized the enormous sacrifices that his parents had made in
promoting his education, but he now began to feel the pinch and to
grow unfamiliar with the image of Francis Joseph I. There was
considerable lag between his dispatches and the corresponding
remittance from home; and when the mathematical expression for the
value of the lag assumed the shape of an eight laid flat on its back,
Mr. Tesla became a very fair example of high thinking and plain
living, but he made up his mind to the struggle and determined to go
through depending solely on his own resources. Not desiring the fame
of a faster, he cast about for a livelihood, and through the help of
friends he secured a berth as assistant in the engineering department
of the government telegraphs. The salary was five dollars a week. This
brought him into direct contact with practical electrical work and
ideas, but it is needless to say that his means did not admit of much
experimenting. By the time he had extracted several hundred thousand
square and cube roots for the public benefit, the limitations,
financial and otherwise, of the position had become painfully
apparent, and he concluded that the best thing to do was to make a
valuable invention. He proceeded at once to make inventions, but their
value was visible only to the eye of faith, and they brought no grist
to the mill. Just at this time the telephone made its appearance in
Hungary, and the success of that great invention determined his
career, hopeless as the profession had thus far seemed to him. He
associated himself at once with telephonic work, and made various
telephonic inventions, including an operative repeater; but it did not
take him long to discover that, being so remote from the scenes of
electrical activity, he was apt to spend time on aims and results
already reached by others, and to lose touch. Longing for new
opportunities and anxious for the development of which he felt himself
possible, if once he could place himself within the genial and direct
influences of the gulf streams of electrical thought, he broke away
from the ties and traditions of the past, and in 1881 made his way to
Paris. Arriving in that city, the ardent young Likan obtained
employment as an electrical engineer with one of the largest electric
lighting companies. The next year he went to Strasburg to install a
plant, and on returning to Paris sought to carry out a number of ideas
that had now ripened into inventions. About this time, however, the
remarkable progress of America in electrical industry attracted his
attention, and once again staking everything on a single throw, he
crossed the Atlantic.
Mr. Tesla buckled down to work as soon as he landed on these shores,
put his best thought and skill into it, and soon saw openings for his
talent. In a short while a proposition was made to him to start his
own company, and, accepting the terms, he at once worked up a
practical system of arc lighting, as well as a potential method of
dynamo regulation, which in one form is now known as the "third brush
regulation." He also devised a thermo-magnetic motor and other kindred
devices, about which little was published, owing to legal
complications. Early in 1887 the Tesla Electric Company of New York
was formed, and not long after that Mr. Tesla produced his admirable
and epoch-marking motors for multiphase alternating currents, in
which, going back to his ideas of long ago, he evolved machines having
neither commutator nor brushes. It will be remembered that about the
time that Mr. Tesla brought out his motors, and read his thoughtful
paper before the American Institute of Electrical Engineers, Professor
Ferraris, in Europe, published his discovery of principles analogous
to those enunciated by Mr. Tesla. There is no doubt, however, that Mr.
Tesla was an independent inventor of this rotary field motor, for
although anticipated in dates by Ferraris, he could not have known
about Ferraris' work as it had not been published. Professor Ferraris
stated himself, with becoming modesty, that he did not think Tesla
could have known of his (Ferraris') experiments at that time, and adds
that he thinks Tesla was an independent and original inventor of this
principle. With such an acknowledgment from Ferraris there can be
little doubt about Tesla's originality in this matter.
Mr. Tesla's work in this field was wonderfully timely, and its worth
was promptly appreciated in various quarters. The Tesla patents were
acquired by the Westinghouse Electric Company, who undertook to
develop his motor and to apply it to work of different kinds. Its use
in mining, and its employment in printing, ventilation, etc., was
described and illustrated in _The Electrical World_ some years ago.
The immense stimulus that the announcement of Mr. Tesla's work gave to
the study of alternating current motors would, in itself, be enough to
stamp him as a leader.
Mr. Tesla is only 35 years of age. He is tall and spare with a
clean-cut, thin, refined face, and eyes that recall all the stories
one has read of keenness of vision and phenomenal ability to see
through things. He is an omnivorous reader, who never forgets; and he
possesses the peculiar facility in languages that enables the least
educated native of eastern Europe to talk and write in at least half a
dozen tongues. A more congenial companion cannot be desired for the
hours when one "pours out heart affluence in discursive talk," and
when the conversation, dealing at first with things near at hand and
next to us, reaches out and rises to the greater questions of life,
duty and destiny.
In the year 1890 he severed his connection with the Westinghouse
Company, since which time he has devoted himself entirely to the study
of alternating currents of high frequencies and very high potentials,
with which study he is at present engaged. No comment is necessary on
his interesting achievements in this field; the famous London lecture
published in this volume is a proof in itself. His first lecture on
his researches in this new branch of electricity, which he may be said
to have created, was delivered before the American Institute of
Electrical Engineers on May 20, 1891, and remains one of the most
interesting papers read before that society. It will be found
reprinted in full in _The Electrical World_, July 11, 1891. Its
publication excited such interest abroad that he received numerous
requests from English and French electrical engineers and scientists
to repeat it in those countries, the result of which has been the
interesting lecture published in this volume.
The present lecture presupposes a knowledge of the former, but it may
be read and understood by any one even though he has not read the
earlier one. It forms a sort of continuation of the latter, and
includes chiefly the results of his researches since that time.
EXPERIMENTS WITH ALTERNATE CURRENTS OF HIGH POTENTIAL AND HIGH FREQUENCY
I cannot find words to express how deeply I feel the honor of
addressing some of the foremost thinkers of the present time, and so
many able scientific men, engineers and electricians, of the country
greatest in scientific achievements.
The results which I have the honor to present before such a gathering
I cannot call my own. There are among you not a few who can lay better
claim than myself on any feature of merit which this work may contain.
I need not mention many names which are world-known--names of those
among you who are recognized as the leaders in this enchanting
science; but one, at least, I must mention--a name which could not be
omitted in a demonstration of this kind. It is a name associated with
the most beautiful invention ever made: it is Crookes!
When I was at college, a good time ago, I read, in a translation (for
then I was not familiar with your magnificent language), the
description of his experiments on radiant matter. I read it only once
in my life--that time--yet every detail about that charming work I
can remember this day. Few are the books, let me say, which can make
such an impression upon the mind of a student.
But if, on the present occasion, I mention this name as one of many
your institution can boast of, it is because I have more than one
reason to do so. For what I have to tell you and to show you this
evening concerns, in a large measure, that same vague world which
Professor Crookes has so ably explored; and, more than this, when I
trace back the mental process which led me to these advances--which
even by myself cannot be considered trifling, since they are so
appreciated by you--I believe that their real origin, that which
started me to work in this direction, and brought me to them, after a
long period of constant thought, was that fascinating little book
which I read many years ago.
And now that I have made a feeble effort to express my homage and
acknowledge my indebtedness to him and others among you, I will make a
second effort, which I hope you will not find so feeble as the first,
to entertain you.
Give me leave to introduce the subject in a few words.
A short time ago I had the honor to bring before our American
Institute of Electrical Engineers[A] some results then arrived at by
me in a novel line of work. I need not assure you that the many
evidences which I have received that English scientific men and
engineers were interested in this work have been for me a great
reward and encouragement. I will not dwell upon the experiments
already described, except with the view of completing, or more clearly
expressing, some ideas advanced by me before, and also with the view
of rendering the study here presented self-contained, and my remarks
on the subject of this evening's lecture consistent.
[Footnote A: For Mr. Tesla's American lecture on this subject see THE
ELECTRICAL WORLD of July 11, 1891, and for a report of his French
lecture see THE ELECTRICAL WORLD of March 26, 1892.]
This investigation, then, it goes without saying, deals with
alternating currents, and, to be more precise, with alternating
currents of high potential and high frequency. Just in how much a very
high frequency is essential for the production of the results
presented is a question which even with my present experience, would
embarrass me to answer. Some of the experiments may be performed with
low frequencies; but very high frequencies are desirable, not only on
account of the many effects secured by their use, but also as a
convenient means of obtaining, in the induction apparatus employed,
the high potentials, which in their turn are necessary to the
demonstration of most of the experiments here contemplated.
Of the various branches of electrical investigation, perhaps the most
interesting and immediately the most promising is that dealing with
alternating currents. The progress in this branch of applied science
has been so great in recent years that it justifies the most sanguine
hopes. Hardly have we become familiar with one fact, when novel
experiences are met with and new avenues of research are opened. Even
at this hour possibilities not dreamed of before are, by the use of
these currents, partly realized. As in nature all is ebb and tide, all
is wave motion, so it seems that; in all branches of industry
alternating currents--electric wave motion--will have the sway.
One reason, perhaps, why this branch of science is being so rapidly
developed is to be found in the interest which is attached to its
experimental study. We wind a simple ring of iron with coils; we
establish the connections to the generator, and with wonder and
delight we note the effects of strange forces which we bring into
play, which allow us to transform, to transmit and direct energy at
will. We arrange the circuits properly, and we see the mass of iron
and wires behave as though it were endowed with life, spinning a heavy
armature, through invisible connections, with great speed and
power--with the energy possibly conveyed from a great distance. We
observe how the energy of an alternating current traversing the wire
manifests itself--not so much in the wire as in the surrounding
space--in the most surprising manner, taking the forms of heat, light,
mechanical energy, and, most surprising of all, even chemical
affinity. All these observations fascinate us, and fill us with an
intense desire to know more about the nature of these phenomena. Each
day we go to our work in the hope of discovering,--in the hope that
some one, no matter who, may find a solution of one of the pending
great problems,--and each succeeding day we return to our task with
renewed ardor; and even if we _are_ unsuccessful, our work has not
been in vain, for in these strivings, in these efforts, we have found
hours of untold pleasure, and we have directed our energies to the
benefit of mankind.
We may take--at random, if you choose--any of the many experiments
which may be performed with alternating currents; a few of which
only, and by no means the most striking, form the subject of this
evening's demonstration: they are all equally interesting, equally
inciting to thought.
Here is a simple glass tube from which the air has been partially
exhausted. I take hold of it; I bring my body in contact with a wire
conveying alternating currents of high potential, and the tube in my
hand is brilliantly lighted. In whatever position I may put it,
wherever I may move it in space, as far as I can reach, its soft,
pleasing light persists with undiminished brightness.
Here is an exhausted bulb suspended from a single wire. Standing on an
insulated support. I grasp it, and a platinum button mounted in it is
brought to vivid incandescence.
Here, attached to a leading wire, is another bulb, which, as I touch
its metallic socket, is filled with magnificent colors of
phosphorescent light.
Here still another, which by my fingers' touch casts a shadow--the
Crookes shadow, of the stem inside of it.
Here, again, insulated as I stand on this platform, I bring my body in
contact with one of the terminals of the secondary of this induction
coil--with the end of a wire many miles long--and you see streams of
light break forth from its distant end, which is set in violent
vibration.
Here, once more, I attach these two plates of wire gauze to the
terminals of the coil. I set them a distance apart, and I set the coil
to work. You may see a small spark pass between the plates. I insert a
thick plate of one of the best dielectrics between them, and instead
of rendering altogether impossible, as we are used to expect, I _aid_
the passage of the discharge, which, as I insert the plate, merely
changes in appearance and assumes the form of luminous streams.
Is there, I ask, can there be, a more interesting study than that of
alternating currents?
In all these investigations, in all these experiments, which are so
very, very interesting, for many years past--ever since the greatest
experimenter who lectured in this hall discovered its principle--we
have had a steady companion, an appliance familiar to every one, a
plaything once, a thing of momentous importance now--the induction
coil. There is no dearer appliance to the electrician. From the ablest
among you, I dare say, down to the inexperienced student, to your
lecturer, we all have passed many delightful hours in experimenting
with the induction coil. We have watched its play, and thought and
pondered over the beautiful phenomena which it disclosed to our
ravished eyes. So well known is this apparatus, so familiar are these
phenomena to every one, that my courage nearly fails me when I think
that I have ventured to address so able an audience, that I have
ventured to entertain you with that same old subject. Here in reality
is the same apparatus, and here are the same phenomena, only the
apparatus is operated somewhat differently, the phenomena are
presented in a different aspect. Some of the results we find as
expected, others surprise us, but all captivate our attention, for in
scientific investigation each novel result achieved may be the centre
of a new departure, each novel fact learned may lead to important
developments.
Usually in operating an induction coil we have set up a vibration of
moderate frequency in the primary, either by means of an interrupter
or break, or by the use of an alternator. Earlier English
investigators, to mention only Spottiswoode and J.E.H. Gordon, have
used a rapid break in connection with the coil. Our knowledge and
experience of to-day enables us to see clearly why these coils under
the conditions of the tests did not disclose any remarkable phenomena,
and why able experimenters failed to perceive many of the curious
effects which have since been observed.
In the experiments such as performed this evening, we operate the coil
either from a specially constructed alternator capable of giving many
thousands of reversals of current per second, or, by disruptively
discharging a condenser through the primary, we set up a vibration in
the secondary circuit of a frequency of many hundred thousand or
millions per second, if we so desire; and in using either of these
means we enter a field as yet unexplored.
It is impossible to pursue an investigation in any novel line without
finally making some interesting observation or learning some useful
fact. That this statement is applicable to the subject of this lecture
the many curious and unexpected phenomena which we observe afford a
convincing proof. By way of illustration, take for instance the most
obvious phenomena, those of the discharge of the induction coil.
Here is a coil which is operated by currents vibrating with extreme
rapidity, obtained by disruptively discharging a Leyden jar. It would
not surprise a student were the lecturer to say that the secondary of
this coil consists of a small length of comparatively stout wire; it
would not surprise him were the lecturer to state that, in spite of
this, the coil is capable of giving any potential which the best
insulation of the turns is able to withstand: but although he may be
prepared, and even be indifferent as to the anticipated result, yet
the aspect of the discharge of the coil will surprise and interest
him. Every one is familiar with the discharge of an ordinary coil; it
need not be reproduced here. But, by way of contrast, here is a form
of discharge of a coil, the primary current of which is vibrating
several hundred thousand times per second. The discharge of an
ordinary coil appears as a simple line or band of light. The discharge
of this coil appears in the form of powerful brushes and luminous
streams issuing from all points of the two straight wires attached to
the terminals of the secondary. (Fig. 1.)
[Illustration: FIG. 1.--DISCHARGE BETWEEN TWO WIRES WITH FREQUENCIES
OF A FEW HUNDRED THOUSAND PER SECOND.]
Now compare this phenomenon which you have just witnessed with the
discharge of a Holtz or Wimshurst machine--that other interesting
appliance so dear to the experimenter. What a difference there is
between these phenomena! And yet, had I made the necessary
arrangements--which could have been made easily, were it not that they
would interfere with other experiments--I could have produced with
this coil sparks which, had I the coil hidden from your view and only
two knobs exposed, even the keenest observer among you would find it
difficult, if not impossible, to distinguish from those of an
influence or friction machine. This may be done in many ways--for
instance, by operating the induction coil which charges the condenser
from an alternating-current machine of very low frequency, and
preferably adjusting the discharge circuit so that there are no
oscillations set up in it. We then obtain in the secondary circuit, if
the knobs are of the required size and properly set, a more or less
rapid succession of sparks of great intensity and small quantity,
which possess the same brilliancy, and are accompanied by the same
sharp crackling sound, as those obtained from a friction or influence
machine.
Another way is to pass through two primary circuits, having a common
secondary, two currents of a slightly different period, which produce
in the secondary circuit sparks occurring at comparatively long
intervals. But, even with the means at hand this evening, I may
succeed in imitating the spark of a Holtz machine. For this purpose I
establish between the terminals of the coil which charges the
condenser a long, unsteady arc, which is periodically interrupted by
the upward current of air produced by it. To increase the current of
air I place on each side of the arc, and close to it, a large plate of
mica. The condenser charged from this coil discharges into the primary
circuit of a second coil through a small air gap, which is necessary
to produce a sudden rush of current through the primary. The scheme of
connections in the present experiment is indicated in Fig. 2.
[Illustration: FIG. 2.--IMITATING THE SPARK OF A HOLTZ MACHINE.]
G is an ordinarily constructed alternator, supplying the primary P of
an induction coil, the secondary S of which charges the condensers or
jars CC. The terminals of the secondary are connected to the inside
coatings of the jars, the outer coatings being connected to the ends
of the primary pp of a second induction coil. This primary pp has a
small air gap ab.
The secondary s of this coil is provided with knobs or spheres KK of
the proper size and set at a distance suitable for the experiment.
A long arc is established between the terminals AB of the first
induction coil. MM are the mica plates.
Each time the arc is broken between A and B the jars are quickly
charged and discharged through the primary pp, producing a snapping
spark between the knobs KK. Upon the arc forming between A and B the
potential falls, and the jars cannot be charged to such high potential
as to break through the air gap ab until the arc is again broken by
the draught.
In this manner sudden impulses, at long intervals, are produced in the
primary pp, which in the secondary s give a corresponding number of
impulses of great intensity. If the secondary knobs or spheres, KK,
are of the proper size, the sparks show much resemblance to those of a
Holtz machine.
But these two effects, which to the eye appear so very different, are
only two of the many discharge phenomena. We only need to change the
conditions of the test, and again we make other observations of
interest.
When, instead of operating the induction coil as in the last two
experiments, we operate it from a high frequency alternator, as in the
next experiment, a systematic study of the phenomena is rendered much
more easy. In such case, in varying the strength and frequency of the
currents through the primary, we may observe five distinct forms of
discharge, which I have described in my former paper on the subject[A]
before the American Institute of Electrical Engineers, May 20, 1891.
[Footnote A: See THE ELECTRICAL WORLD, July 11, 1891.]
It would take too much time, and it would lead us too far from the
subject presented this evening, to reproduce all these forms, but it
seems to me desirable to show you one of them. It is a brush
discharge, which is interesting in more than one respect. Viewed from
a near position it resembles much a jet of gas escaping under great
pressure. We know that the phenomenon is due to the agitation of the
molecules near the terminal, and we anticipate that some heat must be
developed by the impact of the molecules against the terminal or
against each other. Indeed, we find that the brush is hot, and only a
little thought leads us to the conclusion that, could we but reach
sufficiently high frequencies, we could produce a brush which would
give intense light and heat, and which would resemble in every
particular an ordinary flame, save, perhaps, that both phenomena might
not be due to the same agent--save, perhaps, that chemical affinity
might not be _electrical_ in its nature.
As the production of heat and light is here due to the impact of the
molecules, or atoms of air, or something else besides, and, as we can
augment the energy simply by raising the potential, we might, even
with frequencies obtained from a dynamo machine, intensify the action
to such a degree as to bring the terminal to melting heat. But with
such low frequencies we would have to deal always with something of
the nature of an electric current. If I approach a conducting object
to the brush, a thin little spark passes, yet, even with the
frequencies used this evening, the tendency to spark is not very
great. So, for instance, if I hold a metallic sphere at some distance
above the terminal you may see the whole space between the terminal
and sphere illuminated by the streams without the spark passing; and
with the much higher frequencies obtainable by the disruptive
discharge of a condenser, were it not for the sudden impulses, which
are comparatively few in number, sparking would not occur even at very
small distances. However, with incomparably higher frequencies, which
we may yet find means to produce efficiently, and provided that
electric impulses of such high frequencies could be transmitted
through a conductor, the electrical characteristics of the brush
discharge would completely vanish--no spark would pass, no shock would
be felt--yet we would still have to deal with an _electric_
phenomenon, but in the broad, modern interpretation of the word. In my
first paper before referred to I have pointed out the curious
properties of the brush, and described the best manner of producing
it, but I have thought it worth while to endeavor to express myself
more clearly in regard to this phenomenon, because of its absorbing
interest.
When a coil is operated with currents of very high frequency,
beautiful brush effects may be produced, even if the coil be of
comparatively small dimensions. The experimenter may vary them in
many ways, and, if it were nothing else, they afford a pleasing sight.
What adds to their interest is that they may be produced with one
single terminal as well as with two--in fact, often better with one
than with two.
But of all the discharge phenomena observed, the most pleasing to the
eye, and the most instructive, are those observed with a coil which is
operated by means of the disruptive discharge of a condenser. The
power of the brushes, the abundance of the sparks, when the conditions
are patiently adjusted, is often amazing. With even a very small coil,
if it be so well insulated as to stand a difference of potential of
several thousand volts per turn, the sparks may be so abundant that
the whole coil may appear a complete mass of fire.
Curiously enough the sparks, when the terminals of the coil are set at
a considerable distance, seem to dart in every possible direction as
though the terminals were perfectly independent of each other. As the
sparks would soon destroy the insulation it is necessary to prevent
them. This is best done by immersing the coil in a good liquid
insulator, such as boiled-out oil. Immersion in a liquid may be
considered almost an absolute necessity for the continued and
successful working of such a coil.
It is of course out of the question, in an experimental lecture, with
only a few minutes at disposal for the performance of each experiment,
to show these discharge phenomena to advantage, as to produce each
phenomenon at its best a very careful adjustment is required. But even
if imperfectly produced, as they are likely to be this evening, they
are sufficiently striking to interest an intelligent audience.
Before showing some of these curious effects I must, for the sake of
completeness, give a short description of the coil and other apparatus
used in the experiments with the disruptive discharge this evening.
[Illustration: FIG. 3.--DISRUPTIVE DISCHARGE COIL.]
It is contained in a box B (Fig. 3) of thick boards of hard wood,
covered on the outside with zinc sheet Z, which is carefully soldered
all around. It might be advisable, in a strictly scientific
investigation, when accuracy is of great importance, to do away with
the metal cover, as it might introduce many errors, principally on
account of its complex action upon the coil, as a condenser of very
small capacity and as an electrostatic and electromagnetic screen.
When the coil is used for such experiments as are here contemplated,
the employment of the metal cover offers some practical advantages,
but these are not of sufficient importance to be dwelt upon.
The coil should be placed symmetrically to the metal cover, and the
space between should, of course, not be too small, certainly not less
than, say, five centimetres, but much more if possible; especially the
two sides of the zinc box, which are at right angles to the axis of
the coil, should be sufficiently remote from the latter, as otherwise
they might impair its action and be a source of loss.
The coil consists of two spools of hard rubber RR, held apart at a
distance of 10 centimetres by bolts c and nuts n, likewise of hard
rubber. Each spool comprises a tube T of approximately 8 centimetres
inside diameter, and 3 millimetres thick, upon which are screwed two
flanges FF, 24 centimetres square, the space between the flanges being
about 3 centimetres. The secondary, SS, of the best gutta
percha-covered wire, has 26 layers, 10 turns in each, giving for each
half a total of 260 turns. The two halves are wound oppositely and
connected in series, the connection between both being made over the
primary. This disposition, besides being convenient, has the advantage
that when the coil is well balanced--that is, when both of its
terminals T_1 T_1 are connected to bodies or devices of equal
capacity--there is not much danger of breaking through to the primary,
and the insulation between the primary and the secondary need not be
thick. In using the coil it is advisable to attach to _both_ terminals
devices of nearly equal capacity, as, when the capacity of the
terminals is not equal, sparks will be apt to pass to the primary. To
avoid this, the middle point of the secondary may be connected to the
primary, but this is not always practicable.
The primary PP is wound in two parts, and oppositely, upon a wooden
spool W, and the four ends are led out of the oil through hard rubber
tubes tt. The ends of the secondary T_1 T_1 are also led out of the
oil through rubber tubes t_1 t_1 of great thickness. The primary and
secondary layers are insulated by cotton cloth, the thickness of the
insulation, of course, bearing some proportion to the difference of
potential between the turns of the different layers. Each half of the
primary has four layers, 24 turns in each, this giving a total of 96
turns. When both the parts are connected in series, this gives a
ratio of conversion of about 1:2.7, and with the primaries in
multiple, 1:5.4; but in operating with very rapidly alternating
currents this ratio does not convey even an approximate idea of the
ratio of the E.M.Fs. in the primary and secondary circuits. The coil
is held in position in the oil on wooden supports, there being about 5
centimetres thickness of oil all round. Where the oil is not specially
needed, the space is filled with pieces of wood, and for this purpose
principally the wooden box B surrounding the whole is used.
The construction here shown is, of course, not the best on general
principles, but I believe it is a good and convenient one for the
production of effects in which an excessive potential and a very small
current are needed.
In connection with the coil I use either the ordinary form of
discharger or a modified form. In the former I have introduced two
changes which secure some advantages, and which are obvious. If they
are mentioned, it is only in the hope that some experimenter may find
them of use.
[Illustration: FIG. 4.--ARRANGEMENT OF IMPROVED DISCHARGER AND
MAGNET.]
One of the changes is that the adjustable knobs A and B (Fig. 4),
of the discharger are held in jaws of brass, JJ, by spring pressure,
this allowing of turning them successively into different positions,
and so doing away with the tedious process of frequent polishing up.
The other change consists in the employment of a strong electromagnet
NS, which is placed with its axis at right angles to the line joining
the knobs A and B, and produces a strong magnetic field between them.
The pole pieces of the magnet are movable and properly formed so as to
protrude between the brass knobs, in order to make the field as
intense as possible; but to prevent the discharge from jumping to the
magnet the pole pieces are protected by a layer of mica, MM, of
sufficient thickness. s_1 s_1 and s_2 s_2 are screws for fastening the
wires. On each side one of the screws is for large and the other for
small wires. LL are screws for fixing in position the rods RR, which
support the knobs.
In another arrangement with the magnet I take the discharge between
the rounded pole pieces themselves, which in such case are insulated
and preferably provided with polished brass caps.
The employment of an intense magnetic field is of advantage
principally when the induction coil or transformer which charges the
condenser is operated by currents of very low frequency. In such a
case the number of the fundamental discharges between the knobs may be
so small as to render the currents produced in the secondary
unsuitable for many experiments. The intense magnetic field then
serves to blow out the arc between the knobs as soon as it is formed,
and the fundamental discharges occur in quicker succession.
Instead of the magnet, a draught or blast of air may be employed with
some advantage. In this case the arc is preferably established between
the knobs AB, in Fig. 2 (the knobs ab being generally joined, or
entirely done away with), as in this disposition the arc is long and
unsteady, and is easily affected by the draught.
When a magnet is employed to break the arc, it is better to choose
the connection indicated diagrammatically in Fig. 5, as in this case
the currents forming the arc are much more powerful, and the magnetic
field exercises a greater influence. The use of the magnet permits,
however, of the arc being replaced by a vacuum tube, but I have
encountered great difficulties in working with an exhausted tube.
[Illustration: FIG. 5.--ARRANGEMENT WITH LOW-FREQUENCY ALTERNATOR AND
IMPROVED DISCHARGER.]
[Illustration: FIG. 6.--DISCHARGER WITH MULTIPLE GAPS.]
The other form of discharger used in these and similar experiments is
indicated in Figs. 6 and 7. It consists of a number of brass pieces cc
(Fig. 6), each of which comprises a spherical middle portion m with an
extension e below--which is merely used to fasten the piece in a lathe
when polishing up the discharging surface--and a column above, which
consists of a knurled flange f surmounted by a threaded stem l
carrying a nut n, by means of which a wire is fastened to the column.
The flange f conveniently serves for holding the brass piece when
fastening the wire, and also for turning it in any position when it
becomes necessary to present a fresh discharging surface. Two stout
strips of hard rubber RR, with planed grooves gg (Fig. 7) to fit the
middle portion of the pieces cc, serve to clamp the latter and hold
them firmly in position by means of two bolts CC (of which only one is
shown) passing through the ends of the strips.
[Illustration: FIG. 7.--DISCHARGER WITH MULTIPLE GAPS.]
In the use of this kind of discharger I have found three principal
advantages over the ordinary form. First, the dielectric strength of a
given total width of air space is greater when a great many small air
gaps are used instead of one, which permits of working with a smaller
length of air gap, and that means smaller loss and less deterioration
of the metal; secondly by reason of splitting the arc up into smaller
arcs, the polished surfaces are made to last much longer; and,
thirdly, the apparatus affords some gauge in the experiments. I
usually set the pieces by putting between them sheets of uniform
thickness at a certain very small distance which is known from the
experiments of Sir William Thomson to require a certain electromotive
force to be bridged by the spark.
It should, of course, be remembered that the sparking distance is much
diminished as the frequency is increased. By taking any number of
spaces the experimenter has a rough idea of the electromotive force,
and he finds it easier to repeat an experiment, as he has not the
trouble of setting the knobs again and again. With this kind of
discharger I have been able to maintain an oscillating motion without
any spark being visible with the naked eye between the knobs, and they
would not show a very appreciable rise in temperature. This form of
discharge also lends itself to many arrangements of condensers and
circuits which are often very convenient and time-saving. I have used
it preferably in a disposition similar to that indicated in Fig. 2,
when the currents forming the arc are small.
I may here mention that I have also used dischargers with single or
multiple air gaps, in which the discharge surfaces were rotated with
great speed. No particular advantage was, however, gained by this
method, except in cases where the currents from the condenser were
large and the keeping cool of the surfaces was necessary, and in cases
when, the discharge not being oscillating of itself, the arc as soon
as established was broken by the air current, thus starting the
vibration at intervals in rapid succession. I have also used
mechanical interrupters in many ways. To avoid the difficulties with
frictional contacts, the preferred plan adopted was to establish the
arc and rotate through it at great speed a rim of mica provided with
many holes and fastened to a steel plate. It is understood, of course,
that the employment of a magnet, air current, or other interrupter,
produces no effect worth noticing, unless the self-induction, capacity
and resistance are so related that there are oscillations set up upon
each interruption.
I will now endeavor to show you some of the most noteworthy of these
discharge phenomena.
I have stretched across the room two ordinary cotton covered wires,
each about 7 metres in length. They are supported on insulating cords
at a distance of about 30 centimetres. I attach now to each of the
terminals of the coil one of the wires and set the coil in action.
Upon turning the lights off in the room you see the wires strongly
illuminated by the streams issuing abundantly from their whole surface
in spite of the cotton covering, which may even be very thick. When
the experiment is performed under good conditions, the light from the
wires is sufficiently intense to allow distinguishing the objects in a
room. To produce the best result it is, of course, necessary to adjust
carefully the capacity of the jars, the arc between the knobs and the
length of the wires. My experience is that calculation of the length
of the wires leads, in such case, to no result whatever. The
experimenter will do best to take the wires at the start very long,
and then adjust by cutting off first long pieces, and then smaller and
smaller ones as he approaches the right length.
A convenient way is to use an oil condenser of very small capacity,
consisting of two small adjustable metal plates, in connection with
this and similar experiments. In such case I take wires rather short
and set at the beginning the condenser plates at maximum distance. If
the streams for the wires increase by approach of the plates, the
length of the wires is about right; if they diminish the wires are too
long for that frequency and potential. When a condenser is used in
connection with experiments with such a coil, it should be an oil
condenser by all means, as in using an air condenser considerable
energy might be wasted. The wires leading to the plates in the oil
should be very thin, heavily coated with some insulating compound, and
provided with a conducting covering--this preferably extending under
the surface of the oil. The conducting cover should not be too near
the terminals, or ends, of the wire, as a spark would be apt to jump
from the wire to it. The conducting coating is used to diminish the
air losses, in virtue of its action as an electrostatic screen. As to
the size of the vessel containing the oil, and the size of the plates,
the experimenter gains at once an idea from a rough trial. The size of
the plates _in oil_ is, however, calculable, as the dielectric losses
are very small.
In the preceding experiment it is of considerable interest to know
what relation the quantity of the light emitted bears to the frequency
and potential of the electric impulses. My opinion is that the heat as
well as light effects produced should be proportionate, under
otherwise equal conditions of test, to the product of frequency and
square of potential, but the experimental verification of the law,
whatever it may be, would be exceedingly difficult. One thing is
certain, at any rate, and that is, that in augmenting the potential
and frequency we rapidly intensify the streams; and, though it may be
very sanguine, it is surely not altogether hopeless to expect that we
may succeed in producing a practical illuminant on these lines. We
would then be simply using burners or flames, in which there would be
no chemical process, no consumption of material, but merely a transfer
of energy, and which would, in all probability emit more light and
less heat than ordinary flames.
[Illustration: FIG. 8.--EFFECT PRODUCED BY CONCENTRATING STREAMS.]
The luminous intensity of the streams is, of course, considerably
increased when they are focused upon a small surface. This may be
shown by the following experiment:
I attach to one of the terminals of the coil a wire w (Fig. 8), bent
in a circle of about 30 centimetres in diameter, and to the other
terminal I fasten a small brass sphere s, the surface of the wire
being preferably equal to the surface of the sphere, and the centre of
the latter being in a line at right angles to the plane of the wire
circle and passing through its centre. When the discharge is
established under proper conditions, a luminous hollow cone is formed,
and in the dark one-half of the brass sphere is strongly illuminated,
as shown in the cut.
By some artifice or other, it is easy to concentrate the streams upon
small surfaces and to produce very strong light effects. Two thin
wires may thus be rendered intensely luminous.
In order to intensify the streams the wires should be very thin and
short; but as in this case their capacity would be generally too small
for the coil--at least, for such a one as the present--it is necessary
to augment the capacity to the required value, while, at the same
time, the surface of the wires remains very small. This may be done in
many ways.
Here, for instance, I have two plates, RR, of hard rubber (Fig. 9),
upon which I have glued two very thin wires ww, so as to form a name.
The wires may be bare or covered with the best insulation--it is
immaterial for the success of the experiment. Well insulated wires, if
anything, are preferable. On the back of each plate, indicated by the
shaded portion, is a tinfoil coating tt. The plates are placed in line
at a sufficient distance to prevent a spark passing from one to the
other wire. The two tinfoil coatings I have joined by a conductor C,
and the two wires I presently connect to the terminals of the coil. It
is now easy, by varying the strength and frequency of the currents
through the primary, to find a point at which, the capacity of the
system is best suited to the conditions, and the wires become so
strongly luminous that, when the light in the room is turned off the
name formed by them appears in brilliant letters.
[Illustration: FIG. 9.--WIRES RENDERED INTENSELY LUMINOUS.]
It is perhaps preferable to perform this experiment with a coil
operated from an alternator of high frequency, as then, owing to the
harmonic rise and fall, the streams are very uniform, though they are
less abundant then when produced with such a coil as the present. This
experiment, however, may be performed with low frequencies, but much
less satisfactorily.
[Illustration: FIG. 10.--LUMINOUS DISCS.]
When two wires, attached to the terminals of the coil, are set at the
proper distance, the streams between them may be so intense as to
produce a continuous luminous sheet. To show this phenomenon I have
here two circles, C and c (Fig. 10), of rather stout wire, one being
about 80 centimetres and the other 30 centimetres in diameter. To each
of the terminals of the coil I attach one of the circles. The
supporting wires are so bent that the circles may be placed in the
same plane, coinciding as nearly as possible. When the light in the
room is turned off and the coil set to work, you see the whole space
between the wires uniformly filled with streams, forming a luminous
disc, which could be seen from a considerable distance, such is the
intensity of the streams. The outer circle could have been much larger
than the present one; in fact, with this coil I have used much larger
circles, and I have been able to produce a strongly luminous sheet,
covering an area of more than one square metre, which is a remarkable
effect with this very small coil. To avoid uncertainty, the circle has
been taken smaller, and the area is now about 0.43 square metre.
The frequency of the vibration, and the quickness of succession of the
sparks between the knobs, affect to a marked degree the appearance of
the streams. When the frequency is very low, the air gives way in more
or less the same manner, as by a steady difference of potential, and
the streams consist of distinct threads, generally mingled with thin
sparks, which probably correspond to the successive discharges
occurring between the knobs. But when the frequency is extremely high,
and the arc of the discharge produces a very _loud_ but _smooth_
sound--showing both that oscillation takes place and that the sparks
succeed each other with great rapidity--then the luminous streams
formed are perfectly uniform. To reach this result very small coils
and jars of small capacity should be used. I take two tubes of thick
Bohemian glass, about 5 centimetres in diameter and 20 centimetres
long. In each of the tubes I slip a primary of very thick copper wire.
On the top of each tube I wind a secondary of much thinner
gutta-percha covered wire. The two secondaries I connect in series,
the primaries preferably in multiple arc. The tubes are then placed in
a large glass vessel, at a distance of 10 to 15 centimetres from each
other, on insulating supports, and the vessel is filled with boiled
out oil, the oil reaching about an inch above the tubes. The free ends
of the secondary are lifted out of the oil and placed parallel to each
other at a distance of about 10 centimetres. The ends which are
scraped should be dipped in the oil. Two four-pint jars joined in
series may be used to discharge through the primary. When the
necessary adjustments in the length and distance of the wires above
the oil and in the arc of discharge are made, a luminous sheet is
produced between the wires which is perfectly smooth and textureless,
like the ordinary discharge through a moderately exhausted tube.
I have purposely dwelt upon this apparently insignificant experiment.
In trials of this kind the experimenter arrives at the startling
conclusion that, to pass ordinary luminous discharges through gases,
no particular degree of exhaustion is needed, but that the gas may be
at ordinary or even greater pressure. To accomplish this, a very high
frequency is essential; a high potential is likewise required, but
this is a merely incidental necessity. These experiments teach us
that, in endeavoring to discover novel methods of producing light by
the agitation of atoms, or molecules, of a gas, we need not limit our
research to the vacuum tube, but may look forward quite seriously to
the possibility of obtaining the light effects without the use of any
vessel whatever, with air at ordinary pressure.
Such discharges of very high frequency, which render luminous the air
at ordinary pressures, we have probably often occasion to witness in
Nature. I have no doubt that if, as many believe, the aurora borealis
is produced by sudden cosmic disturbances, such as eruptions at the
sun's surface, which set the electrostatic charge of the earth in an
extremely rapid vibration, the red glow observed is not confined to
the upper rarefied strata of the air, but the discharge traverses, by
reason of its very high frequency, also the dense atmosphere in the
form of a _glow_, such as we ordinarily produce in a slightly
exhausted tube. If the frequency were very low, or even more so, if
the charge were not at all vibrating, the dense air would break down
as in a lightning discharge. Indications of such breaking down of the
lower dense strata of the air have been repeatedly observed at the
occurrence of this marvelous phenomenon; but if it does occur, it can
only be attributed to the fundamental disturbances, which are few in
number, for the vibration produced by them would be far too rapid to
allow a disruptive break. It is the original and irregular impulses
which affect the instruments; the superimposed vibrations probably
pass unnoticed.
When an ordinary low frequency discharge is passed through moderately
rarefied air, the air assumes a purplish hue. If by some means or
other we increase the intensity of the molecular, or atomic,
vibration, the gas changes to a white color. A similar change occurs
at ordinary pressures with electric impulses of very high frequency.
If the molecules of the air around a wire are moderately agitated, the
brush formed is reddish or violet; if the vibration is rendered
sufficiently intense, the streams become white. We may accomplish this
in various ways. In the experiment before shown with the two wires
across the room, I have endeavored to secure the result by pushing to
a high value both the frequency and potential: in the experiment with
the thin wires glued on the rubber plate I have concentrated the
action upon a very small surface--in other words, I have worked with a
great electric density.
A most curious form of discharge is observed with such a coil when the
frequency and potential are pushed to the extreme limit. To perform
the experiment, every part of the coil should be heavily insulated,
and only two small spheres--or, better still, two sharp-edged metal
discs (dd, Fig. 11) of no more than a few centimetres in
diameter--should be exposed to the air. The coil here used is immersed
in oil, and the ends of the secondary reaching out of the oil are
covered with an air-tight cover of hard rubber of great thickness.
All cracks, if there are any, should be carefully stopped up, so that
the brush discharge cannot form anywhere except on the small spheres
or plates which are exposed to the air. In this case, since there are
no large plates or other bodies of capacity attached to the terminals,
the coil is capable of an extremely rapid vibration. The potential may
be raised by increasing, as far as the experimenter judges proper, the
rate of change of the primary current. With a coil not widely
differing from the present, it is best to connect the two primaries in
multiple arc; but if the secondary should have a much greater number
of turns the primaries should preferably be used in series, as
otherwise the vibration might be too fast for the secondary. It occurs
under these conditions that misty white streams break forth from the
edges of the discs and spread out phantom-like into space. With this
coil, when fairly well produced, they are about 25 to 30 centimetres
long. When the hand is held against them no sensation is produced, and
a spark, causing a shock, jumps from the terminal only upon the hand
being brought much nearer. If the oscillation of the primary current
is rendered intermittent by some means or other, there is a
corresponding throbbing of the streams, and now the hand or other
conducting object may be brought in still greater proximity to the
terminal without a spark being caused to jump.
[Illustration: FIG. 11.--PHANTOM STREAMS.]
Among the many beautiful phenomena which may be produced with such a
coil I have here selected only those which appear to possess some
features of novelty, and lead us to some conclusions of interest. One
will not find it at all difficult to produce in the laboratory, by
means of it, many other phenomena which appeal to the eye even more
than these here shown, but present no particular feature of novelty.
Early experimenters describe the display of sparks produced by an
ordinary large induction coil upon an insulating plate separating the
terminals. Quite recently Siemens performed some experiments in which
fine effects were obtained, which were seen by many with interest. No
doubt large coils, even if operated with currents of low frequencies,
are capable of producing beautiful effects. But the largest coil ever
made could not, by far, equal the magnificent display of streams and
sparks obtained from such a disruptive discharge coil when properly
adjusted. To give an idea, a coil such as the present one will cover
easily a plate of 1 metre in diameter completely with the streams. The
best way to perform such experiments is to take a very thin rubber or
a glass plate and glue on one side of it a narrow ring of tinfoil of
very large diameter, and on the other a circular washer, the centre of
the latter coinciding with that of the ring, and the surfaces of both
being preferably equal, so as to keep the coil well balanced. The
washer and ring should be connected to the terminals by heavily
insulated thin wires. It is easy in observing the effect of the
capacity to produce a sheet of uniform streams, or a fine network of
thin silvery threads, or a mass of loud brilliant sparks, which
completely cover the plate.
Since I have advanced the idea of the conversion by means of the
disruptive discharge, in my paper before the American Institute of
Electrical Engineers at the beginning of the past year, the interest
excited in it has been considerable. It affords us a means for
producing any potentials by the aid of inexpensive coils operated from
ordinary systems of distribution, and--what is perhaps more
appreciated--it enables us to convert currents of any frequency into
currents of any other lower or higher frequency. But its chief value
will perhaps be found in the help which it will afford us in the
investigations of the phenomena of phosphorescence, which a disruptive
discharge coil is capable of exciting in innumerable cases where
ordinary coils, even the largest, would utterly fail.
Considering its probable uses for many practical purposes, and its
possible introduction into laboratories for scientific research, a few
additional remarks as to the construction of such a coil will perhaps
not be found superfluous.
It is, of course, absolutely necessary to employ in such a coil wires
provided with the best insulation.
Good coils may be produced by employing wires covered with several
layers of cotton, boiling the coil a long time in pure wax, and
cooling under moderate pressure. The advantage of such a coil is that
it can be easily handled, but it cannot probably give as satisfactory
results as a coil immersed in pure oil. Besides, it seems that the
presence of a large body of wax affects the coil disadvantageously,
whereas this does not seem to be the case with oil. Perhaps it is
because the dielectric losses in the liquid are smaller.
I have tried at first silk and cotton covered wires with oil
immersion, but I have been gradually led to use gutta-percha covered
wires, which proved most satisfactory. Gutta-percha insulation adds,
of course, to the capacity of the coil, and this, especially if the
coil be large, is a great disadvantage when extreme frequencies are
desired; but on the other hand, gutta-percha will withstand much more
than an equal thickness of oil, and this advantage should be secured
at any price. Once the coil has been immersed, it should never be
taken out of the oil for more than a few hours, else the gutta-percha
will crack up and the coil will not be worth half as much as before.
Gutta-percha is probably slowly attacked by the oil, but after an
immersion of eight to nine months I have found no ill effects.
I have obtained in commerce two kinds of gutta-percha wire: in one the
insulation sticks tightly to the metal, in the other it does not.
Unless a special method is followed to expel all air, it is much safer
to use the first kind. I wind the coil within an oil tank so that all
interstices are filled up with the oil. Between the layers I use cloth
boiled out thoroughly in oil, calculating the thickness according to
the difference of potential between the turns. There seems not to be a
very great difference whatever kind of oil is used; I use paraffine or
linseed oil.
To exclude more perfectly the air, an excellent way to proceed, and
easily practicable with small coils, is the following: Construct a box
of hard wood of very thick boards which have been for a long time
boiled in oil. The boards should be so joined as to safely withstand
the external air pressure. The coil being placed and fastened in
position within the box, the latter is closed with a strong lid, and
covered with closely fitting metal sheets, the joints of which are
soldered very carefully. On the top two small holes are drilled,
passing through the metal sheet and the wood, and in these holes two
small glass tubes are inserted and the joints made air-tight. One of
the tubes is connected to a vacuum pump, and the other with a vessel
containing a sufficient quantity of boiled-out oil. The latter tube
has a very small hole at the bottom, and is provided with a stopcock.
When a fairly good vacuum has been obtained, the stopcock is opened
and the oil slowly fed in. Proceeding in this manner, it is impossible
that any big bubbles, which are the principal danger, should remain
between the turns. The air is most completely excluded, probably
better than by boiling out, which, however, when gutta-percha coated
wires are used, is not practicable.
For the primaries I use ordinary line wire with a thick cotton
coating. Strands of very thin insulated wires properly interlaced
would, of course, be the best to employ for the primaries, but they
are not to be had.
In an experimental coil the size of the wires is not of great
importance. In the coil here used the primary is No. 12 and the
secondary No. 24 Brown & Sharpe gauge wire; but the sections may be
varied considerably. It would only imply different adjustments; the
results aimed at would not be materially affected.
I have dwelt at some length upon the various forms of brush discharge
because, in studying them, we not only observe phenomena which please
our eye, but also afford us food for thought, and lead us to
conclusions of practical importance. In the use of alternating
currents of very high tension, too much precaution cannot be taken to
prevent the brush discharge. In a main conveying such currents, in an
induction coil or transformer, or in a condenser, the brush discharge
is a source of great danger to the insulation. In a condenser
especially the gaseous matter must be most carefully expelled, for in
it the charged surfaces are near each other, and if the potentials are
high, just as sure as a weight will fall if let go, so the insulation
will give way if a single gaseous bubble of some size be present,
whereas, if all gaseous matter were carefully excluded, the condenser
would safely withstand a much higher difference of potential. A main
conveying alternating currents of very high tension may be injured
merely by a blow hole or small crack in the insulation, the more so as
a blowhole is apt to contain gas at low pressure; and as it appears
almost impossible to completely obviate such little imperfections, I
am led to believe that in our future distribution of electrical energy
by currents of very high tension liquid insulation will be used. The
cost is a great drawback, but if we employ an oil as an insulator the
distribution of electrical energy with something like 100,000 volts,
and even more, become, at least with higher frequencies, so easy that
they could be hardly called engineering feats. With oil insulation
and alternate current motors transmissions of power can be effected
with safety and upon an industrial basis at distances of as much as a
thousand miles.
A peculiar property of oils, and liquid insulation in general, when
subjected to rapidly changing electric stresses, is to disperse any
gaseous bubbles which may be present, and diffuse them through its
mass, generally long before any injurious break can occur. This
feature may be easily observed with an ordinary induction coil by
taking the primary out, plugging up the end of the tube upon which the
secondary is wound, and filling it with some fairly transparent
insulator, such as paraffine oil. A primary of a diameter something
like six millimetres smaller than the inside of the tube may be
inserted in the oil. When the coil is set to work one may see, looking
from the top through the oil, many luminous points--air bubbles which
are caught by inserting the primary, and which are rendered luminous
in consequence of the violent bombardment. The occluded air, by its
impact against the oil, heats it; the oil begins to circulate,
carrying some of the air along with it, until the bubbles are
dispersed and the luminous points disappear. In this manner, unless
large bubbles are occluded in such way that circulation is rendered
impossible, a damaging break is averted, the only effect being a
moderate warming up of the oil. If, instead of the liquid, a solid
insulation, no matter how thick, were used, a breaking through and
injury of the apparatus would be inevitable.
The exclusion of gaseous matter from any apparatus in which the
dielectric is subjected to more or less rapidly changing electric
forces is, however, not only desirable in order to avoid a possible
injury of the apparatus, but also on account of economy. In a
condenser, for instance, as long as only a solid or only a liquid
dielectric is used, the loss is small; but if a gas under ordinary or
small pressure be present the loss may be very great. Whatever the
nature of the force acting in the dielectric may be, it seems that in
a solid or liquid the molecular displacement produced by the force is
small; hence the product of force and displacement is insignificant,
unless the force be very great; but in a gas the displacement, and
therefore this product, is considerable; the molecules are free to
move, they reach high speeds, and the energy of their impact is lost
in heat or otherwise. If the gas be strongly compressed, the
displacement due to the force is made smaller, and the losses are
reduced.
In most of the succeeding experiments I prefer, chiefly on account of
the regular and positive action, to employ the alternator before
referred to. This is one of the several machines constructed by me for
the purposes of these investigations. It has 384 pole projections, and
is capable of giving currents of a frequency of about 10,000 per
second. This machine has been illustrated and briefly described in my
first paper before the American Institute of Electrical Engineers, May
20, 1891, to which I have already referred. A more detailed
description, sufficient to enable any engineer to build a similar
machine, will be found in several electrical journals of that period.
The induction coils operated from the machine are rather small,
containing from 5,000 to 15,000 turns in the secondary. They are
immersed in boiled-out linseed oil, contained in wooden boxes covered
with zinc sheet.
I have found it advantageous to reverse the usual position of the
wires, and to wind, in these coils, the primaries on the top; this
allowing the use of a much bigger primary, which, of course, reduces
the danger of overheating and increases the output of the coil. I make
the primary on each side at least one centimetre shorter than the
secondary, to prevent the breaking through on the ends, which would
surely occur unless the insulation on the top of the secondary be very
thick, and this, of course, would be disadvantageous.
When the primary is made movable, which is necessary in some
experiments, and many times convenient for the purposes of adjustment,
I cover the secondary with wax, and turn it off in a lathe to a
diameter slightly smaller than the inside of the primary coil. The
latter I provide with a handle reaching out of the oil, which serves
to shift it in any position along the secondary.
I will now venture to make, in regard to the general manipulation of
induction coils, a few observations bearing upon points which have not
been fully appreciated in earlier experiments with such coils, and are
even now often overlooked.
The secondary of the coil possesses usually such a high self-induction
that the current through the wire is inappreciable, and may be so even
when the terminals are joined by a conductor of small resistance. If
capacity is added to the terminals, the self-induction is
counteracted, and a stronger current is made to flow through the
secondary, though its terminals are insulated from each other. To one
entirely unacquainted with the properties of alternating currents
nothing will look more puzzling. This feature was illustrated in the
experiment performed at the beginning with the top plates of wire
gauze attached to the terminals and the rubber plate. When the plates
of wire gauze were close together, and a small arc passed between
them, the arc _prevented_ a strong current from passing through the
secondary, because it did away with the capacity on the terminals;
when the rubber plate was inserted between, the capacity of the
condenser formed counteracted the self-induction of the secondary, a
stronger current passed now, the coil performed more work, and the
discharge was by far more powerful.
The first thing, then, in operating the induction coil is to combine
capacity with the secondary to overcome the self-induction. If the
frequencies and potentials are very high gaseous matter should be
carefully kept away from the charged surfaces. If Leyden jars are
used, they should be immersed in oil, as otherwise considerable
dissipation may occur if the jars are greatly strained. When high
frequencies are used, it is of equal importance to combine a condenser
with the primary. One may use a condenser connected to the ends of the
primary or to the terminals of the alternator, but the latter is not
to be recommended, as the machine might be injured. The best way is
undoubtedly to use the condenser in series with the primary and with
the alternator, and to adjust its capacity so as to annul the
self-induction of both the latter. The condenser should be adjustable
by very small steps, and for a finer adjustment a small oil condenser
with movable plates may be used conveniently.
I think it best at this juncture to bring before you a phenomenon,
observed by me some time ago, which to the purely scientific
investigator may perhaps appear more interesting than any of the
results which I have the privilege to present to you this evening.
It may be quite properly ranked among the brush phenomena--in fact, it
is a brush, formed at, or near, a single terminal in high vacuum.
In bulbs provided with a conducting terminal, though it be of
aluminium, the brush has but an ephemeral existence, and cannot,
unfortunately, be indefinitely preserved in its most sensitive state,
even in a bulb devoid of any conducting electrode. In studying the
phenomenon, by all means a bulb having no leading-in wire should be
used. I have found it best to use bulbs constructed as indicated in
Figs. 12 and 13.
In Fig. 12 the bulb comprises an incandescent lamp globe L, in the
neck of which is sealed a barometer tube b, the end of which is blown
out to form a small sphere s. This sphere should be sealed as closely
as possible in the centre of the large globe. Before sealing, a thin
tube t, of aluminium sheet, may be slipped in the barometer tube, but
it is not important to employ it.
The small hollow sphere s is filled with some conducting powder, and a
wire w is cemented in the neck for the purpose of connecting the
conducting powder with the generator.
[Illustration: FIG. 12. FIG. 13. BULBS FOR PRODUCING ROTATING BRUSH.]
The construction shown in Fig. 13 was chosen in order to remove from
the brush any conducting body which might possibly affect it. The bulb
consists in this case of a lamp globe L, which has a neck n, provided
with a tube b and small sphere s, sealed to it, so that two entirely
independent compartments are formed, as indicated in the drawing. When
the bulb is in use, the neck n is provided with a tinfoil coating,
which is connected to the generator and acts inductively upon the
moderately rarefied and highly conducting gas inclosed in the neck.
From there the current passes through the tube b into the small sphere
s to act by induction upon the gas contained in the globe L.
It is of advantage to make the tube t very thick, the hole through it
very small, and to blow the sphere s very thin. It is of the greatest
importance that the sphere s be placed in the centre of the globe L.
[Illustration: FIG. 14.--FORMS AND PHASES OF THE ROTATING BRUSH.]
Figs. 14, 15 and 16 indicate different forms, or stages, of the brush.
Fig. 14 shows the brush as it first appears in a bulb provided with a
conducting terminal; but, as in such a bulb it very soon
disappears--often after a few minutes--I will confine myself to the
description of the phenomenon as seen in a bulb without conducting
electrode. It is observed under the following conditions:
When the globe L (Figs. 12 and 13) is exhausted to a very high
degree, generally the bulb is not excited upon connecting the wire w
(Fig. 12) or the tinfoil coating of the bulb (Fig. 13) to the terminal
of the induction coil. To excite it, it is usually sufficient to grasp
the globe L with the hand. An intense phosphorescence then spreads at
first over the globe, but soon gives place to a white, misty light.
Shortly afterward one may notice that the luminosity is unevenly
distributed in the globe, and after passing the current for some time
the bulb appears as in Fig. 15. From this stage the phenomenon will
gradually pass to that indicated in Fig. 16, after some minutes,
hours, days or weeks, according as the bulb is worked. Warming the
bulb or increasing the potential hastens the transit.
[Illustration: FIG. 15. FIG. 16. FORMS AND PHASES OF THE ROTATING
BRUSH.]
When the brush assumes the form indicated in Fig. 16, it maybe brought
to a state of extreme sensitiveness to electrostatic and magnetic
influence. The bulb hanging straight down from a wire, and all objects
being remote from it, the approach of the observer at a few paces from
the bulb will cause the brush to fly to the opposite side, and if he
walks around the bulb it will always keep on the opposite side. It may
begin to spin around the terminal long before it reaches that
sensitive stage. When it begins to turn around principally, but also
before, it is affected by a magnet, and at a certain stage it is
susceptible to magnetic influence to an astonishing degree. A small
permanent magnet, with its poles at a distance of no more than two
centimetres, will affect it visibly at a distance of two metres,
slowing down or accelerating the rotation according to how it is held
relatively to the brush. I think I have observed that at the stage
when it is most sensitive to magnetic, it is not most sensitive to
electrostatic, influence. My explanation is, that the electrostatic
attraction between the brush and the glass of the bulb, which retards
the rotation, grows much quicker than the magnetic influence when the
intensity of the stream is increased.
When the bulb hangs with the globe L down, the rotation is always
clockwise. In the southern hemisphere it would occur in the opposite
direction and on the equator the brush should not turn at all. The
rotation may be reversed by a magnet kept at some distance. The brush
rotates best, seemingly, when it is at right angles to the lines of
force of the earth. It very likely rotates, when at its maximum speed,
in synchronism with the alternations, say 10,000 times a second. The
rotation can be slowed down or accelerated by the approach or receding
of the observer, or any conducting body, but it cannot be reversed by
putting the bulb in any position. When it is in the state of the
highest sensitiveness and the potential or frequency be varied the
sensitiveness is rapidly diminished. Changing either of these but
little will generally stop the rotation. The sensitiveness is likewise
affected by the variations of temperature. To attain great
sensitiveness it is necessary to have the small sphere s in the centre
of the globe L, as otherwise the electrostatic action of the glass of
the globe will tend to stop the rotation. The sphere s should be small
and of uniform thickness; any dissymmetry of course has the effect to
diminish the sensitiveness.
The fact that the brush rotates in a definite direction in a permanent
magnetic field seems to show that in alternating currents of very high
frequency the positive and negative impulses are not equal, but that
one always preponderates over the other.
Of course, this rotation in one direction may be due to the action of
two elements of the same current upon each other, or to the action of
the field produced by one of the elements upon the other, as in a
series motor, without necessarily one impulse being stronger than the
other. The fact that the brush turns, as far as I could observe, in
any position, would speak for this view. In such case it would turn
at any point of the earth's surface. But, on the other hand, it is
then hard to explain why a permanent magnet should reverse the
rotation, and one must assume the preponderance of impulses of one
kind.
As to the causes of the formation of the brush or stream, I think it
is due to the electrostatic action of the globe and the dissymmetry of
the parts. If the small bulb s and the globe L were perfect concentric
spheres, and the glass throughout of the same thickness and quality, I
think the brush would not form, as the tendency to pass would be equal
on all sides. That the formation of the stream is due to an
irregularity is apparent from the fact that it has the tendency to
remain in one position, and rotation occurs most generally only when
it is brought out of this position by electrostatic or magnetic
influence. When in an extremely sensitive state it rests in one
position, most curious experiments may be performed with it. For
instance, the experimenter may, by selecting a proper position,
approach the hand at a certain considerable distance to the bulb, and
he may cause the brush to pass off by merely stiffening the muscles of
the arm. When it begins to rotate slowly, and the hands are held at a
proper distance, it is impossible to make even the slightest motion
without producing a visible effect upon the brush. A metal plate
connected to the other terminal of the coil affects it at a great
distance, slowing down the rotation often to one turn a second.
I am firmly convinced that such a brush, when we learn how to produce
it properly, will prove a valuable aid in the investigation of the
nature of the forces acting in an electrostatic or magnetic field. If
there is any motion which is measurable going on in the space, such a
brush ought to reveal it. It is, so to speak, a beam of light,
frictionless, devoid of inertia.
I think that it may find practical applications in telegraphy. With
such a brush it would be possible to send dispatches across the
Atlantic, for instance, with any speed, since its sensitiveness may be
so great that the slightest changes will affect it. If it were
possible to make the stream more intense and very narrow, its
deflections could be easily photographed.
I have been interested to find whether there is a rotation of the
stream itself, or whether there is simply a stress traveling around in
the bulb. For this purpose I mounted a light mica fan so that its
vanes were in the path of the brush. If the stream itself was rotating
the fan would be spun around. I could produce no distinct rotation of
the fan, although I tried the experiment repeatedly; but as the fan
exerted a noticeable influence on the stream, and the apparent
rotation of the latter was, in this case, never quite satisfactory,
the experiment did not appear to be conclusive.
I have been unable to produce the phenomenon with the disruptive
discharge coil, although every other of these phenomena can be well
produced by it--many, in fact, much better than with coils operated
from an alternator.
It may be possible to produce the brush by impulses of one direction,
or even by a steady potential, in which case it would be still more
sensitive to magnetic influence.
In operating an induction coil with rapidly alternating currents, we
realize with astonishment, for the first time, the great importance
of the relation of capacity, self-induction and frequency as regards
the general result. The effects of capacity are the most striking, for
in these experiments, since the self-induction and frequency both are
high, the critical capacity is very small, and need be but slightly
varied to produce a very considerable change. The experimenter may
bring his body in contact with the terminals of the secondary of the
coil, or attach to one or both terminals insulated bodies of very
small bulk, such as bulbs, and he may produce a considerable rise or
fall of potential, and greatly affect the flow of the current through
the primary. In the experiment before shown, in which a brush appears
at a wire attached to one terminal, and the wire is vibrated when the
experimenter brings his insulated body in contact with the other
terminal of the coil, the sudden rise of potential was made evident.
I may show you the behavior of the coil in another manner which
possesses a feature of some interest. I have here a little light fan
of aluminium sheet, fastened to a needle and arranged to rotate freely
in a metal piece screwed to one of the terminals of the coil. When the
coil is set to work, the molecules of the air are rhythmically
attracted and repelled. As the force with which they are repelled is
greater than that with which they are attracted, it results that there
is a repulsion exerted on the surfaces of the fan. If the fan were
made simply of a metal sheet, the repulsion would be equal on the
opposite sides, and would produce no effect. But if one of the
opposing surfaces is screened, or if, generally speaking, the
bombardment on this side is weakened in some way or other, there
remains the repulsion exerted upon the other, and the fan is set in
rotation. The screening is best effected by fastening upon one of the
opposing sides of the fan insulated conducting coatings, or, if the
fan is made in the shape of an ordinary propeller screw, by fastening
on one side, and close to it, an insulated metal plate. The static
screen may, however, be omitted, and simply a thickness of insulating
material fastened to one of the sides of the fan.
To show the behavior of the coil, the fan may be placed upon the
terminal and it will readily rotate when the coil is operated by
currents of very high frequency. With a steady potential, of course,
and even with alternating currents of very low frequency, it would not
turn, because of the very slow exchange of air and, consequently,
smaller bombardment; but in the latter case it might turn if the
potential were excessive. With a pin wheel, quite the opposite rule
holds good; it rotates best with a steady potential, and the effort is
the smaller the higher the frequency. Now, it is very easy to adjust
the conditions so that the potential is normally not sufficient to
turn the fan, but that by connecting the other terminal of the coil
with an insulated body it rises to a much greater value, so as to
rotate the fan, and it is likewise possible to stop the rotation by
connecting to the terminal a body of different size, thereby
diminishing the potential.
Instead of using the fan in this experiment, we may use the "electric"
radiometer with similar effect. But in this case it will be found that
the vanes will rotate only at high exhaustion or at ordinary
pressures; they will not rotate at moderate pressures, when the air is
highly conducting. This curious observation was made conjointly by
Professor Crookes and myself. I attribute the result to the high
conductivity of the air, the molecules of which then do not act as
independent carriers of electric charges, but act all together as a
single conducting body. In such case, of course, if there is any
repulsion at all of the molecules from the vanes, it must be very
small. It is possible, however, that the result is in part due to the
fact that the greater part of the discharge passes from the leading-in
wire through the highly conducting gas, instead of passing off from
the conducting vanes.
In trying the preceding experiment with the electric radiometer the
potential should not exceed a certain limit, as then the electrostatic
attraction between the vanes and the glass of the bulb may be so great
as to stop the rotation.
A most curious feature of alternate currents of high frequencies and
potentials is that they enable us to perform many experiments by the
use of one wire only. In many respects this feature is of great
interest.
In a type of alternate current motor invented by me some years ago I
produced rotation by inducing, by means of a single alternating
current passed through a motor circuit, in the mass or other circuits
of the motor, secondary currents, which, jointly with the primary or
inducing current, created a moving field of force. A simple but crude
form of such a motor is obtained by winding upon an iron core a
primary, and close to it a secondary coil, joining the ends of the
latter and placing a freely movable metal disc within the influence of
the field produced by both. The iron core is employed for obvious
reasons, but it is not essential to the operation. To improve the
motor, the iron core is made to encircle the armature. Again to
improve, the secondary coil is made to overlap partly the primary, so
that it cannot free itself from a strong inductive action of the
latter, repel its lines as it may. Once more to improve, the proper
difference of phase is obtained between the primary and secondary
currents by a condenser, self-induction, resistance or equivalent
windings.
I had discovered, however, that rotation is produced by means of a
single coil and core; my explanation of the phenomenon, and leading
thought in trying the experiment, being that there must be a true time
lag in the magnetization of the core. I remember the pleasure I had
when, in the writings of Professor Ayrton, which came later to my
hand, I found the idea of the time lag advocated. Whether there is a
true time lag, or whether the retardation is due to eddy currents
circulating in minute paths, must remain an open question, but the
fact is that a coil wound upon an iron core and traversed by an
alternating current creates a moving field of force, capable of
setting an armature in rotation. It is of some interest, in
conjunction with the historical Arago experiment, to mention that in
lag or phase motors I have produced rotation in the opposite direction
to the moving field, which means that in that experiment the magnet
may not rotate, or may even rotate in the opposite direction to the
moving disc. Here, then, is a motor (diagrammatically illustrated in
Fig. 17), comprising a coil and iron core, and a freely movable copper
disc in proximity to the latter.
[Illustration: FIG. 17.--SINGLE WIRE AND "NO-WIRE" MOTOR.]
