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OSMANIA UNIVERSITY LIBRARY
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Author
This book should
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TREATISE
u N LIGHT
In which are explained
The causes of that which occurs
In REFLEXK)N,&in REFRACTION.
And particularly
In the strange REFRACTION
OF ICELAND CRYSTAL
By CHRISTIAAN HUYGENS.
Rendered into English
By SILVANUS P. THOMPSON.
MACMILLAN AND CO., LIMITED
ST. MARTIN'S STREET, LONDON
MCMXII.
CHISWICK PRESS : CHARLES WHITTINGHAM AND CO,
TOOKS COURT, CHANCERY LANE, LONDON.
PREFACE
WROTE this Treatise during my sojourn
in France twelve years ago, and I com-
municated it in the year 1 678 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 par-
ticularly to the study of Mathematics ; of whom I cannot
cite more than the celebrated gentlemen Cassini, Romer,
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 no-
thing 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 detraCt-
ing from the merit of those who, without having seen any-
thing that I have written, may be found to have treated
of
vi PREFACE
of like matters : as has in fa<5t occurred to two eminent
Geometricians, Messieurs Newton and Leibnitz, with re-
spe6l 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 pro-
posed to myself to give it out along with another Treatise
on Dioptrics, in which I explain the effeds 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. Consider-
ing 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 veri-
fied 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 prob-
ability 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 ob-
servation ; especially when there are a great number of
them,
PREFACE vii
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 fad: 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 in-
quiry; and it must be ill if the fa<5ts 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 know-
ledge. The Hague. The 8 January 1690.
NOTE BY THE TRANSLATOR
ONSIDERING the great influence which
this Treatise has exercised in the develop-
ment 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 every-
thing 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 refraftion
of crystals, and of the refra6tion 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 vibra-
tions, 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 prin-
b ciples
x NOTE BY THE TRANSLATOR
ciples of wave-propagation; and his exposition of the sub-
jeCt marks an epoch in the treatment of Optical problems.
It has been needful in preparing this translation to exer-
cise 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 re-
fraCted ray he habitually terms " the refraction " of the
incident ray. When a wave-front, or, as he terms it, a
u wave," has passed from some initial position to a subse-
quent 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 ter-
mination" of the wave; and the elementary wave-fronts
he terms " particular " waves. Owing to the circumstance
that the French word rayon possesses the double signifi-
cation 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 modern-
ness. 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
CHAP. I. On Rays Propagated in
Straight Lines.
That Light is produced by a certain
movement. p. 3
That no substance passes from the lumin-
ous objeft to the eyes. p. 3
That Light spreads spheric ally ^ almost
as Sound does. p. 4
Whether Light takes time to spread.
P-*
Experience seeming to prove that it passes
instantaneously. p. 5
Experience proving that it takes time.
p. 8
How much its speed is greater than that
of Sound. p. 10
In what the emission of Light differs
from that of Sound. p. 10
That it is not the same medium which
serves for Light and Sound, p. 1 1
How Sound is propagated. p. 12
How Light is propagated. p. 14
Detailed Remarks on the propagation
of Light. p. 15
Why Rays are propagated only in
straight lines. p. 2O
How Light coming in different directions
can cross itself. p. 22
CHAP. II. On Reflexion.
Demonstration of equality of angles of
incidence and reflexion. p. 23
Why the incident and reflected rays are
in the same plane perpendicular to the
reflecting surface. p. 25
That it is not needful for the reflecting
surface to be perfectly flat to attain
equality of the angles of incidence and
reflexion. p. 27
CHAP. III. On Refradlion.
That bodies may be transparent without
any substance passing through them.
p. 29
Proof that the ethereal matter passes
through transparent bodies. p. 30
How this matter passing through can
render them transparent. p. 31
That the most solid bodies in appearance
are of a very loose texture. p. 31
That Light spreads more slowly in water
and in glass than in air. p. 32
Third hypothesis to explain transpar-
ency^ and the retardation which Light
suffers. .A3 2
On that which makes bodies opaque.
P- 34
Demonstration why Refraction obeys the
known proportion of Sines. p. 35
Why the incident and refrafted Rays
produce one another reciprocally, p. 39
Why Reflexion within a triangular
glass prism is suddenly augmented
when the Light can no longer pene-
trate, p. 40
That bodies which cause greater Refrac-
tion also cause stronger Reflexion.
p. 42
Demonstration of the Theorem of Mr.
Fermat. p. 43
CHAP. IV. On the Refradion of
the Air.
That the emanations of Light in the air
are not spherical. p t 45
How consequently some objects appear
higher than they are. p. 47
How the Sun may appear on the Hori-
zon before he has risen. p. 49
xii TABLE OF
That the rays of light become curved in
the Air of the Atmosphere, and what
effects this produces. p. 50
CHAP. V. On the Strange Refrac-
tion of Iceland Crystal.
That this Crystal grows also in other
countries. p. 5 2
Wlw first wrote about it. ^'53
Description of Iceland Crystal; its sub-
stance, shape, and properties, p. 53
That it has two different Refractions.
P- 54
That the ray perpendicular to the surface
suffers refraction, and that some rays
inclined to the surface pass without
suffering refraction. p. 55
Observation of the refractions in this
Crystal. p. 56
That there is a Regular and an Ir-
regular Refraction. P- 57
The way of measuring the two Refrac-
tions of Iceland Crystal. P* 57
Remarkable properties of the Irregular
Refraction. p. 60
Hypothesis to explain the double Refrac-
tion, p. 6 1
That Rock Crystal has also a double
Refraction, p. 62
Hypothesis of emanations ofLight, with-
in Iceland Crystal, of spheroidal form^
for the Irregular Refraction, p. 63
How a perpendicular ray can suffer
Refraction. p. 64
How the position and form of the spher-
oidal emanations in this Crystal can
be defined. p. 65
Explanation of the Irregular Refraction
by these spheroidal emanations, p. 67
Easy way to find the Irregular Re-
fraction of each incident ray. p. 70
Demonstration of the oblique ray which
MATTERS
traverses the Crystal without being
refracted. p. 73
Other irregularities of Refraction ex-
plained. />. 76
That an obj eft placed beneath the Crystal
appears double^in two images of differ-
ent heights, p. Si
Why the apparent heights of one of the
images change on changing the position
of the eyes above the Crystal, p. 85
Of the different sections of this Crystal
which produce yet other refractions^
and confirm all this Theory, p. 88
Particular way of polishing the surfaces
after it has been cut. p. 9 1
Surpr is ing phenomenon touching the rays
which pass through two separated
pieces; the cause of which is not ex-
plained, p. 92
Probable conjecture on the internal com-
position of Iceland Crystal^ and of
what figure its particles are. p. 95
Tests to confirm this conjecture, p. 97
Calculations which have been supposed
in this Chapter. p. 99
CHAP. VI. On the Figures of
transparent bodies which serve
for Refraction and for Reflexion.
