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+*** START OF THE PROJECT GUTENBERG EBOOK 14725 ***
+
+Note: Project Gutenberg also has an HTML version of this
+      file which includes the original illustrations.
+      See 14725-h.htm or 14725-h.zip:
+      (https://www.gutenberg.org/dirs/1/4/7/2/14725/14725-h/14725-h.htm)
+      or
+      (https://www.gutenberg.org/dirs/1/4/7/2/14725/14725-h.zip)
+
+
+
+
+
+TREATISE ON LIGHT
+
+In which are explained
+The causes of that which occurs
+In REFLEXION, & in REFRACTION
+
+And particularly
+In the strange REFRACTION
+OF ICELAND CRYSTAL
+
+by
+
+CHRISTIAAN HUYGENS
+
+Rendered into English by
+
+SILVANUS P. THOMPSON
+
+University of Chicago Press
+
+
+
+
+
+
+
+PREFACE
+
+
+I wrote this Treatise during my sojourn in France twelve years ago,
+and I communicated it in the year 1678 to the learned persons who then
+composed the Royal Academy of Science, to the membership of which the
+King had done me the honour of calling, me. Several of that body who
+are still alive will remember having been present when I read it, and
+above the rest those amongst them who applied themselves particularly
+to the study of Mathematics; of whom I cannot cite more than the
+celebrated gentlemen Cassini, Römer, and De la Hire. And, although I
+have since corrected and changed some parts, the copies which I had
+made of it at that time may serve for proof that I have yet added
+nothing to it save some conjectures touching the formation of Iceland
+Crystal, and a novel observation on the refraction of Rock Crystal. I
+have desired to relate these particulars to make known how long I have
+meditated the things which now I publish, and not for the purpose of
+detracting from the merit of those who, without having seen anything
+that I have written, may be found to have treated of like matters: as
+has in fact occurred to two eminent Geometricians, Messieurs Newton
+and Leibnitz, with respect to the Problem of the figure of glasses for
+collecting rays when one of the surfaces is given.
+
+One may ask why I have so long delayed to bring this work to the
+light. The reason is that I wrote it rather carelessly in the Language
+in which it appears, with the intention of translating it into Latin,
+so doing in order to obtain greater attention to the thing. After
+which I proposed to myself to give it out along with another Treatise
+on Dioptrics, in which I explain the effects of Telescopes and those
+things which belong more to that Science. But the pleasure of novelty
+being past, I have put off from time to time the execution of this
+design, and I know not when I shall ever come to an end if it, being
+often turned aside either by business or by some new study.
+Considering which I have finally judged that it was better worth while
+to publish this writing, such as it is, than to let it run the risk,
+by waiting longer, of remaining lost.
+
+There will be seen in it demonstrations of those kinds which do not
+produce as great a certitude as those of Geometry, and which even
+differ much therefrom, since whereas the Geometers prove their
+Propositions by fixed and incontestable Principles, here the
+Principles are verified by the conclusions to be drawn from them; the
+nature of these things not allowing of this being done otherwise.
+
+It is always possible to attain thereby to a degree of probability
+which very often is scarcely less than complete proof. To wit, when
+things which have been demonstrated by the Principles that have been
+assumed correspond perfectly to the phenomena which experiment has
+brought under observation; especially when there are a great number of
+them, and further, principally, when one can imagine and foresee new
+phenomena which ought to follow from the hypotheses which one employs,
+and when one finds that therein the fact corresponds to our prevision.
+But if all these proofs of probability are met with in that which I
+propose to discuss, as it seems to me they are, this ought to be a
+very strong confirmation of the success of my inquiry; and it must be
+ill if the facts are not pretty much as I represent them. I would
+believe then that those who love to know the Causes of things and who
+are able to admire the marvels of Light, will find some satisfaction
+in these various speculations regarding it, and in the new explanation
+of its famous property which is the main foundation of the
+construction of our eyes and of those great inventions which extend so
+vastly the use of them.
+
+I hope also that there will be some who by following these beginnings
+will penetrate much further into this question than I have been able
+to do, since the subject must be far from being exhausted. This
+appears from the passages which I have indicated where I leave certain
+difficulties without having resolved them, and still more from matters
+which I have not touched at all, such as Luminous Bodies of several
+sorts, and all that concerns Colours; in which no one until now can
+boast of having succeeded. Finally, there remains much more to be
+investigated touching the nature of Light which I do not pretend to
+have disclosed, and I shall owe much in return to him who shall be
+able to supplement that which is here lacking to me in knowledge. The
+Hague. The 8 January 1690.