To demonstrate a novel and interesting feature, I have, for a reason
which I will explain, selected this type of motor. When the ends of
the coil are connected to the terminals of an alternator the disc is
set in rotation. But it is not this experiment, now well known, which
I desire to perform. What I wish to show you is that this motor
rotates with _one single_ connection between it and the generator;
that is to say, one terminal of the motor is connected to one terminal
of the generator--in this case the secondary of a high-tension
induction coil--the other terminals of motor and generator being
insulated in space. To produce rotation it is generally (but not
absolutely) necessary to connect the free end of the motor coil to an
insulated body of some size. The experimenter's body is more than
sufficient. If he touches the free terminal with an object held in the
hand, a current passes through the coil and the copper disc is set in
rotation. If an exhausted tube is put in series with the coil, the
tube lights brilliantly, showing the passage of a strong current.
Instead of the experimenter's body, a small metal sheet suspended on a
cord may be used with the same result. In this case the plate acts as
a condenser in series with the coil. It counteracts the self-induction
of the latter and allows a strong current to pass. In such a
combination, the greater the self-induction of the coil the smaller
need be the plate, and this means that a lower frequency, or
eventually a lower potential, is required to operate the motor. A
single coil wound upon a core has a high self-induction; for this
reason principally, this type of motor was chosen to perform the
experiment. Were a secondary closed coil wound upon the core, it would
tend to diminish the self-induction, and then it would be necessary to
employ a much higher frequency and potential. Neither would be
advisable, for a higher potential would endanger the insulation of the
small primary coil, and a higher frequency would result in a
materially diminished torque.
It should be remarked that when such a motor with a closed secondary
is used, it is not at all easy to obtain rotation with excessive
frequencies, as the secondary cuts off almost completely the lines of
the primary--and this, of course, the more, the higher the
frequency--and allows the passage of but a minute current. In such a
case, unless the secondary is closed through a condenser, it is almost
essential, in order to produce rotation, to make the primary and
secondary coils overlap each other more or less.
But there is an additional feature of interest about this motor,
namely, it is not necessary to have even a single connection between
the motor and generator, except, perhaps, through the ground: for not
only is an insulated plate capable of giving off energy into space,
but it is likewise capable of deriving it from an alternating
electrostatic field, though in the latter case the available energy is
much smaller. In this instance one of the motor terminals is connected
to the insulated plate or body located within the alternating
electrostatic field, and the other terminal preferably to the ground.
It is quite possible, however, that such "no-wire" motors, as they
might be called, could be operated by conduction through the rarefied
air at considerable distances. Alternate currents, especially of high
frequencies, pass with astonishing freedom through even slightly
rarefied gases. The upper strata of the air are rarefied. To reach a
number of miles out into space requires the overcoming of difficulties
of a merely mechanical nature. There is no doubt that with the
enormous potentials obtainable by the use of high frequencies and oil
insulation luminous discharges might be passed through many miles of
rarefied air, and that, by thus directing the energy of many hundreds
or thousands of horse-power, motors or lamps might be operated at
considerable distances from stationary sources. But such schemes are
mentioned merely as possibilities. We shall have no need to transmit
power in this way. We shall have no need to _transmit_ power at all.
Ere many generations pass, our machinery will be driven by a power
obtainable at any point of the universe. This idea is not novel. Men
have been led to it long ago by instinct or reason. It has been
expressed in many ways, and in many places, in the history of old and
new. We find it in the delightful myth of Antheus, who derives power
from the earth; we find it among the subtile speculations of one of
your splendid mathematicians, and in many hints and statements of
thinkers of the present time. Throughout space there is energy. Is
this energy static or kinetic? If static our hopes are in vain; if
kinetic--and this we know it is, for certain--then it is a mere
question of time when men will succeed in attaching their machinery to
the very wheelwork of nature. Of all, living or dead, Crookes came
nearest to doing it. His radiometer will turn in the light of day and
in the darkness of the night; it will turn everywhere where there is
heat, and heat is everywhere. But, unfortunately, this beautiful
little machine, while it goes down to posterity as the most
interesting, must likewise be put on record as the most inefficient
machine ever invented!
The preceding experiment is only one of many equally interesting
experiments which may be performed by the use of only one wire with
alternate currents of high potential and frequency. We may connect an
insulated line to a source of such currents, we may pass an
inappreciable current over the line, and on any point of the same we
are able to obtain a heavy current, capable of fusing a thick copper
wire. Or we may, by the help of some artifice, decompose a solution in
any electrolytic cell by connecting only one pole of the cell to the
line or source of energy. Or we may, by attaching to the line, or only
bringing into its vicinity, light up an incandescent lamp, an
exhausted tube, or a phosphorescent bulb.
However impracticable this plan of working may appear in many cases,
it certainly seems practicable, and even recommendable, in the
production of light. A perfected lamp would require but little energy,
and if wires were used at all we ought to be able to supply that
energy without a return wire.
It is now a fact that a body may be rendered incandescent or
phosphorescent by bringing it either in single contact or merely in
the vicinity of a source of electric impulses of the proper character,
and that in this manner a quantity of light sufficient to afford a
practical illuminant may be produced. It is, therefore, to say the
least, worth while to attempt to determine the best conditions and to
invent the best appliances for attaining this object.
Some experiences have already been gained in this direction, and I
will dwell on them briefly, in the hope that they might prove useful.
The heating of a conducting body inclosed in a bulb, and connected to
a source of rapidly alternating electric impulses, is dependent on so
many things of a different nature, that it would be difficult to give
a generally applicable rule under which the maximum heating occurs. As
regards the size of the vessel, I have lately found that at ordinary
or only slightly differing atmospheric pressures, when air is a good
insulator, and hence practically the same amount of energy by a
certain potential and frequency is given off from the body, whether
the bulb be small or large, the body is brought to a higher
temperature if inclosed in a small bulb, because of the better
confinement of heat in this case.
At lower pressures, when air becomes more or less conducting, or if
the air be sufficiently warmed as to become conducting, the body is
rendered more intensely incandescent in a large bulb, obviously
because, under otherwise equal conditions of test, more energy may be
given off from the body when the bulb is large.
At very high degrees of exhaustion, when the matter in the bulb
becomes "radiant," a large bulb has still an advantage, but a
comparatively slight one, over the small bulb.
Finally, at excessively high degrees of exhaustion, which cannot be
reached except by the employment of special means, there seems to be,
beyond a certain and rather small size of vessel, no perceptible
difference in the heating.
These observations were the result of a number of experiments, of
which one, showing the effect of the size of the bulb at a high degree
of exhaustion, may be described and shown here, as it presents a
feature of interest. Three spherical bulbs of 2 inches, 3 inches and 4
inches diameter were taken, and in the centre of each was mounted an
equal length of an ordinary incandescent lamp filament of uniform
thickness. In each bulb the piece of filament was fastened to the
leading-in wire of platinum, contained in a glass stem sealed in the
bulb; care being taken, of course, to make everything as nearly alike
as possible. On each glass stem in the inside of the bulb was slipped
a highly polished tube made of aluminium sheet, which fitted the stem
and was held on it by spring pressure. The function of this aluminium
tube will be explained subsequently. In each bulb an equal length of
filament protruded above the metal tube. It is sufficient to say now
that under these conditions equal lengths of filament of the same
thickness--in other words, bodies of equal bulk--were brought to
incandescence. The three bulbs were sealed to a glass tube, which was
connected to a Sprengel pump. When a high vacuum had been reached, the
glass tube carrying the bulbs was sealed off. A current was then
turned on successively on each bulb, and it was found that the
filaments came to about the same brightness, and, if anything, the
smallest bulb, which was placed midway between the two larger ones,
may have been slightly brighter. This result was expected, for when
either of the bulbs was connected to the coil the luminosity spread
through the other two, hence the three bulbs constituted really one
vessel. When all the three bulbs were connected in multiple arc to the
coil, in the largest of them the filament glowed brightest, in the
next smaller it was a little less bright, and in the smallest it only
came to redness. The bulbs were then sealed off and separately tried.
The brightness of the filaments was now such as would have been
expected on the supposition that the energy given off was
proportionate to the surface of the bulb, this surface in each case
representing one of the coatings of a condenser. Accordingly, time
was less difference between the largest and the middle sized than
between the latter and the smallest bulb.
An interesting observation was made in this experiment. The three
bulbs were suspended from a straight bare wire connected to a terminal
of the coil, the largest bulb being placed at the end of the wire, at
some distance from it the smallest bulb, and an equal distance from
the latter the middle-sized one. The carbons glowed then in both the
larger bulbs about as expected, but the smallest did not get its share
by far. This observation led me to exchange the position of the bulbs,
and I then observed that whichever of the bulbs was in the middle it
was by far less bright than it was in any other position. This
mystifying result was, of course, found to be due to the electrostatic
action between the bulbs. When they were placed at a considerable
distance, or when they were attached to the corners of an equilateral
triangle of copper wire, they glowed about in the order determined by
their surfaces.
As to the shape of the vessel, it is also of some importance,
especially at high degrees of exhaustion. Of all the possible
constructions, it seems that a spherical globe with the refractory
body mounted in its centre is the best to employ. In experience it has
been demonstrated that in such a globe a refractory body of a given
bulk is more easily brought to incandescence than when otherwise
shaped bulbs are used. There is also an advantage in giving to the
incandescent body the shape of a sphere, for self-evident reasons. In
any case the body should be mounted in the centre, where the atoms
rebounding from the glass collide. This object is best attained in
the spherical bulb; but it is also attained in a cylindrical vessel
with one or two straight filaments coinciding with its axis, and
possibly also in parabolical or spherical bulbs with the refractory
body or bodies placed in the focus or foci of the same; though the
latter is not probable, as the electrified atoms should in all cases
rebound normally from the surface they strike, unless the speed were
excessive, in which case they _would_ probably follow the general law
of reflection. No matter what shape the vessel may have, if the
exhaustion be low, a filament mounted in the globe is brought to the
same degree of incandescence in all parts; but if the exhaustion be
high and the bulb be spherical or pear-shaped, as usual, focal points
form and the filament is heated to a higher degree at or near such
points.
To illustrate the effect, I have here two small bulbs which are alike,
only one is exhausted to a low and the other to a very high degree.
When connected to the coil, the filament in the former glows uniformly
throughout all its length; whereas in the latter, that portion of the
filament which is in the centre of the bulb glows far more intensely
than the rest. A curious point is that the phenomenon occurs even if
two filaments are mounted in a bulb, each being connected to one
terminal of the coil, and, what is still more curious, if they be very
near together, provided the vacuum be very high. I noted in
experiments with such bulbs that the filaments would give way usually
at a certain point, and in the first trials I attributed it to a
defect in the carbon. But when the phenomenon occurred many times in
succession I recognized its real cause.
In order to bring a refractory body inclosed in a bulb to
incandescence, it is desirable, on account of economy, that all the
energy supplied to the bulb from the source should reach without loss
the body to be heated; from there, and from nowhere else, it should be
radiated. It is, of course, out of the question to reach this
theoretical result, but it is possible by a proper construction of the
illuminating device to approximate it more or less.
For many reasons, the refractory body is placed in the centre of the
bulb, and it is usually supported on a glass stem containing the
leading-in wire. As the potential of this wire is alternated, the
rarefied gas surrounding the stem is acted upon inductively, and the
glass stem is violently bombarded and heated. In this manner by far
the greater portion of the energy supplied to the bulb--especially
when exceedingly high frequencies are used--may be lost for the
purpose contemplated. To obviate this loss, or at least to reduce it
to a minimum, I usually screen the rarefied gas surrounding the stem
from the inductive action of the leading-in wire by providing the stem
with a tube or coating of conducting material. It seems beyond doubt
that the best among metals to employ for this purpose is aluminium, on
account of its many remarkable properties. Its only fault is that it
is easily fusible, and, therefore, its distance from the incandescing
body should be properly estimated. Usually, a thin tube, of a diameter
somewhat smaller than that of the glass stem, is made of the finest
aluminium sheet, and slipped on the stem. The tube is conveniently
prepared by wrapping around a rod fastened in a lathe a piece of
aluminium sheet of the proper size, grasping the sheet firmly with
clean chamois leather or blotting paper, and spinning the rod very
fast. The sheet is wound tightly around the rod, and a highly polished
tube of one or three layers of the sheet is obtained. When slipped on
the stem, the pressure is generally sufficient to prevent it from
slipping off, but, for safety, the lower edge of the sheet may be
turned inside. The upper inside corner of the sheet--that is, the one
which is nearest to the refractory incandescent body--should be cut
out diagonally, as it often happens that, in consequence of the
intense heat, this corner turns toward the inside and comes very near
to, or in contact with, the wire, or filament, supporting the
refractory body. The greater part of the energy supplied to the bulb
is then used up in heating the metal tube, and the bulb is rendered
useless for the purpose. The aluminium sheet should project above the
glass stem more or less--one inch or so--or else, if the glass be too
close to the incandescing body, it may be strongly heated and become
more or less conducting, whereupon it may be ruptured, or may, by its
conductivity, establish a good electrical connection between the metal
tube and the leading-in wire, in which case, again, most of the energy
will be lost in heating the former. Perhaps the best way is to make
the top of the glass tube, for about an inch, of a much smaller
diameter. To still further reduce the danger arising from the heating
of the glass stem, and also with the view of preventing an electrical
connection between the metal tube and the electrode, I preferably wrap
the stem with several layers of thin mica, which extends at least as
far as the metal tube. In some bulbs I have also used an outside
insulating cover.
The preceding remarks are only made to aid the experimenter in the
first trials, for the difficulties which he encounters he may soon
find means to overcome in his own way.
To illustrate the effect of the screen, and the advantage of using it,
I have here two bulbs of the same size, with their stems, leading-in
wires and incandescent lamp filaments tied to the latter, as nearly
alike as possible. The stem of one bulb is provided with an aluminium
tube, the stem of the other has none. Originally the two bulbs were
joined by a tube which was connected to a Sprengel pump. When a high
vacuum had been reached, first the connecting tube, and then the
bulbs, were sealed off; they are therefore of the same degree of
exhaustion. When they are separately connected to the coil giving a
certain potential, the carbon filament in the bulb provided with the
aluminium screen is rendered highly incandescent, while the filament
in the other bulb may, with the same potential, not even come to
redness, although in reality the latter bulb takes generally more
energy than the former. When they are both connected together to the
terminal, the difference is even more apparent, showing the importance
of the screening. The metal tube placed on the stem containing the
leading-in wire performs really two distinct functions: First: it acts
more or less as an electrostatic screen, thus economizing the energy
supplied to the bulb; and, second, to whatever extent it may fail to
act electrostatically, it acts mechanically, preventing the
bombardment, and consequently intense heating and possible
deterioration of the slender support of the refractory incandescent
body, or of the glass stem containing the leading-in wire. I say
_slender_ support, for it is evident that in order to confine the heat
more completely to the incandescing body its support should be very
thin, so as to carry away the smallest possible amount of heat by
conduction. Of all the supports used I have found an ordinary
incandescent lamp filament to be the best, principally because among
conductors it can withstand the highest degrees of heat.
The effectiveness of the metal tube as an electrostatic screen depends
largely on the degree of exhaustion.
At excessively high degrees of exhaustion--which are reached by using
great care and special means in connection with the Sprengel
pump--when the matter in the globe is in the ultra-radiant state, it
acts most perfectly. The shadow of the upper edge of the tube is then
sharply defined upon the bulb.
At a somewhat lower degree of exhaustion, which is about the ordinary
"non-striking" vacuum, and generally as long as the matter moves
predominantly in straight lines, the screen still does well. In
elucidation of the preceding remark it is necessary to state that what
is a "non-striking" vacuum for a coil operated, as ordinarily, by
impulses, or currents, of low-frequency, is not, by far, so when the
coil is operated by currents of very high frequency. In such case the
discharge may pass with great freedom through the rarefied gas through
which a low-frequency discharge may not pass, even though the
potential be much higher. At ordinary atmospheric pressures just the
reverse rule holds good: the higher the frequency, the less the spark
discharge is able to jump between the terminals, especially if they
are knobs or spheres of some size.
Finally, at very low degrees of exhaustion, when the gas is well
conducting, the metal tube not only does not act as an electrostatic
screen, but even is a drawback, aiding to a considerable extent the
dissipation of the energy laterally from the leading-in wire. This, of
course, is to be expected. In this case, namely, the metal tube is in
good electrical connection with the leading-in wire, and most of the
bombardment is directed upon the tube. As long as the electrical
connection is not good, the conducting tube is always of some
advantage, for although it may not greatly economize energy, still it
protects the support of the refractory button, and is a means for
concentrating more energy upon the same.
To whatever extent the aluminium tube performs the function of a
screen, its usefulness is therefore limited to very high degrees of
exhaustion when it is insulated from the electrode--that is, when the
gas as a whole is non-conducting, and the molecules, or atoms, act as
independent carriers of electric charges.
In addition to acting as a more or less effective screen, in the true
meaning of the word, the conducting tube or coating may also act, by
reason of its conductivity, as a sort of equalizer or dampener of the
bombardment against the stem. To be explicit, I assume the action as
follows: Suppose a rhythmical bombardment to occur against the
conducting tube by reason of its imperfect action as a screen, it
certainly must happen that some molecules, or atoms, strike the tube
sooner than others. Those which come first in contact with it give up
their superfluous charge, and the tube is electrified, the
electrification instantly spreading over its surface. But this must
diminish the energy lost in the bombardment for two reasons: first,
the charge given up by the atoms spreads over a great area, and hence
the electric density at any point is small, and the atoms are repelled
with less energy than they would be if they would strike against a
good insulator: secondly, as the tube is electrified by the atoms
which first come in contact with it, the progress of the following
atoms against the tube is more or less checked by the repulsion which
the electrified tube must exert upon the similarly electrified atoms.
This repulsion may perhaps be sufficient to prevent a large portion of
the atoms from striking the tube, but at any rate it must diminish the
energy of their impact. It is clear that when the exhaustion is very
low, and the rarefied gas well conducting, neither of the above
effects can occur, and, on the other hand, the fewer the atoms, with
the greater freedom they move; in other words, the higher the degree
of exhaustion, up to a limit, the more telling will be both the
effects.
What I have just said may afford an explanation of the phenomenon
observed by Prof. Crookes, namely, that a discharge through a bulb is
established with much greater facility when an insulator than when a
conductor is present in the same. In my opinion, the conductor acts as
a dampener of the motion of the atoms in the two ways pointed out;
hence, to cause a visible discharge to pass through the bulb, a much
higher potential is needed if a conductor, especially of much surface,
be present.
For the sake of clearness of some of the remarks before made, I must
now refer to Figs. 18, 19 and 20, which illustrate various
arrangements with a type of bulb most generally used.
[Illustration: FIG. 18.--BULB WITH MICA TUBE AND ALUMINIUM SCREEN.]
[Illustration: FIG. 19.--IMPROVED BULB WITH SOCKET AND SCREEN.]
Fig. 18 is a section through a spherical bulb L, with the glass stem
s, containing the leading-in wire w; which has a lamp filament l
fastened to it, serving to support the refractory button m in the
centre. M is a sheet of thin mica wound in several layers around the
stem s, and a is the aluminium tube.
Fig. 19 illustrates such a bulb in a somewhat more advanced stage of
perfection. A metallic tube S is fastened by means of some cement to
the neck of the tube. In the tube is screwed a plug P, of insulating
material, in the centre of which is fastened a metallic terminal t,
for the connection to the leading-in wire w. This terminal must be
well insulated from the metal tube S, therefore, if the cement used is
conducting--and most generally it is sufficiently so--the space
between the plug P and the neck of the bulb should be filled with some
good insulating material, as mica powder.
[Illustration: FIG. 20.--BULB FOR EXPERIMENTS WITH CONDUCTING TUBE.]
Fig. 20 shows a bulb made for experimental purposes. In this bulb the
aluminium tube is provided with an external connection, which serves
to investigate the effect of the tube under various conditions. It is
referred to chiefly to suggest a line of experiment followed.
Since the bombardment against the stem containing the leading-in wire
is due to the inductive action of the latter upon the rarefied gas, it
is of advantage to reduce this action as far as practicable by
employing a very thin wire, surrounded by a very thick insulation of
glass or other material, and by making the wire passing through the
rarefied gas as short as practicable. To combine these features I
employ a large tube T (Fig. 21), which protrudes into the bulb to some
distance, and carries on the top a very short glass stem s, into which
is sealed the leading-in wire w, and I protect the top of the glass
stem against the heat by a small, aluminium tube a and a layer of mica
underneath the same, as usual. The wire w, passing through the large
tube to the outside of the bulb, should be well insulated--with a
glass tube, for instance--and the space between ought to be filled out
with some excellent insulator. Among many insulating powders I have
tried, I have found that mica powder is the best to employ. If this
precaution is not taken, the tube T, protruding into the bulb, will
surely be cracked in consequence of the heating by the brushes which
are apt to form in the upper part of the tube, near the exhausted
globe, especially if the vacuum be excellent, and therefore the
potential necessary to operate the lamp very high.
Fig. 22 illustrates a similar arrangement, with a large tube T
protruding in to the part of the bulb containing the refractors button
m. In this case the wire leading from the outside into the bulb is
omitted, the energy required being supplied through condenser coatings
CC. The insulating packing P should in this construction be tightly
fitting to the glass, and rather wide, or otherwise the discharge
might avoid passing through the wire w, which connects the inside
condenser coating to the incandescent button m. The molecular
bombardment against the glass stem in the bulb is a source of great
trouble. As illustration I will cite a phenomenon only too frequently
and unwillingly observed. A bulb, preferably a large one, may be
taken, and a good conducting body, such as a piece of carbon, may be
mounted in it upon a platinum wire sealed in the glass stem. The bulb
may be exhausted to a fairly high degree, nearly to the point when
phosphorescence begins to appear.
[Illustration: FIG. 21.--IMPROVED BULB WITH NON-CONDUCTING BUTTON.]
[Illustration: FIG. 22.--TYPE OF BULB WITHOUT LEADING-IN WIRE.]
When the bulb is connected with the coil, the piece of carbon, if
small, may become highly incandescent at first, but its brightness
immediately diminishes, and then the discharge may break through the
glass somewhere in the middle of the stem, in the form of bright
sparks, in spite of the fact that the platinum wire is in good
electrical connection with the rarefied gas through the piece of
carbon or metal at the top. The first sparks are singularly bright,
recalling those drawn from a clear surface of mercury. But, as they
heat the glass rapidly, they, of course, lose their brightness, and
cease when the glass at the ruptured place becomes incandescent, or
generally sufficiently hot to conduct. When observed for the first
time the phenomenon must appear very curious, and shows in a striking
manner how radically different alternate currents, or impulses, of
high frequency behave, as compared with steady currents, or currents
of low frequency. With such currents--namely, the latter--the
phenomenon would of course not occur. When frequencies such as are
obtained by mechanical means are used, I think that the rupture of the
glass is more or less the consequence of the bombardment, which warms
it up and impairs its insulating power; but with frequencies
obtainable with condensers I have no doubt that the glass may give way
without previous heating. Although this appears most singular at
first, it is in reality what we might expect to occur. The energy
supplied to the wire leading into the bulb is given off partly by
direct action through the carbon button, and partly by inductive
action through the glass surrounding the wire. The case is thus
analogous to that in which a condenser shunted by a conductor of low
resistance is connected to a source of alternating currents. As long
as the frequencies are low, the conductor gets the most, and the
condenser is perfectly safe: but when the frequency becomes excessive,
the _role_ of the conductor may become quite insignificant. In the
latter case the difference of potential at the terminals of the
condenser may become so great as to rupture the dielectric,
notwithstanding the fact that the terminals are joined by a conductor
of low resistance.
[Illustration: FIG. 23.--EFFECT PRODUCED BY A RUBY DROP.]
It is, of course, not necessary, when it is desired to produce the
incandescence of a body inclosed in a bulb by means of these currents,
that the body should be a conductor, for even a perfect non-conductor
may be quite as readily heated. For this purpose it is sufficient to
surround a conducting electrode with a non-conducting material, as,
for instance, in the bulb described before in Fig. 21, in which a thin
incandescent lamp filament is coated with a non-conductor, and
supports a button of the same material on the top. At the start the
bombardment goes on by inductive action through the non-conductor,
until the same is sufficiently heated to become conducting, when the
bombardment continues in the ordinary way.
A different arrangement used in some of the bulbs constructed is
illustrated in Fig. 23. In this instance a non-conductor m is
mounted in a piece of common arc light carbon so as to project some
small distance above the latter. The carbon piece is connected to the
leading-in wire passing through a glass stem, which is wrapped with
several layers of mica. An aluminium tube a is employed as usual for
screening. It is so arranged that it reaches very nearly as high as
the carbon and only the non-conductor m projects a little above it.
The bombardment goes at first against the upper surface of carbon, the
lower parts being protected by the aluminium tube. As soon, however,
as the non-conductor m is heated it is rendered good conducting, and
then it becomes the centre of the bombardment, being most exposed to
the same.
I have also constructed during these experiments many such single-wire
bulbs with or without internal electrode, in which the radiant matter
was projected against, or focused upon, the body to be rendered
incandescent. Fig. 24 illustrates one of the bulbs used. It consists
of a spherical globe L, provided with a long neck n, on the top, for
increasing the action in some cases by the application of an external
conducting coating. The globe L is blown out on the bottom into a very
small bulb b, which serves to hold it firmly in a socket S of
insulating material into which it is cemented. A fine lamp filament f,
supported on a wire w, passes through the centre of the globe L. The
filament is rendered incandescent in the middle portion, where the
bombardment proceeding from the lower inside surface of the globe is
most intense. The lower portion of the globe, as far as the socket S
reaches, is rendered conducting, either by a tinfoil coating or
otherwise, and the external electrode is connected to a terminal of
the coil.
The arrangement diagrammatically indicated in Fig. 24 was found to be
an inferior one when it was desired to render incandescent a filament
or button supported in the centre of the globe, but it was convenient
when the object was to excite phosphorescence.
In many experiments in which bodies of a different kind were mounted
in the bulb as, for instance, indicated in Fig. 23, some observations
of interest were made.
It was found, among other things, that in such cases, no matter where
the bombardment began, just as soon as a high temperature was reached
there was generally one of the bodies which seemed to take most of the
bombardment upon itself, the other, or others, being thereby relieved.
This quality appeared to depend principally on the point of fusion,
and on the facility with which the body was "evaporated," or,
generally speaking, disintegrated--meaning by the latter term not only
the throwing off of atoms, but likewise of larger lumps. The
observation made was in accordance with generally accepted notions. In
a highly exhausted bulb electricity is carried off from the electrode
by independent carriers, which are partly the atoms, or molecules, of
the residual atmosphere, and partly the atoms, molecules, or lumps
thrown off from the electrode. If the electrode is composed of bodies
of different character, and if one of these is more easily
disintegrated than the others, most of the electricity supplied is
carried off from that body, which is then brought to a higher
temperature than the others, and this the more, as upon an increase of
the temperature the body is still more easily disintegrated.
It seems to me quite probable that a similar process takes place in
the bulb even with a homogeneous electrode, and I think it to be the
principal cause of the disintegration. There is bound to be some
irregularity, even if the surface is highly polished, which, of
course, is impossible with most of the refractory bodies employed as
electrodes. Assume that a point of the electrode gets hotter,
instantly most of the discharge passes through that point, and a
minute patch is probably fused and evaporated. It is now possible that
in consequence of the violent disintegration the spot attacked sinks
in temperature, or that a counter force is created, as in an arc; at
any rate, the local tearing off meets with the limitations incident to
the experiment, whereupon the same process occurs on another place. To
the eye the electrode appears uniformly brilliant, but there are upon
it points constantly shifting and wandering around, of a temperature
far above the mean, and this materially hastens the process of
deterioration. That some such thing occurs, at least when the
electrode is at a lower temperature, sufficient experimental evidence
can be obtained in the following manner: Exhaust a bulb to a very high
degree, so that with a fairly high potential the discharge cannot
pass--that is, not a _luminous_ one, for a weak invisible discharge
occurs always, in all probability. Now raise slowly and carefully the
potential, leaving the primary current on no more than for an instant.
At a certain point, two, three, or half a dozen phosphorescent spots
will appear on the globe. These places of the glass are evidently more
violently bombarded than others, this being due to the unevenly
distributed electric density, necessitated, of course, by sharp
projections, or, generally speaking, irregularities of the electrode.
But the luminous patches are constantly changing in position, which is
especially well observable if one manages to produce very few, and
this indicates that the configuration of the electrode is rapidly
changing.
From experiences of this kind I am led to infer that, in order to be
most durable, the refractory button in the bulb should be in the form
of a sphere with a highly polished surface. Such a small sphere could
be manufactured from a diamond or some other crystal, but a better way
would be to fuse, by the employment of extreme degrees of temperature,
some oxide--as, for instance, zirconia--into a small drop, and then
keep it in the bulb at a temperature somewhat below its point of
fusion.
Interesting and useful results can no doubt be reached in the
direction of extreme degrees of heat. How can such high temperatures
be arrived at? How are the highest degrees of heat reached in nature?
By the impact of stars, by high speeds and collisions. In a collision
any rate of heat generation may be attained. In a chemical process we
are limited. When oxygen and hydrogen combine, they fall,
metaphorically speaking, from a definite height. We cannot go very far
with a blast, nor by confining heat in a furnace, but in an exhausted
bulb we can concentrate any amount of energy upon a minute button.
Leaving practicability out of consideration, this, then, would be the
means which, in my opinion, would enable us to reach the highest
temperature. But a great difficulty when proceeding in this way is
encountered, namely, in most cases the body is carried off before it
can fuse and form a drop. This difficulty exists principally with an
oxide such as zirconia, because it cannot be compressed in so hard a
cake that it would not be carried off quickly. I endeavored repeatedly
to fuse zirconia, placing it in a cup or arc light carbon as indicated
in Fig. 23. It glowed with a most intense light, and the stream of the
particles projected out of the carbon cup was of a vivid white: but
whether it was compressed in a cake or made into a paste with carbon,
it was carried off before it could be fused. The carbon cup containing
the zirconia had to be mounted very low in the neck of a large bulb,
as the heating of the glass by the projected particles of the oxide
was so rapid that in the first trial the bulb was cracked almost in an
instant when the current was turned on. The heating of the glass by
the projected particles was found to be always greater when the carbon
cup contained a body which was rapidly carried off--I presume because
in such cases, with the same potential, higher speeds were reached,
and also because, per unit of time, more matter was projected--that
is, more particles would strike the glass.
The before mentioned difficulty did not exist, however, when the body
mounted in the carbon cup offered great resistance to deterioration.
For instance, when an oxide was first fused in an oxygen blast and
then mounted in the bulb, it melted very readily into a drop.
Generally during the process of fusion magnificent light effects were
noted, of which it would be difficult to give an adequate idea. Fig.
23 is intended to illustrate the effect observed with a ruby drop. At
first one may see a narrow funnel of white light projected against the
top of the globe, where it produces an irregularly outlined
phosphorescent patch. When the point of the ruby fuses the
phosphorescence becomes very powerful; but as the atoms are projected
with much greater speed from the surface of the drop, soon the glass
gets hot and "tired," and now only the outer edge of the patch glows.
In this manner an intensely phosphorescent, sharply defined line, _l_,
corresponding to the outline of the drop, is produced, which spreads
slowly over the globe as the drop gets larger. When the mass begins to
boil, small bubbles and cavities are formed, which cause dark colored
spots to sweep across the globe. The bulb may be turned downward
without fear of the drop falling off, as the mass possesses
considerable viscosity.
I may mention here another feature of some interest, which I believe
to have noted in the course of these experiments, though the
observations do not amount to a certitude. It _appeared_ that under
the molecular impact caused by the rapidly alternating potential the
body was fused and maintained in that state at a lower temperature in
a highly exhausted bulb than was the case at normal pressure and
application of heat in the ordinary way--that is, at least, judging
from the quantity of the light emitted. One of the experiments
performed may be mentioned here by way of illustration. A small piece
of pumice stone was stuck on a platinum wire, and first melted to it
in a gas burner. The wire was next placed between two pieces of
charcoal and a burner applied so as to produce an intense heat,
sufficient to melt down the pumice stone into a small glass-like
button. The platinum wire had to be taken of sufficient thickness to
prevent its melting in the fire. While in the charcoal fire, or when
held in a burner to get a better idea of the degree of heat, the
button glowed with great brilliancy. The wire with the button was then
mounted in a bulb, and upon exhausting the same to a high degree, the
current was turned on slowly so as to prevent the cracking of the
button. The button was heated to the point of fusion, and when it
melted it did not, apparently, glow with the same brilliancy as
before, and this would indicate a lower temperature. Leaving out of
consideration the observer's possible, and even probable, error, the
question is, can a body under these conditions be brought from a solid
to a liquid state with evolution of _less_ light?
When the potential of a body is rapidly alternated it is certain that
the structure is jarred. When the potential is very high, although the
vibrations may be few--say 20,000 per second--the effect upon the
structure may be considerable. Suppose, for example, that a ruby is
melted into a drop by a steady application of energy. When it forms a
drop it will emit visible and invisible waves, which will be in a
definite ratio, and to the eye the drop will appear to be of a certain
brilliancy. Next, suppose we diminish to any degree we choose the
energy steadily supplied, and, instead, supply energy which rises and
falls according to a certain law. Now, when the drop is formed, there
will be emitted from it three different kinds of vibrations--the
ordinary visible, and two kinds of invisible waves: that is, the
ordinary dark waves of all lengths, and, in addition, waves of a well
defined character. The latter would not exist by a steady supply of
the energy; still they help to jar and loosen the structure. If this
really be the case, then the ruby drop will emit relatively less
visible and more invisible waves than before. Thus it would seem that
when a platinum wire, for instance, is fused by currents alternating
with extreme rapidity, it emits at the point of fusion less light and
more invisible radiation than it does when melted by a steady current,
though the total energy used up in the process of fusion is the same
in both cases. Or, to cite another example, a lamp filament is not
capable of withstanding as long with currents of extreme frequency as
it does with steady currents, assuming that it be worked at the same
luminous intensity. This means that for rapidly alternating currents
the filament should be shorter and thicker. The higher the
frequency--that is, the greater the departure from the steady
flow--the worse it would be for the filament. But if the truth of this
remark were demonstrated, it would be erroneous to conclude that such
a refractory button as used in these bulbs would be deteriorated
quicker by currents of extremely high frequency than by steady or low
frequency currents. From experience I may say that just the opposite
holds good: the button withstands the bombardment better with currents
of very high frequency. But this is due to the fact that a high
frequency discharge passes through a rarefied gas with much greater
freedom than a steady or low frequency discharge, and this will say
that with the former we can work with a lower potential or with a less
violent impact. As long, then, as the gas is of no consequence, a
steady or low frequency current is better; but as soon as the action
of the gas is desired and important, high frequencies are preferable.
In the course of these experiments a great many trials were made with
all kinds of carbon buttons. Electrodes made of ordinary carbon
buttons were decidedly more durable when the buttons were obtained by
the application of enormous pressure. Electrodes prepared by
depositing carbon in well known ways did not show up well; they
blackened the globe very quickly. From many experiences I conclude
that lamp filaments obtained in this manner can be advantageously used
only with low potentials and low frequency currents. Some kinds of
carbon withstand so well that, in order to bring them to the point of
fusion, it is necessary to employ very small buttons. In this case the
observation is rendered very difficult on account of the intense heat
produced. Nevertheless there can be no doubt that all kinds of carbon
are fused under the molecular bombardment, but the liquid state must
be one of great instability. Of all the bodies tried there were two
which withstood best--diamond and carborundum. These two showed up
about equally, but the latter was preferable, for many reasons. As it
is more than likely that this body is not yet generally known, I will
venture to call your attention to it.
It has been recently produced by Mr. E.G. Acheson, of Monongahela
City, Pa., U.S.A. It is intended to replace ordinary diamond powder
for polishing precious stones, etc., and I have been informed that it
accomplishes this object quite successfully. I do not know why the
name "carborundum" has been given to it, unless there is something in
the process of its manufacture which justifies this selection. Through
the kindness of the inventor, I obtained a short while ago some
samples which I desired to test in regard to their qualities of
phosphorescence and capability of withstanding high degrees of heat.
Carborundum can be obtained in two forms--in the form of "crystals"
and of powder. The former appear to the naked eye dark colored, but
are very brilliant; the latter is of nearly the same color as ordinary
diamond powder, but very much finer. When viewed under a microscope
the samples of crystals given to me did not appear to have any
definite form, but rather resembled pieces of broken up egg coal of
fine quality. The majority were opaque, but there were some which were
transparent and colored. The crystals are a kind of carbon containing
some impurities; they are extremely hard, and withstand for a long
time even an oxygen blast. When the blast is directed against them
they at first form a cake of some compactness, probably in consequence
of the fusion of impurities they contain. The mass withstands for a
very long time the blast without further fusion; but a slow carrying
off, or burning, occurs, and, finally, a small quantity of a
glass-like residue is left, which, I suppose, is melted alumina. When
compressed strongly they conduct very well, but not as well as
ordinary carbon. The powder, which is obtained from the crystals in
some way, is practically non-conducting. It affords a magnificent
polishing material for stones.
The time has been too short to make a satisfactory study of the
properties of this product, but enough experience has been gained in a
few weeks I have experimented upon it to say that it does possess some
remarkable properties in many respects. It withstands excessively high
degrees of heat, it is little deteriorated by molecular bombardment,
and it does not blacken the globe as ordinary carbon does. The only
difficulty which I have found in its use in connection with these
experiments was to find some binding material which would resist the
heat and the effect of the bombardment as successfully as carborundum
itself does.
I have here a number of bulbs which I have provided with buttons of
carborundum. To make such a button of carborundum crystals I proceed
in the following manner: I take an ordinary lamp filament and dip its
point in tar, or some other thick substance or paint which may be
readily carbonized. I next pass the point of the filament through the
crystals, and then hold it vertically over a hot plate. The tar
softens and forms a drop on the point of the filament, the crystals
adhering to the surface of the drop. By regulating the distance from
the plate the tar is slowly dried out and the button becomes solid. I
then once more dip the button in tar and hold it again over a plate
until the tar is evaporated, leaving only a hard mass which firmly
binds the crystals. When a larger button is required I repeat the
process several times, and I generally also cover the filament a
certain distance below the button with crystals. The button being
mounted in a bulb, when a good vacuum has been reached, first a weak
and then a strong discharge is passed through the bulb to carbonize
the tar and expel all gases, and later it is brought to a very intense
incandescence.
When the powder is used I have found it best to proceed as follows: I
make a thick paint of carborundum and tar, and pass a lamp filament
through the paint. Taking then most of the paint off by rubbing the
filament against a piece of chamois leather, I hold it over a hot
plate until the tar evaporates and the coating becomes firm. I repeat
this process as many times as it is necessary to obtain a certain
thickness of coating. On the point of the coated filament I form a
button in the same manner.
There is no doubt that such a button--properly prepared under great
pressure--of carborundum, especially of powder of the best quality,
will withstand the effect of the bombardment fully as well as anything
we know. The difficulty is that the binding material gives way, and
the carborundum is slowly thrown off after some time. As it does not
seem to blacken the globe in the least, it might be found useful for
coating the filaments of ordinary incandescent lamps, and I think that
it is even possible to produce thin threads or sticks of carborundum
which will replace the ordinary filaments in an incandescent lamp. A
carborundum coating seems to be more durable than other coatings, not
only because the carborundum can withstand high degrees of heat, but
also because it seems to unite with the carbon better than any other
material I have tried. A coating of zirconia or any other oxide, for
instance, is far more quickly destroyed. I prepared buttons of diamond
dust in the same manner as of carborundum, and these came in
durability nearest to those prepared of carborundum, but the binding
paste gave way much more quickly in the diamond buttons: this,
however, I attributed to the size and irregularity of the grains of
the diamond.
It was of interest to find whether carborundum possesses the quality
of phosphorescence. One is, of course, prepared to encounter two
difficulties: first, as regards the rough product, the "crystals,"
they are good conducting, and it is a fact that conductors do not
phosphoresce; second, the powder, being exceedingly fine, would not be
apt to exhibit very prominently this quality, since we know that when
crystals, even such as diamond or ruby, are finely powdered, they lose
the property of phosphorescence to a considerable degree.
The question presents itself here, can a conductor phosphoresce? What
is there in such a body as a metal, for instance, that would deprive
it of the quality of phosphorescence, unless it is that property which
characterizes it as a conductor? for it is a fact that most of the
phosphorescent bodies lose that quality when they are sufficiently
heated to become more or less conducting. Then, if a metal be in a
large measure, or perhaps entirely, deprived of that property, it
should be capable of phosphorescence. Therefore it is quite possible
that at some extremely high frequency, when behaving practically as a
non-conductor, a metal or any other conductor might exhibit the
quality of phosphorescence, even though it be entirely incapable of
phosphorescing under the impact of a low-frequency discharge. There
is, however, another possible way how a conductor might at least
_appear_ to phosphoresce.
Considerable doubt still exists as to what really is phosphorescence,
and as to whether the various phenomena comprised under this head are
due to the same causes. Suppose that in an exhausted bulb, under the
molecular impact, the surface of a piece of metal or other conductor
is rendered strongly luminous, but at the same time it is found that
it remains comparatively cool, would not this luminosity be called
phosphorescence? Now such a result, theoretically at least, is
possible, for it is a mere question of potential or speed. Assume the
potential of the electrode, and consequently the speed of the
projected atoms, to be sufficiently high, the surface of the metal
piece against which the atoms are projected would be rendered highly
incandescent, since the process of heat generation would be
incomparably faster than that of radiating or conducting away from the
surface of the collision. In the eye of the observer a single impact
of the atoms would cause an instantaneous flash, but if the impacts
were repeated with sufficient rapidity they would produce a
continuous impression upon his retina. To him then the surface of the
metal would appear continuously incandescent and of constant luminous
intensity, while in reality the light would be either intermittent or
at least changing periodically in intensity. The metal piece would
rise in temperature until equilibrium was attained--that is until the
energy continuously radiated would equal that intermittently supplied.
But the supplied energy might under such conditions not be sufficient
to bring the body to any more than a very moderate mean temperature,
especially if the frequency of the atomic impacts be very low--just
enough that the fluctuation of the intensity of the light emitted
could not be detected by the eye. The body would now, owing to the
manner in which the energy is supplied, emit a strong light, and yet
be at a comparatively very low mean temperature. How could the
observer call the luminosity thus produced? Even if the analysis of
the light would teach him something definite, still he would probably
rank it under the phenomena of phosphorescence. It is conceivable that
in such a way both conducting and non-conducting bodies may be
maintained at a certain luminous intensity, but the energy required
would very greatly vary with the nature and properties of the bodies.
These and some foregoing remarks of a speculative nature were made
merely to bring out curious features of alternate currents or electric
impulses. By their help we may cause a body to emit _more_ light,
while at a certain mean temperature, than it would emit if brought to
that temperature by a steady supply; and, again, we may bring a body
to the point of fusion, and cause it to emit _less_ light than when
fused by the application of energy in ordinary ways. It all depends on
how we supply the energy, and what kind of vibrations we set up: in
one case the vibrations are more, in the other less, adapted to affect
our sense of vision.
Some effects, which I had not observed before, obtained with
carborundum in the first trials, I attributed to phosphorescence, but
in subsequent experiments it appeared that it was devoid of that
quality. The crystals possess a noteworthy feature. In a bulb provided
with a single electrode in the shape of a small circular metal disc,
for instance, at a certain degree of exhaustion the electrode is
covered with a milky film, which is separated by a dark space from the
glow filling the bulb. When the metal disc is covered with carborundum
crystals, the film is far more intense, and snow-white. This I found
later to be merely an effect of the bright surface of the crystals,
for when an aluminium electrode was highly polished it exhibited more
or less the same phenomenon. I made a number of experiments with the
samples of crystals obtained, principally because it would have been
of special interest to find that they are capable of phosphorescence,
on account of their being conducting. I could not produce
phosphorescence distinctly, but I must remark that a decisive opinion
cannot be formed until other experimenters have gone over the same
ground.
The powder behaved in some experiments as though it contained alumina,
but it did not exhibit with sufficient distinctness the red of the
latter. Its dead color brightens considerably under the molecular
impact, but I am now convinced it does not phosphoresce. Still, the
tests with the powder are not conclusive, because powdered carborundum
probably does not behave like a phosphorescent sulphide, for example,
which could be finely powdered without impairing the phosphorescence,
but rather like powdered ruby or diamond, and therefore it would be
necessary, in order to make a decisive test, to obtain it in a large
lump and polish up the surface.
If the carborundum proves useful in connection with these and similar
experiments, its chief value will be found in the production of
coatings, thin conductors, buttons, or other electrodes capable of
withstanding extremely high degrees of heat.
The production of a small electrode capable of withstanding enormous
temperatures I regard as of the greatest importance in the manufacture
of light. It would enable us to obtain, by means of currents of very
high frequencies, certainly 20 times, if not more, the quantity of
light which is obtained in the present incandescent lamp by the same
expenditure of energy. This estimate may appear to many exaggerated,
but in reality I think it is far from being so. As this statement
might be misunderstood I think it necessary to expose clearly the
problem with which in this line of work we are confronted, and the
manner in which, in my opinion, a solution will be arrived at.
Any one who begins a study of the problem will be apt to think that
what is wanted in a lamp with an electrode is a very high degree of
incandescence of the electrode. There he will be mistaken. The high
incandescence of the button is a necessary evil, but what is really
wanted is the high incandescence of the gas surrounding the button. In
other words, the problem in such a lamp is to bring a mass of gas to
the highest possible incandescence. The higher the incandescence, the
quicker the mean vibration, the greater is the economy of the light
production. But to maintain a mass of gas at a high degree of
incandescence in a glass vessel, it will always be necessary to keep
the incandescent mass away from the glass; that is, to confine it as
much as possible to the central portion of the globe.
In one of the experiments this evening a brush was produced at the end
of a wire. This brush was a flame, a source of heat and light. It did
not emit much perceptible heat, nor did it glow with an intense light;
but is it the less a flame because it does not scorch my hand? Is it
the less a flame because it does not hurt my eye by its brilliancy?
The problem is precisely to produce in the bulb such a flame, much
smaller in size, but incomparably more powerful. Were there means at
hand for producing electric impulses of a sufficiently high frequency,
and for transmitting them, the bulb could be done away with, unless it
were used to protect the electrode, or to economize the energy by
confining the heat. But as such means are not at disposal, it becomes
necessary to place the terminal in a bulb and rarefy the air in the
same. This is done merely to enable the apparatus to perform the work
which it is not capable of performing at ordinary air pressure. In the
bulb we are able to intensify the action to any degree--so far that
the brush emits a powerful light.
The intensity of the light emitted depends principally on the
frequency and potential of the impulses, and on the electric density
of the surface of the electrode. It is of the greatest importance to
employ the smallest possible button, in order to push the density very
far. Under the violent impact of the molecules of the gas surrounding
it, the small electrode is of course brought to an extremely high
temperature, but around it is a mass of highly incandescent gas, a
flame photosphere, many hundred times the volume of the electrode.
With a diamond, carborundum or zirconia button the photosphere can be
as much as one thousand times the volume of the button. Without much
reflecting one would think that in pushing so far the incandescence of
the electrode it would be instantly volatilized. But after a careful
consideration he would find that, theoretically, it should not occur,
and in this fact--which, however, is experimentally demonstrated--lies
principally the future value of such a lamp.
At first, when the bombardment begins, most of the work is performed
on the surface of the button, but when a highly conducting photosphere
is formed the button is comparatively relieved. The higher the
incandescence of the photosphere the more it approaches in
conductivity to that of the electrode, and the more, therefore, the
solid and the gas form one conducting body. The consequence is that
the further is forced the incandescence the more work, comparatively,
is performed on the gas, and the less on the electrode. The formation
of a powerful photosphere is consequently the very means for
protecting the electrode. This protection, of course, is a relative
one, and it should not be thought that by pushing the incandescence
higher the electrode is actually less deteriorated. Still,
theoretically, with extreme frequencies, this result must be reached,
but probably at a temperature too high for most of the refractory
bodies known. Given, then, an electrode which can withstand to a very
high limit the effect of the bombardment and outward strain, it would
be safe no matter how much it is forced beyond that limit. In an
incandescent lamp quite different considerations apply. There the gas
is not at all concerned: the whole of the work is performed on the
filament; and the life of the lamp diminishes so rapidly with the
increase of the degree of incandescence that economical reasons compel
us to work it at a low incandescence. But if an incandescent lamp is
operated with currents of very high frequency, the action of the gas
cannot be neglected, and the rules for the most economical working
must be considerably modified.
In order to bring such a lamp with one or two electrodes to a great
perfection, it is necessary to employ impulses of very high frequency.
The high frequency secures, among others, two chief advantages, which
have a most important bearing upon the economy of the light
production. First, the deterioration of the electrode is reduced by
reason of the fact that we employ a great many small impacts, instead
of a few violent ones, which shatter quickly the structure; secondly,
the formation of a large photosphere is facilitated.
In order to reduce the deterioration of the electrode to the minimum,
it is desirable that the vibration be harmonic, for any suddenness
hastens the process of destruction. An electrode lasts much longer
when kept at incandescence by currents, or impulses, obtained from a
high-frequency alternator, which rise and fall more or less
harmonically, than by impulses obtained from a disruptive discharge
coil. In the latter case there is no doubt that most of the damage is
done by the fundamental sudden discharges.
One of the elements of loss in such a lamp is the bombardment of the
globe. As the potential is very high, the molecules are projected with
great speed; they strike the glass, and usually excite a strong
phosphorescence. The effect produced is very pretty, but for
economical reasons it would be perhaps preferable to prevent, or at
least reduce to the minimum, the bombardment against the globe, as in
such case it is, as a rule, not the object to excite phosphorescence,
and as some loss of energy results from the bombardment. This loss in
the bulb is principally dependent on the potential of the impulses and
on the electric density on the surface of the electrode. In employing
very high frequencies the loss of energy by the bombardment is greatly
reduced, for, first, the potential needed to perform a given amount of
work is much smaller; and, secondly, by producing a highly conducting
photosphere around the electrode, the same result is obtained as
though the electrode were much larger, which is equivalent to a
smaller electric density. But be it by the diminution of the maximum
potential or of the density, the gain is effected in the same manner,
namely, by avoiding violent shocks, which strain the glass much beyond
its limit of elasticity. If the frequency could be brought high
enough, the loss due to the imperfect elasticity of the glass would be
entirely negligible. The loss due to bombardment of the globe may,
however, be reduced by using two electrodes instead of one. In such
case each of the electrodes may be connected to one of the terminals;
or else, if it is preferable to use only one wire, one electrode may
be connected to one terminal and the other to the ground or to an
insulated body of some surface, as, for instance, a shade on the lamp.
In the latter case, unless some judgment is used, one of the
electrodes might glow more intensely than the other.
But on the whole I find it preferable when using such high frequencies
to employ only one electrode and one connecting wire. I am convinced
that the illuminating device of the near future will not require for
its operation more than one lead, and, at any rate, it will have no
leading-in wire, since the energy required can be as well transmitted
through the glass. In experimental bulbs the leading-in wire is most
generally used on account of convenience, as in employing condenser
coatings in the manner indicated in Fig. 22, for example, there is
some difficulty in fitting the parts, but these difficulties would not
exist if a great many bulbs were manufactured; otherwise the energy
can be conveyed through the glass as well as through a wire, and with
these high frequencies the losses are very small. Such illuminating
devices will necessarily involve the use of very high potentials, and
this, in the eyes of practical men, might be an objectionable feature.
Yet, in reality, high potentials are not objectionable--certainly not
in the least as far as the safety of the devices is concerned.
There are two ways of rendering an electric appliance safe. One is to
use low potentials, the other is to determine the dimensions of the
apparatus so that it is safe no matter how high a potential is used.
Of the two the latter seems to me the better way, for then the safety
is absolute, unaffected by any possible combination of circumstances
which might render even a low-potential appliance dangerous to life
and property. But the practical conditions require not only the
judicious determination of the dimensions of the apparatus; they
likewise necessitate the employment of energy of the proper kind. It
is easy, for instance, to construct a transformer capable of giving,
when operated from an ordinary alternate current machine of low
tension, say 50,000 volts, which might be required to light a highly
exhausted phosphorescent tube, so that, in spite of the high
potential, it is perfectly safe, the shock from it producing no
inconvenience. Still, such a transformer would be expensive, and in
itself inefficient; and, besides, what energy was obtained from it
would not be economically used for the production of light. The
economy demands the employment of energy in the form of extremely
rapid vibrations. The problem of producing light has been likened to
that of maintaining a certain high-pitch note by means of a bell. It
should be said a _barely audible_ note; and even these words would not
express it, so wonderful is the sensitiveness of the eye. We may
deliver powerful blows at long intervals, waste a good deal of energy,
and still not get what we want; or we may keep up the note by
delivering frequent gentle taps, and get nearer to the object sought
by the expenditure of much less energy. In the production of light, as
far as the illuminating device is concerned, there can be only one
rule--that is, to use as high frequencies as can be obtained; but the
means for the production and conveyance of impulses of such character
impose, at present at least, great limitations. Once it is decided to
use very high frequencies, the return wire becomes unnecessary, and
all the appliances are simplified. By the use of obvious means the
same result is obtained as though the return wire were used. It is
sufficient for this purpose to bring in contact with the bulb, or
merely in the vicinity of the same, an insulated body of some surface.
The surface need, of course, be the smaller, the higher the frequency
and potential used, and necessarily, also, the higher the economy of
the lamp or other device.
This plan of working has been resorted to on several occasions this
evening. So, for instance, when the incandescence of a button was
produced by grasping the bulb with the hand, the body of the
experimenter merely served to intensify the action. The bulb used was
similar to that illustrated in Fig. 19, and the coil was excited to a
small potential, not sufficient to bring the button to incandescence
when the bulb was hanging from the wire; and incidentally, in order to
perform the experiment in a more suitable manner, the button was taken
so large that a perceptible time had to elapse before, upon grasping
the bulb, it could be rendered incandescent. The contact with the bulb
was, of course, quite unnecessary. It is easy, by using a rather large
bulb with an exceedingly small electrode, to adjust the conditions so
that the latter is brought to bright incandescence by the mere
approach of the experimenter within a few feet of the bulb, and that
the incandescence subsides upon his receding.
[Illustration: FIG. 24.--BULB WITHOUT LEADING-IN WIRE, SHOWING EFFECT
OF PROJECTED MATTER.]
In another experiment, when phosphorescence was excited, a similar
bulb was used. Here again, originally, the potential was not
sufficient to excite phosphorescence until the action was
intensified--in this case, however, to present a different feature, by
touching the socket with a metallic object held in the hand. The
electrode in the bulb was a carbon button so large that it could not
be brought to incandescence, and thereby spoil the effect produced by
phosphorescence.
[Illustration: FIG. 25.--IMPROVED EXPERIMENTAL BULB.]
Again, in another of the early experiments, a bulb was used as
illustrated in Fig. 12. In this instance, by touching the bulb with
one or two fingers, one or two shadows of the stem inside were
projected against the glass, the touch of the finger producing the
same result as the application of an external negative electrode under
ordinary circumstances.
In all these experiments the action was intensified by augmenting the
capacity at the end of the lead connected to the terminal. As a rule,
it is not necessary to resort to such means, and would be quite
unnecessary with still higher frequencies; but when it _is_ desired,
the bulb, or tube, can be easily adapted to the purpose.
[Illustration: FIG. 26.--IMPROVED BULB WITH INTENSIFYING REFLECTOR.]
In Fig. 24, for example, an experimental bulb L is shown, which is
provided with a neck n on the top for the application of an external
tinfoil coating, which may be connected to a body of larger surface.
Such a lamp as illustrated in Fig. 25 may also be lighted by
connecting the tinfoil coating on the neck n to the terminal, and the
leading-in wire w to an insulated plate. If the bulb stands in a
socket upright, as shown in the cut, a shade of conducting material
may be slipped in the neck n, and the action thus magnified.
A more perfected arrangement used in some of these bulbs is
illustrated in Fig. 26. In this case the construction of the bulb is
as shown and described before, when reference was made to Fig. 19. A
zinc sheet Z, with a tubular extension T, is slipped over the metallic
socket S. The bulb hangs downward from the terminal t, the zinc sheet
Z, performing the double office of intensifier and reflector. The
reflector is separated from the terminal t by an extension of the
insulating plug P.
[Illustration: FIG. 27.--PHOSPHORESCENT TUBE WITH INTENSIFYING
REFLECTOR.]
A similar disposition with a phosphorescent tube is illustrated in
Fig. 27. The tube T is prepared from two short tubes of a different
diameter, which are sealed on the ends. On the lower end is placed an
outside conducting coating C, which connects to the wire w. The wire
has a hook on the upper end for suspension, and passes through the
centre of the inside tube, which is filled with some good and tightly
packed insulator. On the outside of the upper end of the tube T is
another conducting coating C_1 upon which is slipped a metallic
reflector Z, which should be separated by a thick insulation from the
end of wire w.
The economical use of such a reflector or intensifier would require
that all energy supplied to an air condenser should be recoverable,
or, in other words, that there should not be any losses, neither in
the gaseous medium nor through its action elsewhere. This is far from
being so, but, fortunately, the losses may be reduced to anything
desired. A few remarks are necessary on this subject, in order to make
the experiences gathered in the course of these investigations
perfectly clear.
Suppose a small helix with many well insulated turns, as in experiment
Fig. 17, has one of its ends connected to one of the terminals of the
induction coil, and the other to a metal plate, or, for the sake of
simplicity, a sphere, insulated in space. When the coil is set to
work, the potential of the sphere is alternated, and the small helix
now behaves as though its free end were connected to the other
terminal of the induction coil. If an iron rod be held within the
small helix it is quickly brought to a high temperature, indicating
the passage of a strong current through the helix. How does the
insulated sphere act in this case? It can be a condenser, storing and
returning the energy supplied to it, or it can be a mere sink of
energy, and the conditions of the experiment determine whether it is
more one or the other. The sphere being charged to a high potential,
it acts inductively upon the surrounding air, or whatever gaseous
medium there might be. The molecules, or atoms, which are near the
sphere are of course more attracted, and move through a greater
distance than the farther ones. When the nearest molecules strike the
sphere they are repelled, and collisions occur at all distances within
the inductive action of the sphere. It is now clear that, if the
potential be steady, but little loss of energy can be caused in this
way, for the molecules which are nearest to the sphere, having had an
additional charge imparted to them by contact, are not attracted until
they have parted, if not with all, at least with most of the
additional charge, which can be accomplished only after a great many
collisions. From the fact that with a steady potential there is but
little loss in dry air, one must come to such a conclusion. When the
potential of the sphere, instead of being steady, is alternating, the
conditions are entirely different. In this case a rhythmical
bombardment occurs, no matter whether the molecules after coming in
contact with the sphere lose the imparted charge or not; what is more,
if the charge is not lost, the impacts are only the more violent.
Still if the frequency of the impulses be very small, the loss caused
by the impacts and collisions would not be serious unless the
potential were excessive. But when extremely high frequencies and more
or less high potentials are used, the loss may be very great. The
total energy lost per unit of time is proportionate to the product of
the number of impacts per second, or the frequency and the energy lost
in each impact. But the energy of an impact must be proportionate to
the square of the electric density of the sphere, since the charge
imparted to the molecule is proportionate to that density. I conclude
from this that the total energy lost must be proportionate to the
product of the frequency and the square of the electric density; but
this law needs experimental confirmation. Assuming the preceding
considerations to be true, then, by rapidly alternating the potential
of a body immersed in an insulating gaseous medium, any amount of
energy may be dissipated into space. Most of that energy then, I
believe, is not dissipated in the form of long ether waves, propagated
to considerable distance, as is thought most generally, but is
consumed--in the case of an insulated sphere, for example--in impact
and collisional losses--that is, heat vibrations--on the surface and
in the vicinity of the sphere. To reduce the dissipation it is
necessary to work with a small electric density--the smaller the
higher the frequency.
But since, on the assumption before made, the loss is diminished with
the square of the density, and since currents of very high frequencies
involve considerable waste when transmitted through conductors, it
follows that, on the whole, it is better to employ one wire than two.
Therefore, if motors, lamps, or devices of any kind are perfected,
capable of being advantageously operated by currents of extremely high
frequency, economical reasons will make it advisable to use only one
wire, especially if the distances are great.
When energy is absorbed in a condenser the same behaves as though its
capacity were increased. Absorption always exists more or less, but
generally it is small and of no consequence as long as the frequencies
are not very great. In using extremely high frequencies, and,
necessarily in such case, also high potentials, the absorption--or,
what is here meant more particularly by this term, the loss of energy
due to the presence of a gaseous medium--is an important factor to be
considered, as the energy absorbed in the air condenser may be any
fraction of the supplied energy. This would seem to make it very
difficult to tell from the measured or computed capacity of an air
condenser its actual capacity or vibration period, especially if the
condenser is of very small surface and is charged to a very high
potential. As many important results are dependent upon the
correctness of the estimation of the vibration period, this subject
demands the most careful scrutiny of other investigators. To reduce
the probable error as much as possible in experiments of the kind
alluded to, it is advisable to use spheres or plates of large surface,
so as to make the density exceedingly small. Otherwise, when it is
practicable, an oil condenser should be used in preference. In oil or
other liquid dielectrics there are seemingly no such losses as in
gaseous media. It being impossible to exclude entirely the gas in
condensers with solid dielectrics, such condensers should be immersed
in oil, for economical reasons if nothing else; they can then be
strained to the utmost and will remain cool. In Leyden jars the loss
due to air is comparatively small, as the tinfoil coatings are large,
close together, and the charged surfaces not directly exposed; but
when the potentials are very high, the loss may be more or less
considerable at, or near, the upper edge of the foil, where the air is
principally acted upon. If the jar be immersed in boiled-out oil, it
will be capable of performing four times the amount of work which it
can for any length of time when used in the ordinary way, and the loss
will be inappreciable.
It should not be thought that the loss in heat in an air condenser is
necessarily associated with the formation of _visible_ streams or
brushes. If a small electrode, inclosed in an unexhausted bulb, is
connected to one of the terminals of the coil, streams can be seen to
issue from the electrode and the air in the bulb is heated; if,
instead of a small electrode, a large sphere is inclosed in the bulb,
no streams are observed, still the air is heated.
Nor should it be thought that the temperature of an air condenser
would give even an approximate idea of the loss in heat incurred, as
in such case heat must be given off much more quickly, since there is,
in addition to the ordinary radiation, a very active carrying away of
heat by independent carriers going on, and since not only the
apparatus, but the air at some distance from it is heated in
consequence of the collisions which must occur.
Owing to this, in experiments with such a coil, a rise of temperature
can be distinctly observed only when the body connected to the coil is
very small. But with apparatus on a larger scale, even a body of
considerable bulk would be heated, as, for instance, the body of a
person; and I think that skilled physicians might make observations of
utility in such experiments, which, if the apparatus were judiciously
designed, would not present the slightest danger.
A question of some interest, principally to meteorologists, presents
itself here. How does the earth behave? The earth is an air condenser,
but is it a perfect or a very imperfect one--a mere sink of energy?
There can be little doubt that to such small disturbance as might be
caused in an experiment the earth behaves as an almost perfect
condenser. But it might be different when its charge is set in
vibration by some sudden disturbance occurring in the heavens. In such
case, as before stated, probably only little of the energy of the
vibrations set up would be lost into space in the form of long ether
radiations, but most of the energy, I think, would spend itself in
molecular impacts and collisions, and pass off into space in the form
of short heat, and possibly light, waves. As both the frequency of the
vibrations of the charge and the potential are in all probability
excessive, the energy converted into heat may be considerable. Since
the density must be unevenly distributed, either in consequence of the
irregularity of the earth's surface, or on account of the condition of
the atmosphere in various places, the effect produced would
accordingly vary from place to place. Considerable variations in the
temperature and pressure of the atmosphere may in this manner be
caused at any point of the surface of the earth. The variations may be
gradual or very sudden, according to the nature of the general
disturbance, and may produce rain and storms, or locally modify the
weather in any way.
From the remarks before made one may see what an important factor of
loss the air in the neighborhood of a charged surface becomes when the
electric density is great and the frequency of the impulses excessive.
But the action as explained implies that the air is insulating--that
is, that it is composed of independent carriers immersed in an
insulating medium. This is the case only when the air is at something
like ordinary or greater, or at extremely small, pressure. When the
air is slightly rarefied and conducting, then true conduction losses
occur also. In such case, of course, considerable energy may be
dissipated into space even with a steady potential, or with impulses
of low frequency, if the density is very great.
When the gas is at very low pressure, an electrode is heated more
because higher speeds can be reached. If the gas around the electrode
is strongly compressed, the displacements, and consequently the
speeds, are very small, and the heating is insignificant. But if in
such case the frequency could be sufficiently increased, the electrode
would be brought to a high temperature as well as if the gas were at
very low pressure; in fact, exhausting the bulb is only necessary
because we cannot produce (and possibly not convey) currents of the
required frequency.
Returning to the subject of electrode lamps, it is obviously of
advantage in such a lamp to confine as much as possible the heat to
the electrode by preventing the circulation of the gas in the bulb. If
a very small bulb be taken, it would confine the heat better than a
large one, but it might not be of sufficient capacity to be operated
from the coil, or, if so, the glass might get too hot. A simple way to
improve in this direction is to employ a globe of the required size,
but to place a small bulb, the diameter of which is properly
estimated, over the refractory button contained in the globe. This
arrangement is illustrated in Fig. 28.
[Illustration: FIG. 28.--LAMP WITH AUXILIARY BULB FOR CONFINING THE
ACTION TO THE CENTRE.]