General and easy rule to find these
Figures. p. 106
Invention of the Ovals of Mr. Des
Cartes for Dioptrics. p. 109
How he was able to find these Lines.
p. 114
{lass for
Way of finding the surface of a git
perfeft refracJion, when the other
surface is given. p. 116
Remark on what happens to rays re-
frafted at a spherical surface, p. 123
Remark on the curved line which is
formed by reflexion in a spherical con-
cave mirror. p. 1 26
TREATISE ON LIGHT
CHAPTER I
ON RAYS PROPAGATED IN STRAIGHT LINES
S happens in all the sciences in which
Geometry is applied to matter, the demon-
strations 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 re-
flexion and of incidence are equal; and that in refraftion
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 effe<5ls 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 thesub-
2 TREATISE
jet, 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 pre-
sent day, some clearer and more probable reasons, firstly of
these properties of light propagated re&ilinearly ; secondly
of light which is reflected on meeting other bodies. Then
I shall explain the phenomena of those rays which are said
to suffer refraftion on passing through transparent bodies
of different sorts; and in this part I shall also explain the
effects of the refradlion of the air by the different densities
of the Atmosphere.
Thereafter I shall examine the causes of the strange re-
fra6tion of a certain kind of Crystal which is brought from
Iceland. And finally I shall treat of the various shapes of
transparent and refle6ting 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
ON LIGHT. CHAP. I 3
other curves which Mr. Des Cartes has ingeniously in-
vented 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 produ<5tion, one sees that here upon the Earth it is
chiefly engendered by fire and flame which contain with-
out 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 effe6ls 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 cer-
tain that the sensation of sight is excited only by the
impression of some movement of a kind of matter which
a6ls 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 diredtly opposite,
the rays traverse one another without hindrance, one may
well understand that when we see aluminous object, it cannot
be by any transport of matter coming to us from this objeft,
in
4 TREATISE
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 fadls 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
ON LIGHT. CHAP. I 5
reason that they are too small, and that the only conclusion
to be drawn from them is that the passage of light is ex-
tremely 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.
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
6 TREATISE
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 B A be-
tween 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 assufned 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
ON LIGHT. CHAP. I 7
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 to me that even Mr. Des Cartes, whose
aim has been to treat all the subje6ts 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 pro-
perties.
But that which I employed only as a hypothesis, has
recently received great seemingness as an established truth
by the ingenious proof of Mr. Romer 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
8
TREATISE
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.
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^ hours, one would
again see a similar emergence, be-
cause that is the time in which it
makes the round of its orbit, and
when it would come again into op-
position 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-^ 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
ONLIGHT. CHAP. I 9
C than it would have been at B, and that there must be
added to this time of 30 times 42^ 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 i6j- diameters in one second or in one beat
of the pulse, which makes more than 1 1 hundred times a
hundred thousand toises; since the diameter of the Earth
contains 2,865 leagues, reckoned at 25 to the degree, and
c each
io TREATISE
each league is 2,282 Toises, according to the exa6t measure-
ment which Mr. Picard made by order of the King in 1 669.
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 be-
tween 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 respe<5t, 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 move-
ment of the Light must originate as from each point of the
luminous obje<5t, else we should not be able to perceive all
the different parts of that obje6t, as will be more evident in
that which follows. And I do not believe that this move-
ment 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
ON LIGHT. CHAP. I n
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 commu-
nicate its tremors either to the glass vessel which encloses
it, or to the machine; a precaution which has hitherto
been negle&ed. 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
12 TREATISE
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 con-
cludes 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 pro-
portion 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 com-
pression, 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 whe^n they are a little more squeezed to-
gether 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
ON LIGHT. CHAP. I 13
and I am about to show here the way in which I conceive
it must occur. For this, it is needful to explain the pro-
perty which hard bodies must possess to transmit move-
ment 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 move-
ment, 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 demon-
strate that all the bodies which we reckon of the hardest
kind, such as quenched steel, glass, and agate, aft 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 them-
selves at the place where they are struck, and immediately
regain their former figute. For I have found that on strik-
ing with a ball of glass or of agate against a large and quite
thick
i 4 TREATISE
thick piece of the same substance whicn nau 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 estimat-
ing the particles of the ether to be of a substance as nearly
approaching to perfeft hardness and possessing a springi-
ness 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 subjeft. 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 pass-
age 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
effefts.
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
ON LIGHT. CHAP. I 15
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 com-
municated, one will find none which agrees better, with
uniform progression, as seems to be necessary, than the pro-
perty 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.^
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 ( B )
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 sup-
posing 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 im-
pulse
16 TREATISE
pulse does not fail to contribute to the aforesaid propaga-
tion 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 propaga-
tion of light as for rendering that propagation easier and
more powerful ; for it is not beyond the limits of prob-
ability 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 atmo-
sphere 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 con-
tinual 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 trans-
port of those particles but merely in a small agitation which
they cannot help communicating to those surrounding,
notwithstanding any movement which may aft 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 pro-
duftion
ON LIGHT. CHAP. I 17
du6Hon 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.
But as the percussions at the centres
of these waves possess no regular suc-
cession, 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 effac-
ing 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 dire<5tions, not only if it is struck by blows
which follow one another closely but even for those which
a<5t 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 D,
i8 TREATISE
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 at as a spring
at both sides, and so will serve at the same instant to trans-
mit these two movements.