The globe L has in this case a large neck n, allowing the small bulb b
to slip through. Otherwise the construction is the same as shown in
Fig. 18, for example. The small bulb is conveniently supported upon
the stem s, carrying the refractory button m. It is separated from the
aluminium tube a by several layers of mica M, in order to prevent the
cracking of the neck by the rapid heating of the aluminium tube upon a
sudden turning on of the current. The inside bulb should be as small
as possible when it is desired to obtain light only by incandescence
of the electrode. If it is desired to produce phosphorescence, the
bulb should be larger, else it would be apt to get too hot, and the
phosphorescence would cease. In this arrangement usually only the
small bulb shows phosphorescence, as there is practically no
bombardment against the outer globe. In some of these bulbs
constructed as illustrated in Fig. 28 the small tube was coated with
phosphorescent paint, and beautiful effects were obtained. Instead of
making the inside bulb large, in order to avoid undue heating, it
answers the purpose to make the electrode m larger. In this case the
bombardment is weakened by reason of the smaller electric density.
Many bulbs were constructed on the plan illustrated in Fig. 29. Here a
small bulb b, containing the refractory button m, upon being exhausted
to a very high degree was sealed in a large globe L, which was then
moderately exhausted and sealed off. The principal advantage of this
construction was that it allowed of reaching extremely high vacua,
and, at the same time use a large bulb. It was found, in the course of
experiences with bulbs such as illustrated in Fig. 29, that it was
well to make the stem s near the seal at e very thick, and the
leading-in wire w thin, as it occurred sometimes that the stem at e
was heated and the bulb was cracked. Often the outer globe L was
exhausted only just enough to allow the discharge to pass through, and
the space between the bulbs appeared crimson, producing a curious
effect. In some cases, when the exhaustion in globe L was very low,
and the air good conducting, it was found necessary, in order to bring
the button m to high incandescence, to place, preferably on the upper
part of the neck of the globe, a tinfoil coating which was connected
to an insulated body, to the ground, or to the other terminal of the
coil, as the highly conducting air weakened the effect somewhat,
probably by being acted upon inductively from the wire w, where it
entered the bulb at e. Another difficulty--which, however, is always
present when the refractory button is mounted in a very small
bulb--existed in the construction illustrated in Fig. 29, namely, the
vacuum in the bulb b would be impaired in a comparatively short time.
[Illustration: FIG. 29.--LAMP WITH INDEPENDENT AUXILIARY BULB.]
The chief idea in the two last described constructions was to confine
the heat to the central portion of the globe by preventing the
exchange of air. An advantage is secured, but owing to the heating of
the inside bulb and slow evaporation of the glass the vacuum is hard
to maintain, even if the construction illustrated in Fig. 28 be
chosen, in which both bulbs communicate.
But by far the better way--the ideal way--would be to reach
sufficiently high frequencies. The higher the frequency the slower
would be the exchange of the air, and I think that a frequency may be
reached at which there would be no exchange whatever of the air
molecules around the terminal. We would then produce a flame in which
there would be no carrying away of material, and a queer flame it
would be, for it would be rigid! With such high frequencies the
inertia of the particles would come into play. As the brush, or flame,
would gain rigidity in virtue of the inertia of the particles, the
exchange of the latter would be prevented. This would necessarily
occur, for, the number of the impulses being augmented, the potential
energy of each would diminish, so that finally only atomic vibrations
could be set up, and the motion of translation through measurable
space would cease. Thus an ordinary gas burner connected to a source
of rapidly alternating potential might have its efficiency augmented
to a certain limit, and this for two reasons--because of the
additional vibration imparted, and because of a slowing down of the
process of carrying off. But the renewal being rendered difficult, and
renewal being necessary to maintain the _burner_, a continued increase
of the frequency of the impulses, assuming they could be transmitted
to and impressed upon the flame, would result in the "extinction" of
the latter, meaning by this term only the cessation of the chemical
process.
I think, however, that in the case of an electrode immersed in a fluid
insulating medium, and surrounded by independent carriers of electric
charges, which can be acted upon inductively, a sufficiently high
frequency of the impulses would probably result in a gravitation of
the gas all around toward the electrode. For this it would be only
necessary to assume that the independent bodies are irregularly
shaped; they would then turn toward the electrode their side of the
greatest electric density, and this would be a position in which the
fluid resistance to approach would be smaller than that offered to the
receding.
The general opinion, I do not doubt, is that it is out of the question
to reach any such frequencies as might--assuming some of the views
before expressed to be true--produce any of the results which I have
pointed out as mere possibilities. This may be so, but in the course
of these investigations, from the observation of many phenomena I have
gained the conviction that these frequencies would be much lower than
one is apt to estimate at first. In a flame we set up light vibrations
by causing molecules, or atoms, to collide. But what is the ratio of
the frequency of the collisions and that of the vibrations set up?
Certainly it must be incomparably smaller than that of the knocks of
the bell and the sound vibrations, or that of the discharges and the
oscillations of the condenser. We may cause the molecules of the gas
to collide by the use of alternate electric impulses of high
frequency, and so we may imitate the process in a flame; and from
experiments with frequencies which we are now able to obtain, I think
that the result is producible with impulses which are transmissible
through a conductor.
In connection with thoughts of a similar nature, it appeared to me of
great interest to demonstrate the rigidity of a vibrating gaseous
column. Although with such low frequencies as, say 10,000 per second,
which I was able to obtain without difficulty from a specially
constructed alternator, the task looked discouraging at first, I made
a series of experiments. The trials with air at ordinary pressure led
to no result, but with air moderately rarefied I obtain what I think
to be an unmistakable experimental evidence of the property sought
for. As a result of this kind might lead able investigators to
conclusions of importance I will describe one of the experiments
performed.
It is well known that when a tube is slightly exhausted the discharge
may be passed through it in the form of a thin luminous thread. When
produced with currents of low frequency, obtained from a coil operated
as usual, this thread is inert. If a magnet be approached to it, the
part near the same is attracted or repelled, according to the
direction of the lines of force of the magnet. It occurred to me that
if such a thread would be produced with currents of very high
frequency, it should be more or less rigid, and as it was visible it
could be easily studied. Accordingly I prepared a tube about 1 inch in
diameter and 1 metre long, with outside coating at each end. The tube
was exhausted to a point at which by a little working the thread
discharge could be obtained. It must be remarked here that the general
aspect of the tube, and the degree of exhaustion, are quite different
than when ordinary low frequency currents are used. As it was found
preferable to work with one terminal, the tube prepared was suspended
from the end of a wire connected to the terminal, the tinfoil coating
being connected to the wire, and to the lower coating sometimes a
small insulated plate was attached. When the thread was formed it
extended through the upper part of the tube and lost itself in the
lower end. If it possessed rigidity it resembled, not exactly an
elastic cord stretched tight between two supports, but a cord
suspended from a height with a small weight attached at the end. When
the finger or a magnet was approached to the upper end of the luminous
thread, it could be brought locally out of position by electrostatic
or magnetic action; and when the disturbing object was very quickly
removed, an analogous result was produced, as though a suspended cord
would be displaced and quickly released near the point of suspension.
In doing this the luminous thread was set in vibration, and two very
sharply marked nodes, and a third indistinct one, were formed. The
vibration, once set up, continued for fully eight minutes, dying
gradually out. The speed of the vibration often varied perceptibly,
and it could be observed that the electrostatic attraction of the
glass affected the vibrating thread; but it was clear that the
electrostatic action was not the cause of the vibration, for the
thread was most generally stationary, and could always be set in
vibration by passing the finger quickly near the upper part of the
tube. With a magnet the thread could be split in two and both parts
vibrated. By approaching the hand to the lower coating of the tube, or
insulated plate if attached, the vibration was quickened; also, as far
as I could see, by raising the potential or frequency. Thus, either
increasing the frequency or passing a stronger discharge of the same
frequency corresponded to a tightening of the cord. I did not obtain
any experimental evidence with condenser discharges. A luminous band
excited in a bulb by repeated discharges of a Leyden jar must possess
rigidity, and if deformed and suddenly released should vibrate. But
probably the amount of vibrating matter is so small that in spite of
the extreme speed the inertia cannot prominently assert itself.
Besides, the observation in such a case is rendered extremely
difficult on account of the fundamental vibration.
The demonstration of the fact--which still needs better experimental
confirmation--that a vibrating gaseous column possesses rigidity,
might greatly modify the views of thinkers. When with low frequencies
and insignificant potentials indications of that property may be
noted, how must a gaseous medium behave under the influence of
enormous electrostatic stresses which may be active in the
interstellar space, and which may alternate with inconceivable
rapidity? The existence of such an electrostatic, rhythmically
throbbing force--of a vibrating electrostatic field--would show a
possible way how solids might have formed from the ultra-gaseous
uterus, and how transverse and all kinds of vibrations may be
transmitted through a gaseous medium filling all space. Then, ether
might be a true fluid, devoid of rigidity, and at rest, it being
merely necessary as a connecting link to enable interaction. What
determines the rigidity of a body? It must be the speed and the amount
of moving matter. In a gas the speed may be considerable, but the
density is exceedingly small; in a liquid the speed would be likely to
be small, though the density may be considerable; and in both cases
the inertia resistance offered to displacement is practically _nil_.
But place a gaseous (or liquid) column in an intense, rapidly
alternating electrostatic field, set the particles vibrating with
enormous speeds, then the inertia resistance asserts itself. A body
might move with more or less freedom through the vibrating mass, but
as a whole it would be rigid.
There is a subject which I must mention in connection with these
experiments: it is that of high vacua. This is a subject the study of
which is not only interesting, but useful, for it may lead to results
of great practical importance. In commercial apparatus, such as
incandescent lamps, operated from ordinary systems of distribution, a
much higher vacuum than obtained at present would not secure a very
great advantage. In such a case the work is performed on the filament
and the gas is little concerned; the improvement, therefore, would be
but trifling. But when we begin to use very high frequencies and
potentials, the action of the gas becomes all important, and the
degree of exhaustion materially modifies the results. As long as
ordinary coils, even very large ones, were used, the study of the
subject was limited, because just at a point when it became most
interesting it had to be interrupted on account of the "non-striking"
vacuum being reached. But presently we are able to obtain from a small
disruptive discharge coil potentials much higher than even the largest
coil was capable of giving, and, what is more, we can make the
potential alternate with great rapidity. Both of these results enable
us now to pass a luminous discharge through almost any vacua
obtainable, and the field of our investigations is greatly extended.
Think we as we may, of all the possible directions to develop a
practical illuminant, the line of high vacua seems to be the most
promising at present. But to reach extreme vacua the appliances must
be much more improved, and ultimate perfection will not be attained
until we shall have discarded the mechanical and perfected an
_electrical_ vacuum pump. Molecules and atoms can be thrown out of a
bulb under the action of an enormous potential: _this_ will be the
principle of the vacuum pump of the future. For the present, we must
secure the best results we can with mechanical appliances. In this
respect, it might not be out of the way to say a few words about the
method of, and apparatus for, producing excessively high degrees of
exhaustion of which I have availed myself in the course of these
investigations. It is very probable that other experimenters have used
similar arrangements; but as it is possible that there may be an item
of interest in their description, a few remarks, which will render
this investigation more complete, might be permitted.
[Illustration: FIG. 30.--APPARATUS USED FOR OBTAINING HIGH DEGREES OF
EXHAUSTION.]
The apparatus is illustrated in a drawing shown in Fig. 30. S
represents a Sprengel pump, which has been specially constructed to
better suit the work required. The stop-cock which is usually employed
has been omitted, and instead of it a hollow stopper s has been fitted
in the neck of the reservoir R. This stopper has a small hole h,
through which the mercury descends; the size of the outlet o being
properly determined with respect to the section of the fall tube t,
which is sealed to the reservoir instead of being connected to it in
the usual manner. This arrangement overcomes the imperfections and
troubles which often arise from the use of the stopcock on the
reservoir and the connection of the latter with the fall tube.
The pump is connected through a U-shaped tube t to a very large
reservoir R_1. Especial care was taken in fitting the grinding
surfaces of the stoppers p and p_1, and both of these and the mercury
caps above them were made exceptionally long. After the U-shaped tube
was fitted and put in place, it was heated, so as to soften and take
off the strain resulting from imperfect fitting. The U-shaped tube was
provided with a stopcock C, and two ground connections g and g_1--one
for a small bulb b, usually containing caustic potash, and the other
for the receiver r, to be exhausted.
The reservoir R_1 was connected by means of a rubber tube to a
slightly larger reservoir R_2, each of the two reservoirs being
provided with a stopcock C_1 and C_2, respectively. The reservoir R_2
could be raised and lowered by a wheel and rack, and the range of its
motion was so determined that when it was filled with mercury and the
stopcock C_2 closed, so as to form a Torricellian vacuum in it when
raised, it could be lifted so high that the mercury in reservoir R_1
would stand a little above stopcock C_1; and when this stopcock was
closed and the reservoir R_2 descended, so as to form a Torricellian
vacuum in reservoir R_1, it could be lowered so far as to completely
empty the latter, the mercury filling the reservoir R_2 up to a little
above stopcock C_2.
The capacity of the pump and of the connections was taken as small as
possible relatively to the volume of reservoir R_1, since, of course,
the degree of exhaustion depended upon the ratio of these quantities.
With this apparatus I combined the usual means indicated by former
experiments for the production of very high vacua. In most of the
experiments it was convenient to use caustic potash. I may venture to
say, in regard to its use, that much time is saved and a more perfect
action of the pump insured by fusing and boiling the potash as soon
as, or even before, the pump settles down. If this course is not
followed the sticks, as ordinarily employed, may give moisture off at
a certain very slow rate, and the pump may work for many hours without
reaching a very high vacuum. The potash was heated either by a spirit
lamp or by passing a discharge through it, or by passing a current
through a wire contained in it. The advantage in the latter case was
that the heating could be more rapidly repeated.
Generally the process of exhaustion was the following:--At the start,
the stop-cocks C and C_1 being open, and all other connections closed,
the reservoir R_2 was raised so far that the mercury filled the
reservoir R_1 and a part of the narrow connecting U-shaped tube. When
the pump was set to work, the mercury would, of course, quickly rise
in the tube, and reservoir R_2 was lowered, the experimenter keeping
the mercury at about the same level. The reservoir R_2 was balanced
by a long spring which facilitated the operation, and the friction of
the parts was generally sufficient to keep it almost in any position.
When the Sprengel pump had done its work, the reservoir R_2 was
further lowered and the mercury descended in R_1 and filled R_2,
whereupon stopcock C_2 was closed. The air adhering to the walls of
R_1 and that absorbed by the mercury was carried off, and to free the
mercury of all air the reservoir R_2 was for a long time worked up and
down. During this process some air, which would gather below stopcock
C_2, was expelled from R_2 by lowering it far enough and opening the
stopcock, closing the latter again before raising the reservoir. When
all the air had been expelled from the mercury, and no air would
gather in R_2 when it was lowered, the caustic potash was resorted to.
The reservoir R_2 was now again raised until the mercury in R_1 stood
above stopcock C_1. The caustic potash was fused and boiled, and the
moisture partly carried off by the pump and partly re-absorbed; and
this process of heating and cooling was repeated many times, and each
time, upon the moisture being absorbed or carried off, the reservoir
R_2 was for a long time raised and lowered. In this manner all the
moisture was carried off from the mercury, and both the reservoirs
were in proper condition to be used. The reservoir R_2 was then again
raised to the top, and the pump was kept working for a long time. When
the highest vacuum obtainable with the pump had been reached the
potash bulb was usually wrapped with cotton which was sprinkled with
ether so as to keep the potash at a very low temperature, then the
reservoir R_2 was lowered, and upon reservoir R_1 being emptied the
receiver r was quickly sealed up.
When a new bulb was put on, the mercury was always raised above
stopcock C_1 which was closed, so as to always keep the mercury and
both the reservoirs in fine condition, and the mercury was never
withdrawn from R_1 except when the pump had reached the highest degree
of exhaustion. It is necessary to observe this rule if it is desired
to use the apparatus to advantage.
By means of this arrangement I was able to proceed very quickly, and
when the apparatus was in perfect order it was possible to reach the
phosphorescent stage in a small bulb in less than 15 minutes, which is
certainly very quick work for a small laboratory arrangement requiring
all in all about 100 pounds of mercury. With ordinary small bulbs the
ratio of the capacity of the pump, receiver, and connections, and that
of reservoir R was about 1-20, and the degrees of exhaustion reached
were necessarily very high, though I am unable to make a precise and
reliable statement how far the exhaustion was carried.
What impresses the investigator most in the course of these
experiences is the behavior of gases when subjected to great rapidly
alternating electrostatic stresses. But he must remain in doubt as to
whether the effects observed are due wholly to the molecules, or
atoms, of the gas which chemical analysis discloses to us, or whether
there enters into play another medium of a gaseous nature, comprising
atoms, or molecules, immersed in a fluid pervading the space. Such a
medium surely must exist, and I am convinced that, for instance, even
if air were absent, the surface and neighborhood of a body in space
would be heated by rapidly alternating the potential of the body; but
no such heating of the surface or neighborhood could occur if all free
atoms were removed and only a homogeneous, incompressible, and elastic
fluid--such as ether is supposed to be--would remain, for then there
would be no impacts, no collisions. In such a case, as far as the body
itself is concerned, only frictional losses in the inside could occur.
It is a striking fact that the discharge through a gas is established
with ever increasing freedom as the frequency of the impulses is
augmented. It behaves in this respect quite contrarily to a metallic
conductor. In the latter the impedance enters prominently into play as
the frequency is increased, but the gas acts much as a series of
condensers would: the facility with which the discharge passes through
seems to depend on the rate of change of potential. If it act so, then
in a vacuum tube even of great length, and no matter how strong the
current, self-induction could not assert itself to any appreciable
degree. We have, then, as far as we can now see, in the gas a
conductor which is capable of transmitting electric impulses of any
frequency which we may be able to produce. Could the frequency be
brought high enough, then a queer system of electric distribution,
which would be likely to interest gas companies, might be realized:
metal pipes filled with gas--the metal being the insulator, the gas
the conductor--supplying phosphorescent bulbs, or perhaps devices as
yet uninvented. It is certainly possible to take a hollow core of
copper, rarefy the gas in the same, and by passing impulses of
sufficiently high frequency through a circuit around it, bring the gas
inside to a high degree of incandescence; but as to the nature of the
forces there would be considerable uncertainty, for it would be
doubtful whether with such impulses the copper core would act as a
static screen. Such paradoxes and apparent impossibilities we
encounter at every step in this line of work, and therein lies, to a
great extent, the claim of the study.
I have here a short and wide tube which is exhausted to a high degree
and covered with a substantial coating of bronze, the coating allowing
barely the light to shine through. A metallic clasp, with a hook for
suspending the tube, is fastened around the middle portion of the
latter, the clasp being in contact with the bronze coating. I now want
to light the gas inside by suspending the tube on a wire connected to
the coil. Any one who would try the experiment for the first time, not
having any previous experience, would probably take care to be quite
alone when making the trial, for fear that he might become the joke of
his assistants. Still, the bulb lights in spite of the metal coating,
and the light can be distinctly perceived through the latter. A long
tube covered with aluminium bronze lights when held in one hand--the
other touching the terminal of the coil--quite powerfully. It might be
objected that the coatings are not sufficiently conducting; still,
even if they were highly resistant, they ought to screen the gas. They
certainly screen it perfectly in a condition of rest, but not by far
perfectly when the charge is surging in the coating. But the loss of
energy which occurs within the tube, notwithstanding the screen, is
occasioned principally by the presence of the gas. Were we to take a
large hollow metallic sphere and fill it with a perfect incompressible
fluid dielectric, there would be no loss inside of the sphere, and
consequently the inside might be considered as perfectly screened,
though the potential be very rapidly alternating. Even were the sphere
filled with oil, the loss would be incomparably smaller than when the
fluid is replaced by a gas, for in the latter case the force produces
displacements; that means impact and collisions in the inside.
No matter what the pressure of the gas may be, it becomes an important
factor in the heating of a conductor when the electric density is
great and the frequency very high. That in the heating of conductors
by lightning discharges air is an element of great importance, is
almost as certain as an experimental fact. I may illustrate the action
of the air by the following experiment: I take a short tube which is
exhausted to a moderate degree and has a platinum wire running through
the middle from one end to the other. I pass a steady or low frequency
current through the wire, and it is heated uniformly in all parts. The
heating here is due to conduction, or frictional losses, and the gas
around the wire has--as far as we can see--no function to perform. But
now let me pass sudden discharges, or a high frequency current,
through the wire. Again the wire is heated, this time principally on
the ends and least in the middle portion; and if the frequency of the
impulses, or the rate of change, is high enough, the wire might as
well be cut in the middle as not, for practically all the heating is
due to the rarefied gas. Here the gas might only act as a conductor of
no impedance diverting the current from the wire as the impedance of
the latter is enormously increased, and merely heating the ends of the
wire by reason of their resistance to the passage of the discharge.
But it is not at all necessary that the gas in the tube should he
conducting; it might be at an extremely low pressure, still the ends
of the wire would be heated--as, however, is ascertained by
experience--only the two ends would in such, case not be electrically
connected through the gaseous medium. Now what with these frequencies
and potentials occurs in an exhausted tube occurs in the lightning
discharges at ordinary pressure. We only need remember one of the
facts arrived at in the course of these investigations, namely, that
to impulses of very high frequency the gas at ordinary pressure
behaves much in the same manner as though it were at moderately low
pressure. I think that in lightning discharges frequently wires or
conducting objects are volatilized merely because air is present and
that, were the conductor immersed in an insulating liquid, it would be
safe, for then the energy would have to spend itself somewhere else.
From the behavior of gases to sudden impulses of high potential I am
led to conclude that there can be no surer way of diverting a
lightning discharge than by affording it a passage through a volume of
gas, if such a thing can be done in a practical manner.
There are two more features upon which I think it necessary to dwell
in connection with these experiments--the "radiant state" and the
"non-striking vacuum."
Any one who has studied Crookes' work must have received the
impression that the "radiant state" is a property of the gas
inseparably connected with an extremely high degree of exhaustion. But
it should be remembered that the phenomena observed in an exhausted
vessel are limited to the character and capacity of the apparatus
which is made use of. I think that in a bulb a molecule, or atom, does
not precisely move in a straight line because it meets no obstacle,
but because the velocity imparted to it is sufficient to propel it in
a sensibly straight line. The mean free path is one thing, but the
velocity--the energy associated with the moving body--is another, and
under ordinary circumstances I believe that it is a mere question of
potential or speed. A disruptive discharge coil, when the potential is
pushed very far, excites phosphorescence and projects shadows, at
comparatively low degrees of exhaustion. In a lightning discharge,
matter moves in straight lines as ordinary pressure when the mean free
path is exceedingly small, and frequently images of wires or other
metallic objects have been produced by the particles thrown off in
straight lines.
[Illustration: FIG. 31.--BULB SHOWING RADIANT LIME STREAM AT LOW
EXHAUSTION.]
I have prepared a bulb to illustrate by an experiment the correctness
of these assertions. In a globe L (Fig. 31) I have mounted upon a lamp
filament f a piece of lime l. The lamp filament is connected with a
wire which leads into the bulb, and the general construction of the
latter is as indicated in Fig. 19, before described. The bulb being
suspended from a wire connected to the terminal of the coil, and the
latter being set to work, the lime piece l and the projecting parts of
the filament f are bombarded. The degree of exhaustion is just such
that with the potential the coil is capable of giving phosphorescence
of the glass is produced, but disappears as soon as the vacuum is
impaired. The lime containing moisture, and moisture being given off
as soon as heating occurs, the phosphorescence lasts only for a few
moments. When the lime has been sufficiently heated, enough moisture
has been given off to impair materially the vacuum of the bulb. As the
bombardment goes on, one point of the lime piece is more heated than
other points, and the result is that finally practically all the
discharge passes through that point which is intensely heated, and a
white stream of lime particles (Fig. 31) then breaks forth from that
point. This stream is composed of "radiant" matter, yet the degree of
exhaustion is low. But the particles move in straight lines because
the velocity imparted to them is great, and this is due to three
causes--to the great electric density, the high temperature of the
small point, and the fact that the particles of the lime are easily
torn and thrown off--far more easily than those of carbon. With
frequencies such as we are able to obtain, the particles are bodily
thrown off and projected to a considerable distance; but with
sufficiently high frequencies no such thing would occur: in such case
only a stress would spread or a vibration would be propagated through
the bulb. It would be out of the question to reach any such frequency
on the assumption that the atoms move with the speed of light; but I
believe that such a thing is impossible; for this an enormous
potential would be required. With potentials which we are able to
obtain, even with a disruptive discharge coil, the speed must be quite
insignificant.
As to the "non-striking vacuum," the point to be noted is that it can
occur only with low frequency impulses, and it is necessitated by the
impossibility of carrying off enough energy with such impulses in high
vacuum since the few atoms which are around the terminal upon coming
in contact with the same are repelled and kept at a distance for a
comparatively long period of time, and not enough work can be
performed to render the effect perceptible to the eye. If the
difference of potential between the terminals is raised, the
dielectric breaks down. But with very high frequency impulses there
is no necessity for such breaking down, since any amount of work can
be performed by continually agitating the atoms in the exhausted
vessel, provided the frequency is high enough. It is easy to
reach--even with frequencies obtained from an alternator as here
used--a stage at which the discharge does not pass between two
electrodes in a narrow tube, each of these being connected to one of
the terminals of the coil, but it is difficult to reach a point at
which a luminous discharge would not occur around each electrode.
A thought which naturally presents itself in connection with high
frequency currents, is to make use of their powerful electro-dynamic
inductive action to produce light effects in a sealed glass globe. The
leading-in wire is one of the defects of the present incandescent
lamp, and if no other improvement were made, that imperfection at
least should be done away with. Following this thought, I have carried
on experiments in various directions, of which some were indicated in
my former paper. I may here mention one or two more lines of
experiment which have been followed up.
Many bulbs were constructed as shown in Fig. 32 and Fig. 33.
In Fig. 32 a wide tube T was sealed to a smaller W-shaped tube U, of
phosphorescent glass. In the tube T was placed a coil C of aluminium
wire, the ends of which were provided with small spheres t and t_1 of
aluminium, and reached into the U tube. The tube T was slipped into a
socket containing a primary coil through which usually the discharges
of Leyden jars were directed, and the rarefied gas in the small U tube
was excited to strong luminosity by the high-tension currents induced
in the coil C. When Leyden jar discharges were used to induce currents
in the coil C, it was found necessary to pack the tube T tightly with
insulating powder, as a discharge would occur frequently between the
turns of the coil, especially when the primary was thick and the air
gap, through which the jars discharged, large, and no little trouble
was experienced in this way.
[Illustration: FIG. 32.--ELECTRO-DYNAMIC INDUCTION TUBE.]
[Illustration: FIG. 33--ELECTRO-DYNAMIC INDUCTION LAMP.]
In Fig. 33 is illustrated another form of the bulb constructed. In
this case a tube T is sealed to a globe L. The tube contains a coil C,
the ends of which pass through two small glass tubes t and t_1, which
are sealed to the tube T. Two refractory buttons m and m_1 are mounted
on lamp filaments which are fastened to the ends of the wires passing
through the glass tubes t and t_1. Generally in bulbs made on this
plan the globe L communicated with the tube T. For this purpose the
ends of the small tubes t and t_1 were just a trifle heated in the
burner, merely to hold the wires, but not to interfere with the
communication. The tube T, with the small tubes, wires through the
same, and the refractory buttons m and m_1, was first prepared, and
then sealed to globe L, whereupon the coil C was slipped in and the
connections made to its ends. The tube was then packed with insulating
powder, jamming the latter as tight as possible up to very nearly the
end, then it was closed and only a small hole left through which the
remainder of the powder was introduced, and finally the end of the
tube was closed. Usually in bulbs constructed as shown in Fig. 33 an
aluminium tube a was fastened to the upper end s of each of the tubes
t and t_1, in order to protect that end against the heat. The buttons
m and m_1 could be brought to any degree of incandescence by passing
the discharges of Leyden jars around the coil C. In such bulbs with
two buttons a very curious effect is produced by the formation of the
shadows of each of the two buttons.
Another line of experiment, which has been assiduously followed, was
to induce by electro-dynamic induction a current or luminous discharge
in an exhausted tube or bulb. This matter has received such able
treatment at the hands of Prof. J.J. Thomson that I could add but
little to what he has made known, even had I made it the special
subject of this lecture. Still, since experiences in this line have
gradually led me to the present views and results, a few words must be
devoted here to this subject.
It has occurred, no doubt, to many that as a vacuum tube is made
longer the electromotive force per unit length of the tube, necessary
to pass a luminous discharge through the latter, gets continually
smaller; therefore, if the exhausted tube be made long enough, even
with low frequencies a luminous discharge could be induced in such a
tube closed upon itself. Such a tube might be placed around a ball or
on a ceiling, and at once a simple appliance capable of giving
considerable light would be obtained. But this would be an appliance
hard to manufacture and extremely unmanageable. It would not do to
make the tube up of small lengths, because there would be with
ordinary frequencies considerable loss in the coatings, and besides,
if coatings were used, it would be better to supply the current
directly to the tube by connecting the coatings to a transformer. But
even if all objections of such nature were removed, still, with low
frequencies the light conversion itself would be inefficient, as I
have before stated. In using extremely high frequencies the length of
the secondary--in other words, the size of the vessel--can be reduced
as far as desired, and the efficiency of the light conversion is
increased, provided that means are invented for efficiently obtaining
such high frequencies. Thus one is led, from theoretical and practical
considerations, to the use of high frequencies, and this means high
electromotive forces and small currents in the primary. When he works
with condenser charges--and they are the only means up to the present
known for reaching these extreme frequencies--he gets to electromotive
forces of several thousands of volts per turn of the primary. He
cannot multiply the electro-dynamic inductive effect by taking more
turns in the primary, for he arrives at the conclusion that the best
way is to work with one single turn--though he must sometimes depart
from this rule--and he must get along with whatever inductive effect
he can obtain with one turn. But before he has long experimented with
the extreme frequencies required to set up in a small bulb an
electromotive force of several thousands of volts he realizes the
great importance of electrostatic effects, and these effects grow
relatively to the electro-dynamic in significance as the frequency is
increased.
Now, if anything is desirable in this case, it is to increase the
frequency, and this would make it still worse for the electro-dynamic
effects. On the other hand, it is easy to exalt the electrostatic
action as far as one likes by taking more turns on the secondary, or
combining self-induction and capacity to raise the potential. It
should also be remembered that, in reducing the current to the
smallest value and increasing the potential, the electric impulses of
high frequency can be more easily transmitted through a conductor.
These and similar thoughts determined me to devote more attention to
the electrostatic phenomena, and to endeavor to produce potentials as
high as possible, and alternating as fast as they could be made to
alternate. I then found that I could excite vacuum tubes at
considerable distance from a conductor connected to a properly
constructed coil, and that I could, by converting the oscillatory
current of a condenser to a higher potential, establish electrostatic
alternating fields which acted through the whole extent of a room,
lighting up a tube no matter where it was held in space. I thought I
recognized that I had made a step in advance, and I have persevered in
this line; but I wish to say that I share with all lovers of science
and progress the one and only desire--to reach a result of utility to
men in any direction to which thought or experiment may lead me. I
think that this departure is the right one, for I cannot see, from the
observation of the phenomena which manifest themselves as the
frequency is increased, what there would remain to act between two
circuits conveying, for instance, impulses of several hundred millions
per second, except electrostatic forces. Even with such trifling
frequencies the energy would be practically all potential, and my
conviction has grown strong that, to whatever kind of motion light may
be due, it is produced by tremendous electrostatic stresses vibrating
with extreme rapidity.
Of all these phenomena observed with currents, or electric impulses,
of high frequency, the most fascinating for an audience are certainly
those which are noted in an electrostatic field acting through
considerable distance, and the best an unskilled lecturer can do is
to begin and finish with the exhibition of these singular effects. I
take a tube in the hand and move it about, and it is lighted wherever
I may hold it; throughout space the invisible forces act. But I may
take another tube and it might not light, the vacuum being very high.
I excite it by means of a disruptive discharge coil, and now it will
light in the electrostatic field. I may put it away for a few weeks or
months, still it retains the faculty of being excited. What change
have I produced in the tube in the act of exciting it? If a motion
imparted to the atoms, it is difficult to perceive how it can persist
so long without being arrested by frictional losses; and if a strain
exerted in the dielectric, such as a simple electrification would
produce, it is easy to see how it may persist indefinitely, but very
difficult to understand why such a condition should aid the excitation
when we have to deal with potentials which are rapidly alternating.
Since I have exhibited these phenomena for the first time, I have
obtained some other interesting effects. For instance, I have produced
the incandescence of a button, filament, or wire enclosed in a tube.
To get to this result it was necessary to economize the energy which
is obtained from the field and direct most of it on the small body to
be rendered incandescent. At the beginning the task appeared
difficult, but the experiences gathered permitted me to reach the
result easily. In Fig. 34 and Fig. 35 two such tubes are illustrated
which are prepared for the occasion. In Fig. 34 a short tube T_1,
sealed to another long tube T, is provided with a stem s, with a
platinum wire sealed in the latter. A very thin lamp filament l is
fastened to this wire, and connection to the outside is made through a
thin copper wire w. The tube is provided with outside and inside
coatings, C and C_1 respectively, and is filled as far as the coatings
reach with conducting, and the space above with insulating powder.
These coatings are merely used to enable me to perform two experiments
with the tube--namely, to produce the effect desired either by direct
connection of the body of the experimenter or of another body to the
wire w, or by acting inductively through the glass. The stem s is
provided with an aluminium tube a, for purposes before explained, and
only a small part of the filament reaches out of this tube. By holding
the tube T_1 anywhere in the electrostatic field the filament is
rendered incandescent.
[Illustration: FIG. 34.--TUBE WITH FILAMENT RENDERED INCANDESCENT IN
AN ELECTROSTATIC FIELD.]
[Illustration: FIG. 35.--CROOKES' EXPERIMENT IN ELECTROSTATIC FIELD.]
A more interesting piece of apparatus is illustrated in Fig. 35. The
construction is the same as before, only instead of the lamp filament
a small platinum wire p, sealed in a stem s, and bent above it in a
circle, is connected to the copper wire w, which is joined to an
inside coating C. A small stem s_1 is provided with a needle, on the
point of which is arranged to rotate very freely a very light fan of
mica v. To prevent the fan from falling out, a thin stem of glass g is
bent properly and fastened to the aluminium tube. When the glass tube
is held anywhere in the electrostatic field the platinum wire becomes
incandescent, and the mica vanes are rotated very fast.
Intense phosphorescence may be excited in a bulb by merely connecting
it to a plate within the field, and the plate need not be any larger
than an ordinary lamp shade. The phosphorescence excited with these
currents is incomparably more powerful than with ordinary apparatus. A
small phosphorescent bulb, when attached to a wire connected to a
coil, emits sufficient light to allow reading ordinary print at a
distance of five to six paces. It was of interest to see how some of
the phosphorescent bulbs of Professor Crookes would behave with these
currents, and he has had the kindness to lend me a few for the
occasion. The effects produced are magnificent, especially by the
sulphide of calcium and sulphide of zinc. From the disruptive
discharge coil they glow intensely merely by holding them in the hand
and connecting the body to the terminal of the coil.
To whatever results investigations of this kind may lead, their chief
interest lies for the present in the possibilities they offer for the
production of an efficient illuminating device. In no branch of
electric industry is an advance more desired than in the manufacture
of light. Every thinker, when considering the barbarous methods
employed, the deplorable losses incurred in our best systems of light
production, must have asked himself, What is likely to be the light of
the future? Is it to be an incandescent solid, as in the present lamp,
or an incandescent gas, or a phosphorescent body, or something like a
burner, but incomparably more efficient?
There is little chance to perfect a gas burner; not, perhaps, because
human ingenuity has been bent upon that problem for centuries without
a radical departure having been made--though this argument is not
devoid of force-but because in a burner the higher vibrations can
never be reached except by passing through all the low ones. For how
is a flame produced unless by a fall of lifted weights? Such process
cannot be maintained without renewal, and renewal is repeated passing
from low to high vibrations. One way only seems to be open to improve
a burner, and that is by trying to reach higher degrees of
incandescence. Higher incandescence is equivalent to a quicker
vibration; that means more light from the same material, and that,
again, means more economy. In this direction some improvements have
been made, but the progress is hampered by many limitations.
Discarding, then, the burner, there remain the three ways first
mentioned, which are essentially electrical.
Suppose the light of the immediate future to be a solid rendered
incandescent by electricity. Would it not seem that it is better to
employ a small button than a frail filament? From many considerations
it certainly must be concluded that a button is capable of a higher
economy, assuming, of course, the difficulties connected with the
operation of such a lamp to be effectively overcome. But to light such
a lamp we require a high potential; and to get this economically we
must use high frequencies.
Such considerations apply even more to the production of light by the
incandescence of a gas, or by phosphorescence. In all cases we require
high frequencies and high potentials. These thoughts occurred to me a
long time ago.
Incidentally we gain, by the use of very high frequencies, many
advantages, such as a higher economy in the light production, the
possibility of working with one lead, the possibility of doing away
with the leading-in wire, etc.
The question is, how far can we go with frequencies? Ordinary
conductors rapidly lose the facility of transmitting electric impulses
when the frequency is greatly increased. Assume the means for the
production of impulses of very great frequency brought to the utmost
perfection, every one will naturally ask how to transmit them when the
necessity arises. In transmitting such impulses through conductors we
must remember that we have to deal with _pressure_ and _flow_, in the
ordinary interpretation of these terms. Let the pressure increase to
an enormous value, and let the flow correspondingly diminish, then
such impulses--variations merely of pressure, as it were--can no doubt
be transmitted through a wire even if their frequency be many hundreds
of millions per second. It would, of course, be out of question to
transmit such impulses through a wire immersed in a gaseous medium,
even if the wire were provided with a thick and excellent insulation
for most of the energy would be lost in molecular bombardment and
consequent heating. The end of the wire connected to the source would
be heated, and the remote end would receive but a trifling part of the
energy supplied. The prime necessity, then, if such electric impulses
are to be used, is to find means to reduce as much as possible the
dissipation.
The first thought is, employ the thinnest possible wire surrounded by
the thickest practicable insulation. The next thought is to employ
electrostatic screens. The insulation of the wire may be covered with
a thin conducting coating and the latter connected to the ground. But
this would not do, as then all the energy would pass through the
conducting coating to the ground and nothing would get to the end of
the wire. If a ground connection is made it can only be made through a
conductor offering an enormous impedance, or though a condenser of
extremely small capacity. This, however, does not do away with other
difficulties.
If the wave length of the impulses is much smaller than the length of
the wire, then corresponding short waves will be sent up in the
conducting coating, and it will be more or less the same as though the
coating were directly connected to earth. It is therefore necessary to
cut up the coating in sections much shorter than the wave length. Such
an arrangement does not still afford a perfect screen, but it is ten
thousand times better than none. I think it preferable to cut up the
conducting coating in small sections, even if the current waves be
much longer than the coating.
If a wire were provided with a perfect electrostatic screen, it would
be the same as though all objects were removed from it at infinite
distance. The capacity would then be reduced to the capacity of the
wire itself, which would be very small. It would then be possible to
send over the wire current vibrations of very high frequencies at
enormous distance without affecting greatly the character of the
vibrations. A perfect screen is of course out of the question, but I
believe that with a screen such as I have just described telephony
could be rendered practicable across the Atlantic. According to my
ideas, the gutta-percha covered wire should be provided with a third
conducting coating subdivided in sections. On the top of this should
be again placed a layer of gutta-percha and other insulation, and on
the top of the whole the armor. But such cables will not be
constructed, for ere long intelligence--transmitted without
wires--will throb through the earth like a pulse through a living
organism. The wonder is that, with the present state of knowledge and
the experiences gained, no attempt is being made to disturb the
electrostatic or magnetic condition of the earth, and transmit, if
nothing else, intelligence.
It has been my chief aim in presenting these results to point out
phenomena or features of novelty, and to advance ideas which I am
hopeful will serve as starting points of new departures. It has been
my chief desire this evening to entertain you with some novel
experiments. Your applause, so frequently and generously accorded, has
told me that I have succeeded.
In conclusion, let me thank you most heartily for your kindness and
attention, and assure you that the honor I have had in addressing such
a distinguished audience, the pleasure I have had in presenting these
results to a gathering of so many able men--and among them also some
of those in whose work for many years past I have found enlightenment
and constant pleasure--I shall never forget.
[Transcriber's note: Corrected the following typesetting errors:
1) 'preceived' to 'perceived', page 16.
2) 'disharging' to 'discharging', page 30.
3) 'park' to 'spark', page 33.
4) 'pssition' to 'position', page 50.
5) 'to th opposite side' to 'to the opposite side', page 56.
6) 's resses' to 'stresses', page 147.]
TREATISE ON LIGHT
In which are explained
The causes of that which occurs
In REFLEXION, & in REFRACTION
And particularly
In the strange REFRACTION
OF ICELAND CRYSTAL
by
CHRISTIAAN HUYGENS
Rendered into English by
SILVANUS P. THOMPSON
University of Chicago Press
PREFACE
I wrote this Treatise during my sojourn in France twelve years ago,
and I communicated it in the year 1678 to the learned persons who then
composed the Royal Academy of Science, to the membership of which the
King had done me the honour of calling, me. Several of that body who
are still alive will remember having been present when I read it, and
above the rest those amongst them who applied themselves particularly
to the study of Mathematics; of whom I cannot cite more than the
celebrated gentlemen Cassini, Roemer, and De la Hire. And, although I
have since corrected and changed some parts, the copies which I had
made of it at that time may serve for proof that I have yet added
nothing to it save some conjectures touching the formation of Iceland
Crystal, and a novel observation on the refraction of Rock Crystal. I
have desired to relate these particulars to make known how long I have
meditated the things which now I publish, and not for the purpose of
detracting from the merit of those who, without having seen anything
that I have written, may be found to have treated of like matters: as
has in fact occurred to two eminent Geometricians, Messieurs Newton
and Leibnitz, with respect to the Problem of the figure of glasses for
collecting rays when one of the surfaces is given.
One may ask why I have so long delayed to bring this work to the
light. The reason is that I wrote it rather carelessly in the Language
in which it appears, with the intention of translating it into Latin,
so doing in order to obtain greater attention to the thing. After
which I proposed to myself to give it out along with another Treatise
on Dioptrics, in which I explain the effects of Telescopes and those
things which belong more to that Science. But the pleasure of novelty
being past, I have put off from time to time the execution of this
design, and I know not when I shall ever come to an end if it, being
often turned aside either by business or by some new study.
Considering which I have finally judged that it was better worth while
to publish this writing, such as it is, than to let it run the risk,
by waiting longer, of remaining lost.
There will be seen in it demonstrations of those kinds which do not
produce as great a certitude as those of Geometry, and which even
differ much therefrom, since whereas the Geometers prove their
Propositions by fixed and incontestable Principles, here the
Principles are verified by the conclusions to be drawn from them; the
nature of these things not allowing of this being done otherwise.
It is always possible to attain thereby to a degree of probability
which very often is scarcely less than complete proof. To wit, when
things which have been demonstrated by the Principles that have been
assumed correspond perfectly to the phenomena which experiment has
brought under observation; especially when there are a great number of
them, and further, principally, when one can imagine and foresee new
phenomena which ought to follow from the hypotheses which one employs,
and when one finds that therein the fact corresponds to our prevision.
But if all these proofs of probability are met with in that which I
propose to discuss, as it seems to me they are, this ought to be a
very strong confirmation of the success of my inquiry; and it must be
ill if the facts are not pretty much as I represent them. I would
believe then that those who love to know the Causes of things and who
are able to admire the marvels of Light, will find some satisfaction
in these various speculations regarding it, and in the new explanation
of its famous property which is the main foundation of the
construction of our eyes and of those great inventions which extend so
vastly the use of them.
I hope also that there will be some who by following these beginnings
will penetrate much further into this question than I have been able
to do, since the subject must be far from being exhausted. This
appears from the passages which I have indicated where I leave certain
difficulties without having resolved them, and still more from matters
which I have not touched at all, such as Luminous Bodies of several
sorts, and all that concerns Colours; in which no one until now can
boast of having succeeded. Finally, there remains much more to be
investigated touching the nature of Light which I do not pretend to
have disclosed, and I shall owe much in return to him who shall be
able to supplement that which is here lacking to me in knowledge. The
Hague. The 8 January 1690.
NOTE BY THE TRANSLATOR
Considering the great influence which this Treatise has exercised in
the development of the Science of Optics, it seems strange that two
centuries should have passed before an English edition of the work
appeared. Perhaps the circumstance is due to the mistaken zeal with
which formerly everything that conflicted with the cherished ideas of
Newton was denounced by his followers. The Treatise on Light of
Huygens has, however, withstood the test of time: and even now the
exquisite skill with which he applied his conception of the
propagation of waves of light to unravel the intricacies of the
phenomena of the double refraction of crystals, and of the refraction
of the atmosphere, will excite the admiration of the student of
Optics. It is true that his wave theory was far from the complete
doctrine as subsequently developed by Thomas Young and Augustin
Fresnel, and belonged rather to geometrical than to physical Optics.
If Huygens had no conception of transverse vibrations, of the
principle of interference, or of the existence of the ordered sequence
of waves in trains, he nevertheless attained to a remarkably clear
understanding of the principles of wave-propagation; and his
exposition of the subject marks an epoch in the treatment of Optical
problems. It has been needful in preparing this translation to
exercise care lest one should import into the author's text ideas of
subsequent date, by using words that have come to imply modern
conceptions. Hence the adoption of as literal a rendering as possible.
A few of the author's terms need explanation. He uses the word
"refraction," for example, both for the phenomenon or process usually
so denoted, and for the result of that process: thus the refracted ray
he habitually terms "the refraction" of the incident ray. When a
wave-front, or, as he terms it, a "wave," has passed from some initial
position to a subsequent one, he terms the wave-front in its
subsequent position "the continuation" of the wave. He also speaks of
the envelope of a set of elementary waves, formed by coalescence of
those elementary wave-fronts, as "the termination" of the wave; and
the elementary wave-fronts he terms "particular" waves. Owing to the
circumstance that the French word _rayon_ possesses the double
signification of ray of light and radius of a circle, he avoids its
use in the latter sense and speaks always of the semi-diameter, not of
the radius. His speculations as to the ether, his suggestive views of
the structure of crystalline bodies, and his explanation of opacity,
slight as they are, will possibly surprise the reader by their seeming
modernness. And none can read his investigation of the phenomena found
in Iceland spar without marvelling at his insight and sagacity.
S.P.T.
June, 1912.
TABLE OF MATTERS
Contained in this Treatise
CHAPTER I.
On Rays Propagated in Straight Lines.
That Light is produced by a certain movement.
That no substance passes from the luminous object to the eyes.
That Light spreads spherically, almost as Sound does.
Whether Light takes time to spread.
Experience seeming to prove that it passes instantaneously.
Experience proving that it takes time.
How much its speed is greater than that of Sound.
In what the emission of Light differs from that of Sound.
That it is not the same medium which serves for Light and Sound.
How Sound is propagated.
How Light is propagated.
Detailed Remarks on the propagation of Light.
Why Rays are propagated only in straight lines.
How Light coming in different directions can cross itself.
CHAPTER II.
On Reflexion.
Demonstration of equality of angles of incidence and reflexion.
Why the incident and reflected rays are in the same plane
perpendicular to the reflecting surface.
That it is not needful for the reflecting surface to be perfectly
flat to attain equality of the angles of incidence and reflexion.
CHAPTER III.
On Refraction.
That bodies may be transparent without any substance passing through
them.
Proof that the ethereal matter passes through transparent bodies.
How this matter passing through can render them transparent.
That the most solid bodies in appearance are of a very loose texture.
That Light spreads more slowly in water and in glass than in air.
Third hypothesis to explain transparency, and the retardation which
Light suffers.
On that which makes bodies opaque.
Demonstration why Refraction obeys the known proportion of Sines.
Why the incident and refracted Rays produce one another reciprocally.
Why Reflexion within a triangular glass prism is suddenly augmented
when the Light can no longer penetrate.
That bodies which cause greater Refraction also cause stronger
Reflexion.
Demonstration of the Theorem of Mr. Fermat.
CHAPTER IV.
On the Refraction of the Air.
That the emanations of Light in the air are not spherical.
How consequently some objects appear higher than they are.
How the Sun may appear on the Horizon before he has risen.
That the rays of light become curved in the Air of the Atmosphere,
and what effects this produces.
CHAPTER V.
On the Strange Refraction of Iceland Crystal.
That this Crystal grows also in other countries.
Who first-wrote about it.
Description of Iceland Crystal; its substance, shape, and properties.
That it has two different Refractions.
That the ray perpendicular to the surface suffers refraction, and
that some rays inclined to the surface pass without suffering
refraction.
Observation of the refractions in this Crystal.
That there is a Regular and an Irregular Refraction.
The way of measuring the two Refractions of Iceland Crystal.
Remarkable properties of the Irregular Refraction.
Hypothesis to explain the double Refraction.
That Rock Crystal has also a double Refraction.
Hypothesis of emanations of Light, within Iceland Crystal, of
spheroidal form, for the Irregular Refraction.
How a perpendicular ray can suffer Refraction.
How the position and form of the spheroidal emanations in this
Crystal can be defined.
Explanation of the Irregular Refraction by these spheroidal
emanations.
Easy way to find the Irregular Refraction of each incident ray.
Demonstration of the oblique ray which traverses the Crystal without
being refracted.
Other irregularities of Refraction explained.
That an object placed beneath the Crystal appears double, in two
images of different heights.
Why the apparent heights of one of the images change on changing the
position of the eyes above the Crystal.
Of the different sections of this Crystal which produce yet other
refractions, and confirm all this Theory.
Particular way of polishing the surfaces after it has been cut.
Surprising phenomenon touching the rays which pass through two
separated pieces; the cause of which is not explained.
Probable conjecture on the internal composition of Iceland Crystal,
and of what figure its particles are.
Tests to confirm this conjecture.
Calculations which have been supposed in this Chapter.
CHAPTER VI.
On the Figures of transparent bodies which serve for Refraction and
for Reflexion.
General and easy rule to find these Figures.
Invention of the Ovals of Mr. Des Cartes for Dioptrics.
How he was able to find these Lines.
Way of finding the surface of a glass for perfect refraction, when
the other surface is given.
Remark on what happens to rays refracted at a spherical surface.
Remark on the curved line which is formed by reflexion in a spherical
concave mirror.
CHAPTER I
ON RAYS PROPAGATED IN STRAIGHT LINES
As happens in all the sciences in which Geometry is applied to matter,
the demonstrations concerning Optics are founded on truths drawn from
experience. Such are that the rays of light are propagated in straight
lines; that the angles of reflexion and of incidence are equal; and
that in refraction the ray is bent according to the law of sines, now
so well known, and which is no less certain than the preceding laws.
The majority of those who have written touching the various parts of
Optics have contented themselves with presuming these truths. But
some, more inquiring, have desired to investigate the origin and the
causes, considering these to be in themselves wonderful effects of
Nature. In which they advanced some ingenious things, but not however
such that the most intelligent folk do not wish for better and more
satisfactory explanations. Wherefore I here desire to propound what I
have meditated on the subject, so as to contribute as much as I can
to the explanation of this department of Natural Science, which, not
without reason, is reputed to be one of its most difficult parts. I
recognize myself to be much indebted to those who were the first to
begin to dissipate the strange obscurity in which these things were
enveloped, and to give us hope that they might be explained by
intelligible reasoning. But, on the other hand I am astonished also
that even here these have often been willing to offer, as assured and
demonstrative, reasonings which were far from conclusive. For I do not
find that any one has yet given a probable explanation of the first
and most notable phenomena of light, namely why it is not propagated
except in straight lines, and how visible rays, coming from an
infinitude of diverse places, cross one another without hindering one
another in any way.
I shall therefore essay in this book, to give, in accordance with the
principles accepted in the Philosophy of the present day, some clearer
and more probable reasons, firstly of these properties of light
propagated rectilinearly; secondly of light which is reflected on
meeting other bodies. Then I shall explain the phenomena of those rays
which are said to suffer refraction on passing through transparent
bodies of different sorts; and in this part I shall also explain the
effects of the refraction of the air by the different densities of the
Atmosphere.
Thereafter I shall examine the causes of the strange refraction of a
certain kind of Crystal which is brought from Iceland. And finally I
shall treat of the various shapes of transparent and reflecting bodies
by which rays are collected at a point or are turned aside in various
ways. From this it will be seen with what facility, following our new
Theory, we find not only the Ellipses, Hyperbolas, and other curves
which Mr. Des Cartes has ingeniously invented for this purpose; but
also those which the surface of a glass lens ought to possess when its
other surface is given as spherical or plane, or of any other figure
that may be.
It is inconceivable to doubt that light consists in the motion of some
sort of matter. For whether one considers its production, one sees
that here upon the Earth it is chiefly engendered by fire and flame
which contain without doubt bodies that are in rapid motion, since
they dissolve and melt many other bodies, even the most solid; or
whether one considers its effects, one sees that when light is
collected, as by concave mirrors, it has the property of burning as a
fire does, that is to say it disunites the particles of bodies. This
is assuredly the mark of motion, at least in the true Philosophy, in
which one conceives the causes of all natural effects in terms of
mechanical motions. This, in my opinion, we must necessarily do, or
else renounce all hopes of ever comprehending anything in Physics.
And as, according to this Philosophy, one holds as certain that the
sensation of sight is excited only by the impression of some movement
of a kind of matter which acts on the nerves at the back of our eyes,
there is here yet one reason more for believing that light consists in
a movement of the matter which exists between us and the luminous
body.
Further, when one considers the extreme speed with which light spreads
on every side, and how, when it comes from different regions, even
from those directly opposite, the rays traverse one another without
hindrance, one may well understand that when we see a luminous object,
it cannot be by any transport of matter coming to us from this object,
in the way in which a shot or an arrow traverses the air; for
assuredly that would too greatly impugn these two properties of light,
especially the second of them. It is then in some other way that light
spreads; and that which can lead us to comprehend it is the knowledge
which we have of the spreading of Sound in the air.
We know that by means of the air, which is an invisible and impalpable
body, Sound spreads around the spot where it has been produced, by a
movement which is passed on successively from one part of the air to
another; and that the spreading of this movement, taking place equally
rapidly on all sides, ought to form spherical surfaces ever enlarging
and which strike our ears. Now there is no doubt at all that light
also comes from the luminous body to our eyes by some movement
impressed on the matter which is between the two; since, as we have
already seen, it cannot be by the transport of a body which passes
from one to the other. If, in addition, light takes time for its
passage--which we are now going to examine--it will follow that this
movement, impressed on the intervening matter, is successive; and
consequently it spreads, as Sound does, by spherical surfaces and
waves: for I call them waves from their resemblance to those which are
seen to be formed in water when a stone is thrown into it, and which
present a successive spreading as circles, though these arise from
another cause, and are only in a flat surface.
To see then whether the spreading of light takes time, let us consider
first whether there are any facts of experience which can convince us
to the contrary. As to those which can be made here on the Earth, by
striking lights at great distances, although they prove that light
takes no sensible time to pass over these distances, one may say with
good reason that they are too small, and that the only conclusion to
be drawn from them is that the passage of light is extremely rapid.
Mr. Des Cartes, who was of opinion that it is instantaneous, founded
his views, not without reason, upon a better basis of experience,
drawn from the Eclipses of the Moon; which, nevertheless, as I shall
show, is not at all convincing. I will set it forth, in a way a little
different from his, in order to make the conclusion more
comprehensible.
[Illustration]
Let A be the place of the sun, BD a part of the orbit or annual path
of the Earth: ABC a straight line which I suppose to meet the orbit of
the Moon, which is represented by the circle CD, at C.
Now if light requires time, for example one hour, to traverse the
space which is between the Earth and the Moon, it will follow that the
Earth having arrived at B, the shadow which it casts, or the
interruption of the light, will not yet have arrived at the point C,
but will only arrive there an hour after. It will then be one hour
after, reckoning from the moment when the Earth was at B, that the
Moon, arriving at C, will be obscured: but this obscuration or
interruption of the light will not reach the Earth till after another
hour. Let us suppose that the Earth in these two hours will have
arrived at E. The Earth then, being at E, will see the Eclipsed Moon
at C, which it left an hour before, and at the same time will see the
sun at A. For it being immovable, as I suppose with Copernicus, and
the light moving always in straight lines, it must always appear where
it is. But one has always observed, we are told, that the eclipsed
Moon appears at the point of the Ecliptic opposite to the Sun; and yet
here it would appear in arrear of that point by an amount equal to the
angle GEC, the supplement of AEC. This, however, is contrary to
experience, since the angle GEC would be very sensible, and about 33
degrees. Now according to our computation, which is given in the
Treatise on the causes of the phenomena of Saturn, the distance BA
between the Earth and the Sun is about twelve thousand diameters of
the Earth, and hence four hundred times greater than BC the distance
of the Moon, which is 30 diameters. Then the angle ECB will be nearly
four hundred times greater than BAE, which is five minutes; namely,
the path which the earth travels in two hours along its orbit; and
thus the angle BCE will be nearly 33 degrees; and likewise the angle
CEG, which is greater by five minutes.
But it must be noted that the speed of light in this argument has been
assumed such that it takes a time of one hour to make the passage from
here to the Moon. If one supposes that for this it requires only one
minute of time, then it is manifest that the angle CEG will only be 33
minutes; and if it requires only ten seconds of time, the angle will
be less than six minutes. And then it will not be easy to perceive
anything of it in observations of the Eclipse; nor, consequently, will
it be permissible to deduce from it that the movement of light is
instantaneous.
It is true that we are here supposing a strange velocity that would be
a hundred thousand times greater than that of Sound. For Sound,
according to what I have observed, travels about 180 Toises in the
time of one Second, or in about one beat of the pulse. But this
supposition ought not to seem to be an impossibility; since it is not
a question of the transport of a body with so great a speed, but of a
successive movement which is passed on from some bodies to others. I
have then made no difficulty, in meditating on these things, in
supposing that the emanation of light is accomplished with time,
seeing that in this way all its phenomena can be explained, and that
in following the contrary opinion everything is incomprehensible. For
it has always seemed tome that even Mr. Des Cartes, whose aim has been
to treat all the subjects of Physics intelligibly, and who assuredly
has succeeded in this better than any one before him, has said nothing
that is not full of difficulties, or even inconceivable, in dealing
with Light and its properties.
But that which I employed only as a hypothesis, has recently received
great seemingness as an established truth by the ingenious proof of
Mr. Roemer which I am going here to relate, expecting him himself to
give all that is needed for its confirmation. It is founded as is the
preceding argument upon celestial observations, and proves not only
that Light takes time for its passage, but also demonstrates how much
time it takes, and that its velocity is even at least six times
greater than that which I have just stated.
For this he makes use of the Eclipses suffered by the little planets
which revolve around Jupiter, and which often enter his shadow: and
see what is his reasoning. Let A be the Sun, BCDE the annual orbit of
the Earth, F Jupiter, GN the orbit of the nearest of his Satellites,
for it is this one which is more apt for this investigation than any
of the other three, because of the quickness of its revolution. Let G
be this Satellite entering into the shadow of Jupiter, H the same
Satellite emerging from the shadow.
[Illustration]
Let it be then supposed, the Earth being at B some time before the
last quadrature, that one has seen the said Satellite emerge from the
shadow; it must needs be, if the Earth remains at the same place,
that, after 42-1/2 hours, one would again see a similar emergence,
because that is the time in which it makes the round of its orbit, and
when it would come again into opposition to the Sun. And if the Earth,
for instance, were to remain always at B during 30 revolutions of this
Satellite, one would see it again emerge from the shadow after 30
times 42-1/2 hours. But the Earth having been carried along during
this time to C, increasing thus its distance from Jupiter, it follows
that if Light requires time for its passage the illumination of the
little planet will be perceived later at C than it would have been at
B, and that there must be added to this time of 30 times 42-1/2 hours
that which the Light has required to traverse the space MC, the
difference of the spaces CH, BH. Similarly at the other quadrature
when the earth has come to E from D while approaching toward Jupiter,
the immersions of the Satellite ought to be observed at E earlier than
they would have been seen if the Earth had remained at D.
Now in quantities of observations of these Eclipses, made during ten
consecutive years, these differences have been found to be very
considerable, such as ten minutes and more; and from them it has been
concluded that in order to traverse the whole diameter of the annual
orbit KL, which is double the distance from here to the sun, Light
requires about 22 minutes of time.
The movement of Jupiter in his orbit while the Earth passed from B to
C, or from D to E, is included in this calculation; and this makes it
evident that one cannot attribute the retardation of these
illuminations or the anticipation of the eclipses, either to any
irregularity occurring in the movement of the little planet or to its
eccentricity.
If one considers the vast size of the diameter KL, which according to
me is some 24 thousand diameters of the Earth, one will acknowledge
the extreme velocity of Light. For, supposing that KL is no more than
22 thousand of these diameters, it appears that being traversed in 22
minutes this makes the speed a thousand diameters in one minute, that
is 16-2/3 diameters in one second or in one beat of the pulse, which
makes more than 11 hundred times a hundred thousand toises; since the
diameter of the Earth contains 2,865 leagues, reckoned at 25 to the
degree, and each each league is 2,282 Toises, according to the exact
measurement which Mr. Picard made by order of the King in 1669. But
Sound, as I have said above, only travels 180 toises in the same time
of one second: hence the velocity of Light is more than six hundred
thousand times greater than that of Sound. This, however, is quite
another thing from being instantaneous, since there is all the
difference between a finite thing and an infinite. Now the successive
movement of Light being confirmed in this way, it follows, as I have
said, that it spreads by spherical waves, like the movement of Sound.
But if the one resembles the other in this respect, they differ in
many other things; to wit, in the first production of the movement
which causes them; in the matter in which the movement spreads; and in
the manner in which it is propagated. As to that which occurs in the
production of Sound, one knows that it is occasioned by the agitation
undergone by an entire body, or by a considerable part of one, which
shakes all the contiguous air. But the movement of the Light must
originate as from each point of the luminous object, else we should
not be able to perceive all the different parts of that object, as
will be more evident in that which follows. And I do not believe that
this movement can be better explained than by supposing that all those
of the luminous bodies which are liquid, such as flames, and
apparently the sun and the stars, are composed of particles which
float in a much more subtle medium which agitates them with great
rapidity, and makes them strike against the particles of the ether
which surrounds them, and which are much smaller than they. But I hold
also that in luminous solids such as charcoal or metal made red hot in
the fire, this same movement is caused by the violent agitation of
the particles of the metal or of the wood; those of them which are on
the surface striking similarly against the ethereal matter. The
agitation, moreover, of the particles which engender the light ought
to be much more prompt and more rapid than is that of the bodies which
cause sound, since we do not see that the tremors of a body which is
giving out a sound are capable of giving rise to Light, even as the
movement of the hand in the air is not capable of producing Sound.
Now if one examines what this matter may be in which the movement
coming from the luminous body is propagated, which I call Ethereal
matter, one will see that it is not the same that serves for the
propagation of Sound. For one finds that the latter is really that
which we feel and which we breathe, and which being removed from any
place still leaves there the other kind of matter that serves to
convey Light. This may be proved by shutting up a sounding body in a
glass vessel from which the air is withdrawn by the machine which Mr.
Boyle has given us, and with which he has performed so many beautiful
experiments. But in doing this of which I speak, care must be taken to
place the sounding body on cotton or on feathers, in such a way that
it cannot communicate its tremors either to the glass vessel which
encloses it, or to the machine; a precaution which has hitherto been
neglected. For then after having exhausted all the air one hears no
Sound from the metal, though it is struck.
One sees here not only that our air, which does not penetrate through
glass, is the matter by which Sound spreads; but also that it is not
the same air but another kind of matter in which Light spreads; since
if the air is removed from the vessel the Light does not cease to
traverse it as before.
And this last point is demonstrated even more clearly by the
celebrated experiment of Torricelli, in which the tube of glass from
which the quicksilver has withdrawn itself, remaining void of air,
transmits Light just the same as when air is in it. For this proves
that a matter different from air exists in this tube, and that this
matter must have penetrated the glass or the quicksilver, either one
or the other, though they are both impenetrable to the air. And when,
in the same experiment, one makes the vacuum after putting a little
water above the quicksilver, one concludes equally that the said
matter passes through glass or water, or through both.
As regards the different modes in which I have said the movements of
Sound and of Light are communicated, one may sufficiently comprehend
how this occurs in the case of Sound if one considers that the air is
of such a nature that it can be compressed and reduced to a much
smaller space than that which it ordinarily occupies. And in
proportion as it is compressed the more does it exert an effort to
regain its volume; for this property along with its penetrability,
which remains notwithstanding its compression, seems to prove that it
is made up of small bodies which float about and which are agitated
very rapidly in the ethereal matter composed of much smaller parts. So
that the cause of the spreading of Sound is the effort which these
little bodies make in collisions with one another, to regain freedom
when they are a little more squeezed together in the circuit of these
waves than elsewhere.
But the extreme velocity of Light, and other properties which it has,
cannot admit of such a propagation of motion, and I am about to show
here the way in which I conceive it must occur. For this, it is
needful to explain the property which hard bodies must possess to
transmit movement from one to another.
When one takes a number of spheres of equal size, made of some very
hard substance, and arranges them in a straight line, so that they
touch one another, one finds, on striking with a similar sphere
against the first of these spheres, that the motion passes as in an
instant to the last of them, which separates itself from the row,
without one's being able to perceive that the others have been
stirred. And even that one which was used to strike remains motionless
with them. Whence one sees that the movement passes with an extreme
velocity which is the greater, the greater the hardness of the
substance of the spheres.
But it is still certain that this progression of motion is not
instantaneous, but successive, and therefore must take time. For if
the movement, or the disposition to movement, if you will have it so,
did not pass successively through all these spheres, they would all
acquire the movement at the same time, and hence would all advance
together; which does not happen. For the last one leaves the whole row
and acquires the speed of the one which was pushed. Moreover there are
experiments which demonstrate that all the bodies which we reckon of
the hardest kind, such as quenched steel, glass, and agate, act as
springs and bend somehow, not only when extended as rods but also when
they are in the form of spheres or of other shapes. That is to say
they yield a little in themselves at the place where they are struck,
and immediately regain their former figure. For I have found that on
striking with a ball of glass or of agate against a large and quite
thick thick piece of the same substance which had a flat surface,
slightly soiled with breath or in some other way, there remained round
marks, of smaller or larger size according as the blow had been weak
or strong. This makes it evident that these substances yield where
they meet, and spring back: and for this time must be required.
Now in applying this kind of movement to that which produces Light
there is nothing to hinder us from estimating the particles of the
ether to be of a substance as nearly approaching to perfect hardness
and possessing a springiness as prompt as we choose. It is not
necessary to examine here the causes of this hardness, or of that
springiness, the consideration of which would lead us too far from our
subject. I will say, however, in passing that we may conceive that the
particles of the ether, notwithstanding their smallness, are in turn
composed of other parts and that their springiness consists in the
very rapid movement of a subtle matter which penetrates them from
every side and constrains their structure to assume such a disposition
as to give to this fluid matter the most overt and easy passage
possible. This accords with the explanation which Mr. Des Cartes gives
for the spring, though I do not, like him, suppose the pores to be in
the form of round hollow canals. And it must not be thought that in
this there is anything absurd or impossible, it being on the contrary
quite credible that it is this infinite series of different sizes of
corpuscles, having different degrees of velocity, of which Nature
makes use to produce so many marvellous effects.
But though we shall ignore the true cause of springiness we still see
that there are many bodies which possess this property; and thus there
is nothing strange in supposing that it exists also in little
invisible bodies like the particles of the Ether. Also if one wishes
to seek for any other way in which the movement of Light is
successively communicated, one will find none which agrees better,
with uniform progression, as seems to be necessary, than the property
of springiness; because if this movement should grow slower in
proportion as it is shared over a greater quantity of matter, in
moving away from the source of the light, it could not conserve this
great velocity over great distances. But by supposing springiness in
the ethereal matter, its particles will have the property of equally
rapid restitution whether they are pushed strongly or feebly; and thus
the propagation of Light will always go on with an equal velocity.
[Illustration]
And it must be known that although the particles of the ether are not
ranged thus in straight lines, as in our row of spheres, but
confusedly, so that one of them touches several others, this does not
hinder them from transmitting their movement and from spreading it
always forward. As to this it is to be remarked that there is a law of
motion serving for this propagation, and verifiable by experiment. It
is that when a sphere, such as A here, touches several other similar
spheres CCC, if it is struck by another sphere B in such a way as to
exert an impulse against all the spheres CCC which touch it, it
transmits to them the whole of its movement, and remains after that
motionless like the sphere B. And without supposing that the ethereal
particles are of spherical form (for I see indeed no need to suppose
them so) one may well understand that this property of communicating
an impulse does not fail to contribute to the aforesaid propagation
of movement.
Equality of size seems to be more necessary, because otherwise there
ought to be some reflexion of movement backwards when it passes from a
smaller particle to a larger one, according to the Laws of Percussion
which I published some years ago.
However, one will see hereafter that we have to suppose such an
equality not so much as a necessity for the propagation of light as
for rendering that propagation easier and more powerful; for it is not
beyond the limits of probability that the particles of the ether have
been made equal for a purpose so important as that of light, at least
in that vast space which is beyond the region of atmosphere and which
seems to serve only to transmit the light of the Sun and the Stars.
I have then shown in what manner one may conceive Light to spread
successively, by spherical waves, and how it is possible that this
spreading is accomplished with as great a velocity as that which
experiments and celestial observations demand. Whence it may be
further remarked that although the particles are supposed to be in
continual movement (for there are many reasons for this) the
successive propagation of the waves cannot be hindered by this;
because the propagation consists nowise in the transport of those
particles but merely in a small agitation which they cannot help
communicating to those surrounding, notwithstanding any movement which
may act on them causing them to be changing positions amongst
themselves.
But we must consider still more particularly the origin of these
waves, and the manner in which they spread. And, first, it follows
from what has been said on the production of Light, that each little
region of a luminous body, such as the Sun, a candle, or a burning
coal, generates its own waves of which that region is the centre. Thus
in the flame of a candle, having distinguished the points A, B, C,
concentric circles described about each of these points represent the
waves which come from them. And one must imagine the same about every
point of the surface and of the part within the flame.
[Illustration]
But as the percussions at the centres of these waves possess no
regular succession, it must not be supposed that the waves themselves
follow one another at equal distances: and if the distances marked in
the figure appear to be such, it is rather to mark the progression of
one and the same wave at equal intervals of time than to represent
several of them issuing from one and the same centre.
After all, this prodigious quantity of waves which traverse one
another without confusion and without effacing one another must not be
deemed inconceivable; it being certain that one and the same particle
of matter can serve for many waves coming from different sides or even
from contrary directions, not only if it is struck by blows which
follow one another closely but even for those which act on it at the
same instant. It can do so because the spreading of the movement is
successive. This may be proved by the row of equal spheres of hard
matter, spoken of above. If against this row there are pushed from two
opposite sides at the same time two similar spheres A and D, one will
see each of them rebound with the same velocity which it had in
striking, yet the whole row will remain in its place, although the
movement has passed along its whole length twice over. And if these
contrary movements happen to meet one another at the middle sphere, B,
or at some other such as C, that sphere will yield and act as a spring
at both sides, and so will serve at the same instant to transmit these
two movements.
[Illustration]
But what may at first appear full strange and even incredible is that
the undulations produced by such small movements and corpuscles,
should spread to such immense distances; as for example from the Sun
or from the Stars to us. For the force of these waves must grow feeble
in proportion as they move away from their origin, so that the action
of each one in particular will without doubt become incapable of
making itself felt to our sight. But one will cease to be astonished
by considering how at a great distance from the luminous body an
infinitude of waves, though they have issued from different points of
this body, unite together in such a way that they sensibly compose one
single wave only, which, consequently, ought to have enough force to
make itself felt. Thus this infinite number of waves which originate
at the same instant from all points of a fixed star, big it may be as
the Sun, make practically only one single wave which may well have
force enough to produce an impression on our eyes. Moreover from each
luminous point there may come many thousands of waves in the smallest
imaginable time, by the frequent percussion of the corpuscles which
strike the Ether at these points: which further contributes to
rendering their action more sensible.
[Illustration]
There is the further consideration in the emanation of these waves,
that each particle of matter in which a wave spreads, ought not to
communicate its motion only to the next particle which is in the
straight line drawn from the luminous point, but that it also imparts
some of it necessarily to all the others which touch it and which
oppose themselves to its movement. So it arises that around each
particle there is made a wave of which that particle is the centre.
Thus if DCF is a wave emanating from the luminous point A, which is
its centre, the particle B, one of those comprised within the sphere
DCF, will have made its particular or partial wave KCL, which will
touch the wave DCF at C at the same moment that the principal wave
emanating from the point A has arrived at DCF; and it is clear that it
will be only the region C of the wave KCL which will touch the wave
DCF, to wit, that which is in the straight line drawn through AB.
Similarly the other particles of the sphere DCF, such as _bb_, _dd_,
etc., will each make its own wave. But each of these waves can be
infinitely feeble only as compared with the wave DCF, to the
composition of which all the others contribute by the part of their
surface which is most distant from the centre A.
One sees, in addition, that the wave DCF is determined by the
distance attained in a certain space of time by the movement which
started from the point A; there being no movement beyond this wave,
though there will be in the space which it encloses, namely in parts
of the particular waves, those parts which do not touch the sphere
DCF. And all this ought not to seem fraught with too much minuteness
or subtlety, since we shall see in the sequel that all the properties
of Light, and everything pertaining to its reflexion and its
refraction, can be explained in principle by this means. This is a
matter which has been quite unknown to those who hitherto have begun
to consider the waves of light, amongst whom are Mr. Hooke in his
_Micrographia_, and Father Pardies, who, in a treatise of which he let
me see a portion, and which he was unable to complete as he died
shortly afterward, had undertaken to prove by these waves the effects
of reflexion and refraction. But the chief foundation, which consists
in the remark I have just made, was lacking in his demonstrations; and
for the rest he had opinions very different from mine, as may be will
appear some day if his writing has been preserved.
To come to the properties of Light. We remark first that each portion
of a wave ought to spread in such a way that its extremities lie
always between the same straight lines drawn from the luminous point.
Thus the portion BG of the wave, having the luminous point A as its
centre, will spread into the arc CE bounded by the straight lines ABC,
AGE. For although the particular waves produced by the particles
comprised within the space CAE spread also outside this space, they
yet do not concur at the same instant to compose a wave which
terminates the movement, as they do precisely at the circumference
CE, which is their common tangent.
And hence one sees the reason why light, at least if its rays are not
reflected or broken, spreads only by straight lines, so that it
illuminates no object except when the path from its source to that
object is open along such lines.
For if, for example, there were an opening BG, limited by opaque
bodies BH, GI, the wave of light which issues from the point A will
always be terminated by the straight lines AC, AE, as has just been
shown; the parts of the partial waves which spread outside the space
ACE being too feeble to produce light there.
Now, however small we make the opening BG, there is always the same
reason causing the light there to pass between straight lines; since
this opening is always large enough to contain a great number of
particles of the ethereal matter, which are of an inconceivable
smallness; so that it appears that each little portion of the wave
necessarily advances following the straight line which comes from the
luminous point. Thus then we may take the rays of light as if they
were straight lines.
It appears, moreover, by what has been remarked touching the
feebleness of the particular waves, that it is not needful that all
the particles of the Ether should be equal amongst themselves, though
equality is more apt for the propagation of the movement. For it is
true that inequality will cause a particle by pushing against another
larger one to strive to recoil with a part of its movement; but it
will thereby merely generate backwards towards the luminous point some
partial waves incapable of causing light, and not a wave compounded of
many as CE was.
Another property of waves of light, and one of the most marvellous,
is that when some of them come from different or even from opposing
sides, they produce their effect across one another without any
hindrance. Whence also it comes about that a number of spectators may
view different objects at the same time through the same opening, and
that two persons can at the same time see one another's eyes. Now
according to the explanation which has been given of the action of
light, how the waves do not destroy nor interrupt one another when
they cross one another, these effects which I have just mentioned are
easily conceived. But in my judgement they are not at all easy to
explain according to the views of Mr. Des Cartes, who makes Light to
consist in a continuous pressure merely tending to movement. For this
pressure not being able to act from two opposite sides at the same
time, against bodies which have no inclination to approach one
another, it is impossible so to understand what I have been saying
about two persons mutually seeing one another's eyes, or how two
torches can illuminate one another.
CHAPTER II
ON REFLEXION
Having explained the effects of waves of light which spread in a
homogeneous matter, we will examine next that which happens to them on
encountering other bodies. We will first make evident how the
Reflexion of light is explained by these same waves, and why it
preserves equality of angles.
Let there be a surface AB; plane and polished, of some metal, glass,
or other body, which at first I will consider as perfectly uniform
(reserving to myself to deal at the end of this demonstration with the
inequalities from which it cannot be exempt), and let a line AC,
inclined to AD, represent a portion of a wave of light, the centre of
which is so distant that this portion AC may be considered as a
straight line; for I consider all this as in one plane, imagining to
myself that the plane in which this figure is, cuts the sphere of the
wave through its centre and intersects the plane AB at right angles.
This explanation will suffice once for all.
[Illustration]
The piece C of the wave AC, will in a certain space of time advance as
far as the plane AB at B, following the straight line CB, which may be
supposed to come from the luminous centre, and which in consequence is
perpendicular to AC. Now in this same space of time the portion A of
the same wave, which has been hindered from communicating its movement
beyond the plane AB, or at least partly so, ought to have continued
its movement in the matter which is above this plane, and this along a
distance equal to CB, making its own partial spherical wave,
according to what has been said above. Which wave is here represented
by the circumference SNR, the centre of which is A, and its
semi-diameter AN equal to CB.
If one considers further the other pieces H of the wave AC, it appears
that they will not only have reached the surface AB by straight lines
HK parallel to CB, but that in addition they will have generated in
the transparent air, from the centres K, K, K, particular spherical
waves, represented here by circumferences the semi-diameters of which
are equal to KM, that is to say to the continuations of HK as far as
the line BG parallel to AC. But all these circumferences have as a
common tangent the straight line BN, namely the same which is drawn
from B as a tangent to the first of the circles, of which A is the
centre, and AN the semi-diameter equal to BC, as is easy to see.
It is then the line BN (comprised between B and the point N where the
perpendicular from the point A falls) which is as it were formed by
all these circumferences, and which terminates the movement which is
made by the reflexion of the wave AC; and it is also the place where
the movement occurs in much greater quantity than anywhere else.
Wherefore, according to that which has been explained, BN is the
propagation of the wave AC at the moment when the piece C of it has
arrived at B. For there is no other line which like BN is a common
tangent to all the aforesaid circles, except BG below the plane AB;
which line BG would be the propagation of the wave if the movement
could have spread in a medium homogeneous with that which is above the
plane. And if one wishes to see how the wave AC has come successively
to BN, one has only to draw in the same figure the straight lines KO
parallel to BN, and the straight lines KL parallel to AC. Thus one
will see that the straight wave AC has become broken up into all the
OKL parts successively, and that it has become straight again at NB.
Now it is apparent here that the angle of reflexion is made equal to
the angle of incidence. For the triangles ACB, BNA being rectangular
and having the side AB common, and the side CB equal to NA, it follows
that the angles opposite to these sides will be equal, and therefore
also the angles CBA, NAB. But as CB, perpendicular to CA, marks the
direction of the incident ray, so AN, perpendicular to the wave BN,
marks the direction of the reflected ray; hence these rays are equally
inclined to the plane AB.
But in considering the preceding demonstration, one might aver that it
is indeed true that BN is the common tangent of the circular waves in
the plane of this figure, but that these waves, being in truth
spherical, have still an infinitude of similar tangents, namely all
the straight lines which are drawn from the point B in the surface
generated by the straight line BN about the axis BA. It remains,
therefore, to demonstrate that there is no difficulty herein: and by
the same argument one will see why the incident ray and the reflected
ray are always in one and the same plane perpendicular to the
reflecting plane. I say then that the wave AC, being regarded only as
a line, produces no light. For a visible ray of light, however narrow
it may be, has always some width, and consequently it is necessary, in
representing the wave whose progression constitutes the ray, to put
instead of a line AC some plane figure such as the circle HC in the
following figure, by supposing, as we have done, the luminous point to
be infinitely distant. Now it is easy to see, following the preceding
demonstration, that each small piece of this wave HC having arrived at
the plane AB, and there generating each one its particular wave, these
will all have, when C arrives at B, a common plane which will touch
them, namely a circle BN similar to CH; and this will be intersected
at its middle and at right angles by the same plane which likewise
intersects the circle CH and the ellipse AB.
[Illustration]
One sees also that the said spheres of the partial waves cannot have
any common tangent plane other than the circle BN; so that it will be
this plane where there will be more reflected movement than anywhere
else, and which will therefore carry on the light in continuance from
the wave CH.
I have also stated in the preceding demonstration that the movement of
the piece A of the incident wave is not able to communicate itself
beyond the plane AB, or at least not wholly. Whence it is to be
remarked that though the movement of the ethereal matter might
communicate itself partly to that of the reflecting body, this could
in nothing alter the velocity of progression of the waves, on which
the angle of reflexion depends. For a slight percussion ought to
generate waves as rapid as strong percussion in the same matter. This
comes about from the property of bodies which act as springs, of which
we have spoken above; namely that whether compressed little or much
they recoil in equal times. Equally so in every reflexion of the
light, against whatever body it may be, the angles of reflexion and
incidence ought to be equal notwithstanding that the body might be of
such a nature that it takes away a portion of the movement made by the
incident light. And experiment shows that in fact there is no polished
body the reflexion of which does not follow this rule.
But the thing to be above all remarked in our demonstration is that it
does not require that the reflecting surface should be considered as a
uniform plane, as has been supposed by all those who have tried to
explain the effects of reflexion; but only an evenness such as may be
attained by the particles of the matter of the reflecting body being
set near to one another; which particles are larger than those of the
ethereal matter, as will appear by what we shall say in treating of
the transparency and opacity of bodies. For the surface consisting
thus of particles put together, and the ethereal particles being
above, and smaller, it is evident that one could not demonstrate the
equality of the angles of incidence and reflexion by similitude to
that which happens to a ball thrown against a wall, of which writers
have always made use. In our way, on the other hand, the thing is
explained without difficulty. For the smallness of the particles of
quicksilver, for example, being such that one must conceive millions
of them, in the smallest visible surface proposed, arranged like a
heap of grains of sand which has been flattened as much as it is
capable of being, this surface then becomes for our purpose as even
as a polished glass is: and, although it always remains rough with
respect to the particles of the Ether it is evident that the centres
of all the particular spheres of reflexion, of which we have spoken,
are almost in one uniform plane, and that thus the common tangent can
fit to them as perfectly as is requisite for the production of light.
And this alone is requisite, in our method of demonstration, to cause
equality of the said angles without the remainder of the movement
reflected from all parts being able to produce any contrary effect.
CHAPTER III
ON REFRACTION
In the same way as the effects of Reflexion have been explained by
waves of light reflected at the surface of polished bodies, we will
explain transparency and the phenomena of refraction by waves which
spread within and across diaphanous bodies, both solids, such as
glass, and liquids, such as water, oils, etc. But in order that it may
not seem strange to suppose this passage of waves in the interior of
these bodies, I will first show that one may conceive it possible in
more than one mode.
First, then, if the ethereal matter cannot penetrate transparent
bodies at all, their own particles would be able to communicate
successively the movement of the waves, the same as do those of the
Ether, supposing that, like those, they are of a nature to act as a
spring. And this is easy to conceive as regards water and other
transparent liquids, they being composed of detached particles. But it
may seem more difficult as regards glass and other transparent and
hard bodies, because their solidity does not seem to permit them to
receive movement except in their whole mass at the same time. This,
however, is not necessary because this solidity is not such as it
appears to us, it being probable rather that these bodies are composed
of particles merely placed close to one another and held together by
some pressure from without of some other matter, and by the
irregularity of their shapes. For primarily their rarity is shown by
the facility with which there passes through them the matter of the
vortices of the magnet, and that which causes gravity. Further, one
cannot say that these bodies are of a texture similar to that of a
sponge or of light bread, because the heat of the fire makes them flow
and thereby changes the situation of the particles amongst themselves.
It remains then that they are, as has been said, assemblages of
particles which touch one another without constituting a continuous
solid. This being so, the movement which these particles receive to
carry on the waves of light, being merely communicated from some of
them to others, without their going for that purpose out of their
places or without derangement, it may very well produce its effect
without prejudicing in any way the apparent solidity of the compound.
By pressure from without, of which I have spoken, must not be
understood that of the air, which would not be sufficient, but that of
some other more subtle matter, a pressure which I chanced upon by
experiment long ago, namely in the case of water freed from air, which
remains suspended in a tube open at its lower end, notwithstanding
that the air has been removed from the vessel in which this tube is
enclosed.
One can then in this way conceive of transparency in a solid without
any necessity that the ethereal matter which serves for light should
pass through it, or that it should find pores in which to insinuate
itself. But the truth is that this matter not only passes through
solids, but does so even with great facility; of which the experiment
of Torricelli, above cited, is already a proof. Because on the
quicksilver and the water quitting the upper part of the glass tube,
it appears that it is immediately filled with ethereal matter, since
light passes across it. But here is another argument which proves this
ready penetrability, not only in transparent bodies but also in all
others.
When light passes across a hollow sphere of glass, closed on all
sides, it is certain that it is full of ethereal matter, as much as
the spaces outside the sphere. And this ethereal matter, as has been
shown above, consists of particles which just touch one another. If
then it were enclosed in the sphere in such a way that it could not
get out through the pores of the glass, it would be obliged to follow
the movement of the sphere when one changes its place: and it would
require consequently almost the same force to impress a certain
velocity on this sphere, when placed on a horizontal plane, as if it
were full of water or perhaps of quicksilver: because every body
resists the velocity of the motion which one would give to it, in
proportion to the quantity of matter which it contains, and which is
obliged to follow this motion. But on the contrary one finds that the
sphere resists the impress of movement only in proportion to the
quantity of matter of the glass of which it is made. Then it must be
that the ethereal matter which is inside is not shut up, but flows
through it with very great freedom. We shall demonstrate hereafter
that by this process the same penetrability may be inferred also as
relating to opaque bodies.
The second mode then of explaining transparency, and one which appears
more probably true, is by saying that the waves of light are carried
on in the ethereal matter, which continuously occupies the interstices
or pores of transparent bodies. For since it passes through them
continuously and freely, it follows that they are always full of it.
And one may even show that these interstices occupy much more space
than the coherent particles which constitute the bodies. For if what
we have just said is true: that force is required to impress a certain
horizontal velocity on bodies in proportion as they contain coherent
matter; and if the proportion of this force follows the law of
weights, as is confirmed by experiment, then the quantity of the
constituent matter of bodies also follows the proportion of their
weights. Now we see that water weighs only one fourteenth part as much
as an equal portion of quicksilver: therefore the matter of the water
does not occupy the fourteenth part of the space which its mass
obtains. It must even occupy much less of it, since quicksilver is
less heavy than gold, and the matter of gold is by no means dense, as
follows from the fact that the matter of the vortices of the magnet
and of that which is the cause of gravity pass very freely through it.
But it may be objected here that if water is a body of so great
rarity, and if its particles occupy so small a portion of the space of
its apparent bulk, it is very strange how it yet resists Compression
so strongly without permitting itself to be condensed by any force
which one has hitherto essayed to employ, preserving even its entire
liquidity while subjected to this pressure.
This is no small difficulty. It may, however, be resolved by saying
that the very violent and rapid motion of the subtle matter which
renders water liquid, by agitating the particles of which it is
composed, maintains this liquidity in spite of the pressure which
hitherto any one has been minded to apply to it.
The rarity of transparent bodies being then such as we have said, one
easily conceives that the waves might be carried on in the ethereal
matter which fills the interstices of the particles. And, moreover,
one may believe that the progression of these waves ought to be a
little slower in the interior of bodies, by reason of the small
detours which the same particles cause. In which different velocity of
light I shall show the cause of refraction to consist.
Before doing so, I will indicate the third and last mode in which
transparency may be conceived; which is by supposing that the movement
of the waves of light is transmitted indifferently both in the
particles of the ethereal matter which occupy the interstices of
bodies, and in the particles which compose them, so that the movement
passes from one to the other. And it will be seen hereafter that this
hypothesis serves excellently to explain the double refraction of
certain transparent bodies.
Should it be objected that if the particles of the ether are smaller
than those of transparent bodies (since they pass through their
intervals), it would follow that they can communicate to them but
little of their movement, it may be replied that the particles of
these bodies are in turn composed of still smaller particles, and so
it will be these secondary particles which will receive the movement
from those of the ether.
Furthermore, if the particles of transparent bodies have a recoil a
little less prompt than that of the ethereal particles, which nothing
hinders us from supposing, it will again follow that the progression
of the waves of light will be slower in the interior of such bodies
than it is outside in the ethereal matter.
All this I have found as most probable for the mode in which the waves
of light pass across transparent bodies. To which it must further be
added in what respect these bodies differ from those which are opaque;
and the more so since it might seem because of the easy penetration of
bodies by the ethereal matter, of which mention has been made, that
there would not be any body that was not transparent. For by the same
reasoning about the hollow sphere which I have employed to prove the
smallness of the density of glass and its easy penetrability by the
ethereal matter, one might also prove that the same penetrability
obtains for metals and for every other sort of body. For this sphere
being for example of silver, it is certain that it contains some of
the ethereal matter which serves for light, since this was there as
well as in the air when the opening of the sphere was closed. Yet,
being closed and placed upon a horizontal plane, it resists the
movement which one wishes to give to it, merely according to the
quantity of silver of which it is made; so that one must conclude, as
above, that the ethereal matter which is enclosed does not follow the
movement of the sphere; and that therefore silver, as well as glass,
is very easily penetrated by this matter. Some of it is therefore
present continuously and in quantities between the particles of silver
and of all other opaque bodies: and since it serves for the
propagation of light it would seem that these bodies ought also to be
transparent, which however is not the case.
Whence then, one will say, does their opacity come? Is it because the
particles which compose them are soft; that is to say, these particles
being composed of others that are smaller, are they capable of
changing their figure on receiving the pressure of the ethereal
particles, the motion of which they thereby damp, and so hinder the
continuance of the waves of light? That cannot be: for if the
particles of the metals are soft, how is it that polished silver and
mercury reflect light so strongly? What I find to be most probable
herein, is to say that metallic bodies, which are almost the only
really opaque ones, have mixed amongst their hard particles some soft
ones; so that some serve to cause reflexion and the others to hinder
transparency; while, on the other hand, transparent bodies contain
only hard particles which have the faculty of recoil, and serve
together with those of the ethereal matter for the propagation of the
waves of light, as has been said.
[Illustration]
Let us pass now to the explanation of the effects of Refraction,
assuming, as we have done, the passage of waves of light through
transparent bodies, and the diminution of velocity which these same
waves suffer in them.
The chief property of Refraction is that a ray of light, such as AB,
being in the air, and falling obliquely upon the polished surface of a
transparent body, such as FG, is broken at the point of incidence B,
in such a way that with the straight line DBE which cuts the surface
perpendicularly it makes an angle CBE less than ABD which it made with
the same perpendicular when in the air. And the measure of these
angles is found by describing, about the point B, a circle which cuts
the radii AB, BC. For the perpendiculars AD, CE, let fall from the
points of intersection upon the straight line DE, which are called the
Sines of the angles ABD, CBE, have a certain ratio between themselves;
which ratio is always the same for all inclinations of the incident
ray, at least for a given transparent body. This ratio is, in glass,
very nearly as 3 to 2; and in water very nearly as 4 to 3; and is
likewise different in other diaphanous bodies.
Another property, similar to this, is that the refractions are
reciprocal between the rays entering into a transparent body and those
which are leaving it. That is to say that if the ray AB in entering
the transparent body is refracted into BC, then likewise CB being
taken as a ray in the interior of this body will be refracted, on
passing out, into BA.
[Illustration]
To explain then the reasons of these phenomena according to our
principles, let AB be the straight line which represents a plane
surface bounding the transparent substances which lie towards C and
towards N. When I say plane, that does not signify a perfect evenness,
but such as has been understood in treating of reflexion, and for the
same reason. Let the line AC represent a portion of a wave of light,
the centre of which is supposed so distant that this portion may be
considered as a straight line. The piece C, then, of the wave AC, will
in a certain space of time have advanced as far as the plane AB
following the straight line CB, which may be imagined as coming from
the luminous centre, and which consequently will cut AC at right
angles. Now in the same time the piece A would have come to G along
the straight line AG, equal and parallel to CB; and all the portion of
wave AC would be at GB if the matter of the transparent body
transmitted the movement of the wave as quickly as the matter of the
Ether. But let us suppose that it transmits this movement less
quickly, by one-third, for instance. Movement will then be spread from
the point A, in the matter of the transparent body through a distance
equal to two-thirds of CB, making its own particular spherical wave
according to what has been said before. This wave is then represented
by the circumference SNR, the centre of which is A, and its
semi-diameter equal to two-thirds of CB. Then if one considers in
order the other pieces H of the wave AC, it appears that in the same
time that the piece C reaches B they will not only have arrived at the
surface AB along the straight lines HK parallel to CB, but that, in
addition, they will have generated in the diaphanous substance from
the centres K, partial waves, represented here by circumferences the
semi-diameters of which are equal to two-thirds of the lines KM, that
is to say, to two-thirds of the prolongations of HK down to the
straight line BG; for these semi-diameters would have been equal to
entire lengths of KM if the two transparent substances had been of the
same penetrability.
Now all these circumferences have for a common tangent the straight
line BN; namely the same line which is drawn as a tangent from the
point B to the circumference SNR which we considered first. For it is
easy to see that all the other circumferences will touch the same BN,
from B up to the point of contact N, which is the same point where AN
falls perpendicularly on BN.
It is then BN, which is formed by small arcs of these circumferences,
which terminates the movement that the wave AC has communicated within
the transparent body, and where this movement occurs in much greater
amount than anywhere else. And for that reason this line, in
accordance with what has been said more than once, is the propagation
of the wave AC at the moment when its piece C has reached B. For there
is no other line below the plane AB which is, like BN, a common
tangent to all these partial waves. And if one would know how the wave
AC has come progressively to BN, it is necessary only to draw in the
same figure the straight lines KO parallel to BN, and all the lines KL
parallel to AC. Thus one will see that the wave CA, from being a
straight line, has become broken in all the positions LKO
successively, and that it has again become a straight line at BN. This
being evident by what has already been demonstrated, there is no need
to explain it further.
Now, in the same figure, if one draws EAF, which cuts the plane AB at
right angles at the point A, since AD is perpendicular to the wave AC,
it will be DA which will mark the ray of incident light, and AN which
was perpendicular to BN, the refracted ray: since the rays are nothing
else than the straight lines along which the portions of the waves
advance.
Whence it is easy to recognize this chief property of refraction,
namely that the Sine of the angle DAE has always the same ratio to the
Sine of the angle NAF, whatever be the inclination of the ray DA: and
that this ratio is the same as that of the velocity of the waves in
the transparent substance which is towards AE to their velocity in the
transparent substance towards AF. For, considering AB as the radius of
a circle, the Sine of the angle BAC is BC, and the Sine of the angle
ABN is AN. But the angle BAC is equal to DAE, since each of them added
to CAE makes a right angle. And the angle ABN is equal to NAF, since
each of them with BAN makes a right angle. Then also the Sine of the
angle DAE is to the Sine of NAF as BC is to AN. But the ratio of BC to
AN was the same as that of the velocities of light in the substance
which is towards AE and in that which is towards AF; therefore also
the Sine of the angle DAE will be to the Sine of the angle NAF the
same as the said velocities of light.
To see, consequently, what the refraction will be when the waves of
light pass into a substance in which the movement travels more quickly
than in that from which they emerge (let us again assume the ratio of
3 to 2), it is only necessary to repeat all the same construction and
demonstration which we have just used, merely substituting everywhere
3/2 instead of 2/3. And it will be found by the same reasoning, in
this other figure, that when the piece C of the wave AC shall have
reached the surface AB at B, all the portions of the wave AC will
have advanced as far as BN, so that BC the perpendicular on AC is to
AN the perpendicular on BN as 2 to 3. And there will finally be this
same ratio of 2 to 3 between the Sine of the angle BAD and the Sine of
the angle FAN.
Hence one sees the reciprocal relation of the refractions of the ray
on entering and on leaving one and the same transparent body: namely
that if NA falling on the external surface AB is refracted into the
direction AD, so the ray AD will be refracted on leaving the
transparent body into the direction AN.
[Illustration]
One sees also the reason for a noteworthy accident which happens in
this refraction: which is this, that after a certain obliquity of the
incident ray DA, it begins to be quite unable to penetrate into the
other transparent substance. For if the angle DAQ or CBA is such that
in the triangle ACB, CB is equal to 2/3 of AB, or is greater, then AN
cannot form one side of the triangle ANB, since it becomes equal to or
greater than AB: so that the portion of wave BN cannot be found
anywhere, neither consequently can AN, which ought to be perpendicular
to it. And thus the incident ray DA does not then pierce the surface
AB.
When the ratio of the velocities of the waves is as two to three, as
in our example, which is that which obtains for glass and air, the
angle DAQ must be more than 48 degrees 11 minutes in order that the
ray DA may be able to pass by refraction. And when the ratio of the
velocities is as 3 to 4, as it is very nearly in water and air, this
angle DAQ must exceed 41 degrees 24 minutes. And this accords
perfectly with experiment.
But it might here be asked: since the meeting of the wave AC against
the surface AB ought to produce movement in the matter which is on the
other side, why does no light pass there? To which the reply is easy
if one remembers what has been said before. For although it generates
an infinitude of partial waves in the matter which is at the other
side of AB, these waves never have a common tangent line (either
straight or curved) at the same moment; and so there is no line
terminating the propagation of the wave AC beyond the plane AB, nor
any place where the movement is gathered together in sufficiently
great quantity to produce light. And one will easily see the truth of
this, namely that CB being larger than 2/3 of AB, the waves excited
beyond the plane AB will have no common tangent if about the centres K
one then draws circles having radii equal to 3/2 of the lengths LB to
which they correspond. For all these circles will be enclosed in one
another and will all pass beyond the point B.
Now it is to be remarked that from the moment when the angle DAQ is
smaller than is requisite to permit the refracted ray DA to pass into
the other transparent substance, one finds that the interior reflexion
which occurs at the surface AB is much augmented in brightness, as is
easy to realize by experiment with a triangular prism; and for this
our theory can afford this reason. When the angle DAQ is still large
enough to enable the ray DA to pass, it is evident that the light from
the portion AC of the wave is collected in a minimum space when it
reaches BN. It appears also that the wave BN becomes so much the
smaller as the angle CBA or DAQ is made less; until when the latter is
diminished to the limit indicated a little previously, this wave BN is
collected together always at one point. That is to say, that when the
piece C of the wave AC has then arrived at B, the wave BN which is the
propagation of AC is entirely reduced to the same point B. Similarly
when the piece H has reached K, the part AH is entirely reduced to the
same point K. This makes it evident that in proportion as the wave CA
comes to meet the surface AB, there occurs a great quantity of
movement along that surface; which movement ought also to spread
within the transparent body and ought to have much re-enforced the
partial waves which produce the interior reflexion against the surface
AB, according to the laws of reflexion previously explained.
And because a slight diminution of the angle of incidence DAQ causes
the wave BN, however great it was, to be reduced to zero, (for this
angle being 49 degrees 11 minutes in the glass, the angle BAN is still
11 degrees 21 minutes, and the same angle being reduced by one degree
only the angle BAN is reduced to zero, and so the wave BN reduced to a
point) thence it comes about that the interior reflexion from being
obscure becomes suddenly bright, so soon as the angle of incidence is
such that it no longer gives passage to the refraction.
Now as concerns ordinary external reflexion, that is to say which
occurs when the angle of incidence DAQ is still large enough to enable
the refracted ray to penetrate beyond the surface AB, this reflexion
should occur against the particles of the substance which touches the
transparent body on its outside. And it apparently occurs against the
particles of the air or others mingled with the ethereal particles and
larger than they. So on the other hand the external reflexion of these
bodies occurs against the particles which compose them, and which are
also larger than those of the ethereal matter, since the latter flows
in their interstices. It is true that there remains here some
difficulty in those experiments in which this interior reflexion
occurs without the particles of air being able to contribute to it, as
in vessels or tubes from which the air has been extracted.
Experience, moreover, teaches us that these two reflexions are of
nearly equal force, and that in different transparent bodies they are
so much the stronger as the refraction of these bodies is the greater.
Thus one sees manifestly that the reflexion of glass is stronger than
that of water, and that of diamond stronger than that of glass.
I will finish this theory of refraction by demonstrating a remarkable
proposition which depends on it; namely, that a ray of light in order
to go from one point to another, when these points are in different
media, is refracted in such wise at the plane surface which joins
these two media that it employs the least possible time: and exactly
the same happens in the case of reflexion against a plane surface. Mr.
Fermat was the first to propound this property of refraction, holding
with us, and directly counter to the opinion of Mr. Des Cartes, that
light passes more slowly through glass and water than through air.
But he assumed besides this a constant ratio of Sines, which we have
just proved by these different degrees of velocity alone: or rather,
what is equivalent, he assumed not only that the velocities were
different but that the light took the least time possible for its
passage, and thence deduced the constant ratio of the Sines. His
demonstration, which may be seen in his printed works, and in the
volume of letters of Mr. Des Cartes, is very long; wherefore I give
here another which is simpler and easier.
[Illustration]
Let KF be the plane surface; A the point in the medium which the light
traverses more easily, as the air; C the point in the other which is
more difficult to penetrate, as water. And suppose that a ray has come
from A, by B, to C, having been refracted at B according to the law
demonstrated a little before; that is to say that, having drawn PBQ,
which cuts the plane at right angles, let the sine of the angle ABP
have to the sine of the angle CBQ the same ratio as the velocity of
light in the medium where A is to the velocity of light in the medium
where C is. It is to be shown that the time of passage of light along
AB and BC taken together, is the shortest that can be. Let us assume
that it may have come by other lines, and, in the first place, along
AF, FC, so that the point of refraction F may be further from B than
the point A; and let AO be a line perpendicular to AB, and FO parallel
to AB; BH perpendicular to FO, and FG to BC.
Since then the angle HBF is equal to PBA, and the angle BFG equal to
QBC, it follows that the sine of the angle HBF will also have the same
ratio to the sine of BFG, as the velocity of light in the medium A is
to its velocity in the medium C. But these sines are the straight
lines HF, BG, if we take BF as the semi-diameter of a circle. Then
these lines HF, BG, will bear to one another the said ratio of the
velocities. And, therefore, the time of the light along HF, supposing
that the ray had been OF, would be equal to the time along BG in the
interior of the medium C. But the time along AB is equal to the time
along OH; therefore the time along OF is equal to the time along AB,
BG. Again the time along FC is greater than that along GC; then the
time along OFC will be longer than that along ABC. But AF is longer
than OF, then the time along AFC will by just so much more exceed the
time along ABC.
Now let us assume that the ray has come from A to C along AK, KC; the
point of refraction K being nearer to A than the point B is; and let
CN be the perpendicular upon BC, KN parallel to BC: BM perpendicular
upon KN, and KL upon BA.
Here BL and KM are the sines of angles BKL, KBM; that is to say, of
the angles PBA, QBC; and therefore they are to one another as the
velocity of light in the medium A is to the velocity in the medium C.
Then the time along LB is equal to the time along KM; and since the
time along BC is equal to the time along MN, the time along LBC will
be equal to the time along KMN. But the time along AK is longer than
that along AL: hence the time along AKN is longer than that along ABC.
And KC being longer than KN, the time along AKC will exceed, by as
much more, the time along ABC. Hence it appears that the time along
ABC is the shortest possible; which was to be proven.
CHAPTER IV
ON THE REFRACTION OF THE AIR
We have shown how the movement which constitutes light spreads by
spherical waves in any homogeneous matter. And it is evident that when
the matter is not homogeneous, but of such a constitution that the
movement is communicated in it more rapidly toward one side than
toward another, these waves cannot be spherical: but that they must
acquire their figure according to the different distances over which
the successive movement passes in equal times.
It is thus that we shall in the first place explain the refractions
which occur in the air, which extends from here to the clouds and
beyond. The effects of which refractions are very remarkable; for by
them we often see objects which the rotundity of the Earth ought
otherwise to hide; such as Islands, and the tops of mountains when one
is at sea. Because also of them the Sun and the Moon appear as risen
before in fact they have, and appear to set later: so that at times
the Moon has been seen eclipsed while the Sun appeared still above the
horizon. And so also the heights of the Sun and of the Moon, and those
of all the Stars always appear a little greater than they are in
reality, because of these same refractions, as Astronomers know. But
there is one experiment which renders this refraction very evident;
which is that of fixing a telescope on some spot so that it views an
object, such as a steeple or a house, at a distance of half a league
or more. If then you look through it at different hours of the day,
leaving it always fixed in the same way, you will see that the same
spots of the object will not always appear at the middle of the
aperture of the telescope, but that generally in the morning and in
the evening, when there are more vapours near the Earth, these objects
seem to rise higher, so that the half or more of them will no longer
be visible; and so that they seem lower toward mid-day when these
vapours are dissipated.
Those who consider refraction to occur only in the surfaces which
separate transparent bodies of different nature, would find it
difficult to give a reason for all that I have just related; but
according to our Theory the thing is quite easy. It is known that the
air which surrounds us, besides the particles which are proper to it
and which float in the ethereal matter as has been explained, is full
also of particles of water which are raised by the action of heat; and
it has been ascertained further by some very definite experiments that
as one mounts up higher the density of air diminishes in proportion.
Now whether the particles of water and those of air take part, by
means of the particles of ethereal matter, in the movement which
constitutes light, but have a less prompt recoil than these, or
whether the encounter and hindrance which these particles of air and
water offer to the propagation of movement of the ethereal progress,
retard the progression, it follows that both kinds of particles flying
amidst the ethereal particles, must render the air, from a great
height down to the Earth, gradually less easy for the spreading of the
waves of light.
[Illustration]
Whence the configuration of the waves ought to become nearly such as
this figure represents: namely, if A is a light, or the visible point
of a steeple, the waves which start from it ought to spread more
widely upwards and less widely downwards, but in other directions more
or less as they approximate to these two extremes. This being so, it
necessarily follows that every line intersecting one of these waves at
right angles will pass above the point A, always excepting the one
line which is perpendicular to the horizon.
[Illustration]
Let BC be the wave which brings the light to the spectator who is at
B, and let BD be the straight line which intersects this wave at right
angles. Now because the ray or straight line by which we judge the
spot where the object appears to us is nothing else than the
perpendicular to the wave that reaches our eye, as will be understood
by what was said above, it is manifest that the point A will be
perceived as being in the line BD, and therefore higher than in fact it
is.
Similarly if the Earth be AB, and the top of the Atmosphere CD, which
probably is not a well defined spherical surface (since we know that
the air becomes rare in proportion as one ascends, for above there is
so much less of it to press down upon it), the waves of light from the
sun coming, for instance, in such a way that so long as they have not
reached the Atmosphere CD the straight line AE intersects them
perpendicularly, they ought, when they enter the Atmosphere, to
advance more quickly in elevated regions than in regions nearer to the
Earth. So that if CA is the wave which brings the light to the
spectator at A, its region C will be the furthest advanced; and the
straight line AF, which intersects this wave at right angles, and
which determines the apparent place of the Sun, will pass above the
real Sun, which will be seen along the line AE. And so it may occur
that when it ought not to be visible in the absence of vapours,
because the line AE encounters the rotundity of the Earth, it will be
perceived in the line AF by refraction. But this angle EAF is scarcely
ever more than half a degree because the attenuation of the vapours
alters the waves of light but little. Furthermore these refractions
are not altogether constant in all weathers, particularly at small
elevations of 2 or 3 degrees; which results from the different
quantity of aqueous vapours rising above the Earth.
And this same thing is the cause why at certain times a distant object
will be hidden behind another less distant one, and yet may at another
time be able to be seen, although the spot from which it is viewed is
always the same. But the reason for this effect will be still more
evident from what we are going to remark touching the curvature of
rays. It appears from the things explained above that the progression
or propagation of a small part of a wave of light is properly what one
calls a ray. Now these rays, instead of being straight as they are in
homogeneous media, ought to be curved in an atmosphere of unequal
penetrability. For they necessarily follow from the object to the eye
the line which intersects at right angles all the progressions of the
waves, as in the first figure the line AEB does, as will be shown
hereafter; and it is this line which determines what interposed bodies
would or would not hinder us from seeing the object. For although the
point of the steeple A appears raised to D, it would yet not appear to
the eye B if the tower H was between the two, because it crosses the
curve AEB. But the tower E, which is beneath this curve, does not
hinder the point A from being seen. Now according as the air near the
Earth exceeds in density that which is higher, the curvature of the
ray AEB becomes greater: so that at certain times it passes above the
summit E, which allows the point A to be perceived by the eye at B;
and at other times it is intercepted by the same tower E which hides A
from this same eye.
[Illustration]
But to demonstrate this curvature of the rays conformably to all our
preceding Theory, let us imagine that AB is a small portion of a wave
of light coming from the side C, which we may consider as a straight
line. Let us also suppose that it is perpendicular to the Horizon, the
portion B being nearer to the Earth than the portion A; and that
because the vapours are less hindering at A than at B, the particular
wave which comes from the point A spreads through a certain space AD
while the particular wave which starts from the point B spreads
through a shorter space BE; AD and BE being parallel to the Horizon.
Further, supposing the straight lines FG, HI, etc., to be drawn from
an infinitude of points in the straight line AB and to terminate on
the line DE (which is straight or may be considered as such), let the
different penetrabilities at the different heights in the air between
A and B be represented by all these lines; so that the particular
wave, originating from the point F, will spread across the space FG,
and that from the point H across the space HI, while that from the
point A spreads across the space AD.
Now if about the centres A, B, one describes the circles DK, EL, which
represent the spreading of the waves which originate from these two
points, and if one draws the straight line KL which touches these two
circles, it is easy to see that this same line will be the common
tangent to all the other circles drawn about the centres F, H, etc.;
and that all the points of contact will fall within that part of this
line which is comprised between the perpendiculars AK, BL. Then it
will be the line KL which will terminate the movement of the
particular waves originating from the points of the wave AB; and this
movement will be stronger between the points KL, than anywhere else at
the same instant, since an infinitude of circumferences concur to form
this straight line; and consequently KL will be the propagation of the
portion of wave AB, as has been said in explaining reflexion and
ordinary refraction. Now it appears that AK and BL dip down toward the
side where the air is less easy to penetrate: for AK being longer than
BL, and parallel to it, it follows that the lines AB and KL, being
prolonged, would meet at the side L. But the angle K is a right angle:
hence KAB is necessarily acute, and consequently less than DAB. If one
investigates in the same way the progression of the portion of the
wave KL, one will find that after a further time it has arrived at MN
in such a manner that the perpendiculars KM, LN, dip down even more
than do AK, BL. And this suffices to show that the ray will continue
along the curved line which intersects all the waves at right angles,
as has been said.
CHAPTER V
ON THE STRANGE REFRACTION OF ICELAND CRYSTAL
1.
There is brought from Iceland, which is an Island in the North Sea, in
the latitude of 66 degrees, a kind of Crystal or transparent stone,
very remarkable for its figure and other qualities, but above all for
its strange refractions. The causes of this have seemed to me to be
worthy of being carefully investigated, the more so because amongst
transparent bodies this one alone does not follow the ordinary rules
with respect to rays of light. I have even been under some necessity
to make this research, because the refractions of this Crystal seemed
to overturn our preceding explanation of regular refraction; which
explanation, on the contrary, they strongly confirm, as will be seen
after they have been brought under the same principle. In Iceland are
found great lumps of this Crystal, some of which I have seen of 4 or 5
pounds. But it occurs also in other countries, for I have had some of
the same sort which had been found in France near the town of Troyes
in Champagne, and some others which came from the Island of Corsica,
though both were less clear and only in little bits, scarcely capable
of letting any effect of refraction be observed.
2. The first knowledge which the public has had about it is due to Mr.
Erasmus Bartholinus, who has given a description of Iceland Crystal
and of its chief phenomena. But here I shall not desist from giving my
own, both for the instruction of those who may not have seen his book,
and because as respects some of these phenomena there is a slight
difference between his observations and those which I have made: for I
have applied myself with great exactitude to examine these properties
of refraction, in order to be quite sure before undertaking to explain
the causes of them.
3. As regards the hardness of this stone, and the property which it
has of being easily split, it must be considered rather as a species
of Talc than of Crystal. For an iron spike effects an entrance into it
as easily as into any other Talc or Alabaster, to which it is equal in
gravity.
[Illustration]
4. The pieces of it which are found have the figure of an oblique
parallelepiped; each of the six faces being a parallelogram; and it
admits of being split in three directions parallel to two of these
opposed faces. Even in such wise, if you will, that all the six faces
are equal and similar rhombuses. The figure here added represents a
piece of this Crystal. The obtuse angles of all the parallelograms, as
C, D, here, are angles of 101 degrees 52 minutes, and consequently
the acute angles, such as A and B, are of 78 degrees 8 minutes.
5. Of the solid angles there are two opposite to one another, such as
C and E, which are each composed of three equal obtuse plane angles.
The other six are composed of two acute angles and one obtuse. All
that I have just said has been likewise remarked by Mr. Bartholinus in
the aforesaid treatise; if we differ it is only slightly about the
values of the angles. He recounts moreover some other properties of
this Crystal; to wit, that when rubbed against cloth it attracts
straws and other light things as do amber, diamond, glass, and Spanish
wax. Let a piece be covered with water for a day or more, the surface
loses its natural polish. When aquafortis is poured on it it produces
ebullition, especially, as I have found, if the Crystal has been
pulverized. I have also found by experiment that it may be heated to
redness in the fire without being in anywise altered or rendered less
transparent; but a very violent fire calcines it nevertheless. Its
transparency is scarcely less than that of water or of Rock Crystal,
and devoid of colour. But rays of light pass through it in another
fashion and produce those marvellous refractions the causes of which I
am now going to try to explain; reserving for the end of this Treatise
the statement of my conjectures touching the formation and
extraordinary configuration of this Crystal.
6. In all other transparent bodies that we know there is but one sole
and simple refraction; but in this substance there are two different
ones. The effect is that objects seen through it, especially such as
are placed right against it, appear double; and that a ray of
sunlight, falling on one of its surfaces, parts itself into two rays
and traverses the Crystal thus.
7. It is again a general law in all other transparent bodies that the
ray which falls perpendicularly on their surface passes straight on
without suffering refraction, and that an oblique ray is always
refracted. But in this Crystal the perpendicular ray suffers
refraction, and there are oblique rays which pass through it quite
straight.
[Illustration]
8. But in order to explain these phenomena more particularly, let
there be, in the first place, a piece ABFE of the same Crystal, and
let the obtuse angle ACB, one of the three which constitute the
equilateral solid angle C, be divided into two equal parts by the
straight line CG, and let it be conceived that the Crystal is
intersected by a plane which passes through this line and through the
side CF, which plane will necessarily be perpendicular to the surface
AB; and its section in the Crystal will form a parallelogram GCFH. We
will call this section the principal section of the Crystal.
9. Now if one covers the surface AB, leaving there only a small
aperture at the point K, situated in the straight line CG, and if one
exposes it to the sun, so that his rays face it perpendicularly above,
then the ray IK will divide itself at the point K into two, one of
which will continue to go on straight by KL, and the other will
separate itself along the straight line KM, which is in the plane
GCFH, and which makes with KL an angle of about 6 degrees 40 minutes,
tending from the side of the solid angle C; and on emerging from the
other side of the Crystal it will turn again parallel to JK, along MZ.
And as, in this extraordinary refraction, the point M is seen by the
refracted ray MKI, which I consider as going to the eye at I, it
necessarily follows that the point L, by virtue of the same
refraction, will be seen by the refracted ray LRI, so that LR will be
parallel to MK if the distance from the eye KI is supposed very great.
The point L appears then as being in the straight line IRS; but the
same point appears also, by ordinary refraction, to be in the straight
line IK, hence it is necessarily judged to be double. And similarly if
L be a small hole in a sheet of paper or other substance which is laid
against the Crystal, it will appear when turned towards daylight as if
there were two holes, which will seem the wider apart from one another
the greater the thickness of the Crystal.
10. Again, if one turns the Crystal in such wise that an incident ray
NO, of sunlight, which I suppose to be in the plane continued from
GCFH, makes with GC an angle of 73 degrees and 20 minutes, and is
consequently nearly parallel to the edge CF, which makes with FH an
angle of 70 degrees 57 minutes, according to the calculation which I
shall put at the end, it will divide itself at the point O into two
rays, one of which will continue along OP in a straight line with NO,
and will similarly pass out of the other side of the crystal without
any refraction; but the other will be refracted and will go along OQ.
And it must be noted that it is special to the plane through GCF and
to those which are parallel to it, that all incident rays which are in
one of these planes continue to be in it after they have entered the
Crystal and have become double; for it is quite otherwise for rays in
all other planes which intersect the Crystal, as we shall see
afterwards.
11. I recognized at first by these experiments and by some others that
of the two refractions which the ray suffers in this Crystal, there is
one which follows the ordinary rules; and it is this to which the rays
KL and OQ belong. This is why I have distinguished this ordinary
refraction from the other; and having measured it by exact
observation, I found that its proportion, considered as to the Sines
of the angles which the incident and refracted rays make with the
perpendicular, was very precisely that of 5 to 3, as was found also by
Mr. Bartholinus, and consequently much greater than that of Rock
Crystal, or of glass, which is nearly 3 to 2.
[Illustration]
12. The mode of making these observations exactly is as follows. Upon
a leaf of paper fixed on a thoroughly flat table there is traced a
black line AB, and two others, CED and KML, which cut it at right
angles and are more or less distant from one another according as it
is desired to examine a ray that is more or less oblique. Then place
the Crystal upon the intersection E so that the line AB concurs with
that which bisects the obtuse angle of the lower surface, or with some
line parallel to it. Then by placing the eye directly above the line
AB it will appear single only; and one will see that the portion
viewed through the Crystal and the portions which appear outside it,
meet together in a straight line: but the line CD will appear double,
and one can distinguish the image which is due to regular refraction
by the circumstance that when one views it with both eyes it seems
raised up more than the other, or again by the circumstance that, when
the Crystal is turned around on the paper, this image remains
stationary, whereas the other image shifts and moves entirely around.
Afterwards let the eye be placed at I (remaining always in the plane
perpendicular through AB) so that it views the image which is formed
by regular refraction of the line CD making a straight line with the
remainder of that line which is outside the Crystal. And then, marking
on the surface of the Crystal the point H where the intersection E
appears, this point will be directly above E. Then draw back the eye
towards O, keeping always in the plane perpendicular through AB, so
that the image of the line CD, which is formed by ordinary refraction,
may appear in a straight line with the line KL viewed without
refraction; and then mark on the Crystal the point N where the point
of intersection E appears.
13. Then one will know the length and position of the lines NH, EM,
and of HE, which is the thickness of the Crystal: which lines being
traced separately upon a plan, and then joining NE and NM which cuts
HE at P, the proportion of the refraction will be that of EN to NP,
because these lines are to one another as the sines of the angles NPH,
NEP, which are equal to those which the incident ray ON and its
refraction NE make with the perpendicular to the surface. This
proportion, as I have said, is sufficiently precisely as 5 to 3, and
is always the same for all inclinations of the incident ray.
14. The same mode of observation has also served me for examining the
extraordinary or irregular refraction of this Crystal. For, the point
H having been found and marked, as aforesaid, directly above the point
E, I observed the appearance of the line CD, which is made by the
extraordinary refraction; and having placed the eye at Q, so that this
appearance made a straight line with the line KL viewed without
refraction, I ascertained the triangles REH, RES, and consequently the
angles RSH, RES, which the incident and the refracted ray make with
the perpendicular.
15. But I found in this refraction that the ratio of FR to RS was not
constant, like the ordinary refraction, but that it varied with the
varying obliquity of the incident ray.
16. I found also that when QRE made a straight line, that is, when the
incident ray entered the Crystal without being refracted (as I
ascertained by the circumstance that then the point E viewed by the
extraordinary refraction appeared in the line CD, as seen without
refraction) I found, I say, then that the angle QRG was 73 degrees 20
minutes, as has been already remarked; and so it is not the ray
parallel to the edge of the Crystal, which crosses it in a straight
line without being refracted, as Mr. Bartholinus believed, since that
inclination is only 70 degrees 57 minutes, as was stated above. And
this is to be noted, in order that no one may search in vain for the
cause of the singular property of this ray in its parallelism to the
edges mentioned.
[Illustration]
17. Finally, continuing my observations to discover the nature of
this refraction, I learned that it obeyed the following remarkable
rule. Let the parallelogram GCFH, made by the principal section of the
Crystal, as previously determined, be traced separately. I found then
that always, when the inclinations of two rays which come from
opposite sides, as VK, SK here, are equal, their refractions KX and KT
meet the bottom line HF in such wise that points X and T are equally
distant from the point M, where the refraction of the perpendicular
ray IK falls; and this occurs also for refractions in other sections
of this Crystal. But before speaking of those, which have also other
particular properties, we will investigate the causes of the phenomena
which I have already reported.
It was after having explained the refraction of ordinary transparent
bodies by means of the spherical emanations of light, as above, that I
resumed my examination of the nature of this Crystal, wherein I had
previously been unable to discover anything.
18. As there were two different refractions, I conceived that there
were also two different emanations of waves of light, and that one
could occur in the ethereal matter extending through the body of the
Crystal. Which matter, being present in much larger quantity than is
that of the particles which compose it, was alone capable of causing
transparency, according to what has been explained heretofore. I
attributed to this emanation of waves the regular refraction which is
observed in this stone, by supposing these waves to be ordinarily of
spherical form, and having a slower progression within the Crystal
than they have outside it; whence proceeds refraction as I have
demonstrated.
19. As to the other emanation which should produce the irregular
refraction, I wished to try what Elliptical waves, or rather
spheroidal waves, would do; and these I supposed would spread
indifferently both in the ethereal matter diffused throughout the
crystal and in the particles of which it is composed, according to the
last mode in which I have explained transparency. It seemed to me that
the disposition or regular arrangement of these particles could
contribute to form spheroidal waves (nothing more being required for
this than that the successive movement of light should spread a little
more quickly in one direction than in the other) and I scarcely
doubted that there were in this crystal such an arrangement of equal
and similar particles, because of its figure and of its angles with
their determinate and invariable measure. Touching which particles,
and their form and disposition, I shall, at the end of this Treatise,
propound my conjectures and some experiments which confirm them.
20. The double emission of waves of light, which I had imagined,
became more probable to me after I had observed a certain phenomenon
in the ordinary [Rock] Crystal, which occurs in hexagonal form, and
which, because of this regularity, seems also to be composed of
particles, of definite figure, and ranged in order. This was, that
this crystal, as well as that from Iceland, has a double refraction,
though less evident. For having had cut from it some well polished
Prisms of different sections, I remarked in all, in viewing through
them the flame of a candle or the lead of window panes, that
everything appeared double, though with images not very distant from
one another. Whence I understood the reason why this substance, though
so transparent, is useless for Telescopes, when they have ever so
little length.
21. Now this double refraction, according to my Theory hereinbefore
established, seemed to demand a double emission of waves of light,
both of them spherical (for both the refractions are regular) and
those of one series a little slower only than the others. For thus the
phenomenon is quite naturally explained, by postulating substances
which serve as vehicle for these waves, as I have done in the case of
Iceland Crystal. I had then less trouble after that in admitting two
emissions of waves in one and the same body. And since it might have
been objected that in composing these two kinds of crystal of equal
particles of a certain figure, regularly piled, the interstices which
these particles leave and which contain the ethereal matter would
scarcely suffice to transmit the waves of light which I have localized
there, I removed this difficulty by regarding these particles as being
of a very rare texture, or rather as composed of other much smaller
particles, between which the ethereal matter passes quite freely.
This, moreover, necessarily follows from that which has been already
demonstrated touching the small quantity of matter of which the bodies
are built up.
22. Supposing then these spheroidal waves besides the spherical ones,
I began to examine whether they could serve to explain the phenomena
of the irregular refraction, and how by these same phenomena I could
determine the figure and position of the spheroids: as to which I
obtained at last the desired success, by proceeding as follows.
[Illustration]
23. I considered first the effect of waves so formed, as respects the
ray which falls perpendicularly on the flat surface of a transparent
body in which they should spread in this manner. I took AB for the
exposed region of the surface. And, since a ray perpendicular to a
plane, and coming from a very distant source of light, is nothing
else, according to the precedent Theory, than the incidence of a
portion of the wave parallel to that plane, I supposed the straight
line RC, parallel and equal to AB, to be a portion of a wave of light,
in which an infinitude of points such as RH_h_C come to meet the
surface AB at the points AK_k_B. Then instead of the hemispherical
partial waves which in a body of ordinary refraction would spread from
each of these last points, as we have above explained in treating of
refraction, these must here be hemi-spheroids. The axes (or rather the
major diameters) of these I supposed to be oblique to the plane AB, as
is AV the semi-axis or semi-major diameter of the spheroid SVT, which
represents the partial wave coming from the point A, after the wave RC
has reached AB. I say axis or major diameter, because the same ellipse
SVT may be considered as the section of a spheroid of which the axis
is AZ perpendicular to AV. But, for the present, without yet deciding
one or other, we will consider these spheroids only in those sections
of them which make ellipses in the plane of this figure. Now taking a
certain space of time during which the wave SVT has spread from A, it
would needs be that from all the other points K_k_B there should
proceed, in the same time, waves similar to SVT and similarly
situated. And the common tangent NQ of all these semi-ellipses would
be the propagation of the wave RC which fell on AB, and would be the
place where this movement occurs in much greater amount than anywhere
else, being made up of arcs of an infinity of ellipses, the centres of
which are along the line AB.
24. Now it appeared that this common tangent NQ was parallel to AB,
and of the same length, but that it was not directly opposite to it,
since it was comprised between the lines AN, BQ, which are diameters
of ellipses having A and B for centres, conjugate with respect to
diameters which are not in the straight line AB. And in this way I
comprehended, a matter which had seemed to me very difficult, how a
ray perpendicular to a surface could suffer refraction on entering a
transparent body; seeing that the wave RC, having come to the aperture
AB, went on forward thence, spreading between the parallel lines AN,
BQ, yet itself remaining always parallel to AB, so that here the light
does not spread along lines perpendicular to its waves, as in ordinary
refraction, but along lines cutting the waves obliquely.
[Illustration]
25. Inquiring subsequently what might be the position and form of
these spheroids in the crystal, I considered that all the six faces
produced precisely the same refractions. Taking, then, the
parallelopiped AFB, of which the obtuse solid angle C is contained
between the three equal plane angles, and imagining in it the three
principal sections, one of which is perpendicular to the face DC and
passes through the edge CF, another perpendicular to the face BF
passing through the edge CA, and the third perpendicular to the face
AF passing through the edge BC; I knew that the refractions of the
incident rays belonging to these three planes were all similar. But
there could be no position of the spheroid which would have the same
relation to these three sections except that in which the axis was
also the axis of the solid angle C. Consequently I saw that the axis
of this angle, that is to say the straight line which traversed the
crystal from the point C with equal inclination to the edges CF, CA,
CB was the line which determined the position of the axis of all the
spheroidal waves which one imagined to originate from some point,
taken within or on the surface of the crystal, since all these
spheroids ought to be alike, and have their axes parallel to one
another.
26. Considering after this the plane of one of these three sections,
namely that through GCF, the angle of which is 109 degrees 3 minutes,
since the angle F was shown above to be 70 degrees 57 minutes; and,
imagining a spheroidal wave about the centre C, I knew, because I have
just explained it, that its axis must be in the same plane, the half
of which axis I have marked CS in the next figure: and seeking by
calculation (which will be given with others at the end of this
discourse) the value of the angle CGS, I found it 45 degrees 20
minutes.
[Illustration]
27. To know from this the form of this spheroid, that is to say the
proportion of the semi-diameters CS, CP, of its elliptical section,
which are perpendicular to one another, I considered that the point M
where the ellipse is touched by the straight line FH, parallel to CG,
ought to be so situated that CM makes with the perpendicular CL an
angle of 6 degrees 40 minutes; since, this being so, this ellipse
satisfies what has been said about the refraction of the ray
perpendicular to the surface CG, which is inclined to the
perpendicular CL by the same angle. This, then, being thus disposed,
and taking CM at 100,000 parts, I found by the calculation which will
be given at the end, the semi-major diameter CP to be 105,032, and the
semi-axis CS to be 93,410, the ratio of which numbers is very nearly 9
to 8; so that the spheroid was of the kind which resembles a
compressed sphere, being generated by the revolution of an ellipse
about its smaller diameter. I found also the value of CG the
semi-diameter parallel to the tangent ML to be 98,779.
[Illustration]
28. Now passing to the investigation of the refractions which
obliquely incident rays must undergo, according to our hypothesis of
spheroidal waves, I saw that these refractions depended on the ratio
between the velocity of movement of the light outside the crystal in
the ether, and that within the crystal. For supposing, for example,
this proportion to be such that while the light in the crystal forms
the spheroid GSP, as I have just said, it forms outside a sphere the
semi-diameter of which is equal to the line N which will be determined
hereafter, the following is the way of finding the refraction of the
incident rays. Let there be such a ray RC falling upon the surface
CK. Make CO perpendicular to RC, and across the angle KCO adjust OK,
equal to N and perpendicular to CO; then draw KI, which touches the
Ellipse GSP, and from the point of contact I join IC, which will be
the required refraction of the ray RC. The demonstration of this is,
it will be seen, entirely similar to that of which we made use in
explaining ordinary refraction. For the refraction of the ray RC is
nothing else than the progression of the portion C of the wave CO,
continued in the crystal. Now the portions H of this wave, during the
time that O came to K, will have arrived at the surface CK along the
straight lines H_x_, and will moreover have produced in the crystal
around the centres _x_ some hemi-spheroidal partial waves similar to
the hemi-spheroidal GSP_g_, and similarly disposed, and of which the
major and minor diameters will bear the same proportions to the lines
_xv_ (the continuations of the lines H_x_ up to KB parallel to CO)
that the diameters of the spheroid GSP_g_ bear to the line CB, or N.
And it is quite easy to see that the common tangent of all these
spheroids, which are here represented by Ellipses, will be the
straight line IK, which consequently will be the propagation of the
wave CO; and the point I will be that of the point C, conformably with
that which has been demonstrated in ordinary refraction.
Now as to finding the point of contact I, it is known that one must
find CD a third proportional to the lines CK, CG, and draw DI parallel
to CM, previously determined, which is the conjugate diameter to CG;
for then, by drawing KI it touches the Ellipse at I.