But what may at first appear full strange and even in-
credible 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 aftion of each one in particular will without doubt be-
come 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. More-
over 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
ON LIGHT. CHAP. I 19
Ether at these points: which further contributes to render-
ing their aftion more sensible.
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 lu-
minous point A, which
is its centre, the particle
B, one of those comprised
within the sphere DCF,
will have made its par-
ticular 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 com-
position of which all the others contribute by the part of
their surface which is most distant from the centre A.
One
B
20 TREATISE
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 pertain-
ing to its reflexion and its refra6lion, 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 effefts of
reflexion and refra&ion. 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 pro-
duced 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
ON LIGHT. CHAP. I 21
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 reflefted or broken, spreads only by straight
lines, so that it illuminates no object except when the path
from its source to that obje6l 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 par-
tial 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 touch-
ing 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.
22 TREATISE
marvellous, is that when some of them come from different
or even from opposing sides, they produce their effeft across
one another without any hindrance. Whence also it comes
about that a number of spectators may view different objefts
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 accord-
ing to the views of Mr. Des Cartes, who makes Light to
consist in a continuous pressure merely tending to move-
ment. For this pressure not being able to aft from two
opposite sides at the same time, against bodies which
have no inclination to approach one another, it is im-
possible 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
AVING explained the effefts 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
ON LIGHT. CHAP. II 23
Let there be a surface AB, plane and polished, of some
metal, glass, or other body, which at first I will consider
as perfeftly 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 A&,
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 my-
self that the plane
in which this fig-
ure is, cuts the
sphere of the wave
through its centre
and intersects the
plane AB at right
angles. This ex-
planation will suf-
fice once for all.
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 sup-
posed 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 con-
tinued its movement in the matter which is above this
plane, and this along a distance equal to CB, making its
own
24 TREATISE
own partial spherical wave, according to what has been
said above. Which wave is here represented by the cir-
cumference 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, re-
presented 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 any-
where 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 homogene-
ous 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
ON LIGHT. CHAP. II 25
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 re6tangular 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, per-
pendicular to the wave BN, marks the direftion 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.
E Now
26 TREATISE
Now it is easy to see, following the preceding demonstra-
tion, that each small piece of this wave HC having arrived
at the plane AB, and there generating each one its parti-
cular wave, these will all have, when C arrives at B, a com-
mon 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 in-
tersedts the circle CH and the ellipse AB.
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
ON LIGHT. CHAP. II 27
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 aft 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 experi-
ment shows that in fa6t there is no polished body the re-
flexion of which does not follow this rule.
But the thing to be above all remarked in our demon-
stration is that it does not require that the reflecting surface
should be considered as a uniform plane, as has been sup-
posed 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
28 TREATISE
this surface then becomes for our purpose as even as a
polished glass is : and, although it always remains rough
with respeft 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 re-
quisite, in our method of demonstration, to cause equality
of the said angles without the remainder of the movement
refle&ed from all parts being able to produce any contrary
effeft.
CHAPTER III
ON REFRACTION
N the same way as the effects of Reflexion
have been explained by waves of light re-
fledled at the surface of polished bodies, we
will explain transparency and thephenomena
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 aft as a spring. And this is
easy
ON LIGHT. CHAP. Ill 29
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 trans-
parent 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 move-
ment 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 effe6l 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
3 o TREATISE
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
ON LIGHT. CHAP. Ill 31
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 transpar-
ent 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 con-
stituent 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 fa6l 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 objedled 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 per-
mitting itself to be condensed by any force which one has
hitherto
32 TREATISE
hitherto essayed to employ, preserving even its entire
liquidity while subjeded 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 inter-
stices 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 differ-
ent velocity of light I shall show the cause of refraftion
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 refra6tion 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
ON LIGHT. CHAP. Ill 33
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 respe<5l 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 trans-
parent. 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 sil-
ver, as well as glass, is very easily penetrated by this matter.
Some of it is therefore present continuously and in quan-
tities between the particles of silver and of all other opaque
F bodies:
34 TREATISE
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 mo-
tion of which they thereby damp, and so hinder the con-
tinuance 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 refleft 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 trans-
parency; while, on the other hand, transparent bodies con-
tain 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.
Let us pass now to the ex-
planation of the effects of Re-
fraction, 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
ON LIGHT. CHAP. Ill 35
broken at the point of incidence B, in such a way that
with the straight line DBE which cuts the surface per-
pendicularly 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 trans-
parent 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 refra6lions
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
t r ansp arent
body is refradted
into BC, then
likewise CB
being taken as
a ray in the
interior of this
body will be
refrafted, on
passing out, in-
to BA.
To explain then the reasons of these phenomena accord-
ing to our principles, let AB be the straight line which
represents
36 TREATISE
represents a plane surface bounding the transparent sub-
stances which lie towards C and towards N. When I say
plane, that does not signify a perfeft 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
ON LIGHT. CHAP. Ill 37
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 cir-
cumferences, 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 propaga-
tion 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
38 TREATISE
mark the ray of incident light, and AN which was per-
pendicular to BN, the refra6led 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 re-
fradtion, 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 BAG is BC,
and the Sine of the angle ABN is AN. But the angle
BAG 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 refradtion 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 substitut-
ing everywhere f instead off. 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
ON LIGHT. CHAP. Ill 39
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 refra6tions
of the ray on entering and on leaving one and the same
transparent body: namely that if NA falling on the ex-
ternal surface AB
is refradted into the
direction AD, so
the ray AD will be
refrafted on leaving
the transparent
body into the direc-
tion AN.
One sees also
the reason for a
noteworthy acci-
dent which hap-
pens in this re-
fraftion : 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 DAQj>r CBA is
such that in the triangle ACB, CB is equal to f 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 per-
pendicular to it. And thus the incident ray DA does not
then pierce the surface AB.
When
40 TREATISE
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^jrnust be more than
48 degrees 1 1 minutes in order that the ray DA may be
able to pass by refra<5lion. And when the ratio of the
velocities is as 3 to 4, as it is very nearly in water and
air, this angle DAQjnust exceed 41 degrees 24 minutes.
And this accords perfe6tly with experiment.