29. Now as we have found CI the refraction of the ray RC, similarly
one will find C_i_ the refraction of the ray _r_C, which comes from
the opposite side, by making C_o_ perpendicular to _r_C and following
out the rest of the construction as before. Whence one sees that if
the ray _r_C is inclined equally with RC, the line C_d_ will
necessarily be equal to CD, because C_k_ is equal to CK, and C_g_ to
CG. And in consequence I_i_ will be cut at E into equal parts by the
line CM, to which DI and _di_ are parallel. And because CM is the
conjugate diameter to CG, it follows that _i_I will be parallel to
_g_G. Therefore if one prolongs the refracted rays CI, C_i_, until
they meet the tangent ML at T and _t_, the distances MT, M_t_, will
also be equal. And so, by our hypothesis, we explain perfectly the
phenomenon mentioned above; to wit, that when there are two rays
equally inclined, but coming from opposite sides, as here the rays RC,
_rc_, their refractions diverge equally from the line followed by the
refraction of the ray perpendicular to the surface, by considering
these divergences in the direction parallel to the surface of the
crystal.
30. To find the length of the line N, in proportion to CP, CS, CG, it
must be determined by observations of the irregular refraction which
occurs in this section of the crystal; and I find thus that the ratio
of N to GC is just a little less than 8 to 5. And having regard to
some other observations and phenomena of which I shall speak
afterwards, I put N at 156,962 parts, of which the semi-diameter CG is
found to contain 98,779, making this ratio 8 to 5-1/29. Now this
proportion, which there is between the line N and CG, may be called
the Proportion of the Refraction; similarly as in glass that of 3 to
2, as will be manifest when I shall have explained a short process in
the preceding way to find the irregular refractions.
31. Supposing then, in the next figure, as previously, the surface of
the crystal _g_G, the Ellipse GP_g_, and the line N; and CM the
refraction of the perpendicular ray FC, from which it diverges by 6
degrees 40 minutes. Now let there be some other ray RC, the refraction
of which must be found.
About the centre C, with semi-diameter CG, let the circumference _g_RG
be described, cutting the ray RC at R; and let RV be the perpendicular
on CG. Then as the line N is to CG let CV be to CD, and let DI be
drawn parallel to CM, cutting the Ellipse _g_MG at I; then joining CI,
this will be the required refraction of the ray RC. Which is
demonstrated thus.
[Illustration]
Let CO be perpendicular to CR, and across the angle OCG let OK be
adjusted, equal to N and perpendicular to CO, and let there be drawn
the straight line KI, which if it is demonstrated to be a tangent to
the Ellipse at I, it will be evident by the things heretofore
explained that CI is the refraction of the ray RC. Now since the angle
RCO is a right angle, it is easy to see that the right-angled
triangles RCV, KCO, are similar. As then, CK is to KO, so also is RC
to CV. But KO is equal to N, and RC to CG: then as CK is to N so will
CG be to CV. But as N is to CG, so, by construction, is CV to CD. Then
as CK is to CG so is CG to CD. And because DI is parallel to CM, the
conjugate diameter to CG, it follows that KI touches the Ellipse at I;
which remained to be shown.
32. One sees then that as there is in the refraction of ordinary
media a certain constant proportion between the sines of the angles
which the incident ray and the refracted ray make with the
perpendicular, so here there is such a proportion between CV and CD or
IE; that is to say between the Sine of the angle which the incident
ray makes with the perpendicular, and the horizontal intercept, in the
Ellipse, between the refraction of this ray and the diameter CM. For
the ratio of CV to CD is, as has been said, the same as that of N to
the semi-diameter CG.
33. I will add here, before passing away, that in comparing together
the regular and irregular refraction of this crystal, there is this
remarkable fact, that if ABPS be the spheroid by which light spreads
in the Crystal in a certain space of time (which spreading, as has
been said, serves for the irregular refraction), then the inscribed
sphere BVST is the extension in the same space of time of the light
which serves for the regular refraction.
[Illustration]
For we have stated before this, that the line N being the radius of a
spherical wave of light in air, while in the crystal it spread through
the spheroid ABPS, the ratio of N to CS will be 156,962 to 93,410. But
it has also been stated that the proportion of the regular refraction
was 5 to 3; that is to say, that N being the radius of a spherical
wave of light in air, its extension in the crystal would, in the same
space of time, form a sphere the radius of which would be to N as 3 to
5. Now 156,962 is to 93,410 as 5 to 3 less 1/41. So that it is
sufficiently nearly, and may be exactly, the sphere BVST, which the
light describes for the regular refraction in the crystal, while it
describes the spheroid BPSA for the irregular refraction, and while it
describes the sphere of radius N in air outside the crystal.
Although then there are, according to what we have supposed, two
different propagations of light within the crystal, it appears that it
is only in directions perpendicular to the axis BS of the spheroid
that one of these propagations occurs more rapidly than the other; but
that they have an equal velocity in the other direction, namely, in
that parallel to the same axis BS, which is also the axis of the
obtuse angle of the crystal.
[Illustration]
34. The proportion of the refraction being what we have just seen, I
will now show that there necessarily follows thence that notable
property of the ray which falling obliquely on the surface of the
crystal enters it without suffering refraction. For supposing the same
things as before, and that the ray makes with the same surface _g_G
the angle RCG of 73 degrees 20 minutes, inclining to the same side as
the crystal (of which ray mention has been made above); if one
investigates, by the process above explained, the refraction CI, one
will find that it makes exactly a straight line with RC, and that thus
this ray is not deviated at all, conformably with experiment. This is
proved as follows by calculation.
CG or CR being, as precedently, 98,779; CM being 100,000; and the
angle RCV 73 degrees 20 minutes, CV will be 28,330. But because CI is
the refraction of the ray RC, the proportion of CV to CD is 156,962 to
98,779, namely, that of N to CG; then CD is 17,828.
Now the rectangle _g_DC is to the square of DI as the square of CG is
to the square of CM; hence DI or CE will be 98,353. But as CE is to
EI, so will CM be to MT, which will then be 18,127. And being added to
ML, which is 11,609 (namely the sine of the angle LCM, which is 6
degrees 40 minutes, taking CM 100,000 as radius) we get LT 27,936; and
this is to LC 99,324 as CV to VR, that is to say, as 29,938, the
tangent of the complement of the angle RCV, which is 73 degrees 20
minutes, is to the radius of the Tables. Whence it appears that RCIT
is a straight line; which was to be proved.
35. Further it will be seen that the ray CI in emerging through the
opposite surface of the crystal, ought to pass out quite straight,
according to the following demonstration, which proves that the
reciprocal relation of refraction obtains in this crystal the same as
in other transparent bodies; that is to say, that if a ray RC in
meeting the surface of the crystal CG is refracted as CI, the ray CI
emerging through the opposite parallel surface of the crystal, which
I suppose to be IB, will have its refraction IA parallel to the ray
RC.
[Illustration]
Let the same things be supposed as before; that is to say, let CO,
perpendicular to CR, represent a portion of a wave the continuation of
which in the crystal is IK, so that the piece C will be continued on
along the straight line CI, while O comes to K. Now if one takes a
second period of time equal to the first, the piece K of the wave IK
will, in this second period, have advanced along the straight line KB,
equal and parallel to CI, because every piece of the wave CO, on
arriving at the surface CK, ought to go on in the crystal the same as
the piece C; and in this same time there will be formed in the air
from the point I a partial spherical wave having a semi-diameter IA
equal to KO, since KO has been traversed in an equal time. Similarly,
if one considers some other point of the wave IK, such as _h_, it will
go along _hm_, parallel to CI, to meet the surface IB, while the point
K traverses K_l_ equal to _hm_; and while this accomplishes the
remainder _l_B, there will start from the point _m_ a partial wave the
semi-diameter of which, _mn_, will have the same ratio to _l_B as IA
to KB. Whence it is evident that this wave of semi-diameter _mn_, and
the other of semi-diameter IA will have the same tangent BA. And
similarly for all the partial spherical waves which will be formed
outside the crystal by the impact of all the points of the wave IK
against the surface of the Ether IB. It is then precisely the tangent
BA which will be the continuation of the wave IK, outside the crystal,
when the piece K has reached B. And in consequence IA, which is
perpendicular to BA, will be the refraction of the ray CI on emerging
from the crystal. Now it is clear that IA is parallel to the incident
ray RC, since IB is equal to CK, and IA equal to KO, and the angles A
and O are right angles.
It is seen then that, according to our hypothesis, the reciprocal
relation of refraction holds good in this crystal as well as in
ordinary transparent bodies; as is thus in fact found by observation.
36. I pass now to the consideration of other sections of the crystal,
and of the refractions there produced, on which, as will be seen, some
other very remarkable phenomena depend.
Let ABH be a parallelepiped of crystal, and let the top surface AEHF
be a perfect rhombus, the obtuse angles of which are equally divided
by the straight line EF, and the acute angles by the straight line AH
perpendicular to FE.
The section which we have hitherto considered is that which passes
through the lines EF, EB, and which at the same time cuts the plane
AEHF at right angles. Refractions in this section have this in common
with the refractions in ordinary media that the plane which is drawn
through the incident ray and which also intersects the surface of the
crystal at right angles, is that in which the refracted ray also is
found. But the refractions which appertain to every other section of
this crystal have this strange property that the refracted ray always
quits the plane of the incident ray perpendicular to the surface, and
turns away towards the side of the slope of the crystal. For which
fact we shall show the reason, in the first place, for the section
through AH; and we shall show at the same time how one can determine
the refraction, according to our hypothesis. Let there be, then, in
the plane which passes through AH, and which is perpendicular to the
plane AFHE, the incident ray RC; it is required to find its refraction
in the crystal.
[Illustration]
37. About the centre C, which I suppose to be in the intersection of
AH and FE, let there be imagined a hemi-spheroid QG_qg_M, such as the
light would form in spreading in the crystal, and let its section by
the plane AEHF form the Ellipse QG_qg_, the major diameter of which
Q_q_, which is in the line AH, will necessarily be one of the major
diameters of the spheroid; because the axis of the spheroid being in
the plane through FEB, to which QC is perpendicular, it follows that
QC is also perpendicular to the axis of the spheroid, and consequently
QC_q_ one of its major diameters. But the minor diameter of this
Ellipse, G_g_, will bear to Q_q_ the proportion which has been defined
previously, Article 27, between CG and the major semi-diameter of the
spheroid, CP, namely, that of 98,779 to 105,032.
Let the line N be the length of the travel of light in air during the
time in which, within the crystal, it makes, from the centre C, the
spheroid QC_qg_M. Then having drawn CO perpendicular to the ray CR and
situate in the plane through CR and AH, let there be adjusted, across
the angle ACO, the straight line OK equal to N and perpendicular to
CO, and let it meet the straight line AH at K. Supposing consequently
that CL is perpendicular to the surface of the crystal AEHF, and that
CM is the refraction of the ray which falls perpendicularly on this
same surface, let there be drawn a plane through the line CM and
through KCH, making in the spheroid the semi-ellipse QM_q_, which will
be given, since the angle MCL is given of value 6 degrees 40 minutes.
And it is certain, according to what has been explained above, Article
27, that a plane which would touch the spheroid at the point M, where
I suppose the straight line CM to meet the surface, would be parallel
to the plane QG_q_. If then through the point K one now draws KS
parallel to G_g_, which will be parallel also to QX, the tangent to
the Ellipse QG_q_ at Q; and if one conceives a plane passing through
KS and touching the spheroid, the point of contact will necessarily be
in the Ellipse QM_q_, because this plane through KS, as well as the
plane which touches the spheroid at the point M, are parallel to QX,
the tangent of the spheroid: for this consequence will be demonstrated
at the end of this Treatise. Let this point of contact be at I, then
making KC, QC, DC proportionals, draw DI parallel to CM; also join CI.
I say that CI will be the required refraction of the ray RC. This will
be manifest if, in considering CO, which is perpendicular to the ray
RC, as a portion of the wave of light, we can demonstrate that the
continuation of its piece C will be found in the crystal at I, when O
has arrived at K.
38. Now as in the Chapter on Reflexion, in demonstrating that the
incident and reflected rays are always in the same plane perpendicular
to the reflecting surface, we considered the breadth of the wave of
light, so, similarly, we must here consider the breadth of the wave CO
in the diameter G_g_. Taking then the breadth C_c_ on the side toward
the angle E, let the parallelogram CO_oc_ be taken as a portion of a
wave, and let us complete the parallelograms CK_kc_, CI_ic_, Kl_ik_,
OK_ko_. In the time then that the line O_o_ arrives at the surface of
the crystal at K_k_, all the points of the wave CO_oc_ will have
arrived at the rectangle K_c_ along lines parallel to OK; and from the
points of their incidences there will originate, beyond that, in the
crystal partial hemi-spheroids, similar to the hemi-spheroid QM_q_,
and similarly disposed. These hemi-spheroids will necessarily all
touch the plane of the parallelogram KI_ik_ at the same instant that
O_o_ has reached K_k_. Which is easy to comprehend, since, of these
hemi-spheroids, all those which have their centres along the line CK,
touch this plane in the line KI (for this is to be shown in the same
way as we have demonstrated the refraction of the oblique ray in the
principal section through EF) and all those which have their centres
in the line C_c_ will touch the same plane KI in the line I_i_; all
these being similar to the hemi-spheroid QM_q_. Since then the
parallelogram K_i_ is that which touches all these spheroids, this
same parallelogram will be precisely the continuation of the wave
CO_oc_ in the crystal, when O_o_ has arrived at K_k_, because it forms
the termination of the movement and because of the quantity of
movement which occurs more there than anywhere else: and thus it
appears that the piece C of the wave CO_oc_ has its continuation at I;
that is to say, that the ray RC is refracted as CI.
From this it is to be noted that the proportion of the refraction for
this section of the crystal is that of the line N to the semi-diameter
CQ; by which one will easily find the refractions of all incident
rays, in the same way as we have shown previously for the case of the
section through FE; and the demonstration will be the same. But it
appears that the said proportion of the refraction is less here than
in the section through FEB; for it was there the same as the ratio of
N to CG, that is to say, as 156,962 to 98,779, very nearly as 8 to 5;
and here it is the ratio of N to CQ the major semi-diameter of the
spheroid, that is to say, as 156,962 to 105,032, very nearly as 3 to
2, but just a little less. Which still agrees perfectly with what one
finds by observation.
39. For the rest, this diversity of proportion of refraction produces
a very singular effect in this Crystal; which is that when it is
placed upon a sheet of paper on which there are letters or anything
else marked, if one views it from above with the two eyes situated in
the plane of the section through EF, one sees the letters raised up by
this irregular refraction more than when one puts one's eyes in the
plane of section through AH: and the difference of these elevations
appears by comparison with the other ordinary refraction of the
crystal, the proportion of which is as 5 to 3, and which always raises
the letters equally, and higher than the irregular refraction does.
For one sees the letters and the paper on which they are written, as
on two different stages at the same time; and in the first position of
the eyes, namely, when they are in the plane through AH these two
stages are four times more distant from one another than when the eyes
are in the plane through EF.
We will show that this effect follows from the refractions; and it
will enable us at the same time to ascertain the apparent place of a
point of an object placed immediately under the crystal, according to
the different situation of the eyes.
40. Let us see first by how much the irregular refraction of the plane
through AH ought to lift the bottom of the crystal. Let the plane of
this figure represent separately the section through Q_q_ and CL, in
which section there is also the ray RC, and let the semi-elliptic
plane through Q_q_ and CM be inclined to the former, as previously, by
an angle of 6 degrees 40 minutes; and in this plane CI is then the
refraction of the ray RC.
[Illustration]
If now one considers the point I as at the bottom of the crystal, and
that it is viewed by the rays ICR, _Icr_, refracted equally at the
points C_c_, which should be equally distant from D, and that these
rays meet the two eyes at R_r_; it is certain that the point I will
appear raised to S where the straight lines RC, _rc_, meet; which
point S is in DP, perpendicular to Q_q_. And if upon DP there is drawn
the perpendicular IP, which will lie at the bottom of the crystal, the
length SP will be the apparent elevation of the point I above the
bottom.
Let there be described on Q_q_ a semicircle cutting the ray CR at B,
from which BV is drawn perpendicular to Q_q_; and let the proportion
of the refraction for this section be, as before, that of the line N
to the semi-diameter CQ.
Then as N is to CQ so is VC to CD, as appears by the method of finding
the refraction which we have shown above, Article 31; but as VC is to
CD, so is VB to DS. Then as N is to CQ, so is VB to DS. Let ML be
perpendicular to CL. And because I suppose the eyes R_r_ to be distant
about a foot or so from the crystal, and consequently the angle RS_r_
very small, VB may be considered as equal to the semi-diameter CQ, and
DP as equal to CL; then as N is to CQ so is CQ to DS. But N is valued
at 156,962 parts, of which CM contains 100,000 and CQ 105,032. Then DS
will have 70,283. But CL is 99,324, being the sine of the complement
of the angle MCL which is 6 degrees 40 minutes; CM being supposed as
radius. Then DP, considered as equal to CL, will be to DS as 99,324 to
70,283. And so the elevation of the point I by the refraction of this
section is known.
[Illustration]
41. Now let there be represented the other section through EF in the
figure before the preceding one; and let CM_g_ be the semi-ellipse,
considered in Articles 27 and 28, which is made by cutting a
spheroidal wave having centre C. Let the point I, taken in this
ellipse, be imagined again at the bottom of the Crystal; and let it be
viewed by the refracted rays ICR, I_cr_, which go to the two eyes; CR
and _cr_ being equally inclined to the surface of the crystal G_g_.
This being so, if one draws ID parallel to CM, which I suppose to be
the refraction of the perpendicular ray incident at the point C, the
distances DC, D_c_, will be equal, as is easy to see by that which has
been demonstrated in Article 28. Now it is certain that the point I
should appear at S where the straight lines RC, _rc_, meet when
prolonged; and that this point will fall in the line DP perpendicular
to G_g_. If one draws IP perpendicular to this DP, it will be the
distance PS which will mark the apparent elevation of the point I. Let
there be described on G_g_ a semicircle cutting CR at B, from which
let BV be drawn perpendicular to G_g_; and let N to GC be the
proportion of the refraction in this section, as in Article 28. Since
then CI is the refraction of the radius BC, and DI is parallel to CM,
VC must be to CD as N to GC, according to what has been demonstrated
in Article 31. But as VC is to CD so is BV to DS. Let ML be drawn
perpendicular to CL. And because I consider, again, the eyes to be
distant above the crystal, BV is deemed equal to the semi-diameter CG;
and hence DS will be a third proportional to the lines N and CG: also
DP will be deemed equal to CL. Now CG consisting of 98,778 parts, of
which CM contains 100,000, N is taken as 156,962. Then DS will be
62,163. But CL is also determined, and contains 99,324 parts, as has
been said in Articles 34 and 40. Then the ratio of PD to DS will be as
99,324 to 62,163. And thus one knows the elevation of the point at the
bottom I by the refraction of this section; and it appears that this
elevation is greater than that by the refraction of the preceding
section, since the ratio of PD to DS was there as 99,324 to 70,283.
[Illustration]
But by the regular refraction of the crystal, of which we have above
said that the proportion is 5 to 3, the elevation of the point I, or
P, from the bottom, will be 2/5 of the height DP; as appears by this
figure, where the point P being viewed by the rays PCR, P_cr_,
refracted equally at the surface C_c_, this point must needs appear
to be at S, in the perpendicular PD where the lines RC, _rc_, meet
when prolonged: and one knows that the line PC is to CS as 5 to 3,
since they are to one another as the sine of the angle CSP or DSC is
to the sine of the angle SPC. And because the ratio of PD to DS is
deemed the same as that of PC to CS, the two eyes Rr being supposed
very far above the crystal, the elevation PS will thus be 2/5 of PD.
[Illustration]
42. If one takes a straight line AB for the thickness of the crystal,
its point B being at the bottom, and if one divides it at the points
C, D, E, according to the proportions of the elevations found, making
AE 3/5 of AB, AB to AC as 99,324 to 70,283, and AB to AD as 99,324 to
62,163, these points will divide AB as in this figure. And it will be
found that this agrees perfectly with experiment; that is to say by
placing the eyes above in the plane which cuts the crystal according
to the shorter diameter of the rhombus, the regular refraction will
lift up the letters to E; and one will see the bottom, and the letters
over which it is placed, lifted up to D by the irregular refraction.
But by placing the eyes above in the plane which cuts the crystal
according to the longer diameter of the rhombus, the regular
refraction will lift the letters to E as before; but the irregular
refraction will make them, at the same time, appear lifted up only to
C; and in such a way that the interval CE will be quadruple the
interval ED, which one previously saw.
43. I have only to make the remark here that in both the positions of
the eyes the images caused by the irregular refraction do not appear
directly below those which proceed from the regular refraction, but
they are separated from them by being more distant from the
equilateral solid angle of the Crystal. That follows, indeed, from all
that has been hitherto demonstrated about the irregular refraction;
and it is particularly shown by these last demonstrations, from which
one sees that the point I appears by irregular refraction at S in the
perpendicular line DP, in which line also the image of the point P
ought to appear by regular refraction, but not the image of the point
I, which will be almost directly above the same point, and higher than
S.
But as to the apparent elevation of the point I in other positions of
the eyes above the crystal, besides the two positions which we have
just examined, the image of that point by the irregular refraction
will always appear between the two heights of D and C, passing from
one to the other as one turns one's self around about the immovable
crystal, while looking down from above. And all this is still found
conformable to our hypothesis, as any one can assure himself after I
shall have shown here the way of finding the irregular refractions
which appear in all other sections of the crystal, besides the two
which we have considered. Let us suppose one of the faces of the
crystal, in which let there be the Ellipse HDE, the centre C of which
is also the centre of the spheroid HME in which the light spreads, and
of which the said Ellipse is the section. And let the incident ray be
RC, the refraction of which it is required to find.
Let there be taken a plane passing through the ray RC and which is
perpendicular to the plane of the ellipse HDE, cutting it along the
straight line BCK; and having in the same plane through RC made CO
perpendicular to CR, let OK be adjusted across the angle OCK, so as
to be perpendicular to OC and equal to the line N, which I suppose to
measure the travel of the light in air during the time that it spreads
in the crystal through the spheroid HDEM. Then in the plane of the
Ellipse HDE let KT be drawn, through the point K, perpendicular to
BCK. Now if one conceives a plane drawn through the straight line KT
and touching the spheroid HME at I, the straight line CI will be the
refraction of the ray RC, as is easy to deduce from that which has
been demonstrated in Article 36.
[Illustration]
But it must be shown how one can determine the point of contact I. Let
there be drawn parallel to the line KT a line HF which touches the
Ellipse HDE, and let this point of contact be at H. And having drawn a
straight line along CH to meet KT at T, let there be imagined a plane
passing through the same CH and through CM (which I suppose to be the
refraction of the perpendicular ray), which makes in the spheroid the
elliptical section HME. It is certain that the plane which will pass
through the straight line KT, and which will touch the spheroid, will
touch it at a point in the Ellipse HME, according to the Lemma which
will be demonstrated at the end of the Chapter. Now this point is
necessarily the point I which is sought, since the plane drawn through
TK can touch the spheroid at one point only. And this point I is easy
to determine, since it is needful only to draw from the point T, which
is in the plane of this Ellipse, the tangent TI, in the way shown
previously. For the Ellipse HME is given, and its conjugate
semi-diameters are CH and CM; because a straight line drawn through M,
parallel to HE, touches the Ellipse HME, as follows from the fact that
a plane taken through M, and parallel to the plane HDE, touches the
spheroid at that point M, as is seen from Articles 27 and 23. For the
rest, the position of this ellipse, with respect to the plane through
the ray RC and through CK, is also given; from which it will be easy
to find the position of CI, the refraction corresponding to the ray
RC.
Now it must be noted that the same ellipse HME serves to find the
refractions of any other ray which may be in the plane through RC and
CK. Because every plane, parallel to the straight line HF, or TK,
which will touch the spheroid, will touch it in this ellipse,
according to the Lemma quoted a little before.
I have investigated thus, in minute detail, the properties of the
irregular refraction of this Crystal, in order to see whether each
phenomenon that is deduced from our hypothesis accords with that which
is observed in fact. And this being so it affords no slight proof of
the truth of our suppositions and principles. But what I am going to
add here confirms them again marvellously. It is this: that there are
different sections of this Crystal, the surfaces of which, thereby
produced, give rise to refractions precisely such as they ought to be,
and as I had foreseen them, according to the preceding Theory.
In order to explain what these sections are, let ABKF _be_ the
principal section through the axis of the crystal ACK, in which there
will also be the axis SS of a spheroidal wave of light spreading in
the crystal from the centre C; and the straight line which cuts SS
through the middle and at right angles, namely PP, will be one of the
major diameters.
[Illustration: {Section ABKF}]
Now as in the natural section of the crystal, made by a plane parallel
to two opposite faces, which plane is here represented by the line GG,
the refraction of the surfaces which are produced by it will be
governed by the hemi-spheroids GNG, according to what has been
explained in the preceding Theory. Similarly, cutting the Crystal
through NN, by a plane perpendicular to the parallelogram ABKF, the
refraction of the surfaces will be governed by the hemi-spheroids NGN.
And if one cuts it through PP, perpendicularly to the said
parallelogram, the refraction of the surfaces ought to be governed by
the hemi-spheroids PSP, and so for others. But I saw that if the plane
NN was almost perpendicular to the plane GG, making the angle NCG,
which is on the side A, an angle of 90 degrees 40 minutes, the
hemi-spheroids NGN would become similar to the hemi-spheroids GNG,
since the planes NN and GG were equally inclined by an angle of 45
degrees 20 minutes to the axis SS. In consequence it must needs be, if
our theory is true, that the surfaces which the section through NN
produces should effect the same refractions as the surfaces of the
section through GG. And not only the surfaces of the section NN but
all other sections produced by planes which might be inclined to the
axis at an angle equal to 45 degrees 20 minutes. So that there are an
infinitude of planes which ought to produce precisely the same
refractions as the natural surfaces of the crystal, or as the section
parallel to any one of those surfaces which are made by cleavage.
I saw also that by cutting it by a plane taken through PP, and
perpendicular to the axis SS, the refraction of the surfaces ought to
be such that the perpendicular ray should suffer thereby no deviation;
and that for oblique rays there would always be an irregular
refraction, differing from the regular, and by which objects placed
beneath the crystal would be less elevated than by that other
refraction.
That, similarly, by cutting the crystal by any plane through the axis
SS, such as the plane of the figure is, the perpendicular ray ought to
suffer no refraction; and that for oblique rays there were different
measures for the irregular refraction according to the situation of
the plane in which the incident ray was.
Now these things were found in fact so; and, after that, I could not
doubt that a similar success could be met with everywhere. Whence I
concluded that one might form from this crystal solids similar to
those which are its natural forms, which should produce, at all their
surfaces, the same regular and irregular refractions as the natural
surfaces, and which nevertheless would cleave in quite other ways, and
not in directions parallel to any of their faces. That out of it one
would be able to fashion pyramids, having their base square,
pentagonal, hexagonal, or with as many sides as one desired, all the
surfaces of which should have the same refractions as the natural
surfaces of the crystal, except the base, which will not refract the
perpendicular ray. These surfaces will each make an angle of 45
degrees 20 minutes with the axis of the crystal, and the base will be
the section perpendicular to the axis.
That, finally, one could also fashion out of it triangular prisms, or
prisms with as many sides as one would, of which neither the sides nor
the bases would refract the perpendicular ray, although they would yet
all cause double refraction for oblique rays. The cube is included
amongst these prisms, the bases of which are sections perpendicular to
the axis of the crystal, and the sides are sections parallel to the
same axis.
From all this it further appears that it is not at all in the
disposition of the layers of which this crystal seems to be composed,
and according to which it splits in three different senses, that the
cause resides of its irregular refraction; and that it would be in
vain to wish to seek it there.
But in order that any one who has some of this stone may be able to
find, by his own experience, the truth of what I have just advanced, I
will state here the process of which I have made use to cut it, and to
polish it. Cutting is easy by the slicing wheels of lapidaries, or in
the way in which marble is sawn: but polishing is very difficult, and
by employing the ordinary means one more often depolishes the surfaces
than makes them lucent.
After many trials, I have at last found that for this service no plate
of metal must be used, but a piece of mirror glass made matt and
depolished. Upon this, with fine sand and water, one smoothes the
crystal little by little, in the same way as spectacle glasses, and
polishes it simply by continuing the work, but ever reducing the
material. I have not, however, been able to give it perfect clarity
and transparency; but the evenness which the surfaces acquire enables
one to observe in them the effects of refraction better than in those
made by cleaving the stone, which always have some inequality.
Even when the surface is only moderately smoothed, if one rubs it over
with a little oil or white of egg, it becomes quite transparent, so
that the refraction is discerned in it quite distinctly. And this aid
is specially necessary when it is wished to polish the natural
surfaces to remove the inequalities; because one cannot render them
lucent equally with the surfaces of other sections, which take a
polish so much the better the less nearly they approximate to these
natural planes.
Before finishing the treatise on this Crystal, I will add one more
marvellous phenomenon which I discovered after having written all the
foregoing. For though I have not been able till now to find its cause,
I do not for that reason wish to desist from describing it, in order
to give opportunity to others to investigate it. It seems that it will
be necessary to make still further suppositions besides those which I
have made; but these will not for all that cease to keep their
probability after having been confirmed by so many tests.
[Illustration]
The phenomenon is, that by taking two pieces of this crystal and
applying them one over the other, or rather holding them with a space
between the two, if all the sides of one are parallel to those of the
other, then a ray of light, such as AB, is divided into two in the
first piece, namely into BD and BC, following the two refractions,
regular and irregular. On penetrating thence into the other piece
each ray will pass there without further dividing itself in two; but
that one which underwent the regular refraction, as here DG, will
undergo again only a regular refraction at GH; and the other, CE, an
irregular refraction at EF. And the same thing occurs not only in this
disposition, but also in all those cases in which the principal
section of each of the pieces is situated in one and the same plane,
without it being needful for the two neighbouring surfaces to be
parallel. Now it is marvellous why the rays CE and DG, incident from
the air on the lower crystal, do not divide themselves the same as the
first ray AB. One would say that it must be that the ray DG in passing
through the upper piece has lost something which is necessary to move
the matter which serves for the irregular refraction; and that
likewise CE has lost that which was necessary to move the matter
which serves for regular refraction: but there is yet another thing
which upsets this reasoning. It is that when one disposes the two
crystals in such a way that the planes which constitute the principal
sections intersect one another at right angles, whether the
neighbouring surfaces are parallel or not, then the ray which has come
by the regular refraction, as DG, undergoes only an irregular
refraction in the lower piece; and on the contrary the ray which has
come by the irregular refraction, as CE, undergoes only a regular
refraction.
But in all the infinite other positions, besides those which I have
just stated, the rays DG, CE, divide themselves anew each one into
two, by refraction in the lower crystal so that from the single ray AB
there are four, sometimes of equal brightness, sometimes some much
less bright than others, according to the varying agreement in the
positions of the crystals: but they do not appear to have all together
more light than the single ray AB.
When one considers here how, while the rays CE, DG, remain the same,
it depends on the position that one gives to the lower piece, whether
it divides them both in two, or whether it does not divide them, and
yet how the ray AB above is always divided, it seems that one is
obliged to conclude that the waves of light, after having passed
through the first crystal, acquire a certain form or disposition in
virtue of which, when meeting the texture of the second crystal, in
certain positions, they can move the two different kinds of matter
which serve for the two species of refraction; and when meeting the
second crystal in another position are able to move only one of these
kinds of matter. But to tell how this occurs, I have hitherto found
nothing which satisfies me.
Leaving then to others this research, I pass to what I have to say
touching the cause of the extraordinary figure of this crystal, and
why it cleaves easily in three different senses, parallel to any one
of its surfaces.
There are many bodies, vegetable, mineral, and congealed salts, which
are formed with certain regular angles and figures. Thus among flowers
there are many which have their leaves disposed in ordered polygons,
to the number of 3, 4, 5, or 6 sides, but not more. This well deserves
to be investigated, both as to the polygonal figure, and as to why it
does not exceed the number 6.
Rock Crystal grows ordinarily in hexagonal bars, and diamonds are
found which occur with a square point and polished surfaces. There is
a species of small flat stones, piled up directly upon one another,
which are all of pentagonal figure with rounded angles, and the sides
a little folded inwards. The grains of gray salt which are formed from
sea water affect the figure, or at least the angle, of the cube; and
in the congelations of other salts, and in that of sugar, there are
found other solid angles with perfectly flat faces. Small snowflakes
almost always fall in little stars with 6 points, and sometimes in
hexagons with straight sides. And I have often observed, in water
which is beginning to freeze, a kind of flat and thin foliage of ice,
the middle ray of which throws out branches inclined at an angle of 60
degrees. All these things are worthy of being carefully investigated
to ascertain how and by what artifice nature there operates. But it is
not now my intention to treat fully of this matter. It seems that in
general the regularity which occurs in these productions comes from
the arrangement of the small invisible equal particles of which they
are composed. And, coming to our Iceland Crystal, I say that if there
were a pyramid such as ABCD, composed of small rounded corpuscles, not
spherical but flattened spheroids, such as would be made by the
rotation of the ellipse GH around its lesser diameter EF (of which the
ratio to the greater diameter is very nearly that of 1 to the square
root of 8)--I say that then the solid angle of the point D would be
equal to the obtuse and equilateral angle of this Crystal. I say,
further, that if these corpuscles were lightly stuck together, on
breaking this pyramid it would break along faces parallel to those
that make its point: and by this means, as it is easy to see, it would
produce prisms similar to those of the same crystal as this other
figure represents. The reason is that when broken in this fashion a
whole layer separates easily from its neighbouring layer since each
spheroid has to be detached only from the three spheroids of the next
layer; of which three there is but one which touches it on its
flattened surface, and the other two at the edges. And the reason why
the surfaces separate sharp and polished is that if any spheroid of
the neighbouring surface would come out by attaching itself to the
surface which is being separated, it would be needful for it to detach
itself from six other spheroids which hold it locked, and four of
which press it by these flattened surfaces. Since then not only the
angles of our crystal but also the manner in which it splits agree
precisely with what is observed in the assemblage composed of such
spheroids, there is great reason to believe that the particles are
shaped and ranged in the same way.
[Illustration: {Pyramid and section of spheroids}]
There is even probability enough that the prisms of this crystal are
produced by the breaking up of pyramids, since Mr. Bartholinus relates
that he occasionally found some pieces of triangularly pyramidal
figure. But when a mass is composed interiorly only of these little
spheroids thus piled up, whatever form it may have exteriorly, it is
certain, by the same reasoning which I have just explained, that if
broken it would produce similar prisms. It remains to be seen whether
there are other reasons which confirm our conjecture, and whether
there are none which are repugnant to it.
[Illustration: {paralleloid arrangement of spheroids with planes of
potential cleavage}]
It may be objected that this crystal, being so composed, might be
capable of cleavage in yet two more fashions; one of which would be
along planes parallel to the base of the pyramid, that is to say to
the triangle ABC; the other would be parallel to a plane the trace of
which is marked by the lines GH, HK, KL. To which I say that both the
one and the other, though practicable, are more difficult than those
which were parallel to any one of the three planes of the pyramid; and
that therefore, when striking on the crystal in order to break it, it
ought always to split rather along these three planes than along the
two others. When one has a number of spheroids of the form above
described, and ranges them in a pyramid, one sees why the two methods
of division are more difficult. For in the case of that division which
would be parallel to the base, each spheroid would be obliged to
detach itself from three others which it touches upon their flattened
surfaces, which hold more strongly than the contacts at the edges. And
besides that, this division will not occur along entire layers,
because each of the spheroids of a layer is scarcely held at all by
the 6 of the same layer that surround it, since they only touch it at
the edges; so that it adheres readily to the neighbouring layer, and
the others to it, for the same reason; and this causes uneven
surfaces. Also one sees by experiment that when grinding down the
crystal on a rather rough stone, directly on the equilateral solid
angle, one verily finds much facility in reducing it in this
direction, but much difficulty afterwards in polishing the surface
which has been flattened in this manner.
As for the other method of division along the plane GHKL, it will be
seen that each spheroid would have to detach itself from four of the
neighbouring layer, two of which touch it on the flattened surfaces,
and two at the edges. So that this division is likewise more difficult
than that which is made parallel to one of the surfaces of the
crystal; where, as we have said, each spheroid is detached from only
three of the neighbouring layer: of which three there is one only
which touches it on the flattened surface, and the other two at the
edges only.
However, that which has made me know that in the crystal there are
layers in this last fashion, is that in a piece weighing half a pound
which I possess, one sees that it is split along its length, as is the
above-mentioned prism by the plane GHKL; as appears by colours of the
Iris extending throughout this whole plane although the two pieces
still hold together. All this proves then that the composition of the
crystal is such as we have stated. To which I again add this
experiment; that if one passes a knife scraping along any one of the
natural surfaces, and downwards as it were from the equilateral obtuse
angle, that is to say from the apex of the pyramid, one finds it quite
hard; but by scraping in the opposite sense an incision is easily
made. This follows manifestly from the situation of the small
spheroids; over which, in the first manner, the knife glides; but in
the other manner it seizes them from beneath almost as if they were
the scales of a fish.
I will not undertake to say anything touching the way in which so many
corpuscles all equal and similar are generated, nor how they are set
in such beautiful order; whether they are formed first and then
assembled, or whether they arrange themselves thus in coming into
being and as fast as they are produced, which seems to me more
probable. To develop truths so recondite there would be needed a
knowledge of nature much greater than that which we have. I will add
only that these little spheroids could well contribute to form the
spheroids of the waves of light, here above supposed, these as well as
those being similarly situated, and with their axes parallel.
_Calculations which have been supposed in this Chapter_.
Mr. Bartholinus, in his treatise of this Crystal, puts at 101 degrees
the obtuse angles of the faces, which I have stated to be 101 degrees
52 minutes. He states that he measured these angles directly on the
crystal, which is difficult to do with ultimate exactitude, because
the edges such as CA, CB, in this figure, are generally worn, and not
quite straight. For more certainty, therefore, I preferred to measure
actually the obtuse angle by which the faces CBDA, CBVF, are inclined
to one another, namely the angle OCN formed by drawing CN
perpendicular to FV, and CO perpendicular to DA. This angle OCN I
found to be 105 degrees; and its supplement CNP, to be 75 degrees, as
it should be.
[Illustration]
To find from this the obtuse angle BCA, I imagined a sphere having its
centre at C, and on its surface a spherical triangle, formed by the
intersection of three planes which enclose the solid angle C. In this
equilateral triangle, which is ABF in this other figure, I see that
each of the angles should be 105 degrees, namely equal to the angle
OCN; and that each of the sides should be of as many degrees as the
angle ACB, or ACF, or BCF. Having then drawn the arc FQ perpendicular
to the side AB, which it divides equally at Q, the triangle FQA has a
right angle at Q, the angle A 105 degrees, and F half as much, namely
52 degrees 30 minutes; whence the hypotenuse AF is found to be 101
degrees 52 minutes. And this arc AF is the measure of the angle ACF in
the figure of the crystal.
[Illustration]
In the same figure, if the plane CGHF cuts the crystal so that it
divides the obtuse angles ACB, MHV, in the middle, it is stated, in
Article 10, that the angle CFH is 70 degrees 57 minutes. This again is
easily shown in the same spherical triangle ABF, in which it appears
that the arc FQ is as many degrees as the angle GCF in the crystal,
the supplement of which is the angle CFH. Now the arc FQ is found to
be 109 degrees 3 minutes. Then its supplement, 70 degrees 57 minutes,
is the angle CFH.
It was stated, in Article 26, that the straight line CS, which in the
preceding figure is CH, being the axis of the crystal, that is to say
being equally inclined to the three sides CA, CB, CF, the angle GCH is
45 degrees 20 minutes. This is also easily calculated by the same
spherical triangle. For by drawing the other arc AD which cuts BF
equally, and intersects FQ at S, this point will be the centre of the
triangle. And it is easy to see that the arc SQ is the measure of the
angle GCH in the figure which represents the crystal. Now in the
triangle QAS, which is right-angled, one knows also the angle A, which
is 52 degrees 30 minutes, and the side AQ 50 degrees 56 minutes;
whence the side SQ is found to be 45 degrees 20 minutes.
In Article 27 it was required to show that PMS being an ellipse the
centre of which is C, and which touches the straight line MD at M so
that the angle MCL which CM makes with CL, perpendicular on DM, is 6
degrees 40 minutes, and its semi-minor axis CS making with CG (which
is parallel to MD) an angle GCS of 45 degrees 20 minutes, it was
required to show, I say, that, CM being 100,000 parts, PC the
semi-major diameter of this ellipse is 105,032 parts, and CS, the
semi-minor diameter, 93,410.
Let CP and CS be prolonged and meet the tangent DM at D and Z; and
from the point of contact M let MN and MO be drawn as perpendiculars
to CP and CS. Now because the angles SCP, GCL, are right angles, the
angle PCL will be equal to GCS which was 45 degrees 20 minutes. And
deducting the angle LCM, which is 6 degrees 40 minutes, from LCP,
which is 45 degrees 20 minutes, there remains MCP, 38 degrees 40
minutes. Considering then CM as a radius of 100,000 parts, MN, the
sine of 38 degrees 40 minutes, will be 62,479. And in the right-angled
triangle MND, MN will be to ND as the radius of the Tables is to the
tangent of 45 degrees 20 minutes (because the angle NMD is equal to
DCL, or GCS); that is to say as 100,000 to 101,170: whence results ND
63,210. But NC is 78,079 of the same parts, CM being 100,000, because
NC is the sine of the complement of the angle MCP, which was 38
degrees 40 minutes. Then the whole line DC is 141,289; and CP, which
is a mean proportional between DC and CN, since MD touches the
Ellipse, will be 105,032.
[Illustration]
Similarly, because the angle OMZ is equal to CDZ, or LCZ, which is 44
degrees 40 minutes, being the complement of GCS, it follows that, as
the radius of the Tables is to the tangent of 44 degrees 40 minutes,
so will OM 78,079 be to OZ 77,176. But OC is 62,479 of these same
parts of which CM is 100,000, because it is equal to MN, the sine of
the angle MCP, which is 38 degrees 40 minutes. Then the whole line CZ
is 139,655; and CS, which is a mean proportional between CZ and CO
will be 93,410.
At the same place it was stated that GC was found to be 98,779 parts.
To prove this, let PE be drawn in the same figure parallel to DM, and
meeting CM at E. In the right-angled triangle CLD the side CL is
99,324 (CM being 100,000), because CL is the sine of the complement of
the angle LCM, which is 6 degrees 40 minutes. And since the angle LCD
is 45 degrees 20 minutes, being equal to GCS, the side LD is found to
be 100,486: whence deducting ML 11,609 there will remain MD 88,877.
Now as CD (which was 141,289) is to DM 88,877, so will CP 105,032 be
to PE 66,070. But as the rectangle MEH (or rather the difference of
the squares on CM and CE) is to the square on MC, so is the square on
PE to the square on C_g_; then also as the difference of the squares
on DC and CP to the square on CD, so also is the square on PE to the
square on _g_C. But DP, CP, and PE are known; hence also one knows GC,
which is 98,779.
_Lemma which has been supposed_.
If a spheroid is touched by a straight line, and also by two or more
planes which are parallel to this line, though not parallel to one
another, all the points of contact of the line, as well as of the
planes, will be in one and the same ellipse made by a plane which
passes through the centre of the spheroid.
Let LED be the spheroid touched by the line BM at the point B, and
also by the planes parallel to this line at the points O and A. It is
required to demonstrate that the points B, O, and A are in one and the
same Ellipse made in the spheroid by a plane which passes through its
centre.
[Illustration]
Through the line BM, and through the points O and A, let there be
drawn planes parallel to one another, which, in cutting the spheroid
make the ellipses LBD, POP, QAQ; which will all be similar and
similarly disposed, and will have their centres K, N, R, in one and
the same diameter of the spheroid, which will also be the diameter of
the ellipse made by the section of the plane that passes through the
centre of the spheroid, and which cuts the planes of the three said
Ellipses at right angles: for all this is manifest by proposition 15
of the book of Conoids and Spheroids of Archimedes. Further, the two
latter planes, which are drawn through the points O and A, will also,
by cutting the planes which touch the spheroid in these same points,
generate straight lines, as OH and AS, which will, as is easy to see,
be parallel to BM; and all three, BM, OH, AS, will touch the Ellipses
LBD, POP, QAQ in these points, B, O, A; since they are in the planes
of these ellipses, and at the same time in the planes which touch the
spheroid. If now from these points B, O, A, there are drawn the
straight lines BK, ON, AR, through the centres of the same ellipses,
and if through these centres there are drawn also the diameters LD,
PP, QQ, parallel to the tangents BM, OH, AS; these will be conjugate
to the aforesaid BK, ON, AR. And because the three ellipses are
similar and similarly disposed, and have their diameters LD, PP, QQ
parallel, it is certain that their conjugate diameters BK, ON, AR,
will also be parallel. And the centres K, N, R being, as has been
stated, in one and the same diameter of the spheroid, these parallels
BK, ON, AR will necessarily be in one and the same plane, which passes
through this diameter of the spheroid, and, in consequence, the points
R, O, A are in one and the same ellipse made by the intersection of
this plane. Which was to be proved. And it is manifest that the
demonstration would be the same if, besides the points O, A, there had
been others in which the spheroid had been touched by planes parallel
to the straight line BM.
CHAPTER VI
ON THE FIGURES OF THE TRANSPARENT BODIES
Which serve for Refraction and for Reflexion
After having explained how the properties of reflexion and refraction
follow from what we have supposed concerning the nature of light, and
of opaque bodies, and of transparent media, I will here set forth a
very easy and natural way of deducing, from the same principles, the
true figures which serve, either by reflexion or by refraction, to
collect or disperse the rays of light, as may be desired. For though I
do not see yet that there are means of making use of these figures, so
far as relates to Refraction, not only because of the difficulty of
shaping the glasses of Telescopes with the requisite exactitude
according to these figures, but also because there exists in
refraction itself a property which hinders the perfect concurrence of
the rays, as Mr. Newton has very well proved by experiment, I will yet
not desist from relating the invention, since it offers itself, so to
speak, of itself, and because it further confirms our Theory of
refraction, by the agreement which here is found between the refracted
ray and the reflected ray. Besides, it may occur that some one in the
future will discover in it utilities which at present are not seen.
[Illustration]
To proceed then to these figures, let us suppose first that it is
desired to find a surface CDE which shall reassemble at a point B rays
coming from another point A; and that the summit of the surface shall
be the given point D in the straight line AB. I say that, whether by
reflexion or by refraction, it is only necessary to make this surface
such that the path of the light from the point A to all points of the
curved line CDE, and from these to the point of concurrence (as here
the path along the straight lines AC, CB, along AL, LB, and along AD,
DB), shall be everywhere traversed in equal times: by which principle
the finding of these curves becomes very easy.
[Illustration]
So far as relates to the reflecting surface, since the sum of the
lines AC, CB ought to be equal to that of AD, DB, it appears that DCE
ought to be an ellipse; and for refraction, the ratio of the
velocities of waves of light in the media A and B being supposed to be
known, for example that of 3 to 2 (which is the same, as we have
shown, as the ratio of the Sines in the refraction), it is only
necessary to make DH equal to 3/2 of DB; and having after that
described from the centre A some arc FC, cutting DB at F, then
describe another from centre B with its semi-diameter BX equal to 2/3
of FH; and the point of intersection of the two arcs will be one of
the points required, through which the curve should pass. For this
point, having been found in this fashion, it is easy forthwith to
demonstrate that the time along AC, CB, will be equal to the time
along AD, DB.
For assuming that the line AD represents the time which the light
takes to traverse this same distance AD in air, it is evident that DH,
equal to 3/2 of DB, will represent the time of the light along DB in
the medium, because it needs here more time in proportion as its speed
is slower. Therefore the whole line AH will represent the time along
AD, DB. Similarly the line AC or AF will represent the time along AC;
and FH being by construction equal to 3/2 of CB, it will represent the
time along CB in the medium; and in consequence the whole line AH will
represent also the time along AC, CB. Whence it appears that the time
along AC, CB, is equal to the time along AD, DB. And similarly it can
be shown if L and K are other points in the curve CDE, that the times
along AL, LB, and along AK, KB, are always represented by the line AH,
and therefore equal to the said time along AD, DB.
In order to show further that the surfaces, which these curves will
generate by revolution, will direct all the rays which reach them from
the point A in such wise that they tend towards B, let there be
supposed a point K in the curve, farther from D than C is, but such
that the straight line AK falls from outside upon the curve which
serves for the refraction; and from the centre B let the arc KS be
described, cutting BD at S, and the straight line CB at R; and from
the centre A describe the arc DN meeting AK at N.
Since the sums of the times along AK, KB, and along AC, CB are equal,
if from the former sum one deducts the time along KB, and if from the
other one deducts the time along RB, there will remain the time along
AK as equal to the time along the two parts AC, CR. Consequently in
the time that the light has come along AK it will also have come along
AC and will in addition have made, in the medium from the centre C, a
partial spherical wave, having a semi-diameter equal to CR. And this
wave will necessarily touch the circumference KS at R, since CB cuts
this circumference at right angles. Similarly, having taken any other
point L in the curve, one can show that in the same time as the light
passes along AL it will also have come along AL and in addition will
have made a partial wave, from the centre L, which will touch the same
circumference KS. And so with all other points of the curve CDE. Then
at the moment that the light reaches K the arc KRS will be the
termination of the movement, which has spread from A through DCK. And
thus this same arc will constitute in the medium the propagation of
the wave emanating from A; which wave may be represented by the arc
DN, or by any other nearer the centre A. But all the pieces of the arc
KRS are propagated successively along straight lines which are
perpendicular to them, that is to say, which tend to the centre B (for
that can be demonstrated in the same way as we have proved above that
the pieces of spherical waves are propagated along the straight lines
coming from their centre), and these progressions of the pieces of the
waves constitute the rays themselves of light. It appears then that
all these rays tend here towards the point B.
One might also determine the point C, and all the others, in this
curve which serves for the refraction, by dividing DA at G in such a
way that DG is 2/3 of DA, and describing from the centre B any arc CX
which cuts BD at N, and another from the centre A with its
semi-diameter AF equal to 3/2 of GX; or rather, having described, as
before, the arc CX, it is only necessary to make DF equal to 3/2 of
DX, and from-the centre A to strike the arc FC; for these two
constructions, as may be easily known, come back to the first one
which was shown before. And it is manifest by the last method that
this curve is the same that Mr. Des Cartes has given in his Geometry,
and which he calls the first of his Ovals.
It is only a part of this oval which serves for the refraction,
namely, the part DK, ending at K, if AK is the tangent. As to the,
other part, Des Cartes has remarked that it could serve for
reflexions, if there were some material of a mirror of such a nature
that by its means the force of the rays (or, as we should say, the
velocity of the light, which he could not say, since he held that the
movement of light was instantaneous) could be augmented in the
proportion of 3 to 2. But we have shown that in our way of explaining
reflexion, such a thing could not arise from the matter of the mirror,
and it is entirely impossible.
[Illustration]
[Illustration]
From what has been demonstrated about this oval, it will be easy to
find the figure which serves to collect to a point incident parallel
rays. For by supposing just the same construction, but the point A
infinitely distant, giving parallel rays, our oval becomes a true
Ellipse, the construction of which differs in no way from that of the
oval, except that FC, which previously was an arc of a circle, is here
a straight line, perpendicular to DB. For the wave of light DN, being
likewise represented by a straight line, it will be seen that all the
points of this wave, travelling as far as the surface KD along lines
parallel to DB, will advance subsequently towards the point B, and
will arrive there at the same time. As for the Ellipse which served
for reflexion, it is evident that it will here become a parabola,
since its focus A may be regarded as infinitely distant from the
other, B, which is here the focus of the parabola, towards which all
the reflexions of rays parallel to AB tend. And the demonstration of
these effects is just the same as the preceding.
But that this curved line CDE which serves for refraction is an
Ellipse, and is such that its major diameter is to the distance
between its foci as 3 to 2, which is the proportion of the refraction,
can be easily found by the calculus of Algebra. For DB, which is
given, being called _a_; its undetermined perpendicular DT being
called _x_; and TC _y_; FB will be _a - y_; CB will be sqrt(_xx + aa
-2ay + yy_). But the nature of the curve is such that 2/3 of TC
together with CB is equal to DB, as was stated in the last
construction: then the equation will be between _(2/3)y + sqrt(xx + aa
- 2ay + yy)_ and _a_; which being reduced, gives _(6/5)ay - yy_ equal
to _(9/5)xx_; that is to say that having made DO equal to 6/5 of DB,
the rectangle DFO is equal to 9/5 of the square on FC. Whence it is
seen that DC is an ellipse, of which the axis DO is to the parameter
as 9 to 5; and therefore the square on DO is to the square of the
distance between the foci as 9 to 9 - 5, that is to say 4; and finally
the line DO will be to this distance as 3 to 2.
[Illustration]
Again, if one supposes the point B to be infinitely distant, in lieu
of our first oval we shall find that CDE is a true Hyperbola; which
will make those rays become parallel which come from the point A. And
in consequence also those which are parallel within the transparent
body will be collected outside at the point A. Now it must be remarked
that CX and KS become straight lines perpendicular to BA, because they
represent arcs of circles the centre of which is infinitely distant.
And the intersection of the perpendicular CX with the arc FC will give
the point C, one of those through which the curve ought to pass. And
this operates so that all the parts of the wave of light DN, coming to
meet the surface KDE, will advance thence along parallels to KS and
will arrive at this straight line at the same time; of which the proof
is again the same as that which served for the first oval. Besides one
finds by a calculation as easy as the preceding one, that CDE is here
a hyperbola of which the axis DO is 4/5 of AD, and the parameter
equal to AD. Whence it is easily proved that DO is to the distance
between the foci as 3 to 2.
[Illustration]
These are the two cases in which Conic sections serve for refraction,
and are the same which are explained, in his _Dioptrique_, by Des
Cartes, who first found out the use of these lines in relation to
refraction, as also that of the Ovals the first of which we have
already set forth. The second oval is that which serves for rays that
tend to a given point; in which oval, if the apex of the surface which
receives the rays is D, it will happen that the other apex will be
situated between B and A, or beyond A, according as the ratio of AD to
DB is given of greater or lesser value. And in this latter case it is
the same as that which Des Cartes calls his 3rd oval.
Now the finding and construction of this second oval is the same as
that of the first, and the demonstration of its effect likewise. But
it is worthy of remark that in one case this oval becomes a perfect
circle, namely when the ratio of AD to DB is the same as the ratio of
the refractions, here as 3 to 2, as I observed a long time ago. The
4th oval, serving only for impossible reflexions, there is no need to
set it forth.
[Illustration]
As for the manner in which Mr. Des Cartes discovered these lines,
since he has given no explanation of it, nor any one else since that I
know of, I will say here, in passing, what it seems to me it must have
been. Let it be proposed to find the surface generated by the
revolution of the curve KDE, which, receiving the incident rays coming
to it from the point A, shall deviate them toward the point B. Then
considering this other curve as already known, and that its apex D is
in the straight line AB, let us divide it up into an infinitude of
small pieces by the points G, C, F; and having drawn from each of
these points, straight lines towards A to represent the incident rays,
and other straight lines towards B, let there also be described with
centre A the arcs GL, CM, FN, DO, cutting the rays that come from A at
L, M, N, O; and from the points K, G, C, F, let there be described
the arcs KQ, GR, CS, FT cutting the rays towards B at Q, R, S, T; and
let us suppose that the straight line HKZ cuts the curve at K at
right-angles.
[Illustration]
Then AK being an incident ray, and KB its refraction within the
medium, it needs must be, according to the law of refraction which was
known to Mr. Des Cartes, that the sine of the angle ZKA should be to
the sine of the angle HKB as 3 to 2, supposing that this is the
proportion of the refraction of glass; or rather, that the sine of the
angle KGL should have this same ratio to the sine of the angle GKQ,
considering KG, GL, KQ as straight lines because of their smallness.
But these sines are the lines KL and GQ, if GK is taken as the radius
of the circle. Then LK ought to be to GQ as 3 to 2; and in the same
ratio MG to CR, NC to FS, OF to DT. Then also the sum of all the
antecedents to all the consequents would be as 3 to 2. Now by
prolonging the arc DO until it meets AK at X, KX is the sum of the
antecedents. And by prolonging the arc KQ till it meets AD at Y, the
sum of the consequents is DY. Then KX ought to be to DY as 3 to 2.
Whence it would appear that the curve KDE was of such a nature that
having drawn from some point which had been assumed, such as K, the
straight lines KA, KB, the excess by which AK surpasses AD should be
to the excess of DB over KB, as 3 to 2. For it can similarly be
demonstrated, by taking any other point in the curve, such as G, that
the excess of AG over AD, namely VG, is to the excess of BD over DG,
namely DP, in this same ratio of 3 to 2. And following this principle
Mr. Des Cartes constructed these curves in his _Geometric_; and he
easily recognized that in the case of parallel rays, these curves
became Hyperbolas and Ellipses.
Let us now return to our method and let us see how it leads without
difficulty to the finding of the curves which one side of the glass
requires when the other side is of a given figure; a figure not only
plane or spherical, or made by one of the conic sections (which is the
restriction with which Des Cartes proposed this problem, leaving the
solution to those who should come after him) but generally any figure
whatever: that is to say, one made by the revolution of any given
curved line to which one must merely know how to draw straight lines
as tangents.
Let the given figure be that made by the revolution of some curve such
as AK about the axis AV, and that this side of the glass receives rays
coming from the point L. Furthermore, let the thickness AB of the
middle of the glass be given, and the point F at which one desires the
rays to be all perfectly reunited, whatever be the first refraction
occurring at the surface AK.
I say that for this the sole requirement is that the outline BDK which
constitutes the other surface shall be such that the path of the
light from the point L to the surface AK, and from thence to the
surface BDK, and from thence to the point F, shall be traversed
everywhere in equal times, and in each case in a time equal to that
which the light employs, to pass along the straight line LF of which
the part AB is within the glass.
[Illustration]
Let LG be a ray falling on the arc AK. Its refraction GV will be given
by means of the tangent which will be drawn at the point G. Now in GV
the point D must be found such that FD together with 3/2 of DG and the
straight line GL, may be equal to FB together with 3/2 of BA and the
straight line AL; which, as is clear, make up a given length. Or
rather, by deducting from each the length of LG, which is also given,
it will merely be needful to adjust FD up to the straight line VG in
such a way that FD together with 3/2 of DG is equal to a given
straight line, which is a quite easy plane problem: and the point D
will be one of those through which the curve BDK ought to pass. And
similarly, having drawn another ray LM, and found its refraction MO,
the point N will be found in this line, and so on as many times as one
desires.
To demonstrate the effect of the curve, let there be described about
the centre L the circular arc AH, cutting LG at H; and about the
centre F the arc BP; and in AB let AS be taken equal to 2/3 of HG; and
SE equal to GD. Then considering AH as a wave of light emanating from
the point L, it is certain that during the time in which its piece H
arrives at G the piece A will have advanced within the transparent
body only along AS; for I suppose, as above, the proportion of the
refraction to be as 3 to 2. Now we know that the piece of wave which
is incident on G, advances thence along the line GD, since GV is the
refraction of the ray LG. Then during the time that this piece of wave
has taken from G to D, the other piece which was at S has reached E,
since GD, SE are equal. But while the latter will advance from E to B,
the piece of wave which was at D will have spread into the air its
partial wave, the semi-diameter of which, DC (supposing this wave to
cut the line DF at C), will be 3/2 of EB, since the velocity of light
outside the medium is to that inside as 3 to 2. Now it is easy to show
that this wave will touch the arc BP at this point C. For since, by
construction, FD + 3/2 DG + GL are equal to FB + 3/2 BA + AL; on
deducting the equals LH, LA, there will remain FD + 3/2 DG + GH equal
to FB + 3/2 BA. And, again, deducting from one side GH, and from the
other side 3/2 of AS, which are equal, there will remain FD with 3/2
DG equal to FB with 3/2 of BS. But 3/2 of DG are equal to 3/2 of ES;
then FD is equal to FB with 3/2 of BE. But DC was equal to 3/2 of EB;
then deducting these equal lengths from one side and from the other,
there will remain CF equal to FB. And thus it appears that the wave,
the semi-diameter of which is DC, touches the arc BP at the moment
when the light coming from the point L has arrived at B along the line
LB. It can be demonstrated similarly that at this same moment the
light that has come along any other ray, such as LM, MN, will have
propagated the movement which is terminated at the arc BP. Whence it
follows, as has been often said, that the propagation of the wave AH,
after it has passed through the thickness of the glass, will be the
spherical wave BP, all the pieces of which ought to advance along
straight lines, which are the rays of light, to the centre F. Which
was to be proved. Similarly these curved lines can be found in all the
cases which can be proposed, as will be sufficiently shown by one or
two examples which I will add.
Let there be given the surface of the glass AK, made by the revolution
about the axis BA of the line AK, which may be straight or curved. Let
there be also given in the axis the point L and the thickness BA of
the glass; and let it be required to find the other surface KDB, which
receiving rays that are parallel to AB will direct them in such wise
that after being again refracted at the given surface AK they will all
be reassembled at the point L.
[Illustration]
From the point L let there be drawn to some point of the given line
AK the straight line LG, which, being considered as a ray of light,
its refraction GD will then be found. And this line being then
prolonged at one side or the other will meet the straight line BL, as
here at V. Let there then be erected on AB the perpendicular BC, which
will represent a wave of light coming from the infinitely distant
point F, since we have supposed the rays to be parallel. Then all the
parts of this wave BC must arrive at the same time at the point L; or
rather all the parts of a wave emanating from the point L must arrive
at the same time at the straight line BC. And for that, it is
necessary to find in the line VGD the point D such that having drawn
DC parallel to AB, the sum of CD, plus 3/2 of DG, plus GL may be equal
to 3/2 of AB, plus AL: or rather, on deducting from both sides GL,
which is given, CD plus 3/2 of DG must be equal to a given length;
which is a still easier problem than the preceding construction. The
point D thus found will be one of those through which the curve ought
to pass; and the proof will be the same as before. And by this it will
be proved that the waves which come from the point L, after having
passed through the glass KAKB, will take the form of straight lines,
as BC; which is the same thing as saying that the rays will become
parallel. Whence it follows reciprocally that parallel rays falling on
the surface KDB will be reassembled at the point L.
[Illustration]
Again, let there be given the surface AK, of any desired form,
generated by revolution about the axis AB, and let the thickness of
the glass at the middle be AB. Also let the point L be given in the
axis behind the glass; and let it be supposed that the rays which fall
on the surface AK tend to this point, and that it is required to find
the surface BD, which on their emergence from the glass turns them as
if they came from the point F in front of the glass.
Having taken any point G in the line AK, and drawing the straight line
IGL, its part GI will represent one of the incident rays, the
refraction of which, GV, will then be found: and it is in this line
that we must find the point D, one of those through which the curve DG
ought to pass. Let us suppose that it has been found: and about L as
centre let there be described GT, the arc of a circle cutting the
straight line AB at T, in case the distance LG is greater than LA; for
otherwise the arc AH must be described about the same centre, cutting
the straight line LG at H. This arc GT (or AH, in the other case) will
represent an incident wave of light, the rays of which tend towards
L. Similarly, about the centre F let there be described the circular
arc DQ, which will represent a wave emanating from the point F.
Then the wave TG, after having passed through the glass, must form the
wave QD; and for this I observe that the time taken by the light along
GD in the glass must be equal to that taken along the three, TA, AB,
and BQ, of which AB alone is within the glass. Or rather, having taken
AS equal to 2/3 of AT, I observe that 3/2 of GD ought to be equal to
3/2 of SB, plus BQ; and, deducting both of them from FD or FQ, that FD
less 3/2 of GD ought to be equal to FB less 3/2 of SB. And this last
difference is a given length: and all that is required is to draw the
straight line FD from the given point F to meet VG so that it may be
thus. Which is a problem quite similar to that which served for the
first of these constructions, where FD plus 3/2 of GD had to be equal
to a given length.
In the demonstration it is to be observed that, since the arc BC falls
within the glass, there must be conceived an arc RX, concentric with
it and on the other side of QD. Then after it shall have been shown
that the piece G of the wave GT arrives at D at the same time that the
piece T arrives at Q, which is easily deduced from the construction,
it will be evident as a consequence that the partial wave generated at
the point D will touch the arc RX at the moment when the piece Q shall
have come to R, and that thus this arc will at the same moment be the
termination of the movement that comes from the wave TG; whence all
the rest may be concluded.
Having shown the method of finding these curved lines which serve for
the perfect concurrence of the rays, there remains to be explained a
notable thing touching the uncoordinated refraction of spherical,
plane, and other surfaces: an effect which if ignored might cause some
doubt concerning what we have several times said, that rays of light
are straight lines which intersect at right angles the waves which
travel along them.
[Illustration]
For in the case of rays which, for example, fall parallel upon a
spherical surface AFE, intersecting one another, after refraction, at
different points, as this figure represents; what can the waves of
light be, in this transparent body, which are cut at right angles by
the converging rays? For they can not be spherical. And what will
these waves become after the said rays begin to intersect one another?
It will be seen in the solution of this difficulty that something very
remarkable comes to pass herein, and that the waves do not cease to
persist though they do not continue entire, as when they cross the
glasses designed according to the construction we have seen.