But it might here be asked : since the meeting of the
wave AC against the surface AB ought to produce move-
ment 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 J- 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 -f- 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 DAQJs smaller than is requisite to permit the
refra&ed ray DA to pass into the other transparent sub-
stance, one finds that the interior reflexion which occurs
at the surface AB is much augmented in brightness, as
is
ON LIGHT. CHAP. Ill 41
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 colle6ted in a minimum space when
it reaches BN. It appears also that the wave BN be-
comes 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 pro-
portion 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 incid-
ence DAQ^causes the wave BN, however great it was,
to be reduced to zero, (for this angle being 49 degrees 1 1
minutes in the glass, the angle BAN is still 1 1 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 refra6tion.
G Now
42 TREATISE
Now as concerns ordinary external reflexion, that is to
say which occurs when the angle of incidence DAQjs
still large enough to enable the refrafted ray to penetrate
beyond the surface AB, this reflexion should occur against
the particles of the substance which touches the trans-
parent 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 par-
ticles 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 re-
flexions are of nearly equal force, and that in different
transparent bodies they are so much the stronger as the
refra&ion 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 re-
fradled 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
ON LIGHT. CHAP. Ill 43
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 demonstra-
tion, which may be seen in his printed works, and in the
volume of letters of Mr. Des Cartes, is very long ; where-
fore I give here another
which is simpler and
easier.
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 pene-
trate, as water. And sup-
pose that a ray has come
from A, by B, to C, hav-
ing been refraded at B according to the law demon-
strated 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
44 TREATISE
that the point of refradtion 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 perpendi-
cular 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
ON LIGHT. CHAP. IV 45
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
E 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 move-
ment 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 differ-
ent 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 effefts of which re-
fradions 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 fa<ft they have, and appear to set
later:
46 TREATISE
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 refrations, as Astronomers know.
But there is one experiment which renders this refraftion
very evident; which is that of fixing a telescope on some
spot so that it views an objet, 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 objefts 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 acftion 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
ON LIGHT. CHAP. IV 47
or whether the encounter and hindrance which these par-
ticles 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.
Whence the configuration of the waves ought to be-
come 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.
Let
48 TREATISE
Let BC be the wave which brings the light to the
spe6lator 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 perpendic-
ular to the wave that reaches our eye, as will be under-
stood by what was said above, it is manifest that the point
A will be per-
c e i ved as
being in the
line BD, and
therefore
higher than
in faft it is.
Similarly
if the Earth
be AB, and
the top of the
Atmosphere
CD, which
probably is
not a well de-
fined spheri-
cal 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
interse<5ts 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
ON LIGHT. CHAP. IV 49
CA is the wave which brings the light to the speftator 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 refra6lion. But this angle EAF is
scarcely ever more than half a degree because the attenua-
tion of the vapours alters the waves of light but little.
Furthermore these refraftions 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 obje6t 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 effeft 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 homo-
geneous media, ought to be curved in an atmosphere of
unequal penetrability. For they necessarily follow from
the objeft to the eye the line which intersefts 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 objed. For
H although
50 TREATISE
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.
: 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
ON LIGHT. CHAP. IV 51
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 penetra-
bilities 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
52 TREATISE
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 intersefts all the waves at right angles,
as has been said.
CHAPTER V
ON THE STRANGE REFRACTION OF ICELAND CRYSTAL
IJHERE is brought from Iceland, which is
an Island in the North Sea, in the latitude
of 66 degrees, a kind of Crystal or trans-
parent 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 in-
vestigated, the more so because amongst transparent bodies
this one alone does not follow the ordinary rules with re-
sped: to rays of light. I have even been under some neces-
sity to make this research, because the refractions of this
Crystal seemed to overturn our preceding explanation of
regular refration; 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
ON LIGHT. CHAP. V 53
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.
4. The pieces of it which are found have the figure of an
oblique parallelepiped ; each of the six faces being a parallel-
ogram; and it admits of
being split in three directions
parallel to two of these op-
posed 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 parallelo- "~p
grams, as C, D, here, are angles of 101 degrees 52 minutes,
and
54 TREATISE
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 com-
posed 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 any-
wise 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 conje6lures 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 effeft is that obje6ts 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- ^
ONLIGHT. CHAP. V 55
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 refrafted. But in this Crystal the
perpendicular ray suffers refra&ion, and there are oblique
rays which pass through it quite straight.
8. But in order to explain these phenomena more par-
ticularly, 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
56 TREATISE
the surface AB ; and its se<5tion in the Crystal will form a
parallelogram GCFH. We will call this sedtion 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 IK, along MZ.
And as, in this extraordinary refra6tion, 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
refra&ed 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 re-
fraftion, to be in the straight line IK, hence it is neces-
sarily 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
ON LIGHT. CHAP. V 57
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 calcula-
tion 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 re-
fraction; 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.
1 1 . 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
refra6ted rays make with the perpendicular, was very pre-
cisely 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.
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
i as
58 TREATISE
as it is desired to examine a ray that is more or less
oblique. Then place the Crystal upon the intersedlion
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 direftly
above the line AB it will appear single only; and one
will see that the portion viewed through jjthe 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 station-
ary, whereas the other image shifts and moves entirely
around. Afterwards let the eye be placed at I (remaining
always
ONLIGHT. CHAP. V 59
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,
6o
TREATISE
RES, which the incident and the refra&ed ray make
with the perpendicular.
15. But I found in this refraftion that the ratio of
FR to RS was not constant, like the ordinary refradtion,
but that it varied with the varying obliquity of the in-
cident ray.
1 6. I found also that when QRE made a straight line,
that is, when the incident ray entered the Crystal with-
out being refradted (as I ascertained by the circumstance
that then the point E viewed by the extraordinary refrac-
tion appeared in the line CD, as seen without refradtion)
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 with-
outbeingrefradted,
as Mr. Bartholinus
believed, since that
inclination is only
70 degrees 57
minutes, as was
stated above. And
this is to be noted,
"F 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,
17. Finally, continuing my observations to discover the
nature
H
ONLIGHT. CHAP. V 61
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 re-
fraCtions 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.
1 8. 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 ex-
tending 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 here-
tofore. 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 demon-
strated.