According to what has been shown above, the straight line AD, which
has been drawn at the summit of the sphere, at right angles to the
axis parallel to which the rays come, represents the wave of light;
and in the time taken by its piece D to reach the spherical surface
AGE at E, its other parts will have met the same surface at F, G, H,
etc., and will have also formed spherical partial waves of which these
points are the centres. And the surface EK which all those waves will
touch, will be the continuation of the wave AD in the sphere at the
moment when the piece D has reached E. Now the line EK is not an arc
of a circle, but is a curved line formed as the evolute of another
curve ENC, which touches all the rays HL, GM, FO, etc., that are the
refractions of the parallel rays, if we imagine laid over the
convexity ENC a thread which in unwinding describes at its end E the
said curve EK. For, supposing that this curve has been thus described,
we will show that the said waves formed from the centres F, G, H,
etc., will all touch it.
It is certain that the curve EK and all the others described by the
evolution of the curve ENC, with different lengths of thread, will cut
all the rays HL, GM, FO, etc., at right angles, and in such wise that
the parts of them intercepted between two such curves will all be
equal; for this follows from what has been demonstrated in our
treatise _de Motu Pendulorum_. Now imagining the incident rays as
being infinitely near to one another, if we consider two of them, as
RG, TF, and draw GQ perpendicular to RG, and if we suppose the curve
FS which intersects GM at P to have been described by evolution from
the curve NC, beginning at F, as far as which the thread is supposed
to extend, we may assume the small piece FP as a straight line
perpendicular to the ray GM, and similarly the arc GF as a straight
line. But GM being the refraction of the ray RG, and FP being
perpendicular to it, QF must be to GP as 3 to 2, that is to say in the
proportion of the refraction; as was shown above in explaining the
discovery of Des Cartes. And the same thing occurs in all the small
arcs GH, HA, etc., namely that in the quadrilaterals which enclose
them the side parallel to the axis is to the opposite side as 3 to 2.
Then also as 3 to 2 will the sum of the one set be to the sum of the
other; that is to say, TF to AS, and DE to AK, and BE to SK or DV,
supposing V to be the intersection of the curve EK and the ray FO.
But, making FB perpendicular to DE, the ratio of 3 to 2 is also that
of BE to the semi-diameter of the spherical wave which emanated from
the point F while the light outside the transparent body traversed the
space BE. Then it appears that this wave will intersect the ray FM at
the same point V where it is intersected at right angles by the curve
EK, and consequently that the wave will touch this curve. In the same
way it can be proved that the same will apply to all the other waves
above mentioned, originating at the points G, H, etc.; to wit, that
they will touch the curve EK at the moment when the piece D of the
wave ED shall have reached E.
Now to say what these waves become after the rays have begun to cross
one another: it is that from thence they fold back and are composed of
two contiguous parts, one being a curve formed as evolute of the curve
ENC in one sense, and the other as evolute of the same curve in the
opposite sense. Thus the wave KE, while advancing toward the meeting
place becomes _abc_, whereof the part _ab_ is made by the evolute
_b_C, a portion of the curve ENC, while the end C remains attached;
and the part _bc_ by the evolute of the portion _b_E while the end E
remains attached. Consequently the same wave becomes _def_, then
_ghk_, and finally CY, from whence it subsequently spreads without any
fold, but always along curved lines which are evolutes of the curve
ENC, increased by some straight line at the end C.
There is even, in this curve, a part EN which is straight, N being the
point where the perpendicular from the centre X of the sphere falls
upon the refraction of the ray DE, which I now suppose to touch the
sphere. The folding of the waves of light begins from the point N up
to the end of the curve C, which point is formed by taking AC to CX in
the proportion of the refraction, as here 3 to 2.
As many other points as may be desired in the curve NC are found by a
Theorem which Mr. Barrow has demonstrated in section 12 of his
_Lectiones Opticae_, though for another purpose. And it is to be noted
that a straight line equal in length to this curve can be given. For
since it together with the line NE is equal to the line CK, which is
known, since DE is to AK in the proportion of the refraction, it
appears that by deducting EN from CK the remainder will be equal to
the curve NC.
Similarly the waves that are folded back in reflexion by a concave
spherical mirror can be found. Let ABC be the section, through the
axis, of a hollow hemisphere, the centre of which is D, its axis being
DB, parallel to which I suppose the rays of light to come. All the
reflexions of those rays which fall upon the quarter-circle AB will
touch a curved line AFE, of which line the end E is at the focus of
the hemisphere, that is to say, at the point which divides the
semi-diameter BD into two equal parts. The points through which this
curve ought to pass are found by taking, beyond A, some arc AO, and
making the arc OP double the length of it; then dividing the chord OP
at F in such wise that the part FP is three times the part FO; for
then F is one of the required points.
[Illustration]
And as the parallel rays are merely perpendiculars to the waves which
fall on the concave surface, which waves are parallel to AD, it will
be found that as they come successively to encounter the surface AB,
they form on reflexion folded waves composed of two curves which
originate from two opposite evolutions of the parts of the curve AFE.
So, taking AD as an incident wave, when the part AG shall have met the
surface AI, that is to say when the piece G shall have reached I, it
will be the curves HF, FI, generated as evolutes of the curves FA, FE,
both beginning at F, which together constitute the propagation of the
part AG. And a little afterwards, when the part AK has met the surface
AM, the piece K having come to M, then the curves LN, NM, will
together constitute the propagation of that part. And thus this folded
wave will continue to advance until the point N has reached the focus
E. The curve AFE can be seen in smoke, or in flying dust, when a
concave mirror is held opposite the sun. And it should be known that
it is none other than that curve which is described by the point E on
the circumference of the circle EB, when that circle is made to roll
within another whose semi-diameter is ED and whose centre is D. So
that it is a kind of Cycloid, of which, however, the points can be
found geometrically.
Its length is exactly equal to 3/4 of the diameter of the sphere, as
can be found and demonstrated by means of these waves, nearly in the
same way as the mensuration of the preceding curve; though it may also
be demonstrated in other ways, which I omit as outside the subject.
The area AOBEFA, comprised between the arc of the quarter-circle, the
straight line BE, and the curve EFA, is equal to the fourth part of
the quadrant DAB.
INDEX
Archimedes, 104.
Atmospheric refraction, 45.
Barrow, Isaac, 126.
Bartholinus, Erasmus, 53, 54, 57, 60, 97, 99.
Boyle, Hon. Robert, 11.
Cassini, Jacques, iii.
Caustic Curves, 123.
Crystals, see Iceland Crystal, Rock Crystal.
Crystals, configuration of, 95.
Descartes, Rene, 3, 5, 7, 14, 22, 42, 43, 109, 113.
Double Refraction, discovery of, 54, 81, 93.
Elasticity, 12, 14.
Ether, the, or Ethereal matter, 11, 14, 16, 28.
Extraordinary refraction, 55, 56.
Fermat, principle of, 42.
Figures of transparent bodies, 105.
Hooke, Robert, 20.
Iceland Crystal, 2, 52 sqq.
Iceland Crystal, Cutting and Polishing of, 91, 92, 98.
Leibnitz, G.W., vi.
Light, nature of, 3.
Light, velocity of, 4, 15.
Molecular texture of bodies, 27, 95.
Newton, Sir Isaac, vi, 106.
Opacity, 34.
Ovals, Cartesian, 107, 113.
Pardies, Rev. Father, 20.
Rays, definition of, 38, 49.
Reflexion, 22.
Refraction, 28, 34.
Rock Crystal, 54, 57, 62, 95.
Roemer, Olaf, v, 7.
Roughness of surfaces, 27.
Sines, law of, 1, 35, 38, 43.
Spheres, elasticity of, 15.
Spheroidal waves in crystals, 63.
Spheroids, lemma about, 103.
Sound, speed of, 7, 10, 12.
Telescopes, lenses for, 62, 105.
Torricelli's experiment, 12, 30.
Transparency, explanation of, 28, 31, 32.
Waves, no regular succession of, 17.
Waves, principle of wave envelopes, 19, 24.
Waves, principle of elementary wave fronts, 19.
Waves, propagation of light as, 16, 63.
[Transcriber's Notes
All apparent printer's errors retained. Variation in punctuation are
as in the original, but missing full stops at end of paragraphs have
been supplied. There are inconsistencies in the use of italics,
spacing of words and use of full stop after 'AXIOME', abbreviations
etc. All are retained to match text. There is a great variation in
spelling including multiple spellings of the same word, all spelling
has been retained to match text. There are several instances of
obviously missing letters or inverted n & u. These have been changed
or obvious letters replaced, with the changes surrounded by {}.
All instances are detailed at the end of the text. It should also be
noted that in the original text there is a missing line at the end of
page 24 in original text.
There are a number of instances in the original text where 'that' is
immediately followed by a second 'that' in the sentence. These could
be potential printer's errors or, since several of them make sense,
part of the author's style. They have been left in the text as they
appear in the original text.
The original text has many sidenotes, some are true sidenotes,
introductions to paragraphs etc, some acting as footnotes with some
marked in original text with *. These have been dealt with in three
ways with the footnotes placed after their relevant paragraph and
sidenotes place before their relevant paragraph.
1) Footnotes marked with capital letter. These were sidenotes in
original text marked with * in the original text and thus acting
like footnotes.
2) Footnotes marked with number. These were sidenotes in original
text that were unmarked but acting like normal footnotes. The anchor
in the text was placed at the most suitable relevant place in
comparison with the placement of the sidenote text in the margin,
but still should be considered only an approximate placement.
3) Sidenotes placed at start of the relevant paragraph. Some
sidenotes were considered not to be relevant as footnotes,
introductions to paragraphs etc, and were left as sidenotes before
their relevant paragraph.]
*       *       *       *       *
A
DISCOURSE
_PRESENTED_
TO THE MOST SERENE
Don Cosimo II.
GREAT DUKE
_OF_
TUSCANY,
CONCERNING
The _NATATION_ of BODIES Vpon,
And _SUBMERSION_ In,
THE
WATER.
By GALILEUS GALILEI: Philosopher and
Mathematician unto His most Serene Highnesse.
Englished from the Second Edition of the ITALIAN,
compared with the Manuscript Copies, and reduced
into PROPOSITIONS:
By _THOMAS SALUSBURY_, Esq;
_LONDON_:
Printed by WILLIAM LEYBOURN:
_M DC LXIII._
*       *       *       *       *
[Decoration]
A DISCOVRSE
Presented to the Most Serene DON COSIMO II.
GREAT DUKE of _TUSCANY_:
CONCERNING
_The Natation of BODIES Upon, or Submersion_
_In, the WATER._
Considering (Most Serene Prince) that the publishing this present
Treatise, of so different an Argument from that which many expect, and
which according to the intentions I proposed in my [A] Astronomicall
_Adviso_, I should before this time have put forth, might peradventure
make some thinke, either that I had wholly relinquished my farther
imployment about the new Celestiall Observations, or that, at least, I
handled them very remissely; I have judged fit to render an account,
aswell of my deferring that, as of my writing, and publishing this
treatise.
[A] His Nuncio Siderio.
As to the first, the last discoveries of _Saturn_ to be
tricorporeall, and of the mutations of Figure in _Venus_, like to
those that are seen in the Moon, together with the Consequents
depending thereupon, have not so much occasioned the demur, as the
investigation of the times of the Conversions of each of the Four
Medicean Planets about _Jupiter_, which I lighted upon in _April_ the
year past, 1611, at my being in _Rome_; where, in the end, I
assertained my selfe, that the first and neerest to _Jupiter_, moved
about 8 _gr._ & 29 _m._ of its Sphere in an houre, makeing its whole
revolution in one naturall day, and 18 hours, and almost an halfe. The
second moves in its Orbe 14 _gr._ 13 _min._ or very neer, in an hour,
and its compleat conversion is consummate in 3 dayes, 13 hours, and
one third, or thereabouts. The third passeth in an hour, 2 _gr._ 6
_min._ little more or less of its Circle, and measures it all in 7
dayes, 4 hours, or very neer. The fourth, and more remote than the
rest, goes in one houre, 0 _gr_ 54 _min._ and almost an halfe of its
Sphere, and finisheth it all in 16 dayes, and very neer 18 hours. But
because the excessive velocity of their returns or restitutions,
requires a most scrupulous precisenesse to calculate their places, in
times past and future, especially if the time be for many Moneths or
Years; I am therefore forced, with other Observations, and more exact
than the former, and in times more remote from one another, to correct
the Tables of such Motions, and limit them even to the shortest
moment: for such exactnesse my first Observations suffice not; not
only in regard of the short intervals of Time, but because I had not
as then found out a way to measure the distances between the said
Planets by any Instrument: I Observed such Intervals with simple
relation to the Diameter of the Body of _Jupiter_; taken, as we have
said, by the eye, the which, though they admit not errors of above a
Minute, yet they suffice not for the determination of the exact
greatness of the Spheres of those Stars. But now that I have hit upon
a way of taking such measures without failing, scarce in a very few
Seconds, I will continue the observation to the very occultation of
_JUPITER_, which shall serve to bring us to the perfect knowledge of
the Motions, and Magnitudes of the Orbes of the said Planets, together
also with some other consequences thence arising. I adde to these
things the observation of some obscure Spots[1], which are discovered in
the Solar Body, which changing, position in that, propounds to our
consideration a great argument either that the Sun revolves in it
selfe, or that perhaps other Starrs, in like manner as _Venus_ and
_Mercury_, revolve about it, invisible in other times, by reason of
their small digressions, lesse than that of _Mercury_, and only
visible when they interpose between the Sun and our eye, or else hint
the truth of both this and that; the certainty of which things ought
not to be contemned, nor omitted.
[1] The Authors Observations of the Solar Spots
_Continuall observation hath at last assured me that these Spots
are matters contiguous to the Body of the Sun, there continually
produced in great number, and afterwards dissolved, some in a
shorter, some in a longer time, and to be by the Conversion or
Revolution of the Sun in it selfe, which in a Lunar Moneth, or
thereabouts, finisheth its Period, caried about in a Circle, an
accident great of it selfe, and greater for its Consequences._
As to the other particular in the next place [B] Many causes have
moved me to write the present Tract, the subject whereof, is the
Dispute which I held some dayes since, with some learned men of this
City, about which, as your Highnesse knows, have followed many
Discourses: The principall of which Causes hath been the Intimation of
your Highnesse, having commended to me Writing, as a singular means to
make true known from false, reall from apparent Reasons, farr better
than by Disputing vocally, where the one or the other, or very often
both the Disputants, through too greate heate, or exalting of the
voyce, either are not understood, or else being transported by
ostentation of not yeilding to one another, farr from the first
Proposition, with the novelty, of the various Proposals, confound both
themselves and their Auditors.
[B] The occasion inducing the Author to write this Treatise.
Moreover, it seemed to me convenient to informe your Highnesse of all
the sequell, concerning the Controversie of which I treat, as it hath
been advertised often already by others: and because the Doctrine
which I follow, in the discussion of the point in hand, is different
from that of _Aristotle_; and interferes with his Principles, I have
considered that against the Authority of that most famous Man, which
amongst many makes all suspected that comes not from the Schooles of
the Peripateticks, its farr better to give ones Reasons by the Pen
than by word of mouth, and therfore I resolved to write the present
discourse: in which yet I hope to demonstrate that it was not out of
capritiousnesse, or for that I had not read or understood _Aristotle_,
that I sometimes swerve from his opinion, but because severall Reasons
perswade me to it, and the same _Aristotle_ hath tought me to fix my
judgment on that which is grounded upon Reason, and not on the bare
Authority of the Master[2]; and it is most certaine according to the
sentence of _Alcinoos_, that philosophating should be free. Nor is the
resolution of our Question in my judgment without some benefit to the
Universall[3], forasmuch as treating whether the figure of Solids
operates, or not, in their going, or not going to the bottome in
Water, in occurrences of building Bridges or other Fabricks on the
Water, which happen commonly in affairs of grand import, it may be of
great availe to know the truth.
[2] _Aristotle_ prefers Reason to the Authority ofan Author.
[3] The benefit of this Argument.
I say therfore, that being the last Summer in company with certain
Learned men, it was said in the argumentation; That Condensation was
the propriety of Cold[4], and there was alledged for instance, the
example of Ice: now I at that time said, that, in my judgment, the Ice
should be rather Water rarified than condensed[5], and my reason was,
because Condensation begets diminution of Mass, and augmentation of
gravity, and Rarifaction causeth greater Lightness, and augmentarion
of Masse: and Water in freezing, encreaseth in Masse, and the Ice made
thereby is lighter than the Water on which it swimmeth.
[4] Condensation the Propriety of Cold, according to the
Peripateticks.
[5] Ice rather water rarified, than condensed, and why:
_What I say, is manifest, because, the medium subtracting from the
whole Gravity of Sollids the weight of such another Masse of the
said Medium; as_ Archimedes _proves in his_ [C] First Booke De
Insidentibus Humido; _when ever the Masse of the said Solid
encreaseth by Distraction, the more shall the_ Medium _detract from
its entire Gravity; and lesse, when by Compression it shall be
condensed and reduced to a lesse Masse._
[C] In lib: 1. of Natation of Bodies Prop. 7.
[Sidenote: Figure operates not in the Natation of Sollids.]
It was answered me, tha{t} that proceeded not from the greater Levity,
but from the Figure, large and flat, which not being able to penetrate
the Resistance of the Water, is the cause that it submergeth not. I
replied, that any piece of Ice, of whatsoever Figure, swims upon the
Water, a manifest signe, that its being never so flat and broad, hath
not any part in its floating: and added, that it was a manifest proofe
hereof to see a piece of Ice of very broad Figure being thrust to the
botome of the Water, suddenly return to flote atoppe, which had it
been more grave, and had its swimming proceeded from its Forme, unable
to penetrate the Resistance of the _Medium_, that would be altogether
impossible; I concluded therefore, that the Figure was in sort a Cause
of the Natation or Submersion of Bodies, but the greater or lesse
Gravity in respect of the Water: and therefore all Bodyes heavier than
it of what Figure soever they be, indifferently go to the bottome, and
the lighter, though of any figure, float indifferently on the top: and
I suppose that those which hold otherwise, were induced to that
beliefe, by seeing how that diversity of Formes or Figures, greatly
altereth the Velosity, and Tardity of Motion; so that Bodies of Figure
broad and thin, descend far more leasurely into the Water, than those
of a more compacted Figure, though both made of the same Matter: by
which some might be induced to believe that the Dilatation of the
Figure might reduce it to such amplenesse that it should not only
retard but wholly impede and take away the Motion, which I hold to be
false. Upon this Conclusion, in many dayes discourse, was spoken much,
and many things, and divers Experiments produced, of which your
Highnesse heard, and saw some, and in this discourse shall have all
that which hath been produced against my Assertion, and what hath been
suggested to my thoughts on this matter, and for confirmation of my
Conclusion: which if it shall suffice to remove that (as I esteem
hitherto false) Opinion, I shall thinke I have not unprofitably spent
my paynes and time. and although that come not to passe, yet ought I
to promise another benefit to my selfe, namely, of attaining the
knowledge of the truth, by hearing my Fallacyes confuted, and true
demonstrations produced by those of the contrary opinion.
And to proceed with the greatest plainness and perspicuity that I can
possible, it is, I conceive, necessary, first of all to declare what
is the true, intrinsecall, and totall Cause, of the ascending of some
Sollid Bodyes in the Water, and therein floating; or on the contrary,
of their sinking and so much the rather in asmuch as I cannot satisfie
myselfe in that which _Aristotle_ hath left written on this Subject.
[Sidenote: The cause of the Natation & submersion of Solids in
the Water.]
I say then the Cause why some Sollid Bodyes descend to the Bottom of
Water, is the excesse of their Gravity, above the Gravity of the
Water; and on the contrary, the excess of the Waters Gravity above the
Gravity of those, is the Cause that others do not descend, rather that
they rise from the Bottom, and ascend to the Surface. This was
subtilly demonstrated by _Archimedes_ in his Book Of the NATATION of
BODIES: Conferred afterwards by a very grave Author, but, if I erre
not invisibly, as below for defence of him, I shall endeavour to
prove.
I, with a different Method, and by other meanes, will endeavour to
demonstrate the same, reducing the Causes of such Effects to more
intrinsecall and immediate Principles, in which also are discovered
the Causes of some admirable and almost incredible Accidents, as that
would be, that a very little quantity of Water, should be able, with
its small weight, to raise and sustain a Solid Body, an hundred or a
thousand times heavier than it.
And because demonstrative Order so requires, I shall define certain
Termes, and afterwards explain some Propositions, of which, as of
things true and obvious, I may make use of to my present purpose.
DEFINITION I.
_I then call equally Grave_ in specie, _those Matters of which equall
Masses weigh equally._
As if for example, two Balls, one of Wax, and the other of some Wood
of equall Masse, were also equall in Weight, we say, that such Wood,
and the Wax are _in specie_ equally grave.
DEFINITION II.
_But equally grave in Absolute Gravity, we call two Sollids,
weighing equally, though of Mass they be unequall._
As for example, a Mass of Lead, and another of Wood, that weigh each
ten pounds, I call equall in Absolute Gravity, though the Mass of the
Wood be much greater then that of the Lead.
_And, consequently, less Grave_ in specie.
DEFINITION III.
_I call a Matter more Grave_ in specie _than another, of which a
Mass, equall to a Mass of the other, shall weigh more._
And so I say, that Lead is more grave _in specie_ than Tinn, because
if you take of them two equall Masses, that of the Lead weigheth more.
DEFINITION IV.
_But I call that Body more grave absolutely than this, if that
weigh more than this, without any respect had to the Masses._
And thus a great piece of Wood is said to weigh more than a little
lump of Lead, though the Lead be _in specie_ more heavy than the Wood.
And the same is to be understood of the less grave _in specie_, and
the less grave absolutely.
These Termes defined, I take from the Mechanicks two Principles: the
first is, that
AXIOME. I.
_Weights absolutely equall, moved with equall Velocity, are of
equall Force and Moment in their operations._
_DEFINITION V._
Moment, amongst Mechanicians, signifieth that Vertue, that Force,
or that Efficacy, with which the Mover moves, and the Moveable
resists.
_Which Vertue dependes not only on the simple Gravity, but on the
Velocity of the Motion, and on the diverse Inclinations of the
Spaces along which the Motion is made: For a descending Weight
makes a greater Impetus in a Space much declining, than in one less
declining; and in summe, what ever is the occasion of such Vertue,
it ever retaines the name of Moment; nor in my Judgement, is this
sence new in our Idiome, for, if I mistake not, I think we often
say; This is a weighty businesse, but the other is of small moment:
and we consider lighter matters and let pass those of Moment; a
Metaphor, I suppose, taken from the Mechanicks._
As for example, two weights equall in absolute Gravity, being put
into a Ballance of equall Arms, they stand in _Equilibrium_, neither
one going down, nor the other up: because the equality of the
Distances of both, from the Centre on which the Ballance is supported,
and about which it moves, causeth that those weights, the said
Ballance moving, shall in the same Time move equall Spaces, that is,
shall move with equall Velocity, so that there is no reason for which
this Weight should descend more than that, or that more than this; and
therefore they make an _Equilibrium_, and their Moments continue of
semblable and equall Vertue.
The second Principle is; That
AXIOME II.
_The Moment and Force of the Gravity, is encreased by the Velocity
of the Motion._
So that Weights absolutely equall, but conjoyned with Velocity
unequall, are of Force, Moment and Vertue unequall: and the more
potent, the more swift, according to the proportion of the Velocity of
the one, to the Velocity of the other. Of this we have a very
pertinent example in the Balance or Stiliard of unequall Arms, at
which Weights absolutely equall being suspended, they do not weigh
down, and gravitate equally, but that which is at a greater distance
from the Centre, about which the Beam moves, descends, raising the
other, and the Motion of this which ascends is slow, and the other
swift: and such is the Force and Vertue, which from the Velocity of
the Mover, is conferred on the Moveable, which receives it, that it
can exquisitely compensate, as much more Weight added to the other
slower Moveable: so that if of the Arms of the Balance, one were ten
times as long as the other, whereupon in the Beames moving about the
Centre, the end of that would go ten times as far as the end of this,
a Weight suspended at the greater distance, may sustain and poyse
another ten times more grave absolutely than it: and that because the
Stiliard moving, the lesser Weight shall move ten times faster than
the bigger. It ought alwayes therefore to be understood, that Motions
are according to the same Inclinations, namely, that if one of the
Moveables move perpendicularly to the Horizon, then the other makes
its Motion by the like Perpendicular; and if the Motion of one were to
be made Horizontally; that then the other is made along the same
Horizontall plain: and in summe, alwayes both in like Inclinations.
This proportion between the Gravity and Velocity is found in all
Mechanicall Instruments: and is considered by _Aristotle_, as a
Principle in his _Mechanicall Questions_; whereupon we also may take
it for a true Assumption, That
AXIOME III.
_Weights absolutely unequall, do alternately counterpoyse and
become of equall Moments, as oft as their Gravities, with
contrary proportion, answer to the Velocity of their Motions._
That is to say, that by how much the one is less grave than the other,
by so much is it in a constitution of moving more swiftly than that.
Having prefatically explicated these things, we may begin to enquire,
what Bodyes those are which totally submerge in Water, and go to the
Bottom, and which those that by constraint float on the top, so that
being thrust by violence under Water, they return to swim, with one
part of their Mass visible above the Surface of the Water: and this we
will do by considering the respective operation of the said Solids,
and of Water: Which operation followes the Submersion and sinking; and
this it is[6], That in the Submersion that the Solid maketh, being
depressed downwards by its proper Gravity, it comes to drive away the
water from the place where it successively subenters, and the water
repulsed riseth and ascends above its first levell, to which Ascent on
the other side it, as being a grave Body of its own nature, resists:
And because the descending Solid more and more immerging, greater and
greater quantity of Water ascends, till the whole Sollid be submerged;
its necessary to compare the Moments of the Resistance of the water to
Ascension, with the Moments of the pressive Gravity of the Solid: And
if the Moments of the Resistance of the water, shall equalize the
Moments of the Solid, before its totall Immersion[7]; in this case
doubtless there shall be made an _Equilibrium_, nor shall the Body
sink any farther. But if the Moment of the Solid, shall alwayes exceed
the Moments wherewith the repulsed water successively makes
Resistance[8], that Solid shall not only wholly submerge under water, but
shall descend to the Bottom. But if, lastly, in the instant of totall
Submersion, the equality shall be made between the Moments of the
prement Solid, and the resisting Water[9]; then shall rest, ensue, and
the said Solid shall be able to rest indifferently, in whatsoever part
of the water. By this time is manifest the necessity of comparing the
Gravity of the water, and of the Solid[10]; and this comparison might at
first sight seem sufficient to conclude and determine which are the
Solids that float a-top, and which those that sink to the Bottom in
the water, asserting that those shall float which are lesse grave _in
specie_ than the water, and those submerge, which are _in specie_ more
grave. For it seems in appearance, that the Sollid in sinking
continually, raiseth so much Water in Mass, as answers to the parts of
its own Bulk submerged: whereupon it is impossible, that a Solid less
grave _in specie_, than water, should wholly sink, as being unable to
raise a weight greater than its own, and such would a Mass of water
equall to its own Mass be. And likewise it seems necessary, that the
graver Solids do go to the Bottom, as being of a Force more than
sufficient for the raising a Masse of water, equall to its own, though
inferiour in weight. Nevertheless the business succeeds otherwise: and
though the Conclusions are true, yet are the Causes thus assigned
deficient, nor is it true, that the Solid in submerging, raiseth and
repulseth Masses of Water, equall to the parts of it self submerged;
but the Water repulsed, is alwayes less than the parts of the Solid
submerged[11]: and so much the more by how much the Vessell in which the
Water is contained is narrower: in such manner that it hinders not,
but that a Solid may submerge all under Water, without raising so much
Water in Mass, as would equall the tenth or twentieth part of its own
Bulk: like as on the contrary, a very small quantity of Water, may
raise a very great Solid Mass[12], though such Solid should weigh
absolutely a hundred times as much, or more, than the said Water, if
so be that the Matter of that same Solid be _in specie_ less grave
than the Water. And thus a great Beam, as suppose of a 1000 weight,
may be raised and born afloat by Water, which weighs not 50: and this
happens when the Moment of the Water is compensated by the Velocity of
its Motion.
[6] How the submersion of Solids in the Water, is effected.
[7] What Solids shall float on the Water.
[8] What Solids shall sinke to the botome.
[9] What Solids shall rest in all places of the Water.
[10] The Gravitie of the Water and Solid must be compared in all
Problems, of Natation of Bodies.
[11] The water repelled is ever less than the parts of the Sollid
submerged.
[12] _A_ small quantity of water, may float a very great Solid
Mass.
But because such things, propounded thus in abstract, are somewhat
difficult to be comprehended, it would be good to demonstrate them by
particular examples; and for facility of demonstration, we will
suppose the Vessels in which we are to put the Water, and place the
Solids, to be inviron'd and included with sides erected perpendicular
to the Plane of the Horizon, and the Solid that is to be put into such
vessell to be either a streight Cylinder, or else an upright Prisme.
_The which proposed and declared, I proceed to demonstrate the truth
of what hath been hinted, forming the ensuing Theoreme._
_THEOREME I._
[Sidenote: The Proportion of the water raised to the Solid
submerged.]
The Mass of the Water which ascends in the submerging of a Solid,
Prisme or Cylinder, or that abaseth in taking it out, is less
than the Mass of the said Solid, so depressed or advanced: and
hath to it the same proportion, that the Surface of the Water
circumfusing the Solid, hath to the same circumfused Surface,
together with the Base of the Solid.
_Let the Vessell be A B C D, and in it the Water raised up to the
Levell E F G, before the Solid Prisme H I K be therein immerged; but
after that it is depressed under Water, let the Water be raised as
high as the Levell L M, the Solid H I K shall then be all under Water,
and the Mass of the elevated Water shall be L G, which is less than
the Masse of the Solid depressed, namely of H I K, being equall to the
only part E I K, which is contained under the first Levell E F G.
Which is manifest, because if the Solid H I K be taken out, the Water
I G shall return into the place occupied by the Mass E I K, where it
was continuate before the submersion of the Prisme. And the Mass L G
being equall to the Mass E K: adde thereto the Mass E N, and it shall
be the whole Mass E M, composed of the parts of the Prisme E N, and of
the Water N F, equall to the whole Solid H I K: And, therefore, the
Mass L G shall have the same proportion to E M, as to the Mass H I K:
But the Mass L G hath the same proportion to the Mass E M, as the
Surface L M hath to the Surface M H: Therefore it is manifest, that
the Mass of Water repulsed L G, is in proportion to the Mass of the
Solid submerged H I K; as the Surface L M, namely, that of the Water
ambient about the Sollid, to the whole Surface H M, compounded of the
said ambient water, and the Base of the Prisme H N. But if we suppose
the first Levell of the Water the according to the Surface H M, and
the Prisme allready submerged H I K; and after to be taken out and
raised to E A O, and the Water to be faln from the first Levell H L M
as low as E F G; It is manifest, that the Prisme E A O being the same
with H I K, its superiour part H O, shall be equall to the inferiour E
I K: and remove the common part E N, and, consequently, the Mass of
the Water L G is equall to the Mass H O; and, therefore, less than the
Solid, which is without the Water, namely, the whole Prisme E A O, to
which likewise, the said Mass of Water abated L G, hath the same
proportion, that the Surface of the Waters circumfused L M hath to the
same circumfused Surface, together with the Base of the Prisme A O:
which hath the same demonstration with the former case above._
[Illustration]
_And from hence is inferred, that the Mass of the Water, that riseth
in the immersion of the Solid, or that ebbeth in elevating it, is not
equall to all the Mass of the Solid, which is submerged or elevated,
but to that part only, which in the immersion is under the first
Levell of the Water, and in the elevation remaines above the first
Levell: Which is that which was to be demonstrated. We will now pursue
the things that remain._
And first we will demonstrate that,
THEOREME II.
[Sidenote: The proportion of the water abated, to the Solid
raised.]
_When in one of the above said Vessels, of what ever breadth,
whether wide or narrow, there is placed such a Prisme or
Cylinder, inviron'd with Water, if we elevate that Solid
perpendicularly, the Water circumfused shall abate, and the
Abatement of the Water, shall have the same proportion to the
Elevation of the Prisme, as one of the Bases of the Prisme, hath
to the Surface of the Water Circumfused._
[Illustration]
Imagine in the Vessell, as is aforesaid, the Prisme A C D B to be
placed, and in the rest of the Space the Water to be diffused as far
as the Levell E A: and raising the Solid, let it be transferred to G
M, and let the Water be abased from E A to N O: I say, that the
descent of the Water, measured by the Line A O, hath the same
proportion to the rise of the Prisme, measured by the Line G A, as the
Base of the Solid G H hath to the Surface of the Water N O. The which
is manifest: because the Mass of the Solid G A B H, raised above the
first Levell E A B, is equall to the Mass of Water that is abased E N
O A. Therefore, E N O A and G A B H are two equall Prismes; for of
equall Prismes, the Bases answer contrarily to their heights:
Therefore, as the Altitude A O is to the Altitude A G, so is the
Superficies or Base G H to the Surface of the Water N O. If therefore,
for example, a Pillar were erected in a waste Pond full of Water, or
else in a Well, capable of little more then the Mass of the said
Pillar, in elevating the said Pillar, and taking it out of the Water,
according as it riseth, the Water that invirons it will gradually
abate, and the abasement of the Water at the instant of lifting out
the Pillar, shall have the same proportion, that the thickness of the
Pillar hath to the excess of the breadth of the said Pond or Well,
above the thickness of the said Pillar: so that if the breadth of the
Well were an eighth part larger than the thickness of the Pillar, and
the breadth of the Pond twenty five times as great as the said
thickness, in the Pillars ascending one foot, the water in the Well
shall descend seven foot, and that in the Pond only 1/25 of a foot.
[Sidenote: Why a Solid less grave _in specie_ than water, stayeth
not under water, in very small depths:]
This Demonstrated, it will not be difficult to show the true cause,
how it comes to pass, that,
THEOREME III.
_A Prisme or regular Cylinder, of a substance specifically less
grave than Water, if it should be totally submerged in Water,
stayes not underneath, but riseth, though the Water circumfused
be very little, and in absolute Gravity, never so much inferiour
to the Gravity of the said Prisme._
Let then the Prisme A E F B, be put into the Vessell C D F B, the same
being less grave _in specie_ than the Water: and let the Water infused
rise to the height of the Prisme: I say, that the Prisme left at
liberty, it shall rise, being born up by the Water circumfused C D E
A. For the Water C E being specifically more grave than the Solid A F,
the absolute weight of the water C E, shall have greater proportion to
the absolute weight of the Prisme A F, than the Mass C E hath to the
Mass A F (in regard the Mass hath the same proportion to the Mass,
that the weight absolute hath to the weight absolute, in case the
Masses are of the same Gravity _in specie_.) But the Mass C E is to
the Mass A F, as the Surface of the water A C, is to the Superficies,
or Base of the Prisme A B; which is the same proportion as the ascent
of the Prisme when it riseth, hath to the descent of the Water
circumfused C E.
[Illustration]
Therefore, the absolute Gravity of the water C E, hath greater
proportion to the absolute Gravity of the Prisme A F; than the Ascent
of the Prisme A F, hath to the descent of the said water C E. The
Moment, therefore, compounded of the absolute Gravity of the water C
E, and of the Velocity of its descent, whilst it forceably repulseth
and raiseth the Solid A F, is greater than the Moment compounded of
the absolute Gravity of the Prisme A F, and of the Tardity of its
ascent, with which Moment it contrasts and resists the repulse and
violence done it by the Moment of the water: Therefore, the Prisme
shall be raised.
[Sidenote: The Proportion according to which the Submersion &
Natation of Solids is made.]
It followes, now, that we proceed forward to demonstrate more
particularly, how much such Solids shall be inferiour in Gravity to
the water elevated; namely, what part of them shall rest submerged,
and what shall be visible above the Surface of the water: but first it
is necessary to demonstrate the subsequent Lemma.
LEMMA I.
[Sidenote: The absolute Gravity of Solids, are in a proportion
compounded of their Specifick Gravities, and of their Masses.]
_The absolute Gravities of Solids, have a proportion compounded of
the proportions of their specificall Gravities, and of their
Masses._
[Illustration]
Let A and B be two Solids. I say, that the Absolute Gravity of A,
hath to the Absolute Gravity of B, a proportion compounded of the
proportions of the specificall Gravity of A, to the Specificall
Gravity of B, and of the Mass A to the Mass B. Let the Line D have the
same proportion to E, that the specifick Gravity of A, hath to the
specifick Gravity of B; and let E be to F, as the Mass A to the Mass
B: It is manifest, that the proportion of D to F, is compounded of the
proportions D and E; and E and F. It is requisite, therefore, to
demonstrate, that as D is to F, so the absolute Gravity of A, is to
the absolute Gravity of B. Take the Solid C, equall in Mass to the
Solid A, and of the same Gravity _in specie_ with the Solid B.
Because, therefore, A and C are equall in Mass, the absolute Gravity
of A, shall have to the absolute Gravity of C, the same proportion, as
the specificall Gravity of A, hath to the specificall Gravity of C, or
of B, which is the same _in specie_; that is, as D is to E. And,
because, C and B are of the same Gravity _in specie_, it shall be,
that as the absolute weight of C, is to the absolute weight of B, so
the Mass C, or the Mass A, is to the Mass B; that is, as the Line E to
the Line F. As therefore, the absolute Gravity of A, is to the
absolute Gravity of C, so is the Line D to the Line E: and, as the
absolute Gravity of C, is to the absolute Gravity of B, so is the Line
E to the Line F: Therefore, by Equality of proportion, the absolute
Gravity of A, is to the absolute Gravity of B, as the Line D to the
Line F: which was to be demonstrated. I proceed now to demonstrate,
how that,
THEOREME IV.
[Sidenote: The proportion of water requisite to make a Solid
swim:]
_If a Solid, Cylinder, or Prisme, lesse grave specifically than the
Water, being put into a Vessel, as above, of whatsoever
greatnesse, and the Water, be afterwards infused, the Solid shall
rest in the bottom, unraised, till the Water arrive to that part
of the Altitude, of the said Prisme, to which its whole Altitude
hath the same proportion, that the Specificall Gravity of the
Water, hath to the Specificall Gravity of the said Solid: but
infusing more Water, the Solid shall ascend._
[Illustration]
Let the Vessell be M L G N of any bigness, and let there be placed in
it the Solid Prisme D F G E, less grave _in specie_ than the water;
and look what proportion the Specificall Gravity of the water, hath to
that of the Prisme, such let the Altitude D F, have to the Altitude F
B. I say, that infusing water to the Altitude F B, the Solid D G shall
not float, but shall stand in _Equilibrium_, so, that that every
little quantity of water, that is infused, shall raise it. Let the
water, therefore, be infused to the Levell A B C; and; because the
Specifick Gravity of the Solid D G, is to the Specifick Gravity of the
water, as the altitude B F is to the altitude F D; that is, as the
Mass B G to the Mass G D; as the proportion of the Mass B G is to the
Mass G D, as the proportion of the Mass G D is to the Mass A F, they
compose the Proportion of the Mass B G to the Mass A F. Therefore, the
Mass B G is to the Mass A F, in a proportion compounded of the
proportions of the Specifick Gravity of the Solid G D, to the
Specifick Gravity of the water, and of the Mass G D to the Mass A F:
But the same proportions of the Specifick Gravity of G D, to the
Specifick Gravity of the water, and of the Mass G D to the Mass A F,
do also by the precedent _Lemma_, compound the proportion of the
absolute Gravity of the Solid D G, to the absolute Gravity of the Mass
of the water A F: Therefore, as the Mass B G is to the Mass A F, so is
the Absolute Gravity of the Solid D G, to the Absolute Gravity of the
Mass of the water A F. But as the Mass B G is to the Mass A F; so is
the Base of the Prisme D E, to the Surface of the water A B; and so is
the descent of the water A B, to the Elevation of the Prisme D G;
Therefore, the descent of the water is to the elevation of the Prisme,
as the absolute Gravity of the Prisme, is to the absolute Gravity of
the water: Therefore, the Moment resulting from the absolute Gravity
of the water A F, and the Velocity of the Motion of declination, with
which Moment it forceth the Prisme D G, to rise and ascend, is equall
to the Moment that results from the absolute Gravity of the Prisme D
G, and from the Velocity of the Motion, wherewith being raised, it
would ascend: with which Moment it resists its being raised: because,
therefore, such Moments are equall, there shall be an _Equilibrium_
between the water and the Solid. And, it is manifest, that putting a
little more water unto the other A F, it will increase the Gravity and
Moment, whereupon the Prisme D G, shall be overcome, and elevated till
that the only part B F remaines submerged. Which is that that was to
be demonstrated.
COROLLARY I.
[Sidenote: _H_ow far Solids less grave _in specie_ than water, do
submerge.]
_By what hath been demonstrated, it is manifest, that Solids less
grave_ in specie _than the water, submerge only so far, that as
much water in Mass, as is the part of the Solid submerged, doth
weigh absolutely as much as the whole Solid._
For, it being supposed, that the Specificall Gravity of the water, is
to the Specificall Gravity of the Prisme D G, as the Altitude D F, is
to the Altitude F B; that is, as the Solid D G is to the Solid B G; we
might easily demonstrate, that as much water in Mass as is equall to
the Solid B G, doth weigh absolutely as much as the whole Solid D G;
For, by the _Lemma_ foregoing, the Absolute Gravity of a Mass of
water, equall to the Mass B G, hath to the Absolute Gravity of the
Prisme D G, a proportion compounded of the proportions, of the Mass B
G to the Mass G D, and of the Specifick Gravit{y} of the water, to the
Specifick Gravity of the Prisme: But the Gravity _in specie_ of the
water, to the Gravity _in specie_ of the Prisme, is supposed to be as
the Mass G D to the Mass G B. Therefore, the Absolute Gravity of a
Mass of water, equall to the Mass B G, is to the Absolute Gravity of
the Solid D G, in a proportion compounded of the proportions, of the
Mass B G to the Mass G D, and of the Mass D G to the Mass G B; which
is a proportion of equalitie. The Absolute Gravity, therefore, of a
Mass of Water equall to the part of the Mass of the Prisme B G, is
equall to the Absolute Gravity of the whole Solid D G.
COROLLARY II.
[Sidenote: _A_ Rule to equilibrate Solids in the water.]
_It followes, moreover, that a Solid less grave than the water,
being put into a Vessell of any imaginable greatness, and water
being circumfused about it to such a height, that as much water
in Mass, as is the part of the Solid submerged, do weigh
absolutely as much as the whole Solid; it shall by that water be
justly sustained, be the circumfused Water in quantity greater or
lesser._
[Illustration]
For, if the Cylinder or Prisme M, less grave than the water, _v. gra._
in Subsequiteriall proportion, shall be put into the capacious Vessell
A B C D, and the water raised about it, to three quarters of its
height, namely, to its Levell A D: it shall be sustained and exactly
poysed in _Equilibrium_. The same will happen; if the Vessell E N S F
were very small, so, that between the Vessell and the Solid M, there
were but a very narrow space, and only capable of so much water, as
the hundredth part of the Mass M, by which it should be likewise
raised and erected, as before it had been elevated to three fourths of
the height of the Solid: which to many at the first sight, may seem a
notable Paradox, and beget a conceit, that the Demonstration of these
effects, were sophisticall and fallacious: but, for those who so
repute it, the Experiment is a means that may fully satisfie them. But
he that shall but comprehend of what Importance Velocity of Motion is,
and how it exactly compensates the defect and want of Gravity, will
cease to wonder, in considering that at the elevation of the Solid M,
the great Mass of water A B C D abateth very little, but the little
Mass of water E N S F decreaseth very much, and in an instant, as the
Solid M before did rise, howbeit for a very short space: Whereupon the
Moment, compounded of the small Absolute Gravity of the water E N S F,
and of its great Velocity in ebbing, equalizeth the Force and and
Moment, that results from the composition of the immense Gravity of
the water A B C D, with its great slownesse of ebbing; since that in
the Elevation of the Sollid M, the abasement of the lesser water E S,
is performed just so much more swiftly than the great Mass of water A
C, as this is more in Mass than that which we thus demonstrate.
[Sidenote: _T_he proportion according to which water riseth and
falls in different Vessels at the Immersion and Elevation of
Solids.]
In the rising of the Solid M, its elevation hath the same proportion
to the circumfused water E N S F, that the Surface of the said water,
hath to the Superficies or Base of the said Solid M; which Base hath
the same proportion to the Surface of the water A D, that the
abasement or ebbing of the water A C, hath to the rise or elevation of
the said Solid M. Therefore, by Perturbation of proportion, in the
ascent of the said Solid M, the abasement of the water A B C D, to the
abasement of the water E N S F, hath the same proportion, that the
Surface of the water E F, hath to the Surface of the water A D; that
is, that the whole Mass of the water E N S F, hath to the whole Mass A
B C D, being equally high: It is manifest, therefore, that in the
expulsion and elevation of the Solid M, the water E N S F shall exceed
in Velocity of _M_otion the water A B C D, asmuch as it on the other
side is exceeded by that in quantity: whereupon their Moments in such
operations, are mutually equall.
[Illustration]
_And, for ampler confirmation, and clearer explication of this, let
us consider the present Figure, (which if I be not deceived, may
serve to detect the errors of some Practick Mechanitians who upon a
false foundation some times attempt impossible enterprizes,) in
which, unto the large Vessell E I D F, the narrow Funnell or Pipe I
C A B is continued, and suppose water infused into them, unto the
Levell L G H, which water shall rest in this position, not without
admiration in some, who cannot conceive how it can be, that the
heavie charge of the great Mass of water G D, pressing downwards,
should not elevate and repulse the little quantity of the other,
contained in the Funnell or Pipe C L, by which the descent of it is
resisted and hindered: But such wonder shall cease, if we begin to
suppose the water G D to be abased only to Q D, and shall
afterwards consider, what the water C L hath done, which to give
place to the other, which is descended from the Levell G H, to the
Levell Q O, shall of necessity have ascended in the same time, from
the Levell L unto A B. And the ascent L B, shall be so much greater
than the descent G Q, by how much the breadth of the Vessell G D,
is greater than that of the Funnell I C; which, in summe, is as
much as the water G D, is more than the water L C: but in regard
that the Moment of the Velocity of the Motion, in one Moveable,
compensates that of the Gravity of another what wonder is it, if
the swift ascent of the lesser Water C L, shall resist the slow
descent of the greater G D?_
The same, therefore, happens in this operation, as in rhe Stilliard,
in which a weight of two pounds counterpoyseth an other of 200,
asoften as that shall move in the same time, a space 100 times greater
than this: which falleth out when one Arme of the Beam is an hundred
times as long as the other. Let the erroneous opinion of those
therefore cease, who hold that a Ship is better, and easier born up in
a great abundance of water, then in a lesser quantity[13], (_this was
believed by_ Aristotle _in his Problems, Sect. 23, Probl. 2._) it being
on the contrary true, that its possible, that a Ship may as well float
in ten Tun of water, as in an Ocean.
[13] A ship flotes as well in ten Tun of Water as in an Ocean.
[Sidenote: A Solid specifiaclly graver than the water, cannot be
born up by any quantity of it.]
But following our matter, I say, that by what hath been hitherto
demonstrated, we may understand how, that
COROLLARY III.
_One of the above named Solids, when more grave_ in specie _than the
water, can never be sustained, by any whatever quantity of it._
For having seen how that the Moment wherewith such a Solid, as grave
_in specie_ as the water, contrasts with the Moment of any Mass of
water whatsoever, is able to retain it, even to its totall Submersion,
without its ever ascending; it remaineth, manifest, that the water is
far less able to raise it up, when it exceeds the same _in specie_:
so, that though you infuse water till its totall Submersion, it shall
still stay at the Bottome, and with such Gravity, and Resistance to
Elevation, as is the excess of its Absolute Gravity, above the
Absolute Gravity of a Mass equall to it, made of water, or of a Matter
_in specie_ equally grave with the water: and, though you should
moreover adde never so much water above the Levell of that which
equalizeth the Altitude of the Solid, it shall not, for all that,
encrease the Pression, or Gravitation, of the parts circumfused about
the said Solid, by which greater pression, it might come to be
repulsed; because, the Resistance is not made, but only by those parts
of the water, which at the Motion of the said Solid do also move, and
these are those only, which are comprehended by the two Superficies
equidistant to the Horizon, and their parallels, that comprehend the
Altitude of the Solid immerged in the water.
I conceive, I have by this time sufficiently declared and opened the
way to the contemplation of the true, intrinsecall and proper Causes
of diverse Motions, and of the Rest of many Solid Bodies in diverse
_Mediums_, and particularly in the water, shewing how all in effect,
depend on the mutuall excesses of the Gravity of the Moveables and of
the _Mediums_: and, that which did highly import, removing the
Objection, which peradventure would have begotten much doubting, and
scruple in some, about the verity of my Conclusion, namely, how that
notwithstanding, that the excess of the Gravity of the water, above
the Gravity of the Solid, demitted into it, be the cause of its
floating and rising from the Bottom to the Surface, yet a quantity of
water, that weighs not ten pounds, can raise a Solid that weighs above
100 pounds: in that we have demonstrated, That it sufficeth, that such
difference be found between the Specificall Gravities of the _Mediums_
and Moveables, let the particular and absolute Gravities be what they
will: insomuch, that a Solid, provided that it be Specifically less
grave than the water, although its absolute weight were 1000 pounds,
yet may it be born up and elevated by ten pounds of water, and less:
and on the contrary, another Solid, so that it be Specifically more
grave than the water, though in absolute Gravity it were not above a
pound, yet all the water in the Sea, cannot raise it from the Bottom,
or float it. This sufficeth me, for my present occasion, to have, by
the above declared Examples, discovered and demonstrated, without
extending such matters farther, and, as I might have done, into a long
Treatise: yea, but that there was a necessity of resolving the above
proposed doubt, I should have contented my self with that only, which
is demonstrated by _Archimedes_, in his first _Book De Insidentibus
humido_[14]: where in generall termes he infers and confirms the same
Conclusions, namely, that Solids (_a_) less grave than water, swim or
float upon it, the (_b_) more grave go to the Bottom, and the (_c_)
equally grave rest indifferently in all places, yea, though they
should be wholly under water.
[14] _Of Natation_ (a) _Lib. 1, Prop. 4._ (b) _Id. Lib. 1. Prop.
3._ (c) _Id. Lib. 1. Prop. 3._
[Sidenote: The Authors defence of _Archimedes_ his Doctrine,
against the oppositions of _Buonamico_.]
But, because that this Doctrine of Archimedes, perused, transcribed
and examined by _Signor Francesco Buonamico_, in his _fifth Book of
Motion, Chap. 29_, and afterwards by him confuted, might by the
Authority of so renowned, and famous a Philosopher, be rendered
dubious, and suspected of falsity; I have judged it necessary to
defend it, if I am able so to do, and to clear _Archimedes_, from
those censures, with which he appeareth to be charged. _Buonamico_
rejecteth the Doctrine of _Archimedes_, first[15], as not consentaneous
with the Opinion of _Aristotle_, adding, that it was a strange thing
to him, that the Water should exceed the Earth in Gravity[16], seeing on
the contrary, that the Gravity of water, increaseth, by means of the
participation of Earth. And he subjoyns presently after[17], that he was
not satisfied with the Reasons of _Archimedes_, as not being able with
that Doctrine, to assign the cause whence it comes, that a Boat and a
Vessell, which otherwise, floats above the water, doth sink to the
Bottom, if once it be filled with water; that by reason of the
equality of Gravity, between the water within it, and the other water
without, it should stay a top; but yet, nevertheless, we see it to go
to the Bottom.
[15] His first Objection against the Doctrine of _Archimedes_.
[16] His Second Objection.
[17] His third Objection.
He farther addes[18], that _Aristotle_ had clearly confuted the Ancients,
who said, that light Bodies moved upwards[19], driven by the impulse of
the more grave Ambient: which if it were so, it should seem of
necessity to follow, that all naturall Bodies are by nature heavy, and
none light: For that the same would befall the Fire and Air, if put in
the Bottom of the water. And, howbeit, _Aristotle_ grants a Pulsion in
the Elements, by which the Earth is reduced into a Sphericall Figure,
yet nevertheless, in his judgement; it is not such that it can remove
grave Bodies from their naturall places, but rather, that it send them
toward the Centre, to which (as he somewhat obscurely continues to
say,) the water principally moves, if it in the interim meet not with
something that resists it, and, by its Gravity, thrusts it out of its
place: in which case, if it cannot directly, yet at least as well as
it can, it tends to the Centre: but it happens, that light Bodies by
such Impulsion, do all ascend upward: but this properly they have by
nature, as also, that other of swimming. He concludes, lastly[20], that he
concurs with _Archimedes_ in his Conclusions; but not in the Causes,
which he would referre to the facile and difficult Separation of the
_Medium_, and to the predominance of the Elements, so that when the
Moveable superates the power of the _Medium_; as for example, Lead
doth the Continuity of water, it shall move thorow it, else not.
[18] His fourth Objection.
[19] The _A_ncients denyed _A_bsolute Levity.
[20] The causes of Natation & Submersion, according to the
Peripateticks.
This is all that I have been able to collect, as produced against
_Archimedes_ by _Signor Buonamico_: who hath not well observed the
Principles and Suppositions of _Archimedes_; which yet must be false,
if the Doctrine be false, which depends upon them; but is contented to
alledge therein some Inconveniences, and some Repugnances to the
Doctrine and Opinion of _Aristotle_. In answer to which Objections, I
say, first[21], That the being of _Archimedes_ Doctrine, simply different
from the Doctrine of _Aristotle_, ought not to move any to suspect it,
there being no cause, why the Authority of this should be preferred to
the Authority of the other: but, because, where the decrees of Nature
are indifferently exposed to the intellectuall eyes of each, the
Authority of the one and the other, loseth all a{u}thenticalness of
Perswasion, the absolute power residing in Reason; therefore I pass to
that which he alledgeth in the second place[22], as an absurd consequent
of the Doctrine of _Archimedes_, namely, That water should be more
grave than Earth. But I really find not, that ever _Archimedes_ said
such a thing, or that it can be rationally deduced from his
Conclusions: and if that were manifest unto me, I verily believe, I
should renounce his Doctrine, as most erroneous. Perhaps this
Deduction of _Buonamico_, is founded upon that which he citeth of the
Vessel, which swims as long as its voyd of water, but once full it
sinks to the Bottom, and understanding it of a Vessel of Earth, he
infers against _Archimedes_ thus: Thou sayst that the Solids which
swim, are less grave than water: this Vessell swimmeth: therefore,
this Vessell is lesse grave than water. If this be the Illation. I
easily answer, granting that this Vessell is lesse grave than water,
and denying the other consequence, namely, that Earth is less Grave
than Water. The Vessel that swims occupieth in the water, not only a
place equall to the Mass of the Earth, of which it is formed; but
equall to the Earth and to the Air together, contained in its
concavity. And, if such a Mass compounded of Earth and Air, shall be
less grave than such another quantity of water, it shall swim, and
shall accord with the Doctrine of _Archimedes_; but if, again,
removing the Air, the Vessell shall be filled with water, so that the
Solid put in the water, be nothing but Earth, nor occupieth other
place, than that which is only possest by Earth, it shall then go to
the Bottom, by reason that the Earth is heavier than the water: and
this corresponds well with the meaning of _Archimedes_. See the same
effect illustrated, with such another Experiment, In pressing a Viall
Glass to the Bottom of the water, when it is full of Air, it will meet
with great resistance, because it is not the Glass alone, that is
pressed under water, but together with the Glass a great Mass of Air,
and such, that if you should take as much water, as the Mass of the
Glass, and of the Air contained in it, you would have a weight much
greater than that of the Viall, and of its Air: and, therefore, it
will not submerge without great violence: but if we demit only the
Glass into the water, which shall be when you shall fill the Glass
with water, then shall the Glass descend to the Bottom; as superiour
in Gravity to the water.
[21] The Authors answer to the first Objection.
[22] The Authors answer to the second Objection.
Returning, therefore, to our first purpose; I say, that Earth is more
grave than water, and that therefore, a Solid of Earth goeth to the
bottom of it; but one may possibly make a composition of Earth and
Air, which shall be less grave than a like Mass of Water; and this
shall swim: and yet both this and the other experiment shall very well
accord with the Doctrine of _Archimedes_. But because that in my
judgment it hath nothing of difficulty in it, I will not positively
affirme that _Signor Buonamico_, would by such a discourse object unto
_Archimedes_ the absurdity of inferring by his doctrine, that Earth
was less grave than Water, though I know not how to conceive what
other accident he could have induced thence.
Perhaps such a Probleme (in my judgement false) was read by _Signor
Buonamico_ in some other Author, by whom peradventure it was
attributed as a singular propertie, of some particular Water, and so
comes now to be used with a double errour in confutation of
_Archimedes_, since he saith no such thing, nor by him that did say it
was it meant of the common Element of Water.
[Sidenote: The Authors answer to the third Objection.]
The third difficulty in the doctrine of _Archimedes_ was, that he
could not render a reason whence it arose, that a piece of Wood, and a
Vessell of Wood, which otherwise floats, goeth to the bottom, if
filled with Water. _Signor Buonamico_ hath supposed that a Vessell of
Wood, and of Wood that by nature swims, as before is said, goes to the
bottom, if it be filled with water; of which he in the following
Chapter, which is the 30 of the fifth Book copiously discourseth: but
I (speaking alwayes without diminution of his singular Learning) dare
in defence of _Archimedes_ deny this experiment, being certain that a
piece of Wood which by its nature sinks not in Water, shall not sinke
though it be turned and converted into the forme of any Vessell
whatsoever, and then filled with Water: and he that would readily see
the Experiment in some other tractable Matter, and that is easily
reduced into several Figures, may take pure Wax, and making it first
into a Ball or other solid Figure, let him adde to it so much Lead as
shall just carry it to the bottome, so that being a graine less it
could not be able to sinke it, and making it afterwards into the forme
of a Dish, and filling it with Water, he shall finde that without the
said Lead it shall not sinke, and that with the Lead it shall descend
with much slowness: & in short he shall satisfie himself, that the
Water included makes no alteration. I say not all this while, but that
its possible of Wood to make Barkes, which being filled with water,
sinke; but that proceeds not through its Gravity, encreased by the
Water, but rather from the Nailes and other Iron Workes, so that it no
longer hath a Body less grave than Water, but one mixt of Iron and
Wood, more grave than a like Masse of Water. Therefore let _Signor
Buonamico_ desist from desiring a reason of an effect, that is not in
nature: yea if the sinking of the Woodden Vessell when its full of
Water, may call in question the Doctrine of _Archimedes_, which he
would not have you to follow, is on the contrary consonant and
agreeable to the Doctrine of the Peripateticks, since it aptly
assignes a reason why such a Vessell must, when its full of Water,
descend to the bottom; converting the Argument the other way, we may
with safety say that the Doctrine of _Archimedes_ is true, since it
aptly agreeth with true experiments, and question the other, whose
Deductions are fastened upon erroneouss Conclusions. As for the other
point hinted in this same Instance, where it seemes that _Benonamico_
understands the same not only of a piece of wood, shaped in the forme
of a Vessell, but also of massie Wood, which filled, _scilicet_, as I
believe, he would say, soaked and steeped in Water, goes finally to
the bottom that happens in some porose Woods, which, while their
Porosity is replenished with Air, or other Matter less grave than
Water, are Masses specificially less grave than the said Water, like
as is that Viall of Glass whilest it is full of Air: but when, such
light Matter departing, there succeedeth Water into the same
Porosities and Cavities, there results a compound of Water and Glass
more grave than a like Mass of Water: but the excess of its Gravity
consists in the Matter of the Glass, and not in the Water, which
cannot be graver than it self: so that which remaines of the Wood, the
Air of its Cavities departing, if it shall be more grave _in specie_
than Water, fil but its Porosities with Water, and you shall have a
Compost of Water and of Wood more grave than Water, but not by vertue
of the Water received into and imbibed by the Porosities, but of that
Matter of the Wood which remains when the Air is departed: and being
such it shall, according to the Doctrine of _Archimedes_, goe to the
bottom, like as before, according to the same Doctrine it did swim.
[Sidenote: The Authors answer to the fourth Objection.]
As to that finally which presents itself in the fourth place,
namely, that the _Ancients_ have been heretofore confuted by
_Aristotle_, who denying Positive and Absolute Levity, and truely
esteeming all Bodies to be grave, said, that that which moved upward
was driven by the circumambient Air, and therefore that also the
Doctrine of _Archimedes_, as an adherent to such an Opinion was
convicted and confuted: I answer first, that _Signor Buonamico_ in my
judgement hath imposed upon _Archimedes_, and deduced from his words
more than ever he intended by them, or may from his Propositions be
collected, in regard that _Archimedes_ neither denies, nor admitteth
Positive Levity, nor doth he so much as mention it: so that much less
ought _Buonamico_ to inferre, that he hath denyed that it might be the
Cause and Principle of the Ascension of Fire, and other Light Bodies[23]:
having but only demonstrated, that Solid Bodies more grave than Water
descend in it, according to the excess of their Gravity above the
Gravity of that, he demonstrates likewise, how the less grave ascend
in the same Water[24], accordng to its excess of Gravity, above the
Gravity of them. So that the most that can be gathered from the
Demonstration of _Archimedes_ is, that like as the excess of the
Gravity of the Moveable above the Gravity of the Water, is the Cause
that it descends therein, so the excess of the Gravity of the water
above that of the Moveable, is a sufficient Cause why it descends not,
but rather betakes it self to swim: not enquiring whether of moving
upwards there is, or is not any other Cause contrary to Gravity: nor
doth _Archimedes_ discourse less properly than if one should say: If
the South Winde shall assault the Barke with greater _Impetus_ than is
the violence with which the Streame of the River carries it towards
the South, the motion of it shall be towards the North: but if the
_Impetus_ of the Water shall overcome that of the Winde, its motion
shall be towards the South. The discourse is excellent and would be
unworthily contradicted by such as should oppose it, saying: Thou
mis-alledgest as Cause of the motion of the Bark towards the South,
the _Impetus_ of the Stream of the Water above that of the South
Winde; mis-alledgest I say, for it is the Force of the North Winde
opposite to the South, that is able to drive the Bark towards the
South. Such an Objection would be superfluous, because he which
alledgeth for Cause of the Motion the Stream of the Water, denies not
but that the Winde opposite to the South may do the same, but only
affirmeth that the force of the Water prevailing over the South Wind,
the Bark shall move towards the South: and saith no more than is true.
And just thus when _Archimedes_ saith, that the Gravity of the Water
prevailing over that by which the moveable descends to the Bottom,
such moveable shall be raised from the Bottom to the Surface alledgeth
a very true Cause of such an Accident, nor doth he affirm or deny that
there is, or is not, a vertue contrary to Gravity, called by some
Levity, that hath also a power of moving some Matters upwards. Let
therefore the Weapons of _Signor Buonamico_ be directed against
_Plato_[25], and other _Ancients_, who totally denying _Levity_, and
taking all Bodies to be grave, say that the Motion upwards is made,
not from an intrinsecal Principle of the Moveable, but only by the
Impulse of the _Medium_; and let _Archimedes_ and his Doctrine escape
him, since he hath given him no Cause of quarelling with him. But if
this Apologie, produced in defence of _Archimedes_, should seem to
some insufficient to free him from the Objections and Arguments,
produced by _Aristotle_ against _Plato_, and the other _Ancients_, as if
they did also fight against _Archimedes_, alledging the Impulse of the
Water as the Cause of the swimming of some Bodies less grave than it[26],
I would not question, but that I should be able to maintaine the
Doctrine of _Plato_ and those others to be most true, who absolutely
deny Levity, and affirm no other Intrinsecal Principle of Motion to be
in Elementary Bodies save only that towards the Centre of the Earth[27],
nor no other Cause of moving upwards, speaking of that which hath the
resemblance of natural Motion, but only the repulse of the _Medium_,
fluid, and exceeding the Gravity of the Moveable[28]: and as to the
Reasons of _Aristotle_ on the contrary, I believe that I could be able
fully to answer them, and I would assay to do it, if it were
absolutely necessary to the present Matter, or were it not too long a
Digression for this short Treatise. I will only say, that if there
were in some of our Ellementary Bodies an Intrinsecall Principle and
Naturall Inclination to shun the Centre of the Earth, and to move
towards the Concave of the Moon, such Bodies, without doubt, would
more swiftly ascend through those _Mediums_ that least oppose the
Velocity of the Moveable, and these are the more tenuous and subtle;
as is, for example, the Air in comparison of the Water, we daily
proving that we can with farre more expeditious Velocity move a Hand
or a Board to and again in one than in the other[29]: nevertheless, we
never could finde any Body, that did not ascend much more swiftly in
the water than in the Air. Yea of Bodies which we see continually to
ascend in the Water, there is none that having arrived to the confines
of the Air, do not wholly lose their Motion[30]; even the Air it self,
which rising with great Celerity through the Water, being once come to
its Region it loseth all
[23] Of Natation, Lib. 1. Prop. 7.
[24] Of Natation, Lib. 1. Prop. 4.
[25] _Plato_ denyeth Positive Levity.
[26] The Authors defence of the doctrine of _Plato_ and the
_Ancients_, who absolutely deny Levity:
[27] According to _Plato_ there is no Principle of the Motion, of
descent in Naturall Bodies, save that to the Centre.
[28] No cause of the motion of Ascent, save the Impulse of the
_Medium_, exceeding the Moveable in Gravitie.
[29] Bodies ascend much swifter in the Water, than in the Air.