19. As to the other emanation which should produce
the
62 TREATISE
the irregular refra6tion, 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 ex-
periments 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
ON LIGHT. CHAP. V 63
2 1 . Now this double refraftion, according to my Theory
hereinbefore established, seemed to demand a double emis-
sion 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 ad-
mitting two emissions of waves in one and the same body.
And since it might have been objefted 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.
23. I considered first the effeft of waves so formed, as
respedts 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
H
S A
X
(
T
B
N
64 TREATISE
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/^C come to meet the
surface AB at the points
AKB. 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 refraftion, these must here be hemispheroids.
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 se6lion 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^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
ON LIGHT. CHAP. V 65
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 respeft 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
refradtion 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 perpen-
dicular to its waves, as in ordinary refraftion, but along
lines cutting the waves obliquely.
25. Inquiring subsequently what might be the position
and form of these spheroids in the crystal, I considered that
all the six faces produced pre-
cisely the same refra&ions.
Taking, then, the parallele-
piped AFB, of which the ob-
tuse solid angle C is contained
between the three equal plane
angles, and imagining in it
the three principal se&ions,
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
K CA,
66 TREATISE
CA, and the third perpendicular to the face AF passing
through the edge BC; I knew that the refra&ions 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 sur-
face 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 se6lions, 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, 1 found it 45 degrees 20 minutes.
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 refra6lion of
the
ON LIGHT. CHAP. V 67
the ray perpendicular to the surface CG, which is inclined
to the perpendicular CL by the same angle. This, then,
being thus dis-
posed, and tak- /
ing CM at f
100,000 parts,
I found by the
cal culat ion
which will be
given at the
end, the semi-
major dia-
meter 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.
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 re-
fraCtions 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 in-
cident rays. Let there be such a ray RC falling upon the
surface
68 TREATISE
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 contaft I join 1C, which will be
the required refraftion 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 refra6tion. For the
refra&ion of the ray RC is nothing else than the progres-
sion 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#, and will moreover have
produced in the crystal around the centres x some hemi-
spheroidal partial waves similar to the hemi-spheroidal
GSPg, and similarly disposed, and of which the major
and
ONLIGHT. CHAP. V 69
and minor diameters will bear the same proportions to
the lines xv (the continuations of the lines H# up to KB
parallel to CO) that the diameters of the spheroid GSPg
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 deter-
mined, 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/ the refraCtion of the ray rC,
which comes from the opposite side, by making Co perpen-
dicular to rC and following out the rest of the construction
as before. Whence one sees that if the ray rC is inclined
equally with RC, the line Cd will necessarily be equal to
CD, because Ck is equal to CK, and Cg to CG. And in
consequence 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 will
be parallel to gG. Therefore if one prolongs the refraCted
rays CI, C/, until they meet the tangent ML at T and /,
the distances MT, M/, will also be equal. And so, by our
hypothesis, we explain perfectly the phenomenon men-
tioned above; to wit, that when there are two rays equally
inclined, but coming from opposite sides, as here the rays
RC, re, their refractions diverge equally from the line
followed
70 TREATISE
followed by the refra6lion of the ray perpendicular to the
surface, by considering these divergences in the direftion
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 refraftion 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 T 1 T . 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.
3 1 . Supposing then, in the next figure, as previously, the
surface of the crystal gG, the Ellipse GPg, and the line
N; and CM the refradlion of the perpendicular ray FC,
from which it diverges by 6 degrees 40 minutes. Now let
there be some other ray RC, the refra&ion of which must
be found.
About the centre C, with semi-diameter CG, let the
circumference gRG 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 ^MG at I; then join-
ing CI, this will be the required refraftion of the ray RC.
Which is demonstrated thus.
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
ON LIGHT. CHAP. V 71
is demonstrated to be a tangent to the Ellipse at I, it will be
evident by the things heretofore explained that CI is the
refradtion 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
N
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 construdion, 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
72 TREATISE
ordinary media a certain constant proportion between the
sines of the angles which the incident ray and the refra&ed
ray make with the perpendicular, so here there is such a
proportion between CV and CD or IE; that is to say be-
tween the Sine of the angle which the incident ray makes
with the perpendicular, and the horizontal intercept, in
the Ellipse, between the refra6lion 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 com-
paring together the regular and irregular refraction of
this crystal, there is this remarkable fat, 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 refra&ion), then the inscribed
sphere BVST is the extension in the same space of time of
the light which serves for the regular refradlion.
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 refraftion 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 ^y. So that it is sufficiently nearly, and may
be
N
N
ON LIGHT. CHAP. V 73
be exaftly, 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 perpendi-
cular 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.
34. The pro-
portion of the re-
fraCtion being
what we have just
seen, I will now
show that there
necessarily follows
thence that not-
able property of
the ray which fall-
ing obliquely on
the surface of the
crystal enters it
without suffering
refraction. For
supposing the same things as before, and that the ray
RC makes with the same surface gG the angle RCG of
L 73 degrees
74 TREATISE
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
refraftion CI, one will find that it makes exaftly 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 reftangle'gDC 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 El, so will CM be to
MT, which will then be 18,127. And being added to
ML, which is 1 1,609 (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 demonstra-
tion, which proves that the reciprocal relation of refrac-
tion obtains in this crystal the same as in other trans-
parent 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
ON LIGHT. CHAP. V 75
crystal, which I suppose to be IB, will have its refraftion
IA parallel to
the ray RC.
Let the same
things be sup-
posed as before ;
that is to say,
let CO, perpen-
dicular to CR,
represent a por-
tion of a wave
the continua-
tion 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 /z, it will go along hm^ parallel to CI, to meet
the surface IB, while the point K traverses K/ equal to hm ;
and while this accomplishes the remainder /B, there will
start from the point m a partial wave the semi-diameter
of which, mn, will have the same ratio to /B as IA to
KB.
76 TREATISE
KB. Whence it is evident that this wave or 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
impa6l 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 refra6lion 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 I A 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 refradtion holds good in this crystal
as well as in ordinary transparent bodies ; as is thus in fa<5t
found by observation.
36. I pass now to the consideration of other sections of
the crystal, and of the refraftions there produced, on
which, as will be seen, some other very remarkable phe-
nomena 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
toFE.