[30] All Bodies ascending through Water, lose their Motion,
comming to the confines of the Air.
[Sidenote: The lighter Bodies ascend more swiftly through Water.]
And, howbeit, Experience shewes, that the Bodies, successively less
grave, do most expeditiously ascend in water, it cannot be doubted,
but that the Ignean Exhalations do ascend more swiftly through the
water, than doth the Air: which Air is seen by Experience to ascend
more swiftly through the Water, than the Fiery Exhalations through the
Air[31]: Therefore, we must of necessity conclude, that the said
Exhalations do much more expeditiously ascend through the Water, than
through the Air; and that, consequently, they are moved by the Impulse
of the Ambient _Medium_, and not by an intrinsick Principle that is in
them, of avoiding the Centre of the Earth; to which other grave Bodies
tend.
[31] Fiery Exhalations ascend thorow the Water more swiftly than
doth the Air; & the Air ascends more swiftly thorow the Water,
than Fire thorow the Air.
[Sidenote: _T_he Authors confutation of the Peripateticks Causes
of Natation & Submersion.]
To that which for a finall conclusion, _Signor Buonamico_ produceth
of going about to reduce the descending or not descending, to the
easie and uneasie Division of the _Medium_, and to the predominancy of
the Elements: I answer, as to the first part, that that cannot in any
manner be admitted as a Cause, being that in none of the Fluid
_Mediums_, as the Air, the Water, and other Liquids, there is any
Resistance against Division[32], but all by every the least Force, are
divided and penetrated, as I will anon demonstrate: so, that of such
Resistance of Division there can be no Act, since it self is not in
being. As to the other part, I say, that the predominancy of the
Elements in Moveables[33], is to be considered, as far as to the excesse
or defect of Gravity, in relation to the _Medium_: for in that Action,
the Elements operate not, but only, so far as they are grave or light:
therefore, to say that the Wood of the Firre sinks not, because Air
predominateth in it, is no more than to say, because it is less grave
than the Water. Yea, even the immediate Cause, is its being less grave
than the Water[34]: and it being under the predominancy of the Air, is the
Cause of its less Gravity: Therefore, he that alledgeth the
predominancy of the Element for a Cause, brings the Cause of the
Cause, and not the neerest and immediate Cause. Now, who knows not
that the true Cause is the immediate, and not the mediate[35]? Moreover,
he that alledgeth Gravity, brings a Cause most perspicuous to Sence[36]:
The cause we may very easily assertain our selves; whether Ebony, for
example, and Firre, be more or less grave than water: but whether
Earth or Air predominates in them, who shall make that manifest?
Certainly, no Experiment can better do it than to observe whether they
swim or sink. So, that he who knows, not whether such a Solid swims,
unless when he knows that Air predominates in it, knows not whether it
swim, unless he sees it swim, for then he knows that it swims, when he
knows that it is Air that predominates, but knows not that Air hath
the predominance, unless he sees it swim: therefore, he knows not if
it swims, till such time as he hath seen it swim.
[32] Water & other fluids void of Resistance against Division.
[33] _T_he predominancy of Elements in Moveables to be considered
only in relation to their excess or defect of Gravity in reference
to the _Medium_.
[34] _T_he immediate Cause of Natation is that the Moveable is
less grave than the Water.
[35] _T_he Peripateticks alledge for the reason of Natation the
Cause of the Cause.
[36] Gravity a Cause most perspicuous to sence.
Let us not then despise those Hints, though very dark, which Reason,
after some contemplation, offereth to our Intelligence, and lets be
content to be taught by _Archimedes_, that then any Body shall
submerge in water[37], when it shall be specifically more grave than it,
and that if it shall be less grave[38], it shall of necessity swim, and
that it will rest indifferently in any place under water, if its
Gravity be perfectly like to that of the water.
[37] Lib 1. of Natation Prop. 7
[38] Id. Lib. 1. Prop. 4.
These things explained and proved[39], I come to consider that which
offers it self, touching what the Diversity of figure given unto the
said Moveable hath to do with these Motions and Rests; and proceed to
affirme, that,
[39] Id. Lib 1. Prop. 3.
THEOREME V.
[Sidenote: Diversity of Figure no Cause of its absolute Natation
or Submersion.]
_The diversity of Figures given to this or that Solid, cannot any
way be a Cause of its absolute Sinking or Swimming._
So that if a Solid being formed, for example, into a Sphericall
Figure, doth sink or swim in the water, I say, that being formed into
any other Figure, the same figure in the same water, shall sink or
swim: nor can such its Motion by the Expansion or by other mutation of
Figure, be impeded or taken away.
[Sidenote: The Expansion of Figure, retards the Velocity of the
ascent or descent of the Moveable in the water; but doth not
deprive it of all Motion.]
The Expansion of the Figure may indeed retard its Velocity, aswell of
ascent as descent, and more and more according as the said Figure is
reduced to a greater breadth and thinness: but that it may be reduced
to such a form as that that same matter be wholly hindred from moving
in the same water, that I hold to be impossible. In this I have met
with great contradictors, who producing some Experiments, and in
perticular a thin Board of Ebony, and a Ball of the same Wood, and
shewing how the Ball in Water descended to the bottom, and the Board
being put lightly upon the Water submerged not, but rested; have held,
and with the Authority of _Aristotle_, confirmed themselves in their
Opinions, that the Cause of that Rest was the breadth of the Figure,
u{n}able by its small weight to pierce and penetrate the Resistance of
the Waters Crassitude, which Resistance is readily overcome by the
other Sphericall Figure.
This is the Principal point in the present Question, in which I
perswade my self to be on the right side.
Therefore, beginning to investigate with the examination of exquisite
Experiments that really the Figure doth not a jot alter the descent or
Ascent of the same Solids, and having already demonstrated that the
greater or less Gravity of the Solid in relation to the Gravity of the
_Medium_ is the cause of Descent or Ascent: when ever we would make
proof of that, which about this Effect the diversity of Figure
worketh, its necessary to make the Experiment with Matter wherein
variety of Gravities hath no place. For making use of Matters which
may be different in their Specifical Gravities, and meeting with
varieties of effects of Ascending and Descending, we shall alwayes be
left unsatisfied whether that diversity derive it self really from the
sole Figure, or else from the divers Gravity also. We may remedy this
by takeing one only Matter, that is tractable and easily reduceable
into every sort of Figure. Moreover, it will be an excellent expedient
to take a kinde of Matter, exactly alike in Gravity unto the Water:
for that Matter, as far as pertaines to the Gravity, is indifferent
either to Ascend or Descend; so that we may presently observe any the
least difference that derives it self from the diversity of Figure.
[Sidenote: An Experiment in Wax, that proveth Figure to have no
Operation in Natation & Submersion.]
Now to do this, Wax is most apt, which, besides its incapacity of
receiveing any sensible alteration from its imbibing of Water, is
ductile or pliant, and the same piece is easily reduceable into all
Figures: and being _in Specie_ a very inconsiderable matter inferiour
in Gravity to the Water, by mixing therewith a little of the fileings
of Lead it is reduced to a Gravity exactly equall to that of the
Water.
This Matter prepared, and, for example, a Ball being made thereof as
bigge as an Orange or biger, and that made so grave as to sink to the
bottom, but so lightly, that takeing thence one only Grain of Lead, it
returns to the top, and being added, it submergeth to the bottom, let
the same Wax afterwards be made into a very broad and thin Flake or
Cake; and then, returning to make the same Experiment, you shall see
that it being put to the bottom, it shall, with the Grain of Lead rest
below, and that Grain deducted, it shall ascend to the very Surface,
and added again it shall dive to the bottom. And this same effect
shall happen alwaies in all sort of Figures, as wel regular as
irregular: nor shall you ever finde any that will swim without the
removall of the Grain of Lead, or sinke to the bottom unless it be
added: and, in short, about the going or not going to the Bottom, you
shall discover no diversity, although, indeed, you shall about the
quick and slow descent: for the more expatiated and distended Figures
move more slowly aswel in the diveing to the bottom as in the rising
to the top; and the other more contracted and compact Figures, more
speedily. Now I know not what may be expected from the diversity of
Figures, if the most contrary to one another operate not so much as
doth a very small Grain of Lead, added or removed.
Me thinkes I hear some of the Adversaries to raise a doubt upon my
produced Experiment[40]. And first that they offer to my consideration,
that the Figure, as a Figure simply, and disjunct from the Matter
workes not any effect, but requires to be conjoyned with the Matter;
and, furthermore, not with every Matter, but with those only,
wherewith it may be able to execute the desired operation. Like as we
see it verified by Experience, that the Acute and sharp Angle is more
apt to cut, than the Obtuse; yet alwaies provided, that both the one
and the other, be joyned with a Matter apt to cut, as for example,
with Steel. Therefore, a Knife with a fine and sharp edge, cuts Bread
or Wood with much ease, which it will not do, if the edge be blunt and
thick: but he that will instead of Steel, take Wax, and mould it into
a Knife, undoubtedly shall never know the effects of sharp and blunt
edges: because neither of them will cut, the Wax being unable by
reason of its flexibility, to overcome the hardness of the Wood and
Bread. And, therefore, applying the like discourse to our purpose,
they say, that the difference of Figure will shew different effects,
touching Natation and Submersion, but not conjoyned with any kind of
Matter, but only with those Matters which, by their Gravity, are apt
to resist the Velocity of the water, whence he that would elect for
the Matter, Cork or other light wood, unable, through its Levity, to
superate the Crassitude of the water, and of that Matter should forme
Solids of divers Figures, would in vain seek to find out what
operation Figure hath in Natation or Submersion; because all would
swim, and that not through any property of this or that Figure, but
through the debility of the Matter, wanting so much Gravity, as is
requisite to superate and overcome the Density and Crassitude of the
water.
[40] An objection against the Experiment in Water.
Its needfull, therefore, if wee would see the effect wrought by the
Diversity of Figure, first to make choice of a Matter of its nature
apt to penetrate the Crassitude of the water. And, for this effect[41],
they have made choice of such a Matter, as fit, that being readily
reduced into Sphericall Figure, goes to the Bottom; and it is Ebony,
of which they afterwards making a small Board or Splinter, as thin as
a Lath, have illustrated how that this, put upon the Surface of the
water, rests there without descending to the Bottom: and making, on
the otherside, of the same wood a Ball, no less than a hazell Nut,
they shew, that this swims not, but descendes. From which Experiment,
they think they may frankly conclude, that the Breadth of the Figure
in the flat Lath or Board, is the cause of its not descending to the
Bottom, for as much as a Ball of the same Matter, not different from
the Board in any thing but in Figure, submergeth in the same water to
the Bottom. The discourse and the Experiment hath really so much of
probability and likelyhood of truth in it, that it would be no wonder,
if many perswaded by a certain cursory observation, should yield
credit to it; nevertheless, I think I am able to discover, how that it
is not free from falacy.
[41] An Experiment in Ebany, brought to disprove the Experiment
in Wax.
Beginning, therefore, to examine one by one, all the particulars that
have been produced, I say, that Figures, as simple Figures, not only
operate not in naturall things, but neither are they ever seperated
from the Corporeall substance[42]: nor have I ever alledged them stript of
sensible Matter, like as also I freely admit, that in our endeavouring
to examine the Diversity of Accidents, dependant upon the variety of
Figures, it is necessary to apply them to Matters, which obstruct not
the various operations of those various Figures: and I admit and
grant, that I should do very ill; if I would experiment the influence
of Acutenesse of edge with a Knife of Wax, applying it to cut an Oak,
because there is no Acuteness in Wax able to cut that very hard wood.
But yet such an Experiment of this Knife, would not be besides the
purpose, to cut curded Milk, or other very yielding Matter: yea, in
such like Matters, the Wax is more commodious than Steel; for finding
the diversity depending upon Angles, more or less Acute, for that Milk
is indifferently cut with a Raisor, and with a Knife, that hath a
blunt edge. It needs, therefore, that regard be had, not only to the
hardness, solidity or Gravity of Bodies, which under divers figures,
are to divide and penetrate some Matters, but it forceth also, that
regard be had, on the other side, to the Resistance of the Matters, to
be divided and penetrated. But since I have in making the Experiment
concerning our Contest; chosen a Matter which penetrates the
Resistance of the water; and in all figures descendes to the Bottome,
the Adversaries can charge me with no defect; yea, I have propounded
so much a more excellent Method than they, in as much as I have
removed all other Causes, of descending or not descending to the
Bottom, and retained the only sole and pure variety of Figures,
demonstrating that the same Figures all descende with the only
alteration of a Grain in weight: which Grain being removed, they
return to float and swim; it is not true, therefore, (resuming the
Example by them introduced) that I have gon{e} about to experiment the
efficacy of Acuteness, in cutting with Matters unable to cut, but with
Matters proportioned to our occasion, since they are subjected to no
other variety, then that alone which depends on the Figure more or
less acute.
[42] Figure is unseperable from Corporeall Substance.
[Sidenote: The answer to the Objection against the Experiment of
the Wax.]
But let us proceed a little farther, and observe, how that indeed the
Consideration, which, they say, ought to be had about the Election of
the Matter, to the end, that it may be proportionate for the making of
our experiment, is needlessly introduced, declaring by the example of
Cutting, that like as Acuteness is inefficient to cut, unless when it
is in a Matter hard and apt to superate the Resistance of the wood or
other Matter, which we intend to cut; so the aptitude of descending or
not descending in water, ought and can only be known in those Matters,
that are able to overcome the Renitence, and superate the Crassitude
of the water. Unto which, I say, that to make distinction and
election, more of this than of that Matter, on which to impress the
Figures for cutting or penetrating this or that Body, as the solidity
or obdurateness of the said Bodies shall be greater or less, is very
necessary: but withall I subjoyn, that such distinction, election and
caution would be superfluous and unprofitable, if the Body to be cut
or penetrated, should have no Resistance, or should not at all
withstand the Cutting or Penitration: and if the Knife were to be used
in cutting a Mist or Smoak, one of Paper would be equally serviceable
with one of _Damascus_ Steel: and so by reason the water hath not any
Resistance against the Penitration of any Solid Body, all choice of
Matter is superfluous and needless, and the Election which I said
above to have been well made of a Matter reciprocall in Gravity to
water, was not because it was necessary, for the overcoming of the
crassitude of the water, but its Gravity, with which only it resists
the sinking of Solid Bodies: and for what concerneth the Resistance of
the crassitude, if we narrowly consider it, we shall find that all
Solid Bodies, as well those that sink, as those that swim, are
indifferently accomodated and apt to bring us to the knowledge of the
truth in question. Nor will I be frighted out of the belief of these
Conclusions, by the Experiments which may be produced against me, of
many severall Woods, Corks, Galls, and, moreover, of subtle slates and
plates of all sorts of Stone and Mettall, apt by means of their
Naturall Gravity, to move towards the Centre of the Earth, the which,
nevertheless, being impotent, either through the Figure (as the
Adversaries thinke) or through Levity, to break and penetrate the
Continuity of the parts of the water, and to distract its union, do
continue to swimm without submerging in the least: nor on the other
side, shall the Authority of _Aristotle_ move me, who in more than one
place, affirmeth the contrary to this, which Experience shews me.
[Sidenote: No Solid of such Levity, nor of such Figure, but that
it doth penetrate the Crassitude of the Water.]
[Sidenote: Bodies of all Figures, laid upon the water, do
penetrate its Crassitude, and in what proportion.]
I return, therefore, to assert, that there is not any Solid of such
Levity, nor of such Figure, that being put upon the water, doth not
divide and penetrate its Crassitude: yea if any with a more
perspicatious eye, shall return to observe more exactly the thin
Boards of Wood, he shall see them to be with part of their thickness
under water, and not only with their inferiour Superficies, to kisse
the Superiour of the water, as they of necessity must have believed,
who have said, that such Boards submerge not, as not being able to
divide the Tenacity of the parts of the water: and, moreover, he shall
see, that subtle shivers of Ebony, Stone or Metall, when they float,
have not only broak the Continuity of the water, but are with all
their thickness, under the Surface of it; and more and more, according
as the Matters are more grave: so that a thin Plate of Lead, shall be
lower than the Surface of the circumfused water, by at least twelve
times the thickness of the Plate, and Gold shall dive below the Levell
of the water, almost twenty times the thickness of the Plate, as I
shall anon declare.
But let us proceed to evince, that the water yields and suffers it
self to be penetrated by every the lightest Body; and therewithall
demonstrate, how, even by Matters that submerge not, we may come to
know that Figure operates nothing about the going or not going to the
Bottom, seeing that the water suffers it self to be penetrated equally
by every Figure.
[Sidenote: The Experiment of a Cone, demitted with its Base, and
after with its Point downwards.]
Make a Cone, or a Piramis of Cypress, of Firre, or of other Wood of
like Gravity, or of pure Wax, and let its height be somewhat great,
namely a handfull, or more, and put it into the water with the Base
downwards: first, you shall see that it will penetrate the water, nor
shall it be at all impeded by the largeness of the Base, nor yet shall
it sink all under water, but the part towards the point shall lye
above it: by which shall be manifest, first, that that Solid forbeares
not to sink out of an inability to divide the Continuity of the water,
having already divided it with its broad part, that in the opinion of
the Adversaries is the less apt to make the division. The Piramid
being thus fixed, note what part of it shall be submerged, and revert
it afterwards with the point downwards, and you shall see that it
shall not dive into the water more than before, but if you observe how
far it shall sink, every person expert in Geometry, may measure, that
those parts that remain out of the water, both in the one and in the
other Experiment are equall to an hair: whence he may manifestly
conclude, that the acute Figure which seemed most apt to part and
penetrate the water, doth not part or penetrate it more than the large
and spacious.
And he that would have a more easie Experiment, let him take two
Cylinders of the same Matter, one long and small, and the other short,
but very broad, and let him put them in the water, not distended, but
erect and endways: he shall see, if he diligently measure the parts of
the one and of the other, that in each of them the part submerged,
retains exactly the same proportion to that out of the water, and that
no greater part is submerged of that long and small one, than of the
other more spacious and broad: howbeit, this rests upon a very large,
and that upon a very little Superficies of water: therefore the
diversity of Figure, occasioneth neither facility, nor difficulty, in
parting and penetrating the Continuity of the water, and,
consequently, cannot be the Cause of the Natation or Submersion. He
may likewise discover the non operating of variety of Figures, in
arising from the Bottom of the water, towards the Surface, by taking
Wax, and tempering it with a competent quantity of the filings of
Lead, so that it may become a considerable matter graver than the
water: then let him make it into a Ball, and thrust it unto the Bottom
of the water; and fasten to it as much Cork, or other light matter, as
just serveth to raise it, and draw it towards the Surface: for
afterwards changing the same Wax into a thin Cake, or into any other
Figure, that same Cork shall raise it in the same manner to a hair.
This silenceth not my Antagonists, but they say, that all the
discourse hitherto made by me little importeth to them, and that it
serves their turn, that they have demonstrated in one only particular,
and in what matter, and under what Figure pleaseth them, namely, in a
Board and in a Ball of Ebony, that this put in the water, descends to
the Bottom, and that stays atop to swim: and the Matter being the
same, and the two Bodies differing in nothing but in Figure, they
affirm, that they have with all perspicuity demonstrated and sensibly
manifested what they undertook; and lastly, that they have obtained
their intent. Nevertheless, I believe, and thinke, I can demonstrate,
that that same Experiment proveth nothing against my Conclusion.
[Sidenote: In Experiments of Natation, the Solid is to be put
into, not upon the water.]
And first, it is false, that the Ball descends, and the Board not: for
the Board shall also descend, if you do to both the Figures, as the
words of our Question requireth; that is, if you put them both into
the water.
[Sidenote: The Question of Natation stated.]
_The words were these. That the Antagonists having an opinion, that
the Figure would alter the Solid Bodies, in relation to the
descending or not descending, ascending or not ascending in the
same_ Medium, _as_ v. gr. _in the same water, in such sort, that, for
Example, a Solid that being of a Sphericall Figure, shall descend
to the Bottom, being reduced into some other Figure, shall not
descend: I holding the contrary, do affirm, that a Corporeall Solid
Body, which reduced into a Sphericall Figure, or any other, shall
go to the Bottom, shall do the like under whatsoever other Figure,
{&}c._
[Sidenote: Place defined according to Aristotle.]
But to be in the water, implies to be placed in the water, and by
_Aristotles_ own Definition of place, to be placed, importeth to be
invironed by the Superficies of the Ambient Body, therefore, then
shall the two Figures be in the water, when the Superficies of the
water, shall imbrace and inviron them: but when the Adversaries shew
the Board of Ebony not descending to the Bottom, they put it not into
the water, but upon the water, where being by a certain impediment (as
by and by we will shew) retained, it is invironed, part by water, and
part by air, which thing is contrary to our agreement, that was, that
the Bodies should be in the water, and not part in water, and part in
air.
_The which is again made manifest, by the questions being put as
well about the things which go to the Bottom, as those which arise
from the Bottom to swimme, and who sees not that things placed in
the Bottom, must have water about them._
[Sidenote: The confutation of the Experiment in the Ebany.]
It is now to be noted, that the Board of Ebany and the Ball, put into
the water, both sink, but the Ball more swiftly, and the Board more
slowly; and slower and slower, according as it shall be more broad and
thin, and of this Tardity the breadth of the Figure is the true Cause:
But these broad Boards that slowly descend, are the same, that being
put lightly upon the water, do swimm: Therefore, if that were true
which the Adversaries affirm, the same numerical Figure, would in the
same numericall water, cause one while Rest, and another while Tardity
of Motion, which is impossible: for every perticular Figure which
descends to the Bottom[43], hath of necessity its own determinate Tardity
and slowness, proper and naturall unto it, according to which it
moveth, so that every other Tardity, greater or lesser is improper to
its nature: if, therefore, a Board, as suppose of a foot square,
descendeth naturally with six degrees of Tardity, it is impossible,
that it should descend with ten or twenty, unless some new impediment
do arrest it. Much less can it, by reason of the same Figure rest, and
wholly cease to move; but it is necessary, that when ever it resteth,
there do some greater impediment intervene than the breadth of the
Figure. Therefore, it must be somewhat else, and not the Figure, that
stayeth the Board of Ebany above water, of which Figure the only
Effect is the retardment of the Motion, according to which it
descendeth more slowly than the Ball. Let it be confessed, therefore,
rationally discoursing, that the true and sole Cause of the Ebanys
going to the Bottom, is the excess of its Gravity above the Gravity of
the water: and the Cause of the greater or less Tardity, the breadth
of this Figure, or the contractedness of that: but of its Rest, it can
by no means be allowed, that the quallity of the Figure, is the Cause
thereof: aswell, because, making the Tardity greater, according as the
Figure more dilateth, there cannot be so immense a Dilatation, to
which there may not be found a correspondent immence Tardity without
redusing it to Nullity of Motion; as, because the Figures produced by
the Antagonists for effecters of Rest, are the self same that do also
go to the Bottom.
[43] Every perticular Figure hath its own peculiar Tardity.
I will not omit another reason, founded also upon Experience, and if
I deceive not my self, manifestly concluding, how that the
Introduction of the breadth or amplitude of Figure, and the Resistance
of the water against penetration, have nothing to do in the Effect of
descending, or ascending, or resting in the water. [D]Take a piece of
wood or other Matter, of which a Ball ascends from the Bottom of the
water to the Surface, more slowly than a Ball of Ebony of the same
bignesse, so that it is manifest, that the Ball of Ebony more readily
divideth the water in descending, than the other in ascending; as for
Example, let the Wood be Walnut-tree. Then take a Board of
Walnut-tree, like and equall to that of Ebony of the Antagonists,
which swims; and if it be true, that this floats above water, by
reason of the Figure, unable through its breadth, to pierce the
Crassitude of the same, the other of Wallnut-tree, without all
question, being thrust unto the Bottom, will stay there, as less apt,
through the same impediment of Figure, to divide the said Resistance
of the water. But if we shall find, and by experience see, that not
only the thin Board, but every other Figure of the same Wallnut-tree
will return to float, as undoubtedly we shall, then I must desier my
opposers to forbear to attribute the floating of the Ebony, unto the
Figure of the Board, in regard that the Resistance of the water is the
same, as well to the ascent, as to the descent, and the force of the
Wallnut-trees ascension, is lesse than the Ebonys force in going to
the Bottom.
[D] The Figure & Resistance of the Medium against Division, have
nothing to do with the Effect of Natation or Submersion, by an
Experiment in Wallnut tree.
[Sidenote: An Experiment in Gold, to prove the non-operating of
Figure in Natation and Submersion.]
Nay, I will say more, that if we shall consider Gold in comparison of
water, we shall find, that it exceeds it in Gravity almost twenty
times, so that the Force and Impetus, wherewith a Ball of Gold goes to
the Bottom, is very great. On the contrary, there want not matters, as
Virgins Wax, and some Woods, which are not above a fiftieth part less
grave than water, whereupon their Ascension therein is very slow, and
a thousand times weaker than the _Impetus_ of the Golds descent: yet
notwithstanding, a plate of Gold swims without descending to the
Bottom, and, on the contrary, we cannot make a Cake of Wax, or thin
Board of Wood, which put in the Bottom of the Water, shall rest there
without ascending. Now if the Figure can obstruct the Penetration, and
impede the descent of Gold, that hath so great an _Impetus_, how can
it choose but suffice to resist the same Penetration of the other
matter in ascending, when as it hath scarce a thousandth part of the
_Impetus_ that the Gold hath in descending? Its therefore, necessary,
that that which suspends the thin Plate of Gold, or Board of Ebony,
upon the water, be some thing that is wanting to the other Cakes and
Boards of Matters less grave than the water; since that being put to
the Bottom, and left at liberty, they rise up to the Surface, without
any obstruction: But they want not for flatness and breadth of Figure:
Therefore, the spaciousnesse of the Figure, is not that which makes
the Gold and Ebony to swim.
And, because, that the excess of their Gravity above the Gravity of
the water, is questionless the Cause of the sinking of the flat piece
of Ebony, and the thin Plate of Gold, when they go to the Bottom,
therefore, of necessity, when they float, the Cause of their staying
above water, proceeds from Levity, which in that case, by some
Accident, peradventure not hitherto observed, cometh to meet with the
said Board, rendering it no longer as it was before, whilst it did
sink more ponderous than the water, but less.
Now, let us return to take the thin Plate of Gold, or of Silver, or
the thin Board of Ebony, and let us lay it lightly upon the water, so
that it stay there without sinking, and diligently observe its effect.
And first, see how false the assertion of _Aristotle_, and our
oponents is, to wit, that it stayeth above water, through its
unability to pierce and penetrate the Resistance of the waters
Crassitude: for it will manifestly appear, not only that the said
Plates have penetrated the water, but also that they are a
considerable matter lower than the Surface of the same, the which
continueth eminent, and maketh as it were a Rampert on all sides,
round about the said Plates, the profundity of which they stay
swimming: and, according as the said Plates shall be more grave than
the water, two, four, ten or twenty times, it is necessary, that their
Superficies do stay below the universall Surface of the water, so much
more, than the thickness of those Plates, as we shal more distinctly
shew anon. In the mean space, for the more easie understanding of what
I say, observe with me a little the present Scheme: in which let us
suppose the Surface of the water to be distended, according to the
Lines F L D B, upon which if one shall put a board of matter
specifically more grave than water, but so lightly that it submerge
not, it shall not rest any thing above, but shall enter with its whole
thickness into the water: and, moreover, shall sink also, as we see by
the Board A I, O I, whose breadth is wholly sunk into the water, the
little Ramperts of water L A and D O incompassing it, whose
Superficies is notably higher than the Superficies of the Board. See
now whether it be true, that the said Board goes not to the Bottom, as
being of Figure unapt to penetrate the Crassitude of the water.
[Illustration]
[Sidenote: Why solids having penitrated the Water, do not proceed
to a totall Submersion.]
But, if it hath already penetrated, and overcome the Continuity of
the water, & is of its own nature more grave than the said water, why
doth it not proceed in its sinking, but stop and suspend its self
within that little dimple or cavitie, which with its ponderosity it
hath made in the water? I answer; because that in submerging it self,
so far as till its Superficies come to the Levell with that of the
water, it loseth a part of its Gravity, and loseth the rest of it as
it submergeth & descends beneath the Surface of the water, which
maketh Ramperts and Banks round about it, and it sustaines this loss
by means of its drawing after it, and carrying along with it, the Air
that is above it, and by Contact adherent to it, which Air succeeds to
fill the Cavity that is invironed by the Ramperts of water; so that
that which in this case descends and is placed in the water, is not
only the Board of Ebony or Plate of Iron, but a composition of Ebony
and Air, from which resulteth a Solid no longer superiour in Gravity
to the water, as was the simple Ebony, or the simple Gold. And, if we
exactly consider, what, and how great the Solid is, that in this
Experiment enters into the water, and contrasts with the Gravity of
the same, it will be found to be all that which we find to be beneath
the Surface of the water, the which is an aggregate and Compound of a
Board of Ebony, and of almost the like quantity of Air, or a Mass
compounded of a Plate of Lead, and ten or twelve times as much Air.
But, Gentlemen, you that are my Antagonists in our Question, we
require the Identity of Matter, and the alteration only of the Figure;
therefore, you must remove that Air, which being conjoyned with the
Board, makes it become another Body less grave than the Water, and put
only the Ebony into the Water, and you shall certainly see the Board
descend to the Bottom; and, if that do not happen, you have got the
day. And to seperate the Air from the Ebony[44], there needs no more but
only to bath the Superficies of the said Board with the same Water:
for the Water being thus interposed between the Board and the Air, the
other circumfused Water shall run together without any impediment, and
shall receive into it the sole and bare Ebony, as it was to do.
[44] How to seperate the Air from Solids in demitting them into
the water.
But, me thinks I hear some of the Adversaries cunningly opposing this,
and telling me, that they will not yield, by any means, that their
Board be wetted, because the weight added thereto by the Water, by
making it heavier than it was before, draws it to the Bottom, and that
the addition of new weight is contrary to our agreement, which was,
that the Matter be the same.
To this, I answer, first; that treating of the operation of Figure
in Bodies put into the Water, none can suppose them to be put into the
Water without being wet; nor do I desire more to be done to the Board,
then I will give you leave to do to the Ball. Moreover, it is untrue,
that the Board sinks by vertue of the new Weight added to it by the
Water, in the single and slight bathing of it: for I will put ten or
twenty drops of Water upon the same Board, whilst it is sustained upon
the water; which drops, because not conjoyned with the other Water
circumfused, shall not so encrease the weight of it, as to make it
sink: but if the Board being taken out, and all the water wiped off
that was added thereto, I should bath all its Superficies with one
only very small drop, and put it again upon the water, without doubt
it shall sink, the other Water running to cover it, not being retained
by the superiour Air; which Air by the interposition of the thin vail
of water, that takes away its Contiguity unto the Ebony, shall without
Renitence be seperated, nor doth it in the least oppose the succession
of the other Water: but rather, to speak better, it shall descend
freely; because it shall be all invironed and covered with water, as
soon as its superiour Superficies, before vailed with water, doth
arrive to the Levell of the universall Surface of the said water. To
say, in the next place, that water can encrease the weight of things
that are demitted into it, is most false; for water hath no Gravity in
water[45], since it descends not: yea, if we would well consider what any
immense Mass of water doth put upon a grave Body; that is placed in
it, we shall find experimentally, that it, on the contrary, will
rather in a great part deminish the weight of it[46], and that we may be
able to lift an huge Stone from the Bottom of the water, which the
water being removed, we are not able to stir. Nor let them tell me by
way of reply, that although the superposed water augment not the
Gravity of things that are in it, yet it increaseth the ponderosity of
those that swim, and are part in the water and part in the Air, as is
seen, for Example, in a Brass Ketle[47], which whilst it is empty of
water, and replenished only with Air shall swim, but pouring of Water
therein, it shall become so grave, that it shall sink to the Bottom,
and that by reason of the new weight added thereto. To this I will
return answer, as above, that the Gravity of the Water, contained in
the Vessel is not that which sinks it to the Bottom, but the proper
Gravity of the Brass, superiour to the Specificall Gravity of the
Water: for if the Vessel were less grave than water, the Ocean would
not suffice to submerge it[48]. And, give me leave to repeat it again, as
the fundamentall and principall point in this Case, that the Air
contained in this Vessel before the infusion of the Water, was that
which kept it a-float[49], since that there was made of it, and of the
Brass, a Composition less grave than an equall quantity of Water: and
the place that the Vessel occupyeth in the Water whilst it floats, is
not equall to the Brass alone, but to the Brass and to the Air
together, which filleth that part of the Vessel that is below the
Levell of the water: Moreover, when the Water is infused, the Air is
removed, and there is a composition made of Brass and of water, more
grave _in specie_ than the simple water, but not by vertue of the
water infused, as having greater Specifick Gravity than the other
water, but through the proper Gravity of the Brass, and through the
alienation of the Air. Now, as he that should say that Brass, that by
its nature goes to the Bottom, being formed into the Figure of a
Ketle[50], acquireth from that Figure a vertue of lying in the Water
without sinking, would say that which is false; because that Brass
fashioned into any whatever Figure, goeth always to the Bottom,
provided, that that which is put into the water be simple Brass; and
it is not the Figure of the Vessel that makes the Brass to float, but
it is because that that is not purely Brass which is put into the
water, but an aggregate of Brass and of Air: so is it neither more nor
less false, that a thin Plate of Brass or of Ebony, swims by vertue of
its dilated & broad Figure: for the truth is, that it bares up without
submerging, because that that which is put in the water, is not pure
Brass or simple Ebony, but an aggregate of Brass and Air, or of Ebony
and Air. And, this is not contrary unto my Conclusion, the which,
(having many a time seen Vessels of Mettall, and thin pieces of
diverse grave Matters float, by vertue of the Air conjoyned with them)
did affirm, That Figure was not the Cause of the Natation or
Submersion of such Solids as were placed in the water. Nay more, I
cannot omit, but must tell my Antagonists, that this new conceit of
denying that the Superficies of the Board should be bathed, may beget
in a third person an opinion of a poverty of Arguments of defence on
their part, since that such bathing was never insisted upon by them in
the beginning of our Dispute, and was not questioned in the least,
being that the Originall of the discourse arose upon the swiming of
Flakes of Ice, wherein it would be simplicity to require that their
Superficies might bedry: besides, that whether these pieces of Ice be
wet or dry they alwayes swim, and as the Adversaries say, by reason of
the Figure.
[45] Water hath no Gravity in Water.
[46] Water deminisheth the Gravity of Solids immerged therein.
[47] The Experiment of a brass Ketle swiming when empty, & sinking
when full, alledged to prove that water gravitates in water,
answered.
[48] An Ocean sufficeth not to sink a Vessel specifically less
grave than water.
[49] Air, the Cause of the Natation of empty Vessels of Matters
graver _in specie_ than the water.
[50] Neither Figure, nor the breadth of Figure, is the Cause of
Natation.
Some peradventure, by way of defence, may say, that wetting the Board
of Ebony, and that in the superiour Superficies, it would, though of
it self unable to pierce and penetrate the water, be born downwards,
if not by the weight of the additionall water, at least by that desire
and propension that the superiour parts of the water have to re-unite
and rejoyn themselves: by the Motion of which parts, the said Board
cometh in a certain manner, to be depressed downwards.
[Sidenote: The Bathed Solid descends not out of any affectation
of union in the upper parts of the water.]
This weak Refuge will be removed, if we do but consider, that the
repugnancy of the inferiour parts of the water, is as great against
Dis-union, as the Inclination of its superiour parts is to union: nor
can the uper unite themselves without depressing the board, nor can it
descend without disuniting the parts of the nether Water: so that it
doth follow, by necessary consequence, that for those respects, it
shall not descend. Moreover, the same that may be said of the upper
parts of the water, may with equall reason be said of the nether,
namely, that desiring to unite, they shall force the said Board
upwards.
Happily, some of these Gentlemen that dissent from me, will wonder,
that I affirm, that the contiguous superiour Air is able to sustain
that Plate of Brass or of Silver, that stayeth above water; as if I
would in a certain sence allow the Air[51], a kind of Magnetick vertue of
sustaining the grave Bodies, with which it is contiguous. To satisfie
all I may, to all doubts, I have been considering how by some other
sensible Experiment I might demonstrate, how truly that little
contiguous and superiour Air sustaines those Solids, which being by
nature apt to descend to the Bottom, being placed lightly on the water
submerge not, unless they be first thorowly bathed; and have found,
that one of these Bodies having descended to the Bottom, by
conveighing to it (without touching it in the least) a little Air,
which conjoyneth with the top of the same, it becometh sufficient, not
only, as before to sustain it, but also to raise it, and to carry it
back to the top, where it stays and abideth in the same manner, till
such time, as the assistance of the conjoyned Air is taken away. And
to this effect, I have taken a Ball of Wax, and made it with a little
Lead, so grave, that it leasurely descends to the Bottom, making with
all its Superficies very smooth and pollite: and this being put gently
into the water, almost wholly submergeth, there remaining vissible
only a little of the very top[52], the which so long as it is conjoyned
with the Air, shall retain the Ball a-top, but the Contiguity of the
Air taken away by wetting it, it shall descend to the Bottom and there
remain. Now to make it by vertue of the Air, that before sustained it
to return again to the top, and stay there, thrust into the water a
Glass reversed with the mouth downwards, the which shall carry with it
the Air it contains, and move this towards the Ball, abasing it till
such time that you see, by the transparency of the Glass, that the
contained Air do arrive to the summity of the _B_all[53]: then gently
withdraw the Glass upwards, and you shall see the _B_all to rise, and
afterwards stay on the top of the water[54], if you carefully part the
Glass and the water without overmuch commoving and disturbing it.
There is, therefore, a certain affinity between the Air and other
Bodies, which holds them unied, so, that they seperate not without a
kind of violence. The same likewise is seen in the water[55]; for if we
shall wholly submerge some Body in it, so that it be thorowly bathed,
in the drawing of it afterwards gently out again, we shall see the
water follow it, and rise notably above its Surface, before it
seperates from it. Solid Bodies, also[56], if they be equall and alike in
Superficies, so, that they make an exact Contact without the
interposition of the least Air, that may part them in the seperation
and yield untill that the ambient _Medium_ succeeds to replenish the
place, do hold very firmly conjoyned, and are not to be seperated
without great force but, because, the Air, Water, and other Liquids,
very expeditiously shape themselves to contact with any Solid
_B_odies, so that their Superficies do exquisitely adopt themselves to
that of the Solids, without any thing remaining between them,
therefore, the effect of this Conjunction and Adherence is more
manifestly and frequently observed in them, than in hard and
inflexible Bodies, whose Superficies do very rarely conjoyn with
exactness of Contact[57]. This is therefore that Magnetick vertue, which
with firm Connection conjoyneth all Bodies, that do touch without the
interposition of flexible fluids; and, who knows, but that that a
Contact, when it is very exact, may be a sufficient Cause of the Union
and Continuity of the parts of a naturall _B_ody?
[51] _A_ Magnetisme in the _A_ir, by which it bears up those
Solids in the water, that are contiguous with it.
[52] The Effect of the Airs Contiguity in the Natation of Solids.
[53] The force of Contact.
[54] _A_n Affectation of Conjunction betwixt Solids and the Air
contiguous to them.
[55] The like affectation of Conjunction betwixt Solids & the
water.
[56] Also the like affectation and Conjunction betwixt Solids
themselves.
[57] Contact may be the Cause of the Continuity of Naturall
Bodies.
Now, pursuing my purpose, I say; that it needs not, that we have
recourse to the Tenacity, that the parts of the water have amongst
themselves, by which they resist and oppose Division, Distraction, and
Seperation, because there is no such Coherence and Resistance of
Division for if there were, it would be no less in the internall parts
than in those nearer the superiour or externall Surface, so that the
same Board, finding alwayes the same Resistance and Renitence, would
no less stop in the middle of the water than about the Surface, which
is false. Moreover, what Resistance can we place in the Continuity of
the water, if we see that it is impossible to find any Body of
whatsoever Matter, Figure or Magnitude, which being put into the
water, shall be obstructed and impeded by the Tenacity of the parts of
the water to one another, so, but that it is moved upwards or
downwards, according as the Cause of their Motion transports it? And,
what greater proof of it can we desier, than that which we daily see
in Muddy waters, which being put into Vessels to be drunk, and being,
after some hours setling[58], still, as we say, thick in the end, after
four or six dayes they are wholly setled, and become pure and clear?
Nor can their Resistance of Penetration stay those impalpable and
insensible Atomes of Sand, which by reason of their exceeding small
force, spend six dayes in descending the space of half a yard.
[58] The settlement of _M_uddy Water, proveth that that Element
hath no aversion to Division.
_Nor let them say, that the seeing of such small Bodies, consume
six dayes in descending so little a way, is a sufficient Argument
of the Waters Resistance of Division; because that is no resisting
of Division, but a retarding of Motion; and it would be simplicity
to say, that a thing opposeth Division[59], and that in the same
instant, it permits it self to be divided: nor doth the Retardation
of Motion at all favour the Adversaries cause, for that they are to
instance in a thing that wholly prohibiteth Motion, and procureth
Rest; it is necessary, therefore, to find out Bodies that stay in
the water, if one would shew its repugnancy to Division, and not
such as move in it, howbeit but slowly._
[59] Water cannot oppose division, and at the same time permit it
self to be divided.
What then is this Crassitude of the water, with which it resisteth
Division? What, I beseech you, should it be, if we (as we have said
above) with all diligence attempting the reduction of a Matter into so
like a Gravity with the water, that forming it into a dilated Plate it
rests suspended as we have said, between the two waters, it be
impossible to effect it, though we bring them to such an
Equiponderance, that as much Lead as the fourth part of a Grain of
Musterd-seed, added to the same expanded Plate, that in Air [_i. e.
out of the water_] shall weigh four or six pounds, sinketh it to the
Bottom, and being substracted, it ascends to the Surface of the water?
I cannot see, (if what I say be true, as it is most certain) what
minute vertue and force we can possibly find or imagine, to which the
Resistance of the water against Division and Penetration is not
inferiour; whereupon, we must of necessity conclude that it is
nothing: because, if it were of any sensible power, some large Plate
might be found or compounded of a Matter alike in Gravity to the
water, which not only would stay between the two waters; but,
moreover, should not be able to descend or ascend without notable
force. We may likewise collect the same from an other Experiment[60],
shewing that the Water gives way also in the same manner to
transversall Division; for if in a setled and standing water we should
place any great Mass that goeth not to the bottom, drawing it with a
single Womans Hair, we might carry it from place to place without any
opposition, and this whatever Figure it hath, though that it possess a
great space of water, as for instance, a great Beam would do moved
side-ways. Perhaps some might oppose me and say, that if the
Resistance of water against Division, as I affirm, were nothing; Ships
should not need such a force of Oars and Sayles for the moving of them
from place to place in a tranquile Sea, or standing Lake. To him that
should make such an objection, I would reply[61], that the water
contrasteth not against, nor simply resisteth Division, but a sudden
Division, and with so much greater Renitence, by how much greater the
Velocity is: and the Cause of this Resistance depends not on
Crassitude, or any other thing that absolutely opposeth Division, but
because that the parts of the water divided, in giving way to that
Solid that is moved in it, are themselves also necessitated locally to
move, some to the one side, and some to the other, and some downwards:
and this must no less be done by the waves before the Ship, or other
Body swimming through the water, than by the posteriour and
subsequent; because, the Ship proceeding forwards, to make it self a
way to receive its Bulk, it is requisite, that with the Prow it
repulse the adjacent parts of the water, as well on one hand as on the
other, and that it move them as much transversly, as is the half of
the breadth of the Hull: and the like removall must those waves make,
that succeeding the Poump do run from the remoter parts of the Ship
towards those of the middle, successively to replenish the places,
which the Ship in advancing forwards, goeth, leaving vacant. Now,
because, all Motitions are made in Time[62], and the longer in greater
time: and it being moreover true, that those Bodies that in a certain
time are moved by a certain power such a certain space, shall not be
moved the same space, and in a shorter Time, unless by a greater
Power: therefore, the broader Ships move slower than the narrower,
being put on by an equall Force: and the same Vessel requires so much
greater force of Wind, or Oars, the faster it is to move.
[60] An hair will draw a great Mass thorow the Water; which
proveth, that it hath no Resistance against transversall
Division.
[61] How ships are moved in the water.
[62] Bodies moved a certain space in a certain Time, by a
certain power, cannot be moved the same space and in a shorter
time, but by a greater power.
_But yet for all this, any great Mass swimming in a standing Lake,
may be moved by any petit force; only it is true, that a lesser
force more slowly moves it: but if the waters Resistance of
Division, were in any manner sensible, it would follow, that the
said Mass, should, notwithstanding the percussion of some sensible
force, continue immoveable, which is not so[63]. Yea, I will say
farther, that should we retire our selves into the more internall
contemplation of the Nature of water and other Fluids, perhaps we
should discover the Constitution of their parts to be such, that
they not only do not oppose Division, but that they have not any
thing in them to be divided: so that the Resistance that is
observed in moving through the water[64], is like to that which we meet
with in passing through a great Throng of People, wherein we find
impediment, and not by any difficulty in the Division, for that
none of those persons are divided whereof the Croud is composed,
but only in moving of those persons side-ways which were before
divided and disjoyned: and thus we find Resistance in thrusting a
Stick into an heap of Sand, not because any part of the Sand is to
be cut in pieces, but only to be moved and raised[65]. Two manners of
Penetration, therefore, offer themselves to us, one in Bodies,
whose parts were continuall, and here Division seemeth necessary,
the other in the aggregates of parts not continuall, but contiguous
only[66], and here there is no necessity of dividing but of moving
only. Now, I am not well resolved, whether water and other Fluids
may be esteemed to be of parts continuall or contiguous only[67]; yet I
find my self indeed inclined to think that they are rather
contiguous (if there be in Nature no other manner of aggregating,
than by the union, or by the touching of the extreams:) and I am
induced thereto by the great difference that I see between the
Conjunction of the parts of an hard or Solid Body[68], and the
Conjunction of the same parts when the same Body shall be made
Liquid and Fluid: for if, for example, I take a Mass of Silver or
other Solid and hard Mettall, I shall in dividing it into two
parts, find not only the resistance that is found in the moving of
it only[69], but an other incomparably greater, dependent on that
vertue, whatever it be, which holds the parts united: and so if we
would divide again those two parts into other two, and successively
into others and others, we should still find a like Resistance, but
ever less by how much smaller the parts to be divided shall be; but
if, lastly, employing most subtile and acute Instruments, such as
are the most tenuous parts of the Fire, we shall resolve it
(perhaps) into its last and least Particles, there shall not be
left in them any longer either Resistance of Division, or so much
as a capacity of being farther divided, especially by Instruments
more grosse than the acuities of Fire: and what Knife or Rasor put
into well melted Silver can we finde, that will divide a thing
which surpasseth the separating power of Fire? Certainly none:
because either the whole shall be reduced to the most minute and
ultimate Divisions, or if there remain parts capable still of other
Subdidivisions, they cannot receive them, but only from acuter
Divisors than Fire; but a Stick or Rod of Iron, moved in the melted
Metall, is not such a one. Of a like Constitution and Consistence,
I account the parts of Water[70], and other Liquids to be, namely,
incapable of Division by reason of their Ienuity; or if not
absolutely indivisible, yet at least not to be divided by a Board,
or other Solid Body, palpable unto the hand, the Sector being
alwayes required to be more sharp than the Solid to be cut. Solid
Bodies, therefore, do only move, and not divide the Water[71], when put
into it; whose parts being before divided to the extreamest
minuity, and therefore capable of being moved, either many of them
at once, or few, or very few, they soon give place to every small
Corpuscle, that descends in the same: for that, it being little and
light, descending in the Air, and arriving to the Surface of the
Water, it meets with Particles of Water more small, and of less
Resistance against Motion and Extrusion, than is its own prement
and extrusive force; whereupon it submergeth, and moveth such a
portion of them, as is proportionate to its Power. There is not,
therefore, any Resistance in Water against Division, nay, there is
not in it any divisible parts. I adde; moreover, that in case yet
there should be any small Resistance found (which is absolutely
false)[72] haply in attempting with an Hair to move a very great natant
Machine, or in essaying by the addition of one small Grain of Lead
to sink, or by removall of it to raise a very broad Plate of
Matter, equall in Gravity with Water, (which likewise will not
happen, in case we proceed with dexterity) we may observe that that
Resistance is a very different thing from that which the
Adversaries produce for the Cause of the Natation of the Plate of
Lead or Board of Ebony, for that one may make a Board of Ebony,
which being put upon the Water swimmeth, and cannot be submerged,
no not by the addition of an hundred Grains of Lead put upon the
same, and afterwards being bathed, not only sinks, though the said
Lead be taken away, but though moreover a quantity of Cork, or of
some other light Body fastened to it, sufficeth not to hinder it
from sinking unto the bottome: so that you see, that although it
were granted that there is a certain small Resistance of Division
found in the substance of the Water, yet this hath nothing to do
with that Cause which supports the Board above the Water, with a
Resistance an hundred times greater than that which men can find in
the parts of the Water: nor let them tell me, that only the Surface
of the Water hath such Resistance[73], and not the internall parts, or
that such Resistance is found greatest in the beginning of the
Submersion, as it also seems that in the beginning, Motion meets
with greater opposition, than in the continuance of it; because,
first, I will permit, that the Water be stirred, and that the
superiour parts be mingled with the middle[74], and inferiour parts, or
that those above be wholly removed, and those in the middle only
made use off, and yet you shall see the effect for all that, to be
still the same: Moreover, that Hair which draws a Beam through the
Water, is likewise to divide the upperparts, and is also to begin
the Motion, and yet it begins it, and yet it divides it: and
finally, let the Board of Ebony be put in the midway, betwixt the
bottome and the top of the Water, and let it there for awhile be
suspended and setled, and afterwards let it be left at liberty, and
it will instantly begin its Motion, and will continue it unto the
bottome. Nay, more, the Board so soon as it is dimitted upon the
Water, hath not only begun to move and divide it, but is for a good
space dimerged into it._
[63] The parts of Liquids, so farre from resisting Division, that
they contain not any thing that may be divided.
[64] The Resistance a Solid findeth in moving through the water,
like to that we meet with in passing through a throng of people:
[65] Or in thrusting a Stick into an heap of Sand.
[66] Two kinds of Penetration, one in Bodies continuall, the other
in Bodies only contiguous.
[67] Water consists not of continuall, but only of contiguous
parts.
[68] _Se{e} what satisfaction he hath given, as to this point, in
Lib. de Motu. Dial. 2._
[69] Great difference betwixt the Conjunction of the parts of a
Body when Solid, and when fluid.
[70] Water consists of parts that admit of no farther division.
[71] Solids dimitted into the water, do onely move, and not divide
it.
[72] If there were any Resistance of Division in water, it must
needs be small, in that it is overcome by an Hair, a Grain of
Lead, or a slight bathing of the Solid.
[73] The uper parts of the Water, do no more resist Division than
the middle or lower parts.
[74] Waters Resistance of division, not greater in the beginning
of the Submersion.
Let us receive it, therefore, for a true and undoubted Conclusion,
That the Water hath not any Renitence against simple Division, and
that it is not possible to find any Solid Body, be it of what Figure
it will, which being put into the Water, its Motion upwards or
downwards, according as it exceedeth, or shall be exceeded by the
Water in Gravity (although such excesse and difference be insensible)
shall be prohibited, and taken away, by the Crassitude of the said
Water. When, therefore, we see the Board of Ebony, or of other Matter,
more grave than the Water, to stay in the Confines of the Water and
Air, without submerging, we must have recourse to some other
Originall, for the investing the Cause of that Effect, than to the
breadth of the Figure, unable to overcome the Renitence with which the
Water opposeth Division, since there is no Resistance; and from that
which is not in being, we can expect no Action. It remains most true,
therefore, as we have said before, that this so succeds, for that that
which in such manner put upon the water, not the same Body with that
which is put _into_ the Water: because this which is put _into_ the
Water, is the pure Board of Ebony, which for that it is more grave
than the Water, sinketh, and that which is put _upon_ the Water, is a
Composition of Ebony, and of so much Air, that both together are
specifically less grave than the Water, and therefore they do not
descend.
I will farther confirm this which I say. Gentlemen, my Antagonists, we
are agreed, that the excess or defect of the Gravity of the Solid,
unto the Gravity of the Water, is the true and proper Cause of
Natation or Submersion.
[Sidenote: Great Caution to be had in experimenting the operation
of Figure in Natation.]
Now, if you will shew that besides the former Cause, there is another
which is so powerfull, that it can hinder and remove the Submersion of
those very Solids, that by their Gravity sink, and if you will say,
that this is the breadth or ampleness of Figure, you are oblieged,
when ever you would shew such an Experiment, first to make the
circumstances certain, that that Solid which you put into the Water,
be not less grave _in specie_ than it, for if you should not do so,
any one might with reason say, that not the Figure, but the Levity was
the cause of that Natation. But I say, that when you shall dimit a
Board of Ebony into the Water, you do not put therein a Solid more
grave _in specie_ than the Water, but one lighter, for besides the
Ebony, there is in the Water a Mass of Air, united with the Ebony, and
such, and so light, that of both there results a Composition less
grave than the Water: See, therefore, that you remove the Air, and put
the Ebony alone into the Water, for so you shall immerge a Solid more
grave then the Water, and if this shall not go to the Bottom, you have
well Philosophized and I ill.
Now, since we have found the true Cause of the Natation of those
Bodies, which otherwise, as being graver than the Water, would descend
to the bottom, I think, that for the perfect and distinct knowledge of
this business, it would be good to proceed in a way of discovering
demonstratively those particular Accidents that do attend these
effects, and,
PROBL. I.
[Sidenote: To finde the proportion Figures ought to have to the
waters Gravity, that by help of the contiguous Air, they may
swim.]
_To finde what proportion severall Figures of different Matters
ought to have, unto the Gravity of the Water, that so they may be
able by vertue of the Contiguous Air to stay afloat._
[Illustration]
Let, therefore, for better illustration, D F N E be a Vessell,
wherein the water is contained, and suppose a Plate or Board, whose
thickness is comprehended between the Lines I C and O S, and let it be
of Matter exceeding the water in Gravity, so that being put upon the
water, it dimergeth and abaseth below the Levell of the said water,
leaving the little Banks A I and B C, which are at the greatest height
they can be, so that if the Plate I S should but descend any little
space farther, the little Banks or Ramparts would no longer consist,
but expulsing the Air A I C B, they would diffuse themselves over the
Superficies I C, and would submerge the Plate. The height A I B C is
therefore the greatest profundity that the little Banks of water admit
of. Now I say, that from this, and from the proportion in Gravity,
that the Matter of the Plate hath to the water, we may easily finde of
what thickness, at most, we may make the said Plates, to the end, they
may be able to bear up above water: for if the Matter of the Plate or
Board I S were, for Example, as heavy again as the water, a Board of
that Matter shall be, at the most of a thickness equall to the
greatest height of the Banks, that is, as thick as A I is high: which
we will thus demonstrate. Let the Solid I S be double in Gravity to
the water, and let it be a regular Prisme, or Cylinder, to wit, that
hath its two flat Superficies, superiour and inferiour, alike and
equall, and at Right Angles with the other laterall Superficies, and
let its thickness I O be equall to the greatest Altitude of the Banks
of water: I say, that if it be put upon the water, it will not
submerge: for the Altitude A I being equall to the Altitude I O, the
Mass of the Air A B C I shall be equall to the Mass of the Solid C I O
S: and the whole Mass A O S B double to the Mass I S; And since the
Mass of the Air A C, neither encreaseth nor diminisheth the Gravity of
the Mass I S, and the Solid I S was supposed double in Gravity to the
water; Therefore as much water as the Mass submerged A O S B,
compounded of the Air A I C B, and of the Solid I O S C, weighs just
as much as the same submerged Mass A O S B: but when such a Mass of
water, as is the submerged part of the Solid, weighs as much as the
said Solid, it descends not farther, but resteth, as by (_a_)
_Archimedes_[75], and above by us, hath been demonstrated: Therefore, I S
shall descend no farther, but shall rest. And if the Solid I S shall
be Sesquialter in Gravity to the water, it shall float, as long as its
thickness be not above twice as much as the greatest Altitude of the
Ramparts of water, that is, of A I. For I S being Sesquialter in
Gravity to the water, and the Altitude O I, being double to I A, the
Solid submerged A O S B, shall be also Sesquialter in Mass to the
Solid I S. And because the Air A C, neither increaseth nor diminisheth
the ponderosity of the Solid I S: Therefore, as much water in quantity
as the submerged Mass A O S B, weighs as much as the said Mass
submerged: And, therefore, that Mass shall rest. And briefly in
generall.
[75] Of Natation Lib. 1. Prop. 3.
[Illustration]
THEOREME. VI.
[Sidenote: The proportion of the greatest thickness of Solids,
beyond which encreased they sink.]
_When ever the excess of the Gravity of the Solid above the Gravity
of the Water, shall have the same proportion to the Gravity of
the Water, that the Altitude of the Rampart, hath to the
thickness of the Solid, that Solid shall not sink, but being
never so little thicker it shall._
[Illustration]
Let the Solid I S be superior in Gravity to the water, and of such
thickness, that the Altitude of the Rampart A I, be in proportion to
the thickness of the Solid I O, as the excess of the Gravity of the
said Solid I S, above the Gravity of a Mass of water equall to the
Mass I S, is to the Gravity of the Mass of water equall to the Mass I
S. I say, that the Solid I S shall not sinke, but being never so
little thicker it shall go to the bottom: For being that as A I is to
I O, so is the Excess of the Gravity of the Solid I S, above the
Gravity of a Mass of water equall to the Mass I S, to the Gravity of
the said Mass of water: Therefore, compounding, as A O is to O I, so
shall the Gravity of the Solid I S, be to the Gravity of a Mass of
water equall to the Mass I S: And, converting, as I O is to O A, so
shall the Gravity of a Mass of water equall to the Mass I S, be to the
Gravity of the Solid I S: But as I O is to O A, so is a Mass of water
I S, to a Mass of water equall to the Mass A B S O: and so is the
Gravity of a Mass of water I S, to the Gravity of a Mass of water A S:
Therefore as the Gravity of a Mass of water, equall to the Mass I S,
is to the Gravity of the Solid I S, so is the same Gravity of a Mass
of water I S, to the Gravity of a Mass of Water A S: Therefore the
Gravity of the Solid I S, is equall to the Gravity of a Mass of water
equall to the Mass A S: But the Gravity of the Solid I S, is the same
with the Gravity of the Solid A S, compounded of the Solid I S, and of
the Air A B C I. Therefore the whole compounded Solid A O S B, weighs
as much as the water that would be comprised in the place of the said
Compound A O S B: And, therefore, it shall make an _Equilibrium_ and
rest, and that same Solid I O S C shall sinke no farther. But if its
thickness I O should be increased, it would be necessary also to
encrease the Altitude of the Rampart A I, to maintain the due
proportion: But by what hath been supposed, the Altitude of the
Rampart A I, is the greatest that the Nature of the Water and Air do
admit, without the waters repulsing the Air adherent to the
Superficies of the Solid I C, and possessing the space A I C B:
Therefore, a Solid of greater thickness than I O, and of the same
Matter with the Solid I S, shall not rest without submerging, but
shall descend to the bottome: which was to be demonstrated. In
consequence of this that hath been demonstrated, sundry and various
Conclusions may be gathered, by which the truth of my principall
Proposition comes to be more and more confirmed, and the imperfection
of all former Argumentations touching the present Question cometh to
be discovered.
_And first we gather from the things demonstrated, that,_
THEOREME VII.
[Sidenote: The heaviest Bodies may swimme.]
_All Matters, how heavy soever, even to Gold it self, the heaviest
of all Bodies, known by us, may float upon the Water._
Because its Gravity being considered to be almost twenty times greater
than that of the water, and, moreover, the greatest Altitude that the
Rampart of water can be extended to, without breaking the Contiguity
of the Air, adherent to the Surface of the Solid, that is put upon the
water being predetermined, if we should make a Plate of Gold so thin,
that it exceeds not the nineteenth part of the Altitude of the said
Rampart, this put lightly upon the water shall rest, without going to
the bottom: and if Ebony shall chance to be in sesquiseptimall
proportion more grave than the water, the greatest thickness that can
be allowed to a Board of Ebony, so that it may be able to stay above
water without sinking, would be seaven times more than the height of
the Rampart Tinn, _v. gr._ eight times more grave than water, shall
swimm as oft as the thickness of its Plate, exceeds not the 7th part
of the Altitude of the Rampart.
[Sidenote: _He elsewhere cites this as a Proposition, therefore I
make it of that number._]
And here I will not omit to note, as a second Corrollary dependent
upon the things demonstrated, that,
THEOREME VIII.
[Sidenote: Natation and Submersion, collected from the thickness,
excluding the length and breadth of Plates.]
_The Expansion of Figure not only is not the Cause of the Natation
of those grave Bodies, which otherwise do submerge, but also the
determining what be those Boards of Ebony, or Plates of Iron or
Gold that will swimme, depends not on it, rather that same
determination is to be collected from the only thickness of those
Figures of Ebony or Gold, wholly excluding the consideration of
length and breadth, as having no wayes any share in this Effect._
It hath already been manifested, that the only cause of the Natation
of the said Plates, is the reduction of them to be less grave than the
water, by means of the connexion of that Air, which descendeth
together with them, and possesseth place in the water; which place so
occupyed, if before the circumfused water diffuseth it self to fill
it, it be capable of as much water, as shall weigh equall with the
Plate, the Plate shall remain suspended, and sinke no farther.
Now let us see on which of these three dimensions of the Solid
depends the terminating, what and how much the Mass of that ought to
be, that so the assistance of the Air contiguous unto it, may suffice
to render it specifically less grave than the water, whereupon it may
rest without Submersion. It shall undoubtedly be found, that the
length and breadth have not any thing to do in the said determination,
but only the height, or if you will the thickness: for, if we take a
Plate or Board, as for Example, of Ebony, whose Altitude hath unto the
greatest possible Altitude of the Rampart, the proportion above
declared, for which cause it swims indeed, but yet not if we never so
little increase its thickness; I say, that retaining its thickness,
and encreasing its Superficies to twice, four times, or ten times its
bigness, or dminishing it by dividing it into four, or six, or twenty,
or a hundred parts, it shall still in the same manner continue to
float: but encreasing its thickness only a Hairs breadth, it will
alwaies submerge, although we should multiply the Superficies a
hundred and a hundred times. Now forasmuch as that this is a Cause,
which being added, we adde also the Effect, and being removed, it is
removed; and by augmenting or lessening the length or breadth in any
manner, the effect of going, or not going to the bottom, is not added
or removed: I conclude, that the greatness and smalness of the
Superficies hath no influence upon the Natation or Submersion. And
that the proportion of the Altitude of the Ramparts of Water, to the
Altitude of the Solid, being constituted in the manner aforesaid, the
greatness or smalness of the Superficies, makes not any variation, is
manifest from that which hath been above demonstrated, and from this,
that, _The Prisms and Cylinders which have the same Base, are in
proportion to one another as their heights._ Whence Cylinders or
Prismes[76], namely, the Board, be they great or little, so that they be
all of equall thickness, have the same proportion to their
Conterminall Air, which hath for Base the said Superficies of the
Board, and for height the Ramparts of water; so that alwayes of that
Air, and of the Board, Solids, are compounded, that in Gravity equall
a Mass of water equall to the Mass of the Solids, compounded of Air,
and of the Board: whereupon all the said Solids do in the same manner
continue afloat. We will conclude in the third place, that,
[76] Prismes and Cylinders having the same Base, are to one
another as their heights.
THEOREME. IX.
[Sidenote: All Figures of all Matters, float by hep of the
Rampart replenished with Air, and some but only touch the water.]
_All sorts of Figures of whatsoever Matter, albeit more grave than
the Water, do by Benefit of the said Rampart, not only float, but
some Figures, though of the gravest Matter, do stay wholly above
Water, wetting only the inferiour Surface that toucheth the
Water._
And these shall be all Figures, which from the inferiour Base upwards,
grow lesser and lesser; the which we shall exemplifie for this time in
Piramides or Cones, of which Figures the passions are common. We will
demonstrate therefore, that,
_It is possible to form a Piramide, of any whatsoever Matter
preposed, which being put with its Base upon the Water, rests not
only without submerging, but without wetting it more then its
Base._
For the explication of which it is requisite, that we first
demonstrate the subsequent Lemma, namely, that,
LEMMA II.
[Sidenote: Solids whose Masses are in contrary proportion to
their Specifick Gravities are equall in absolute Gravity.]
_Solids whose Masses answer in proportion contrarily to their
Specificall Gravities, are equall in Absolute Gravities._
[Illustration]
Let A C and B be two Solids, and let the Mass A C be to the Mass B,
as the Specificall Gravity of the Solid B, is to the Specificall
Gravity of the Solid A C: I say, the Solids A C and B are equall in
absolute weight, that is, equally grave. For if the Mass A C be equall
to the Mass B, then, by the Assumption, the Specificall Gravity of B,
shall be equall to the Specificall Gravity of A C, and being equall in
Mass, and of the same Specificall Gravity they shall absolutely weigh
one as much as another. But if their Masses shall be unequall, let the
Mass A C be greater, and in it take the part C, equall to the Mass B.