The se6lion 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. Refrac-
tions in this sedtion have this in common with the re-
fradtions in ordinary media that the plane which is drawn
through the incident ray and which also intersefts the
surface
ON LIGHT. CHAP. V 77
surface of the crystal at right angles, is that in which the
refra&ed ray also is found. But the refraftions which
appertain to every other seftion of this crystal have this
strange property that the refra&ed 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
H
which fa<5l we shall show the reason, in the first place, for
the seftion through AH ; and we shall show at the same
time how one can determine the refra6lion, 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.
37. About
78 TREATISE
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^g-M, such as the light would form in
spreading in the crystal, and let its section by the plane
AEHF form the Ellipse QG^g-, the major diameter of
which 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 FEE, to
which QC is perpendicular, it follows that QC is also
perpendicular to the axis of the spheroid, and consequently
QCg one of its major diameters. But the minor diameter
of this Ellipse, Gg-, will bear to Qg 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 QGggM. 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 AGO, 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 refra&ion of the ray which falls perpendi-
cularly 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^, 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
ON LIGHT. CHAP. V 79
straight line CM to meet the surface, would be parallel
to the plane QG^. If then through the point K one now
draws KS parallel to Gg, which will be parallel also to
QX, the tangent to the Ellipse QGq 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^, 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 conta6t be at I, then making
KC, QC, DC proportionals, draw DI parallel to CM;
also join CL 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 con-
tinuation 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 demon-
strating 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 Gg. Taking then the breadth Cc on
the side toward the angle E, let the parallelogram COoc
be taken as a portion of a wave, and let us complete the
parallelograms CK&:, CI/r, KI/, QK&o. In the time
then that the line Oo arrives at the surface of the crystal
at K, all the points of the wave COoc will have arrived
at the re6tangle Kr 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-
80 TREATISE
hemi-spheroid QM^, and similarly disposed. These hemi-
spheroids will necessarily all touch the plane of the
parallelogram KI/> at the same instant that Oo has
reached 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 demon-
strated the refra6tion of the oblique ray in the principal
se6tion through EF) and all those which have their
centres in the line Cc will touch the same plane KI in the
line I/; all these being similar to the hemi-spheroid
QMy. Since then the parallelogram K/ is that which
touches all these sphe'roids, this same parallelogram will
be precisely the continuation of the wave COoc in the
crystal, when Oo has arrived at 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
COoc has its continuation at I ; that is to say, that the
ray RC is refracted as CL
From this it is to be noted that the proportion of the
refraction for this se&ion of the crystal is that of the line
N to the semi-diameter CQ; by which one will easily
find the refradtions of all incment 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 refraftion
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 CQj:he major semi-diameter of the
spheroid, that is to say, as 156,962 to 105,032, very nearly
as
ON LIGHT. CHAP. V 81
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 effeft 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 refraftion 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 refra6tion of the crystal, the proportion of which is
as 5 to 3 5 an d which always raises the letters equally, and
higher than the irregular refra&ion 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 effeft follows from the refrac-
tions; and it will enable us at the same time to ascertain
the apparent place of a point of an objet placed immedi-
ately under the crystal, according to the different situation
of the eyes.
40. Let us see first by how much the irregular refraftion
of the plane through AH ought to lift the bottom of the
crystal. Let the plane of this figure represent separately
the section through Qg and CL, in which seflion there is
also the ray RC, and let the semi-elliptic plane through
Qq 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 refralion of the ray RC.
M If
82
TREATISE
If now one considers the point I as at the bottom of
the crystal, and that it is viewed by the rays ICR, Icr,
refradted equally at
the points Cr,
which should be
equally distant
from D, and that
these rays meet the
two eyes at Rr; it
is certain that the
\q point I will appear
raised to S where
the straight lines
RC, rr, meet;
which point S is
in DP, perpendicu-
lar to Qg. 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^ a semicircle cutting the
ray CR at B, from which BV is drawn perpendicular to
Qy; 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 CQj>o is VC to CD, as appears by the
method of finding the refradtion which we have shown
above, Article 31; but as VC is to CD, so is VBto DS. Then
as N is to CQ, so is VB to DS. Let ML be perpendicular to
CL. And because I suppose the eyes Rr to be distant about
a foot or so from the crystal, and consequently the angle
RSr 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
ON LIGHT. CHAP. V 83
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, con-
sidered as equal to CL, will be to DS as 99,324 to 70,283.
And so the elevation of the point I by the refraftion of
this section is known.
41. Now let there be represented the other seftion
through EF in the figure before the preceding one ; and
let CMg 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 refrafted rays ICR, Icr, which go to the two eyes;
CR and cr being
equally inclined to the
surface of the crystal
Gg. This being so, if
one draws ID parallel
to CM, which I sup-
pose to be the refra6tion
of the perpendicular
ray incident at the .
point C, the distances
DC, DC, will be equal,
as is easy to see by that
which has been demon-
strated in Article 28.
Now it is certain that
the point I should appear at S where the straight lines RC,
re, meet when prolonged; and that this point will fall in the
line
84 TREATISE
line DP perpendicular to Gg. If one draws IP perpendicu-
lar to this DP, it will be the distance PS which will
mark the apparent elevation of the point I. Let there be
described on Gg a semicircle cutting CR at B, from
which let BV be drawn perpendicular to Gg; and let N
to GC be the proportion of the refraftion in this seftion,
as in Article 28. Since then CI is the refraftion 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 con-
sider, again, the eyes to be distant above the crystal, BV
is deemed equal to th6 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 refradtion of this section; and it appears
that this elevation is greater than that by
the refra&ion of the preceding se6lion, since
the ratio of PD to DS was there as 99,324
to 70,283.
But by the regular refra&ion 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
of the height DP; as appears by this figure, where the point
P being viewed by the rays PCR, Per, refrafted equally
at
ON LIGHT. CHAP. V 85
at the surface O, this point must needs appear to be at S,
in the perpendicular PD where the lines RC, re, meet
when prolonged: and one knows that the line PC is to
CS as 5 to 3, since they aVe 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 -f- of PD.