And, because the Masses B and C are equall; the Absolute weight of B,
shall have the same proportion to the Absolute weight of C, that the
Specificall Gravity of B, hath to the Specificall Gravity of C; or of
C A, which is the same _in specie_: But look what proportion the
Specificall Gravity of B, hath to the Specificall Gravity of C A, the
like proportion, by the Assumption, hath the Mass C A, to the Mass B,
that is, to the Mass C: Therefore, the absolute weight of B, to the
absolute weight of C, is as the Mass A C to the Mass C: But as the
Mass A C, is to the Mass C, so is the absolute weight of A C, to the
absolute weight of C: Therefore the absolute weight of B, hath the
same proportion to the absolute weight of C, that the absolute weight
of A C, hath to the absolute weight of C: Therefore, the two Solids A
C and B are equall in absolute Gravity: which was to be demonstrated.
Having demonstrated this, I say,
THEOREME X.
[Sidenote: There may be Cones and Piramides of any Matter, which
demitted into the water, rest only their Bases.]
_That it is possible of any assigned Matter, to form a Piramide or
Cone upon any Base, which being put upon the Water shall not
submerge, nor wet any more than its Base._
[Illustration]
Let the greatest possible Altitude of the Rampart be the Line D B,
and the Diameter of the Base of the Cone to be made of any Matter
assigned B C, at right angles to D B: And as the Specificall Gravity
of the Matter of the Piramide or Cone to be made, is to the
Specificall Gravity of the water, so let the Altitude of the Rampart D
B, be to the third part of the Piramide or Cone A B C, described upon
the Base, whose Diameter is B C: I say, that the said Cone A B C, and
any other Cone, lower then the same, shall rest upon the Surface of
the water B C without sinking. Draw D F parallel to B C, and suppose
the Prisme or Cylinder E C, which shall be tripple to the Cone A B C.
And, because the Cylinder D C hath the same proportion to the Cylinder
C E, that the Altitude D B, hath to the Altitude B E: But the Cylinder
C E, is to the Cone A B C, as the Altitude E B is to the third part of
the Altitude of the Cone: Therefore, by Equality of proportion, the
Cylinder D C is to the Cone A B C, as D B is to the third part of the
Altitude B E: But as D B is to the third part of B E, so is the
Specificall Gravity of the Cone A B C, to the Specificall Gravity of
the water: Therefore, as the Mass of the Solid D C, is to the Mass of
the Cone A _B_ C, so is the Specificall Gravity of the said Cone, to
the Specificall Gravity of the water: Therefore, by the precedent
Lemma, the Cone A B C weighs in absolute Gravity, as much as a Mass of
Water equall to the Mass D C: But the water which by the imposition of
the Cone A B C, is driven out of its place, is as much as would
precisely lie in the place D C, and is equall in weight to the Cone
that displaceth it: Therefore, there shall be an _Equilibrium_, and
the Cone shall rest without farther submerging. And its manifest,
COROLARY I.
[Sidenote: Amongst Cones of the same Base, those of least
Altitude shall sink the least.]
_That making upon the same Basis, a Cone of a less Altitude, it
shall be also less grave, and shall so much the more rest without
Submersion._
COROLARY II.
[Sidenote: There may be Cones and Piramides of any Matter, which
demitted with the Point downwards do float atop.]
_It is manifest, also, that one may make Cones and Piramids of any
Matter whatsoever, more grave than the water, which being put
into the water, with the Apix or Point downwards, rest without
Submersion._
Because if we reassume what hath been above demonstrated, of Prisms
and Cylinders, and that on Bases equall to those of the said
Cylinders, we make Cones of the same Matter, and three times as high
as the Cylinders, they shall rest afloat, for that in Mass and Gravity
they shall be equall to those Cylinders, and by having their Bases
equall to those of the Cylinders, they shall leave equall Masses of
Air included within the Ramparts. This, which for Example sake hath
been demonstrated, in Prisms, Cylinders, Cones and Piramids, might be
proved in all other Solid Figures, but it would require a whole Volume
(such is the multitude and variety of their Symptoms and Accidents) to
comprehend the particuler demonstration of them all, and of their
severall Segments: but I will to avoid prolixity in the present
Discourse, content my self, that by what I have declared every one of
ordinary Capacity may comprehend, that there is not any Matter so
grave, no not Gold it self, of which one may not form all sorts of
Figures, which by vertue of the superiour Air adherent to them, and
not by the Waters Resistance of Penetration, do remain afloat, so that
they sink not. Nay, farther, I will shew, for removing that Error,
that,
THEOREME XI.
[Sidenote: A Piramide or Cone, demitted with the Point downwards
shal swim, with its Base downward shall sink.]
_A Piramide or Cone put into the Water, with the Point downward
shall swimme, and the same put with the Base downwards shall
sinke, and it shall be impossible to make it float._
Now the quite contrary would happen, if the difficulty of Penetrating
the water, were that which had hindred the descent, for that the said
Cone is far apter to pierce and penetrate with its sharp Point, than
with its broad and spacious Base.
And, to demonstrate this, let the Cone be _A B C_, twice as grave as
the water, and let its height be tripple to the height of the Rampart
_D A E C_: I say, first, that being put lightly into the water with
the Point downwards, it shall not descend to the bottom: for the
Aeriall Cylinder contained betwixt the Ramparts _D A C E_, is equall
in Mass to the Cone _A B C_; so that the whole Mass of the Solid
compounded of the Air _D A C E_, and of the Cone _A B C_, shall be
double to the Cone _A C B_: And, because the Cone _A B C_ is supposed
to be of Matter double in Gravity to the water, therefore as much
water as the whole Masse _D A B C E_, placed beneath the Levell of the
water, weighs as much as the Cone _A B C_: and, therefore, there shall
be an _Equilibrium_, and the Cone _A B C_ shall descend no lower. Now,
I say farther, that the same Cone placed with the Base downwards,
shall sink to the bottom, without any possibility of returning again,
by any means to swimme.
[Illustration]
Let, therefore, the Cone be _A B D_, double in Gravity to the water,
and let its height be tripple the height of the Rampart of water L B:
It is already manifest, that it shall not stay wholly out of the
water, because the Cylinder being comprehended betwixt the Ramparts _L
B D P_, equall to the Cone _A B D_, and the Matter of the Cone, beig
double in Gravity to the water, it is evident that the weight of the
said Cone shall be double to the weight of the Mass of water equall to
the Cylinder _L B D P_: Therefore it shall not rest in this state, but
shall descend.
[Illustration]
COROLARY I.
[Sidenote: Much less shall the said Cone swim, if one immerge a
part thereof.]
_I say farther; that much lesse shall the said Cone stay afloat, if
one immerge a part thereof._
Which you may see, comparing with the water as well the part that
shall immerge as the other above water. Let us therefore of the Cone A
B D, submergeth part N T O S, and advance the Point N S F above water.
The Altitude of the Cone F N S, shall either be more than half the
whole Altitude of the Cone F T O, or it shall not be more: if it shall
be more than half, the Cone F N S shall be more than half of the
Cylinder E N S C: for the Altitude of the Cone F N S, shall be more
than Sesquialter of the Altitude of the Cylinder E N S C: And, because
the Matter of the Cone is supposed to be double in Specificall Gravity
to the water, the water which would be contained within the Rampart E
N S C, would be less grave absolutely than the Cone F N S; so that the
whole Cone F N S cannot be sustained by the Rampart: But the part
immerged N T O S, by being double in Specificall Gravity to the water,
shall tend to the bottom: Therefore, the whole _C_one F T O, as well
in respect of the part submerged, as the part above water shall
descend to the bottom. But if the Altitude of the Point F N S, shall
be half the Altitude of the whole Cone F T O, the same Altitude of the
said Cone F N S shall be Sesquialter to the Altitude E N: and,
therefore, E N S C shall be double to the Cone F N S; and as much
water in Mass as the _C_ylinder E N S C, would weigh as much as the
part of the _C_one F N S. But, because the other immerged part N T O
S, is double in Gravity to the water, a Mass of water equall to that
compounded of the _C_ylinder E N S C, and of the Solid N T O S, shall
weigh less than the _C_one F T O, by as much as the weight of a Mass
of water equall to the Solid N T O S: Therefore, the _C_one sha{l}l
also descend. Again, because the Solid N T O S, is septuple to the
Cone F N S, to which the _C_ylinder E S is double, the proportion of
the Solid N T O S, shall be to the _C_ylinder E N S C, as seaven to
two: Therefore, the whole Solid compounded of the _C_ylinder E N S C,
and of the Solid N T O S, is much less than double the Solid N T O S:
Therefore, the single Solid N T O S, is much graver than a Mass of
water equall to the Mass, compounded of the _C_ylinder E N S C, and of
N T O S.
COROLARY II.
[Sidenote: Part of the Cones towards the Cuspis removed, it shall
still sink.]
_From whence it followeth, that though one should remove and take
away the part of the Cone F N S, the sole remainder N T O S would
go to the bottom._
COROLARY III.
[Sidenote: The more the Cone is immerged, the more impossible is
its floating.]
_And if we should more depress the Cone F T O, it would be so much
the more impossible that it should sustain it self afloat, the
part submerged N T O S still encreasing, and the Mass of Air
contained in the Rampart diminishing, which ever grows less, the
more the Cone submergeth._
That Cone, therefore, that with its Base upwards, and its _Cuspis_
downwards doth swimme, being dimitted with its Base downward must of
necessity sinke. They have argued farre from the truth, therefore, who
have ascribed the cause of Natation to waters resistance of Division,
as to a passive principle, and to the breadth of the Figure, with
which the division is to be made, as the Efficient.
I come in the fourth place, to collect and conclude the reason of
that which I have proposed to the Adversaries, namely,
THEOREME XII.
[Sidenote: Solids of any Figure & greatnesse, that naturally
sink, may by help of the Air in the Rampart swimme.]
_That it is possible to fo{r}m Solid Bodies, of what Figure and
greatness soever, that of their own Nature goe to the Bottome;
But by the help of the Air contained in the Rampart, rest without
submerging._
[Illustration]
The truth of this Proposition is sufficiently manifest in all those
Solid Figures, that determine in their uppermost part in a plane
Superficies: for making such Figures of some Matter specifically as
grave as the water, putting them into the water, so that the whole
Mass be covered, it is manifest, that they shall rest in all places,
provided, that such a Matter equall in weight to the water, may be
exactly adjusted: and they shall by consequence, rest or lie even with
the Levell of the water, without making any Rampart. If, therefore, in
respect of the Matter, such Figures are apt to rest without
submerging, though deprived of the help of the Rampart, it is
manifest, that they may admit so much encrease of Gravity, (without
encreasing their Masses) as is the weight of as much water as would be
contained within the Rampart, that is made about their upper plane
Surface: by the help of which being sustained, they shall rest afloat,
but being bathed, they shall descend, having been made graver than the
water. In Figures, therefore, that determine above in a plane, we may
cleerly comprehend, that the Rampart added or removed, may prohibit or
permit the descent: but in those Figures that go lessening upwards
towards the top, some Persons may, and that not without much seeming
Reason, doubt whether the same may be done, and especially by those
which terminate in a very acute Point, such as are your Cones and
small Piramids. Touching these, therefore, as more dubious than the
rest, I will endeavour to demonstrate, that they also lie under the
same Accident of going, or not going to the Bottom, be they of any
whatever bigness. Let therefore the Cone be A B D, made of a matter
specifically as grave as the water; it is manifest that being put all
under water, it shall rest in all places (alwayes provided, that it
shall weigh exactly as much as the water, which is almost impossible
to effect) and that any small weight being added to it, it shall sink
to the bottom: but if it shall descend downwards gently, I say, that
it shall make the Rampart E S T O, and that there shall stay out of
the water the point A S T, tripple in height to the Rampart E S: which
is manifest, for the Matter of the Cone weighing equally with the
water, the part submerged _S B D T_, becomes indifferent to move
downwards or upwards; and the Cone _A S T_, being equall in Mass to
the water that would be contained in the concave of the Rampart _E S T
O_, shall be also equall unto it in Gravity: and, therefore, there
shall be a perfect _Equilibrium_, and, consequently, a Rest. Now here
ariseth a doubt, whether the Cone _A B D_ may be made heavier, in such
sort, that when it is put wholly under water, it goes to the bottom,
but yet not in such sort, as to take from the Rampart the vertue of
sustaining it that it sink not, and, the reason of the doubt is this:
that although at such time as the Cone _A B D_ is specifically as
grave as the water, the Rampart _E S T O_ sustaines it, not only when
the point _A S T_ is tripple in height to the Altitude of the Rampart
_E S_, but also when a lesser part is above water; [for although in
the Descent of the Cone the Point _A S T_ by little and little
diminisheth, and so likewise the Rampart _E S T O_, yet the Point
diminisheth in greater proportion than the Rampart, in that it
diminisheth according to all the three Dimensions, but the Rampart
according to two only, the Altitude still remaining the same; or, if
you will, because the Cone _S {A} T_ goes diminishing, according to
the proportion of the cubes of the Lines that do successively become
the Diameters of the Bases of emergent Cones, and the Ramparts
diminish according to the proportion of the Squares of the same Lines;
whereupon the proportions of the Points are alwayes Sesquialter of the
proportions of the Cylinders, contained within the Rampart; so that
if, for Example, the height of the emergent Point were double, or
equall to the height of the Rampart, in these cases, the Cylinder
contained within the Rampart, would be much greater than the said
Point, because it would be either sesquialter or tripple, by reason of
which it would perhaps serve over and above to sustain the whole Cone,
since the part submerged would no longer weigh any thing;] yet,
nevertheless, when any Gravity is added to the whole Mass of the Cone,
so that also the part submerged is not without some excesse of Gravity
above the Gravity of the water, it is not manifest, whether the
Cylinder contained within the Rampart, in the descent that the Cone
shall make, can be reduced to such a proportion unto the emergent
Point, and to such an excesse of Mass above the Mass of it, as to
compensate the excesse of the Cones Specificall Gravity above the
Gravity of the water: and the Scruple ariseth, because that howbeit in
the descent made by the Cone, the emergent Point _A S T_ diminisheth,
whereby there is also a diminution of the excess of the Cones Gravity
above the Gravity of the water, yet the case stands so, that the
Rampart doth also contract it self, and the Cylinder contained in it
doth deminish. Nevertheless it shall be demonstrated, how that the
Cone _A B D_ being of any supposed bignesse, and made at the first of
a Matter exactly equall in Gravity to the Water, if there may be
affixed to it some Weight, by means of which i{t} may descend to the
bottom, when submerged under water, it may also by vertue of the
Rampart stay above without sinking.
[Illustration]
[Illustration]
Let, therefore, the Cone _A B D_ be of any supposed greatnesse, and
alike in specificall Gravity to the water. It is manifest, that being
put lightly into the water, it shall rest without descending; and it
shall advance above water, the Point _A S T_, tripple in height to the
height of the Rampart _E S_: Now, suppose the Cone _A B D_ more
depressed, so that it advance above water, only the Point _A I R_,
higher by half than the Point _A S T_, with the Rampart about it _C I
R N_. And, because, the Cone _A B D_ is to the Cone _A I R_, as the
cube of the Line _S T_ is to the cube of the Line _I R_, but the
Cylinder _E S T O_, is to the Cylinder _C I R N_, as the Square of _S
T_ to the Square of _I R_, the Cone _A S T_ shall be Octuple to the
Cone _A I R_, and the Cylinder _E S T O_, quadruple to the Cylinder _C
I R N_: But the Cone _A S T_, is equall to the Cylinder _E S T O_:
Therefore, the Cylinder _C I R N_, shall be double to the Cone _A I
R_: and the water which might be contained in the Rampart _C I R N_,
would be double in Mass and in Weight to the Cone _A I R_, and,
therefore, would be able to sustain the double of the Weight of the
Cone _A I R_: Therefore, if to the whole Cone _A B D_, there be added
as much Weight as the Gravity of the Cone _A I R_, that is to say, the
eighth part of the weight of the Cone _A S T_, it also shall be
sustained by the Rampart _C I R N_, but without that it shall go to
the bottome: the Cone _A B D_, being, by the addition of the eighth
part of the weight of the Cone _A S T_, made specifically more grave
than the water. But if the Altitude of the Cone _A I R_, were two
thirds of the Altitude of the Cone _A S T_, the Cone _A S T_ would be
to the Cone _A I R_, as twenty seven to eight; and the Cylinder _E S T
O_, to the Cylinder _C I R N_, as nine to four, that is, as twenty
seven to twelve; and, therefore, the Cylinder _C I R N_, to the Cone
_A I R_, as twelve to eight; and the excess of the Cylinder _C I R N_,
above the Cone _A I R_, to the Cone _A S T_, as four to twenty seven:
therefore if to the Cone _A B D_ be added so much weight as is the
four twenty sevenths of the weight of the Cone _A S T_, which is a
little more then its seventh part, it also shall continue to swimme,
and the height of the emergent Point shall be double to the height of
the Rampart. This that hath been demonstrated in Cones, exactly holds
in Piramides, although the one or the other should be very sharp in
their Point or Cuspis[77]: From whence we conclude, that the same Accident
shall so much the more easily happen in all other Figures, by how much
the less sharp the Tops shall be, in which they determine, being
assisted by more spacious Ramparts.
[77] Natatio{n} easiest effected in Figures broad toward the top.
THEOREME XIII.
[Sidenote: All Figures sink or swim, upon bathing or not bathing
of their tops.]
_All Figures, therefore, of whatever greatnesse, may go, and not
go, to the Bottom, according as their Sumities or Tops shall be
bathed or not bathed._
And this Accident being common to all sorts of Figures, without
exception of so much as one. Figure hath, therefore, no part in the
production of this Effect, of sometimes sinking, and sometimes again
not sinking, but only the being sometimes conjoyned to, and sometimes
seperated from, the supereminent Air: which cause, in fine, who so
shall rightly, and, as we say, with both his Eyes, consider this
business, will find that it is reduced to, yea, that it really is the
same with, the true, Naturall and primary cause of Natation or
Submersion; to wit, the excess or deficiency of the Gravity of the
water, in relation to the Gravity of that Solid Magnitude, that is
demitted into the water. For like as a Plate of Lead, as thick as the
back of a Knife, which being put into the water by it self alone goes
to the bottom, if upon it you fasten a piece of Cork four fingers
thick, doth continue afloat, for that now the Solid that is demitted
in the water, is not, as before, more grave than the water, but less,
so the Board of Ebony, of its own nature more grave than water; and,
therefore, descending to the bottom, when it is demitted by it self
alone into the water, if it shall be put upon the water, conjoyned
with an Expanded vail of Air, that together with the Ebony doth
descend, and that it be such, as that it doth make with it a compound
less grave than so much water in Mass, as equalleth the Mass already
submerged and depressed beneath the Levell of the waters Surface, it
shall not descend any farther, but shall rest, for no other than the
universall and most common cause, which is that Solid Magnitudes, less
grave _in specie_ than the water, go not to the bottom.
So that if one should take a Plate of Lead, as for Example, a finger
thick, and an handfull broad every way, and should attempt to make it
swimme, with putting it lightly on the water, he would lose his
Labour, because that if it should be depressed an Hairs breadth beyond
the possible Altitude of the Ramparts of water, it would dive and
sink; but if whilst it is going downwards, one should make certain
Banks or Ramparts about it, that should hinder the defusion of the
water upon the said Plate, the which Banks should rise so high, as
that they might be able to contain as much water, as should weigh
equally with the said Plate, it would, witho{u}t all Question, descend
no lower, but would rest, as being sustained by vertue of the Air
contained within the aforesaid Ramparts: and, in short, there would be
a Vessell by this means formed with the bottom of Lead. But if the
thinness of the Lead shall be such, that a very small height of
Rampart would suffice to contain so much Air, as might keep it afloat,
it shall also rest without the Artificiall Banks or Ramparts, but yet
not without the Air, because the Air by it self makes Banks sufficient
for a small height, to resist the Superfusion of the water: so that
that which in this case swimmes, is as it were a Vessell filled with
Air, by vertue of which it continueth afloat.
I will, in the last place, with an other Experime{n}t, attempt to
remove all difficulties, if so be there should yet be any doubt left
in any one, touching the opperation of this [E]Continuity of the Air,
with the thin Plate which swims, and afterwards put an end to this
part of my discourse.
[E] Or rather Contiguity,
I suppose my self to be questioning with some of my Oponents.
Whether Figure have any influence upon the encrease or diminution of
the Resistance in any Weight against its being raised in the Air[78]; and
I suppose, that I am to maintain the Affirmative, asserting that a
Mass of Lead, reduced to the Figure of a Ball, shall be raised with
less force, then if the same had been made into a thinne and broad
Plate, because that it in this spacious Figure, hath a great quantity
of Air to penetrate, and in that other, more compacted and contracted
very little: and to demonstrate the truth of such my Opinion, I will
hang in a small thred first the Ball or Bullet, and put that into the
water, tying the thred that upholds it to one end of the Ballance that
I hold in the Air, and to the other end I by degrees adde so much
Weight, till that at last it brings up the Ball of Lead out of the
water: to do which, suppose a Gravity of thirty Ounces sufficeth; I
afterwards reduce the said Lead into a flat and thinne Plate, the
which I likewise put into the water, suspended by three threds, which
hold it parallel to the Surface of the water, and putting in the same
manner, Weights to the other end, till such time as the Plate comes to
be raised and drawn out of the water: I finde that thirty six ounces
will not suffice to seperate it from the water, and raise it thorow
the Air: and arguing from this Experiment, I affirm, that I have fully
demonstrated the truth of my Proposition. Here my Oponents desires me
to look down, shewing me a thing which I had not before observed, to
wit, that in the Ascent of the Plate out of the water, it draws after
it another Plate (_if I may so call it_) of water, which before it
divides and parts from the inferiour Surface of the Plate of Lead, is
raised above the Levell of the other water, more than the thickness of
the back of a Knife: Then he goeth to repeat the Experiment with the
Ball, and makes me see, that it is but a very small quantity of water,
which cleaves to its compacted and contracted Figure: and then he
subjoynes, that its no wonder, if in seperating the thinne and broad
Plate from the water, we meet with much greater Resistance, than in
seperating the Ball, since together with the Plate, we are to raise a
great quantity of water, which occurreth not in the Ball: He telleth
me moreover, how that our Question is, whether the Resistance of
Elevation be greater in a dilated Plate of Lead, than in a Ball, and
not whether more resisteth a Plate of Lead with a great quantity of
water, or a Ball with a very little water: He sheweth me in the close,
that the putting the Plate and the Ball first into the water, to make
proofe thereby of their Resistance in the Air, is besides our case,
which treats of Elivating in the Air, and of things placed in the Air,
and not of the Resistance that is made in the Confines of the Air and
water, and by things which are part in Air and part in water: and
lastly, they make me feel with my hand, that when the thinne Plate is
in the Air, and free from the weight of the water, it is raised with
the very same Force that raiseth the Ball. Seeing, and understanding
these things, I know not what to do, unless to grant my self
convinced, and to thank such a Friend, for having made me to see that
which I never till then observed: and, being advertised by this same
Accident, to tell my Adversaries, that our Question is, whether a
Board and a Ball of Ebony, equally go to the bottom in water, and not
a Ball of Ebony and a Board of Ebony, joyned with another flat Body of
Air: and, farthermore, that we speak of sinking, and not sinking to
the bottom, in water, and not of that which happeneth in the Confines
of the water and Air to Bodies that be part in the Air, and part in
the water; nor much less do we treat of the greater or lesser Force
requisite in seperating this or that Body from the Air; not omitting
to tell them, in the last place, that the Air doth resist, and
gravitate downwards in the water, just so much as the water (if I may
so speak) gravitates and resists upwards in the Air, and that the same
Force is required to sinke a Bladder under water, that is full of Air,
as to raise it in the Air, being full of water, removing the
consideration of the weight of that Filme or Skinne, and considering
the water and the Air only. And it is likewise true, that the same
Force is required to sink a Cup or such like Vessell under water,
whilst it is full of Air, as to raise it above the Superficies of the
water, keeping it with the mouth downwards; whilst it is full of
water, which is constrained in the same manner to follow the Cup which
contains it, and to rise above the other water into the Region of the
Air, as the Air is forced to follow the same Vessell under the Surface
of the water, till that in this c{a}se the water, surmounting the
brimme of the Cup, breaks in, driving thence the Air, and in that
case, the said brimme coming out of the water, and arriving to the
Confines of the Air, the water falls down, and the Air sub-enters to
fill the cavity of the Cup: upon which ensues, that he no less
transgresses the Articles of the _Convention_, who produceth a Plate
conjoyned with much Air, to see if it descend to the bottom in water,
then he that makes proof of the Resistance against Elevation in Air
with a Plate of Lead, joyned with a like quantity of water.
[78] An Experiment of the operation of Figures, in encreasing or
lessening of the Airs Resistance of Division.
[Sidenote: _Aristotles_ opinion touching the Operation of Figure
examined.]
I have said all that I could at present think of, to maintain the
Assertion I have undertook. It remains, that I examine that which
_Aristotle_ hath writ of this matter towards the end of his Book De
Caelo[79]; wherein I shall note two things: the one that it being true as
hath been demonstrated, that Figure hath nothing to do about the
moving or not moving it self upwards or downwards, its seemes that
_Aristotle_ at his first falling upon this Speculation, was of the
same opinion, as in my opinion may be collected from the examination
of his words. 'Tis true, indeed, that in essaying afterwards to render
a reason of such effect, as not having in my conceit hit upon the
right, (which in the second place I will examine) it seems that he is
brought to admit the largenesse of Figure, to be interessed in this
operation. As to the first particuler, hear the precise words of
_Aristotle_.
[79] _Aristot. de Caelo_ Lib. 4. Cap 6.
_Figures are not the Causes of moving simply upwards or downwards, but
of moving more slowly or swiftly[80][81], and by what means this comes to
pass, it is not difficult to see._
[80] _Aristotle_ makes not Figure the cause of Motion absolutely,
but of swift or slow motion,
[81] Lib. 4. Cap. 6: Text. 42.
Here first I note, that the terms being four, which fall under the
present consideration, namely, Motion, Rest, Slowly and Swiftly: And
_Aristotle_ naming figures as Causes of Tardity and Velocity,
excluding them from being the Cause of absolute and simple Motion, it
seems necessary, that he exclude them on the other side, from being
the Cause of Rest, so that his meaning is this. Figures are not the
Causes of moving or not moving absolutely, but of moving quickly or
slowly: and, here, if any should say the mind of _Aristotle_ is to
exclude Figures from being Causes of Motion, but yet not from being
Causes of Rest, so that the sence would be to remove from Figures,
there being the Causes of moving simply, but yet not there being
Causes of Rest, I would demand, whether we ought with _Aristotle_ to
understand, that all Figures universally, are, in some manner, the
causes of Rest in those Bodies, which otherwise would move, or else
some particular Figures only, as for Example, broad and thinne
Figures: If all indifferently, then every Body shall rest: because
every Body hath some Figure, which is false; but if some particular
Figures only may be in some manner a Cause of Rest, as, for Example,
the broad, then the others would be in some manner the Causes of
Motion: for if from seeing some Bodies of a contracted Figure move,
which after dilated into Plates rest, may be inferred, that the
Amplitude of Figure hath a part in the Cause of that Rest; so from
seeing such like Figures rest, which afterwards contracted move, it
may with the same reason be affirmed, that the united and contracted
Figure, hath a part in causing Motion, as the remover of that which
impeded it: The which again is directly opposite to what _Aristotle_
saith, namely, that Figures are not the Causes of Motion. Besides, if
_Aristotle_ had admitted and not excluded Figures from being Causes of
not moving in some Bodies, which moulded into another Figure would
move, he would have impertinently propounded in a dubitative manner,
in the words immediately following, whence it is, that the large and
thinne Plates of Lead or Iron, rest upon the water, since the Cause
was apparent, namely, the Amplitude of Figure. Let us conclude,
therefore, that the meaning of _Aristotle_ in this place is to affirm,
that Figures are not the Causes of absolutely moving or not moving,
but only of moving swiftly or slowly: which we ought the rather to
believe, in regard it is indeed a most true conceipt and opinion. Now
the mind of _Aristotle_ being such, and appearing by consequence,
rather contrary at the first sight, then favourable to the assertion
of the Oponents, it is necessary, that their Interpretation be not
exactly the same with that, but such, as being in part understood by
some of them, and in part by others, was set down: and it may easily
be indeed so, being an Interpretation consonent to the sence of the
more famous Interpretors, which is, that the Adverbe _Simply_ or
_Absolutely_, put in the Text, ought not to be joyned to the Verbe to
_Move_, but with the Noun _Causes_: so that the purport of
_Aristotles_ words, is to affirm, That Figures are not the Causes
absolutely of moving or not moving, but yet are Causes _Secundum
quid_, _viz._ in some sort; by which means, they are called Auxiliary
and Concomitant Causes: and this Proposition is received and asserted
as true by _Signor Buonamico Lib. 5. Cap. 28._ where he thus writes.
_There are other Causes concomitant, by which some things float, and
others sink, among which the Figures of Bodies hath the first place_,
&c.
Concerning this Proposition, I meet with many doubts and difficulties,
for which me thinks the words of _Aristotle_ are not capable of such a
construction and sence, and the difficulties are these.
First in the order and disposure of the words of _Aristotle_, the
particle _Simpliciter_, or if you will _absolute_, is conjoyned with
the Verb _to move_, and seperated from the Noun _Causes_, the which is
a great presumption in my favour, seeing that the writing and the Text
saith, Figures are not the Cause of moving simply upwards or
downwards, but of quicker or slower Motion: and, saith not, Figures
are not simply the Causes of moving upwards or downwards, and when the
words of a Text receive, transposed, a sence different from that which
they sound, taken in the order wherein the Author disposeth them, it
is not convenient to inverte them. And who will affirm that
_Aristotle_ desiring to write a Proposition, would dispose the words
in such sort, that they should import a different, nay, a contrary
sence? contrary, I say, because understood as they are written; they
say, that Figures are not the Causes of Motion, but inverted, they
say, that Figures are the Causes of Motion, &c.
Moreover, if the intent of _Aristotle_ had been to say, that Figures
are not simply the Causes of moving upwards or downwards, but only
Causes _Secundum quid_, he would not have adjoyned those words, _but
they are Causes of the more swift or slow Motion_; yea, the subjoining
this would have been not only superfluous but false, for that the
whole tenour of the Proposition would import thus much. Figures are
not the absolute Causes of moving upwards or downwards, but are the
absolute Cause of the swift or slow Motion; which is not true: because
the primary Causes of greater or lesser Velocity, are by _Aristotle_
in the 4th of his _Physicks_, _Text. 71._ attributed to the greater or
lesser Gravity of Moveables, compared among themselves, and to the
greater or lesser Resistance of the _Medium's_, depending on their
greater or less Crassitude: and these are inserted by _Aristotle_ as
the primary Causes; and these two only are in that place nominated:
and Figure comes afterwards to be considered, _Text. 74._ rather as an
Instrumentall Cause of the force of the Gravity, the which divides
either with the Figure, or with the _Impetus_; and, indeed, Figure by
it self without the force of Gravity or Levity, would opperate
nothing.
I adde, that if _Aristotle_ had an opinion that Figure had been in
some sort the Cause of moving or not moving, the inquisition which he
makes immediately in a doubtfull manner, whence it comes, that a Plate
of Lead flotes, would have been impertinent; for if but just before he
had said, that Figure was in a certain sort the Cause of moving or not
moving, he needed not to call in Question, by what Cause the Plate of
Lead swims, and then ascribing the Cause to its Figure; and framing a
discourse in this manner. Figure is a Cause _Secundum quid_ of not
sinking: but, now, if it be doubted, for what Cause a thin Plate of
Lead goes not to the bottom; it shall be answered, that that proceeds
from its Figure: a discourse which would be indecent in a Child, much
more in _Aristotle_; For where is the occasion of doubting? And who
sees not, that if _Aristotle_ had held, that Figure was in some sort a
Cause of Natation, he would without the least Hesitation have writ;
That Figure is in a certain sort the Cause of Natation, and therefore
the Plate of Lead in respect of its large and expatiated Figure swims;
but if we take the proposition of _Aristotle_ as I say, and as it is
written, and as indeed it is true, the ensuing words come in very
oppositely, as well in the introduction of swift and slow, as in the
question, which very pertinently offers it self, and would say thus
much.
Figures are not the Cause of moving or not moving simply upwards or
downwards, but of moving more quickly or slowly: But if it be so, the
Cause is doubtfull, whence it proceeds, that a Plate of Lead or of
Iron broad and thin doth swim, &c. And the occasion of the doubt is
obvious, because it seems at the first glance, that the Figure is the
Cause of this Natation, since the same Lead, or a less quantity, but
in another Figure, goes to the bottom, and we have already affirmed,
that the Figure hath no share in this effect.
Lastly, if the intent of _Aristotle_ in this place had been to say,
that Figures, although not absolutely, are at least in some measure
the Cause of moving or not moving: I would have it considered, that he
names no less the Motion upwards, than the other downwards: and
because in exemplifying it afterwards, he produceth no other
Experiments than of a Plate of Lead, and Board of Ebony, Matters that
of their own Nature go to the bottom, but by vertue (as our
Adversaries say) of their Figure, rest afloat; it is fit that they
should produce some other Experiment of those Matters, which by their
Nature swims, but retained by their Figure rest at the bottom. But
since this is impossible to be done, we conclude, that _Aristotle_ in
this place, hath not attributed any action to the Figure of simply
moving or not moving.
But though he hath exquisitely Philosophiz'd, in investigating the
solution of the doubts he proposeth, yet will I not undertake to
maintain, rather various difficulties, that present themselves unto
me, give me occasion of suspecting that he hath not entirely displaid
unto us, the true Cause of the present Conclusion: which difficulties
I will propound one by one, ready to change opinion, whenever I am
shewed, that the Truth is different from what I say; to the confession
whereof I am much more inclinable than to contradiction.
[Sidenote: _Aristotle_ erred in affirming a Needle dimitted long
wayes to sink.]
_Aristotle_ having propounded the Question, whence it proceeds, that
broad Plates of Iron or Lead, float or swim; he addeth (as it were
strengthening the occasion of doubting) forasmuch as other things,
less, and less grave, be they round or long, as for instance a Needle
go to the bottom. Now I here doubt, or rather am certain that a Needle
put lightly upon the water, rests afloat, no less than the thin Plates
of Iron or Lead. I cannot believe, albeit it hath been told me, that
some to defend _Aristotle_ should say, that he intends a Needle
demitted not longwayes but endwayes, and with the Point downwards;
nevertheless, not to leave them so much as this, though very weak
refuge, and which in my judgement _Aristotle_ himself would refuse, I
say it ought to be understood, that the Needle must be demitted,
according to the Dimension named by _Aristotle_, which is the length:
because, if any other Dimension than that which is named, might or
ought to be taken, I would say, that even the Plates of Iron and Lead,
sink to the bottom, if they be put into the water edgewayes and not
flatwayes. But because _Aristotle_ saith, broad Figures go not to the
bottom, it is to be understood, being demitted broadwayes: and,
therefore, when he saith, long Figures as a Needle, albeit light, rest
not afloat, it ought to be understood of them when demitted longwayes.
_Moreover, to say that_ Aristotle _is to be understood of the
Needle demitted with the Point downwards, is to father upon him a
great impertinency; for in this place he saith, that little
Particles of Lead or Iron, if they be round or long as a Needle, do
sink to the bottome; so that by his Opinion, a Particle or small
Grain of Iron cannot swim: and if he thus believed, what a great
folly would it be to subjoyn, that neither would a Needle demitted
endwayes swim? And what other is such a Needle, but many such like
Graines accumulated one upon another? It was too unworthy of such a
man to say, that one single Grain of Iron could not swim, and that
neither can it swim, though you put a hundred more upon it._
Lastly, either _Aristotle_ believed, that a Needle demitted longwayes
upon the water, would swim, or he believed that it would not swim: If
he believed it would not swim, he might well speak as indeed he did;
but if he believed and knew that it would float, why, together with
the dubious Problem of the Natation of broad Figures, though of
ponderous Matter, hath he not also introduced the Question; whence it
proceeds, that even long and slender Figures, howbeit of Iron or Lead
do swim? And the rather, for that the occasion of doubting seems
greater in long and narrow Figures, than in broad and thin, as from
_Aristotles_ not having doubted of it, is manifested.
No lesser an inconvenience would they fasten upon _Aristotle_, who in
his defence should say, that he means a Needle pretty thick, and not a
small one; for take it for granted to be intended of a small one; and
it shall suffice to reply, that he believed that it would swim; and I
will again charge him with having avoided a more wonderfull and
intricate Probleme, and introduced the more facile and less
wonderfull.
We say freely therefore, that _Aristotle_ did hold, that only the
broad Figure did swim, but the long and slender, such as a Needle,
not. The which nevertheless is false, as it is also false in round
Bodies: because, as from what hath been predemonstrated, may be
gathered, little Balls of Lead and Iron, do in like manner swim.
[Sidenote: _Aristotle_ affirmeth some Bodies volatile for their
Minuity, Text. 42.]
He proposeth likewise another Conclusion, which likewise seems
different from the truth, and it is, That some things, by reason of
their littleness fly in the Air, as the small dust of the Earth, and
the thin leaves of beaten Gold: but in my Opinion, Experience shews
us, that that happens not only in the Air, but also in the water, in
which do descend, even those Particles or Atomes of Earth, that
disturbe it, whose minuity is such, that they are not deservable, save
only when they are many hundreds together. Therefore, the dust of the
Earth, and beaten Gold, do not any way sustain themselves in the Air,
but descend downwards, and only fly to and again in the same, when
strong Windes raise them, or other agitations of the Air commove them:
and this also happens in the commotion of the water, which raiseth its
Sand from the bottom, and makes it muddy. But _Aristotle_ cannot mean
this impediment of the commotion, of which he makes no mention, nor
names other than the lightness of such Minutiae or Atomes, and the
Resistance of the Crassitudes of the Water and Air, by which we see,
that he speakes of a calme, and not disturbed and agitated Air: but in
that case, neither Gold nor Earth, be they never so small, are
sustained, but speedily descend.
[Sidenote: _Democritus_ placed the Cause of Natation in certain
fiery Atomes.]
He passeth next to confute _Democritus_[82], which, by his Testimony would
have it, that some Fiery Atomes, which continually ascend through the
water, do spring upwards, and sustain those grave Bodies, which are
very broad, and that the narrow descend to the bottom, for that but a
small quantity of those Atomes, encounter and resist them.
[82] _Aristot. De Caelo_ lib. 4. cap. 6. text. 43.
I say, _Aristotle_ confutes this position[83], saying, that that should
much more occurre in the Air, as the same _Democritus_ instances
against himself, but after he had moved the objection, he slightly
resolves it, with saying, that those Corpuscles which ascend in the
Air, make not their _Impetus_ conjunctly. Here I will not say, that
the reason alledged by _Democritus_ is true[84], but I will only say, it
seems in my judgement, that it is not wholly confuted by _Aristotle_,
whilst he saith, that were it true, that the calid ascending Atomes,
should sustain Bodies grave, but very broad, it would much more be
done in the Air, than in Water, for that haply in the Opinion of
_Aristotle_, the said calid Atomes ascend with much greater Force and
Velocity through the Air, than through the water. And if this be so,
as I verily believe it is, the Objection of _Aristotle_ in my
judgement seems to give occasion of suspecting, that he may possibly
be deceived in more than one particular: First, because those calid
Atomes, (whether they be Fiery Corpuscles, or whether they be
Exhalations, or in short, whatever other matter they be, that ascends
upwards through the Air) cannot be believed to mount faster through
Air, than through water: but rather on the contrary, they peradventure
move more impetuously through the water, than through the Air, as hath
been in part demonstrated above. And here I cannot finde the reason,
why _Aristotle_ seeing, that the descending Motion of the same
Moveable, is more swift in Air, than in water, hath not advertised us,
that from the contrary Motion, the contrary should necessarily follow;
to wit, that it is more swift in the water, than in the Air: for since
that the Moveable which descendeth, moves swifter through the Air,
than through the water, if we should suppose its Gravity gradually to
diminish, it would first become such, that descending swiftly through
the Air, it would descend but slowly through the water: and then
again, it might be such, that descending in the Air, it should ascend
in the water: and being made yet less grave, it shall ascend swiftly
through the water, and yet descend likewise through the Air: and in
short, before it can begin to ascend, though but slowly through the
Air, it shall ascend swiftly through the water: how then is it true,
that ascending Moveables move swifter through the Air, than through
the water?
[83] _Democritus_ confuted by _Aristotle_, text 43.
[84] _Aristotles_ confutation of _Democritus_ refuted by the
Author.
That which hath made _Aristotle_ believe, the Motion of Ascent to be
swifter in Air, than in water, was first, the having referred the
Causes of slow and quick, as well in the Motion of Ascent, as of
Descent, only to the diversity of the Figures of the Moveable, and to
the more or less Resistance of the greater or lesser Crassitude, or
Rarity of the _Medium_; not regarding the comparison of the Excesses
of the Gravities of the Moveables, and of the _Mediums_: the which
notwithstanding, is the most principal point in this affair: for if
the augmentation and diminution of the Tardity or Velocity, should
have only respect to the Density or Rarity of the _Medium_, every Body
that descends in Air, would descend in water: because whatever
difference is found between the Crassitude of the water, and that of
the Air, may well be found between the Velocity of the same Moveable
in the Air, and some other Velocity: and this should be its proper
Velocity in the water, which is absolutely false. The other occasion
is, that he did believe, that like as there is a positive and
intrinsecall Quality, whereby Elementary Bodies have a propension of
moving towards the Centre of the Earth, so there is another likewise
intrinsecall[85], whereby some of those Bodies have an _Impetus_ of flying
the Centre, and moving upwards: by Vertue of which intrinsecall
Principle, called by him Levity, the Moveables which have that same
Motion more easily penetrate the more subtle _Medium_, than the more
dense: but such a Proposition appears likewise uncertain, as I have
above hinted in part, and as with Reasons and Experiments, I could
demonstrate, did not the present Argument importune me, or could I
dispatch it in few words.
[85] Lib. 4. Cap. 5.
The Objection therefore of _Aristotle_ against _Democritus_, whilst he
saith, that if the Fiery ascending Atomes should sustain Bodies grave,
but of a distended Figure, it would be more observable in the Air than
in the water, because such Corpuscles move swifter in that, than in
this, is not good; yea the contrary would evene, for that they ascend
more slowly through the Air: and, besides their moving slowly, they
ascend, not united together, as in the water, but discontinue, and, as
we say, scatter: And, therefore, as _Democritus_ well replyes,
resolving the instance they make not their push or _Impetus_
conjunctly.
_Aristotle_, in the second place, deceives himself, whilst he will
have the said grave Bodies to be more easily sustained by the said
Fiery ascending Atomes in the Air than in the Water: not observing,
that the said Bodies are much more grave in that, than in this, and
that such a Body weighs ten pounds in the Air, which will not in the
water weigh 1/2 an ounce; how can it then be more easily sustained in
the Air, than in the Water?
[Sidenote: _Democritus_ confuted by the Authour.]
Let us conclude, therefore, that _Democritus_ hath in this particular
better Philosophated than _Aristotle_. But yet will not I affirm, that
_Democritus_ hath reason'd rightly, but I rather say, that there is a
manifest Experiment that overthrows his Reason, and this it is, That
if it were true, that calid ascending Atomes should uphold a Body,
that if they did not hinder, would go to the bottom, it would follow,
that we may find a Matter very little superiour in Gravity to the
water, the which being reduced into a Ball, or other contracted
Figure, should go to the bottom, as encountring but few Fiery Atomes;
and which being distended afterwards into a dilated and thin Plate,
should come to be thrust upwards by the impulsion of a great Multitude
of those Corpuscles, and at last carried to the very Surface of the
water: which wee see not to happen; Experience shewing us, that a Body
_v. gra._ of a Sphericall Figure, which very hardly, and with very
great leasure goeth to the bottom, will rest there, and will also
descend thither, being reduced into whatsoever other distended Figure.
We must needs say then, either that in the water, there are no such
ascending Fiery Atoms, or if that such there be, that they are not
able to raise and lift up any Plate of a Matter, that without them
would go to the bottom: Of which two Positions, I esteem the second to
be true, understanding it of water, constituted in its naturall
Coldness. But if we take a Vessel of Glass, or Brass, or any other
hard matter, full of cold water, within which is put a Solid of a flat
or concave Figure, but that in Gravity exceeds the water so little,
that it goes slowly to the bottom; I say, that putting some burning
Coals under the said Vessel, as soon as the new Fiery Atomes shall
have penetrated the substance of the Vessel, they shall without doubt,
ascend through that of the water, and thrusting against the foresaid
Solid, they shall drive it to the Superficies, and there detain it, as
long as the incursions of the said Corpuscles shall last, which
ceasing after the removall of the Fire, the Solid being abandoned by
its supporters, shall return to the bottom.
But _Democritus_ notes, that this Cause only takes place when we
treat of raising and sustaining of Plates of Matters, but very little
heavier than the water, or extreamly thin: but in Matters very grave,
and of some thickness, as Plates of Lead or other Mettal, that same
Effect wholly ceaseth: In Testimony of which, let's observe that such
Plates, being raised by the Fiery Atomes, ascend through all the depth
of the water, and stop at the Confines of the Air, still staying under
water: but the Plates of the Opponents stay not, but only when they
have their upper Superficies dry, nor is there any means to be used,
that when they are within the water, they may not sink to the bottom.
The cause, therefore, of the Supernatation of the things of which
_Democritus_ speaks is one, and that of the Supernatation of the
things of which we speak is another. But, returning to _Aristotle_[86],
methinks that he hath more weakly confuted _Democritus_, than
_Democritus_ himself hath done: For _Aristotle_ having propounded the
Objection which he maketh against him, and opposed him with saying,
that if the calid ascendent Corpuscles were those that raised the thin
Plate, much more then would such a Solid be raised and born upwards
through the Air, it sheweth that the desire in _Aristotle_ to detect
_Democritus_, was predominate over the exquisiteness of Solid
Philosophizing: which desire of his he hath discovered in other
occasions, and that we may not digress too far from this place, in the
Text precedent to this Chapter which we have in hand[87]; where he
attempts to confute the same _Democritus_ for that he, not contenting
himself with names only, had essayed more particularly to declare what
things Gravity and Levity were; that is, the Causes of descending and
ascending, (and had introduced Repletion and Vacuity) ascribing this
to Fire, by which it moves upwards, and that to the Earth, by which it
descends; afterwards attributing to the Air more of Fire, and to the
water more of Earth. But _Aristotle_ desiring a positive Cause, even
of ascending Motion, and not as _Plato_, or these others, a simple
negation, or privation, such as Vacuity would be in reference to
Repletion[88], argueth against _Democritus_ and saith: If it be true, as
you suppose, then there shall be a great Mass of water, which shall
have more of Fire, than a small Mass of Air, and a great Mass of Air,
which shall have more of Earth than a little Mass of water, whereby it
would ensue, that a great Mass of Air, should come more swiftly
downwards, than a little quantity of water: But that is never in any
case soever: Therefore _Democritus_ discourseth erroneously.
[86] _Aristotle_ shews his desire of finding _Democritus_ in an
Error, to exceed that of discovering Truth.
[87] Cap. 5. Text 41.
[88] Id. ibid.
But in my opinion, the Doctrine of _Democritus_ is not by this
allegation overthrown, but if I erre not, the manner of _Aristotle_
deduction either concludes not, or if it do conclude any thing, it may
with equall force be restored against himself. _Democritus_ will grant
to _Aristotle_, that there may be a great Mass of Air taken, which
contains more Earth, than a small quantity of water, but yet will
deny, that such a Mass of Air, shall go faster downwards than a little
water, and that for many reasons. First, because if the greater
quantity of Earth, contained in the great Mass of Air, ought to cause
a greater Velocity than a less quantity of Earth, contained in a
little quantity of water, it would be necessary, first, that it were
true, that a greater Mass of pure Earth, should move more swiftly than
a less: But this is false, though _Aristotle_ in many places affirms
it to be true: because not the greater absolute, but the greater
specificall Gravity, is the cause of greater Velocity[89]: nor doth a Ball
of Wood, weighing ten pounds, descend more swiftly than one weighing
ten Ounces, and that is of the same Matter: but indeed a Bullet of
Lead of four Ounces, descendeth more swiftly than a Ball of Wood of
twenty Pounds: because the Lead is more _grave in specie_ than the
Wood. Therefore, its not necessary, that a great Mass of Air, by
reason of the much Earth contained in it, do descend more swiftly than
a little Mass of water[90], but on the contrary, any whatsoever Mass of
water, shall move more swiftly than any other of Air, by reason the
participation of the terrene parts _in specie_ is greater in the
water, than in the Air. Let us note, in the second place, how that in
multiplying the Mass of the Air, we not only multiply that which is
therein of terrene, but its Fire also: whence the Cause of ascending,
no less encreaseth, by vertue of the Fire, than that of descending on
the account of its multiplied Earth. It was requisite in increasing
the greatness of the Air, to multiply that which it hath of terrene
only, leaving its Fire in its first state, for then the terrene parts
of the augmented Air, overcoming the terrene parts of the small
quantity of water, it might with more probability have been pretended,
that the great quantity of Air, ought to descend with a greater
_Impetus_, than the little quantity of water.
[89] The greater Specificall, not the greater absolute Gravity,
is the Cause of Velocity.
[90] Any Mass of water shal move more swiftly, than any of Air,
and why.
Therefore, the Fallacy lyes more in the Discourse of _Aristotle_, than
in that of _Democritus_, who with severall other Reasons might oppose
_Aristotle_, and alledge; If it be true, that the extreame Elements be
one simply grave, and the other simply light, and that the mean
Elements participate of the one, and of the other Nature; but the Air
more of Levity, and the water more of Gravity, then there shall be a
great Mass of Air, whose Gravity shall exceed the Gravity of a little
quantity of water, and therefore such a Mass of Air shall descend more
swiftly than that little water: But that is never seen to occurr:
Therefore its not true, that the mean Elements do participate of the
one, and the other quality. This argument is fallacious, no less than
the other against _Democritus_.
Lastly, _Aristotle_ having said, that if the Position of _Democritus_
were true, it would follow, that a great Mass of Air should move more
swiftly than a small Mass of water, and afterwards subjoyned, that
that is never seen in any Case: methinks others may become desirous to
know of him in what place this should evene, which he deduceth against
_Democritus_, and what Experiment teacheth us, that it never falls out
so. To suppose to see it in the Element of water, or in that of the
Air is vain, because neither doth water through water, nor Air through
Air move, nor would they ever by any whatever participation others
assign them, of Earth or of Fire: the Earth, in that it is not a Body
fluid, and yielding to the mobility of other Bodies, is a most
improper place and _Medium_ for such an Experiment: _Vacuum_,
according to the same _Aristotle_ himself, there is none, and were
there, nothing would move in it: there remains the Region of Fire, but
being so far distant from us, what Experiment can assure us, or hath
assertained _Aristotle_ in such sort, that he should as of a thing
most obvious to sence, affirm what he produceth in confutation of
_Democritus_, to wit, that a great Mass of Air, is moved no swifter
than a little one of water? But I will dwell no longer upon this
matter, whereon I have spoke sufficiently: but leaving _Democritus_, I
return to the Text of _Aristotle_, wherein he goes about to render the
true reason, how it comes to pass, that the thin Plates of Iron or
Lead do swim on the water; and, moreover, that Gold it self being
beaten into thin Leaves, not only swims in water, but flyeth too and
again in the Air. He supposeth that of Continualls[91], some are easily
divisible, others not: and that of the easily divisible, some are more
so, and some less: and these he affirms we should esteem the Causes.
He addes that that is easily divisible, which is well terminated, and
the more the more divisible, and that the Air is more so, than the
water, and the water than the Earth. And, lastly, supposeth that in
each kind, the lesse quantity is easlyer divided and broken than the
greater.
[91] _De Caelo_ l. 4. c. 6. t. 44.
Here I note, that the Conclusions of _Aristotle_ in generall are all
true, but methinks, that he applyeth them to particulars, in which
they have no place, as indeed they have in others, as for Example, Wax
is more easily divisible than Lead, and Lead than Silver, inasmuch as
Wax receives all the terms more easlier than Lead, and Lead than
Silver. Its true, moreover, that a little quantity of Silver is
easlier divided than a great Mass: and all these Propositions are
true, because true it is, that in Silver, Lead and Wax, there is
simply a Resistance against Division, and where there is the absolute,
there is also the respective. But if as well in water as in Air, there
be no Renitence against simple Division, how can we say, that the
water is easlier divided than the Air? We know not how to extricate
our selves from the Equivocation: whereupon I return to answer, that
Resistance of absolute Division is one thing, and Resistance of
Division made with such and such Velocity is another. But to produce
Rest, and to abate the Motion, the Resistance of absolute Division is
necessary; and the Resistance of speedy Division, causeth not Rest,
but slowness of Motion. But that as well in the Air, as in water,
there is no Resistance of simple Division, is manifest, for that there
is not found any Solid Body which divides not the Air, and also the
water: and that beaten Gold, or small dust, are not able to superate
the Resistance of the Air, is contrary to that which Experience shews
us, for we see Gold and Dust to go waving to and again in the Air, and
at last to descend downwards, and to do the same in the water, if it
be put therein, and separated from the Air. And, because, as I say,
neither the water, nor the Air do resist simple Division, it cannot be
said, that the water resists more than the Air. Nor let any object
unto me, the Example of most light Bodies, as a Feather, or a little
of the pith of Elder, or water-reed that divides the Air and not the
water, and from this infer, that the Air is easlier divisible than the
water; for I say unto them, that if they do well observe, they shall
see the same Body likewise divide the Continuity of the water[92], and
submerge in part, and in such a part, as that so much water in Mass
would weigh as much as the whole Solid. And if they shal yet persist
in their doubt, that such a Solid sinks not through inability to
divide the water, I will return them this reply, that if they put it
under water, and then let it go, they shall see it divide the water,
and presently ascend with no less celerity, than that with which it
divided the Air in descending: so that to say that this Solid ascends
in the Air, but that coming to the water, it ceaseth its Motion, and
therefore the water is more difficult to be divided, concludes
nothing: for I, on the contrary, will propose them a piece of Wood, or
of Wax, which riseth from the bottom of the water, and easily divides
its Resistance, which afterwards being arrived at the Air, stayeth
there, and hardly toucheth it; whence I may aswell say, that the water
is more easier divided than the Air.
[92] _Archimed. De Insident. humi_ lib. 2. prop. 1.
I will not on this occasion forbear to give warning of another fallacy
of these persons, who attribute the reason of sinking or swimming to
the greater or lesse Resistance of the Crassitude of the water against
Division, making use of the example of an Egg, which in sweet water
goeth to the bottom, but in salt water swims; and alledging for the
cause thereof, the faint Resistance of fresh water against Division,
and the strong Resistance of salt water. But if I mistake not, from
the same Experiment, we may aswell deduce the quite contrary; namely,
that the fresh water is more dense, and the salt more tenuous and
subtle, since an Egg from the bottom of salt water speedily ascends to
the top, and divides its Resistance, which it cannot do in the fresh,
in whose bottom it stays, being unable to rise upwards. Into such like
perplexities, do false Principles Lead men: but he that rightly
Philosophating, shall acknowledge the excesses of the Gravities of the
Moveables and of the Mediums, to be the Causes of those effects, will
say, that the Egg sinks to the bottom in fresh water, for that it is
more grave than it, and swimeth in the salt, for that its less grave
than it: and shall without any absurdity, very solidly establish his
Conclusions.
Therefore the reason totally ceaseth, that _Aristotle_ subjoyns in the
Text saying[93]; The things, therefore, which have great breadth remain
above, because they comprehend much, and that which is greater, is not
easily divided. Such discoursing ceaseth, I say, because its not true,
that there is in water or in Air any Resistance of Division; besides
that the Plate of Lead when it stays, hath already divided and
penetrated the Crassitude of the water, and profounded it self ten or
twelve times more than its own thickness: besides that such Resistance
of Division, were it supposed to be in the water, could not rationally
be affirmed to be more in its superiour parts than in the middle, and
lower: but if there were any difference, the inferiour should be the
more dense, so that the Plate would be no less unable to penetrate the
lower, than the superiour parts of the water; nevertheless we see that
no sooner do we wet the superiour Superficies of the Board or thin
Piece of Wood, but it precipitatly, and without any retension,
descends to the bottom.
[93] Text 45.
I believe not after all this, that any (thinking perhaps thereby to
defend _Aristotle_) will say, that it being true, that the much water
resists more than the little, the said Board being put lower
descendeth, because there remaineth a less Mass of water to be divided
by it: because if after the having seen the same Board swim in four
Inches of water, and also after that in the same to sink, he shall try
the same Experiment upon a profundity of ten or twenty fathom water,
he shall see the very self same effect. And here I will take occasion
to remember, for the removall of an Error that is too common; That
that Ship or other whatsoever Body, that on the depth of an hundred or
a thousand fathom, swims with submerging only six fathom of its own
height, [_or in the Sea dialect, that draws six fathom water_] shall
swim in the same manner in water, that hath but six fathom and half an
Inch of depth[94]. Nor do I on the other side, think that it can be said,
that the superiour parts of the water are the more dense, although a
most grave Authour hath esteemed the superiour water in the Sea to be
so, grounding his opinion upon its being more salt, than that at the
bottom: but I doubt the Experiment, whether hitherto in taking the
water from the bottom, the Observatour did not light upon some spring
of fresh water there spouting up: but we plainly see on the contrary,
the fresh Waters of Rivers to dilate themselves for some miles beyond
their place of meeting with the salt water of the Sea, without
descending in it, or mixing with it, unless by the intervention of
some commotion or turbulency of the Windes.
[94] A Ship that in 100 Fathome water draweth 6 Fathome, shall
float in 6 Fathome and 1/2 an Inch of depth.
But returning to _Aristotle_, I say, that the breadth of Figure hath
nothing to do in this business more or less, because the said Plate of
Lead, or other Matter, cut into long Slices, swim neither more nor
less[95]; and the same shall the Slices do, being cut anew into little
pieces, because its not the breadth but the thickness that operates in
this business. I say farther, that in case it were really true, that
the Renitence to Division were the proper Cause of swimming[96], the
Figures more narrow and short, would much better swim than the more
spacious and broad, so that augmenting the breadth of the Figure, the
facility of supernatation will be deminished, and decreasing, that
this will encrease.
[95] Thickness not breadth of Figure to be respected in Natation.
[96] Were Renitence the cause of Natation, breadth of Figure would
hinder the swiming of Bodies.
And for declaration of what I say, consider that when a thin Plate of
Lead descends, dividing the water, the Division and discontinuation is
made between the parts of the water, invironing the perimeter or
Circumference of the said Plate, and according to the bigness greater
or lesser of that circuit, it hath to divide a greater or lesser
quantity of water, so that if the circuit, suppose of a Board, be ten
Feet in sinking it flatways, it is to make the seperation and
division, and to so speak, an incission upon ten Feet of water; and
likewise a lesser Board that is four Feet in Perimeter, must make an
incession of four Feet. This granted, he that hath any knowledge in
Geometry, will comprehend, not only that a Board sawed in many long
thin pieces, will much better float than when it was entire, but that
all Figures, the more short and narrow they be, shall so much the
better swim. Let the Board A B C D be, for Example, eight Palmes long,
and five broad, its circuit shall be twenty six Palmes; and so many
must the incession be, which it shall make in the water to descend
therein: but if we do saw ir, as suppose into eight little pieces,
according to the Lines E F, G H, {&}c. making seven Segments, we must
adde to the twenty six Palmes of the circuit of the whole Board,
seventy others; whereupon the eight little pieces so cut and
seperated, have to cut ninty six Palmes of water. And, if moreover, we
cut each of the said pieces into five parts, reducing them into
Squares, to the circuit of ninty six Palmes, with four cuts of eight
Palmes apiece; we shall adde also sixty four Palmes, whereupon the
said Squares to descend in the water, must divide one hundred and
sixty Palmes of water, but the Resistance is much greater than that of
twenty six; therefore to the lesser Superficies, we shall reduce them,
so much the more easily will they float: and the same will happen in
all other Figures, whose Superficies are simular amongst themselves,
but different in bigness: because the said Superficies, being either
deminished or encreased, always diminish or encrease their Perimeters
in subduple proportion; to wit, the Resistance that they find in
penetrating the water; therefore the little pieces gradually swim,
with more and more facility as their breadth is lessened.
[Illustration]
_This is manifest; for keeping still the same height of the Solid,
with the same proportion as the Base encreaseth or deminisheth,
doth the said Solid also encrease or diminish; whereupon the Solid
more diminishing than the Circuit, the Cause of Submersion more
diminisheth than the Cause of Natation: And on the contrary, the
Solid more encreasing than the Circuit, the Cause of Submersion
encreaseth more, that of Natation less._
And this may all be deduced out of the Doctrine of _Aristotle_ against
his own Doctrine.
Lastly, to that which we read in the latter part of the Text[97], that is
to say, that we must compare the Gravity of the Moveable with the
Resistance of the Medium against Division, because if the force of the
Gravity exceed the Resistance of the _Medium_, the Moveable will
descend, if not it will float. I need not make any other answer, but
that which hath been already delivered; namely, that its not the
Resistance of absolute Division, (which neither is in Water nor Air)
but the Gravity of the _Medium_ that must be compared with the Gravity
of the Moveables; and if that of the _Medium_ be greater, the Moveable
shall not descend, nor so much as make a totall Submersion, but a
partiall only; because in the place which it would occupy in the
water, there must not remain a Body that weighs less than a like
quantity of water: but if the Moveable be more grave, it shall descend
to the bottom, and possess a place where it is more conformable for it
to remain, than another Body that is less grave. And this is the only
true proper and absolute Cause of Natation and Submersion, so that
nothing else hath part therein: and the Board of the Adversaries
swimmeth, when it is conjoyned with as much Air, as, together with it,
doth form a Body less grave than so much water as would fill the place
that the said Compound occupyes in the water; but when they shall
demit the simple Ebony into the water, according to the Tenour of our
Question, it shall alwayes go to the bottom, though it were as thin as
a Paper.
[97] Lib. 4. c. 6. Text 45.
[Decoration]
FINIS.
[Decoration]
*       *       *       *       *
[Detailed Transcriber's Notes
The text has been made to match the original text as much as
possible including variation in spelling, punctuation, italics etc.
The following, details apparent printer's errors as well as changes
or additions to aid readability of text.
Page 1, missing full stop after abbreviation gr. '0 gr 54 min.'.
Page 3, sidenote, missing space between words 'the Authority ofan
Author.'.
Page 3, printer's error, augmentarion for augmentation 'and
augmentarion of Masse'.
Page 4, missing letter t 'tha{t} that proceeded not'.
Page 4, printer's error or inconsistent punctuation, 'my paynes and
time. and although'.
Page 6, inconsistent punctuation, full stop after axiome where as
there are none after those following 'AXIOME. I.'.
Page 9, missing full stop added to end of paragraph 'or else an
upright Prisme.'.
Page 11, printer's error, missing letter C in illustration, 'the
Prisme A C D B to be placed'.
Page 15, printer's error or archaic lettering, final y looking like
a 7 in original text'and of the Specifick Gravit{y}'.
Page 16, printer's error, letter N for T in text to refer to
illustration, 'if the Vessell E N S F'.
Page 16, printer's error, duplicate word in text 'equalizeth the
Force and and Moment,'.
Page 17, printer's error, rhe for the 'as in rhe Stilliard,'.
Page 17, missing space between words 'asoften as that'.
Page 18, sidenote, printer's error, specifiaclly for specifically
'A Solid specifiaclly graver'.
Page 20, potential printer's error, properly for property, 'but
this properly they have'.
Page 20, printer's error, n for u 'loseth all a{u}thenticalness'.
Page 22, printer's error or variation in spelling, Benonamico for
Buonamico 'it seemes that Benonamico'.
Page 23, printer's error, missing i 'accordng to its excess'.
Page 24, missing line at the end of page in original text 'its
Region it loseth all'.
Page 26, missing letter n 'u{n}able by its small weight'.
Page 29, missing letter e 'that I have gon{e} about'.
Page 32, unclear symbol in original text 'other Figure, {&}c.'.
Page 37, potential printer's error, comma in unusual position
'whatever Figure, goeth always'.
Page 38, missing space between words 'Superficies might bedry:'.
Page 39, missing letter t, unied for united 'which holds them unied'.
Page 41, printer's error, Motitions for Motions 'all Motitions are
made'.
Page 42, sidenote, possible missing letter e, 'Se{e} what satisfaction'.
Page 43, printer's error, Subdidivisions for Subdivisions, 'other
Subdidivisions,'.
Page 49, missing letter i, dminishing for diminishing 'or dminishing
it by dividing'.
Page 50, sidenote, printer's error, missing letter l, hep for help
'float by hep of'.
Page 53, printer's error, missing letter n, beig for being 'beig
double in Gravity'.
Page 54, printer's error, missing letter l, 'sha{l}l also descend.'.
Page 55, printer's error, missing letter r, 'to fo{r}m Solid Bodies'.
Page 56, printer's error, missing letter A, 'Cone S {A} T'.
Page 57, printer's error, missing letter t, 'of which i{t} may
descend'.
Page 58, sidenote, printer's error, inverted n, 'Natatio{n} easiest
effected'.
Page 59, missing letter u, 'witho{u}t all Question,'.
Page 59, printer's error, inverted n, 'with an other Experime{n}t'.
Page 59, potential printer's error, sidenote ends with comma, 'Or
rather Contiguity,'.
Page 61, missing letter a, 'that in this c{a}se the water,'.
Page 74, printer's error, ir for it, 'but if we do saw ir,'.
Page 75, unclear symbol in original text '{&}c. making seven
Segments'.
]