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 f 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 re-
fra6lion 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
refra6tion do not appear direCtly below those which pro-
ceed
B
86 TREATISE
ceed 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 demon-
strations, 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 be-
tween the two heights of D and C, passing from one to
the other as one turns one's self around about the immov-
able 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
N
ON LIGHT. CHAP. V 87
let OK be adjusted across the angle OCK, so as to be
perpendicular to OC and equal to the line N, which I sup-
pose 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 demon-
strated in Article 36.
But it must be shown how one can determine the
point of contadl 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 perpendicu-
lar 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.
88 TREATISE
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
fa6t 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 respe6t 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 refra6tions 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 refra6lion of this Crystal, in order to see
whether each phenomenon that is deduced from our hypo-
thesis accords with that which is observed in fadt. 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 refradtions pre-
cisely such as they ought to be, and as I had foreseen them,
according to the preceding Theory.
In
ON LIGHT. CHAP. V 89
In order to explain what these seftions are, let ABKF
be the principal se<5tion through the axis of the crystal
ACK, in which there will also be the axis SS of a sphe-
roidal 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 dia-
meters.
Now as in the natural
se6tion of the crystal, made by a plane parallel to two
opposite faces, which plane is here represented by the
line GG, the refra6lion of the surfaces which are pro-
duced by it will be governed by the hemi-spheroids
GNG, according to what has been explained in the pre-
ceding 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 seftion through
N NN
9 o TREATISE
NN produces should effe6t the same refractions as the sur-
faces of the section through GG. And not only the surfaces
of the sedtion 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 re-
fractions as the natural surfaces of the crystal, or as the
sedtion 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 refradtion of the
surfaces aught 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 objedts placed beneath the crystal
would be less elevated than by that other refradtion.
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 refradtion ; and that
for oblique rays there were different measures for the
irregular refradtion according to the situation of the plane
in which the incident ray was.
Now these things were found in fadt 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 refradtions 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
ON LIGHT. CHAP. V 91
as one desired, all the surfaces of which should have the
same refra6lions as the natural surfaces of the crystal, except
the base, which will not refraft 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 se6lion 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 refraft the perpen-
dicular ray, although they would yet all cause double re-
fra6lion 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 re-
fradlion ; 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
92 TREATISE
way as spe6lacle glasses, and polishes it simply by continu-
ing the work, but ever reducing the material. I have not,
however, been able to give it perfect clarity and trans-
parency ; but the evenness which the surfaces acquire en-
ables 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 distindtly. And this aid is specially necessary when
it is wished to polish the natural surfaces to remove the in-
equalities ; 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 rea-
son wish to desist from describing it, in order to give op-
portunity 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.
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
ON LIGHT. CHAP. V 93
regular and irregular. On penetrating thence into the
other piece each ray will pass there without further divid-
ing itself in two ; but that one which underwent the regular
refra6tion, as here DG, will undergo again only a regular
refraCtion at GH ; and the other, CE, an irregular re-
fraftion 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 neigh-
bouring 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 irre-
gular refraCtion ; and that likewise CE has lost that which
was
94 TREATISE
was necessary to move the matter whicn 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 interseft one another at right angles, whether the
neighbouring surfaces are parallel or not, then the ray
which has come by the regular refra&ion, as DG, under-
goes only an irregular refraction in the lower piece ; and
on the contrary the ray which has come by the irregular
refraftion, as CE, undergoes only a regular refradtion.
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 con-
clude 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 refraftion ;
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
ON LIGHT. CHAP. V 95
Leaving then to others this research, I pass to what I
have to say touching the cause of the extraordinary figure
Df 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 diredtly upon one another, which are all of pen-
tagonal figure with rounded angles, and the sides a little
folded inwards. The grains of gray salt which are formed
from sea water affe<5t 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 per-
fe6lly 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 produ6lions comes from the arrange-
ment of the small invisible equal particles of which they
are composed. And, coming to our Iceland Crystal, I say
that
96 TREATISE
that if there were a pyramid such as ABCD, composed of
small rounded corpuscles, not spherical but flattened spher-
oids, 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 i 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,
oft 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 simi-
lar to those of the same crystal as this other figure re-
presents. 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 neighbour-
ing surface would come out by attaching itself to the surface
which is being separated, it would be needful for it to de-
tach 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.
There
B
ON LIGHT. CHAP. V 97
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.
It may be objedted
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,
o each
98 TREATISE
each spheroid would be obliged to detach itself from three
others which it touches upon their flattened surfaces, which
hold more strongly than the conta&s at the edges. And be-
sides 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, diretly 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 ex-
tending 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
ON LIGHT. CHAP. V 99
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 inci-
sion 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 exa&itude, 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,
ioo TREATISE
CBDA, CBVF, are inclined to one another, namely the
angle OCN formed by drawing CN perpendicular to FV,
and CO perpendi-
cular to DA. This
angle OCN I found
to be 105 degrees;
and its supplement
CNP, to be 75 de-
M grees, as it should be.
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 A
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.
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
ON LIGHT. CHAP. V 101
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 contadt M let
MN and MO be drawn as perpendiculars to CP and CS.
Now because the angles SCP, GCL, are right angles, the
angle
io2 TREATISE
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. Con-
sidering 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.
Similarly, because the angle OMZ is equal to CDZ, or
LCZ, which is 44 degrees 40 minutes, being the com-
plement 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
ON LIGHT. CHAP. V 103
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 com-
plement 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 deducing ML 1 1,609 there w ^l 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
reftangle 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 Cg-; 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 gC. 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 conta6t 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.
Through
104 TREATISE
Through the line BM, and through the points O and
A, let there be drawn planes parallel to one another,
which, in cutting the spher-
oid make the ellipses LED,
POP,QAQ; which will all
be similar and similarly dis-
posed, 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 LED,
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,
ON LIGHT. CHAP. VI 105
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 in-
terse6tion 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.
FTER 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 refradtion, to colleft 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 re-
p quisite
io6
TREATISE
quisite exa6litude according to these figures, but also be-
cause there exists in refra&ion itself a property which
hinders the perfeft 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 re-
fraftion, by the agreement which here is found between
the refrafted ray and the reflefted ray. Besides, it may
occur that some one in the future will discover in it utili-
ties which at present are not seen.
To proceed then to these figures, let us suppose first
that it is desired to find a surface CDE which shall re-
assemble 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 refradlion, 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 prin-
ciple the finding of these curves becomes very easy.
So
ON LIGHT. CHAP. VI 107
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 -f 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 -| 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 f 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. There-
fore 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 f of CB, it will represent the time along CB in the me-
dium; and in consequence the whole line AH will represent
also the time along AC, CB. Whence it appears that the
time
io8 TREATISE
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 dire6l 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; ancj from the centre B let the arc KS be de-
scribed, 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. Conse-
quently 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. Simi-
larly, 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
ON LIGHT. CHAP. VI 109
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 refraftion, by
dividing DA at G in such a way that DG is -| of DA,
and describing from the centre B any arc CX which cuts
BD at X, and another from the centre A with its semi-
diameter AF equal to f of GX ; or rather, having de-
scribed, as before, the arc CX, it is only necessary to make
DF equal to f of DX, and from he 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 re-
marked that it could serve for reflexions, if there were
some material of a mirror of such a nature that by its
means
no TREATISE
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
K
a thing could not arise from the matter of the mirror, and
it is entirely impossible.
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
ON LIGHT. CHAP. VI in
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 5 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 demon-
stration of these effects is just the same as the preceding.
But that this curved line CDE which serves for refrac-
tion 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 refra6lion, 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 ay\ CB will be \/xx + aa2ay+yy.
But the nature of the curve is such that J- of TC together
with CB is equal to DB, as was stated in the last construction :
then the equation will be between \y + ^/xx + aa 2 ay+yy
and a ; which being reduced, gives \ay-yy equal to-f-x*;
that is to say that having made DO equal to of DB, the
rectangle DFO is equal to |- 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.
Again,
I 12
TREATISE
Again, if one supposes the point B to be infinitely dis-
tant, 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 re-
marked that CX and KS become straight lines perpendi-
cular 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
ON LIGHT. CHAP. VI 113
is -- 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.
These are the two cases in which Conic se6lions serve
for refra6tion, 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 refra&ion, as also that of the
N
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
Q the
ii4 TREATISE
the same as that of the first, and the demonstration of its
effedt likewise. But it is worthy of remark that in one
case this oval be-
comes a perfedt
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
A time ago. The 4th
oval, serving only
for impossible re-
flexions, there is
no need to set it
forth.
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 to-
wards 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
ON LIGHT. CHAP. VI 115
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.
Then AK being an incident ray, and KB its refraftion
within the medium, it needs must be, according to the law
of refraftion 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 refraftion 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
n6 TREATISE
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 demon-
strated, 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 solu-
tion 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 re-
fra6lion occurring at the surface AK.
I say that for this the sole requirement is that the out-
line BDK which constitutes the other surface shall be
such
ON LIGHT. CHAP. VI
117
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 every-
where in equal
times, and in each
case in a time
equal to that which
the light employs K/
to pass along the
straight line LF
of which the part
AB is within the
glass.
Let LG be a ray
falling on the arc
AK. Its refraHon
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 f of DG and
the straight line
H
GL,
n8 TREATISE
GL, may be equal to FB together with f of BA and
the straight line AL ; which, as is clear, make up a given
length. Or rather, by deducing 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 -f 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 effe6t 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 | 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 refra&ion to be as 3 to 2. Now we
know that the piece of wave which is incident on G, ad-
vances thence along the line GD, since GV is the refradtion
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 f 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 +
IDG
ON LIGHT. CHAP. VI 119
I- DG + GL are equal to FB + | BA + AL ; on dedud-
ing the equals LH, LA, there will remain FD + fDG +
GH equal to FB + f BA. And, again, deducting from
one side GH, and from the other side -f- of AS, which are
._
equal, there will remain FD with -f DG equal to FB with
! of BS. But f of DG are equal to f of ES ; then FD is
equal to FB with | of BE. But DC was equal to f of EB ;
then deducing 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 termi-
nated 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 receiv-
ing rays that are parallel to AB will dire<5t them in such
wise that after being again refracted at the given surface
AK they will all be reassembled at the point L.
From the point L let there be drawn to some point of
the
120 TREATISE
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 erefted on AB the
perpendicular BC, which
will represent a wave of
light coming from the in-
finitely 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 of DG, plus GL may be equal to | of AB,
plus AL : or rather, on deducting from both sides GL,
which is given, CD plus f 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
ON LIGHT. CHAP. VI 121
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 paral-
lel rays falling on the surface
KDB will be reassembled at the
point L.
Again, let there be given the
surface AK, of any desired form,
generated by revolution about
the axis AB, and let the thick-
ness 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
R tend
122 TREATISE
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 f of AT, I observe that
| of GD ought to be equal to J~ of SB, plus BQ ; and,
deducting both of them from FD or FQ, that FD less
| of GD ought to be equal to FB less | 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 f 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 sjiall 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 construc-
tion, 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 termi-
nation 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
ON LIGHT. CHAP. VI 123
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 D
angles the waves which
travel along them.
of
B
For in the case
rays which, for ex-
ample, fall parallel
upon a spherical sur-
face AFE, intersecting
one another, after re-
fraCtion, 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 oan not be spheri-
cal. 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 some-
thing 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
124 TREATISE
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 refraftions 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
ON LIGHT. CHAP. VI 125
to the ray GM, and similarly the arc GF as a straight
line. But GM being the refradion 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 refra&ion ; 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 inter-
section 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 intersedl the ray FM at the same point V where
it is interse6led at right angles by the curve EK, and con-
sequently 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 advanc-
ing toward the meeting place becomes abc^ whereof the
part ab is made by the evolute ^C, a portion of the curve
ENC,
ia6 TREATISE
ENC, while the end C remains attached; and the part be
by the evolute of the portion E while the end E remains
attached. Consequently the same wave becomes def^ then
ghkj 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 de-
monstrated in section 1 2 of his LeStiones 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
sedtion, 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
ON LIGHT. CHAP. VI 127
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 A I
points.
And as the parallel
rays are merely per-
pendiculars to the
waves which fall on the concave surface, which waves are
parallel to AD, it will be found that as they come succes-
sively 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, gener-
ated 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 de-
scribed
ia8 TREATISE ON LIGHT
scribed 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 J 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 subjedt. 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.
END.