+
+
+
+
+NOTE BY THE TRANSLATOR
+
+
+Considering the great influence which this Treatise has exercised in
+the development of the Science of Optics, it seems strange that two
+centuries should have passed before an English edition of the work
+appeared. Perhaps the circumstance is due to the mistaken zeal with
+which formerly everything that conflicted with the cherished ideas of
+Newton was denounced by his followers. The Treatise on Light of
+Huygens has, however, withstood the test of time: and even now the
+exquisite skill with which he applied his conception of the
+propagation of waves of light to unravel the intricacies of the
+phenomena of the double refraction of crystals, and of the refraction
+of the atmosphere, will excite the admiration of the student of
+Optics. It is true that his wave theory was far from the complete
+doctrine as subsequently developed by Thomas Young and Augustin
+Fresnel, and belonged rather to geometrical than to physical Optics.
+If Huygens had no conception of transverse vibrations, of the
+principle of interference, or of the existence of the ordered sequence
+of waves in trains, he nevertheless attained to a remarkably clear
+understanding of the principles of wave-propagation; and his
+exposition of the subject marks an epoch in the treatment of Optical
+problems. It has been needful in preparing this translation to
+exercise care lest one should import into the author's text ideas of
+subsequent date, by using words that have come to imply modern
+conceptions. Hence the adoption of as literal a rendering as possible.
+A few of the author's terms need explanation. He uses the word
+"refraction," for example, both for the phenomenon or process usually
+so denoted, and for the result of that process: thus the refracted ray
+he habitually terms "the refraction" of the incident ray. When a
+wave-front, or, as he terms it, a "wave," has passed from some initial
+position to a subsequent one, he terms the wave-front in its
+subsequent position "the continuation" of the wave. He also speaks of
+the envelope of a set of elementary waves, formed by coalescence of
+those elementary wave-fronts, as "the termination" of the wave; and
+the elementary wave-fronts he terms "particular" waves. Owing to the
+circumstance that the French word _rayon_ possesses the double
+signification of ray of light and radius of a circle, he avoids its
+use in the latter sense and speaks always of the semi-diameter, not of
+the radius. His speculations as to the ether, his suggestive views of
+the structure of crystalline bodies, and his explanation of opacity,
+slight as they are, will possibly surprise the reader by their seeming
+modernness. And none can read his investigation of the phenomena found
+in Iceland spar without marvelling at his insight and sagacity.
+
+S.P.T.
+
+June, 1912.
+
+
+
+
+TABLE OF MATTERS
+
+Contained in this Treatise
+
+
+CHAPTER I.
+On Rays Propagated in Straight Lines.
+
+  That Light is produced by a certain movement.
+
+  That no substance passes from the luminous object to the eyes.
+
+  That Light spreads spherically, almost as Sound does.
+
+  Whether Light takes time to spread.
+
+  Experience seeming to prove that it passes instantaneously.
+
+  Experience proving that it takes time.
+
+  How much its speed is greater than that of Sound.
+
+  In what the emission of Light differs from that of Sound.
+
+  That it is not the same medium which serves for Light and Sound.
+
+  How Sound is propagated.
+
+  How Light is propagated.
+
+  Detailed Remarks on the propagation of Light.
+
+  Why Rays are propagated only in straight lines.
+
+  How Light coming in different directions can cross itself.
+
+CHAPTER II.
+On Reflexion.
+
+  Demonstration of equality of angles of incidence and reflexion.
+
+  Why the incident and reflected rays are in the same plane
+  perpendicular to the reflecting surface.
+
+  That it is not needful for the reflecting surface to be perfectly
+  flat to attain equality of the angles of incidence and reflexion.
+
+CHAPTER III.
+On Refraction.
+
+  That bodies may be transparent without any substance passing through
+  them.
+
+  Proof that the ethereal matter passes through transparent bodies.
+
+  How this matter passing through can render them transparent.
+
+  That the most solid bodies in appearance are of a very loose texture.
+
+  That Light spreads more slowly in water and in glass than in air.
+
+  Third hypothesis to explain transparency, and the retardation which
+  Light suffers.
+
+  On that which makes bodies opaque.
+
+  Demonstration why Refraction obeys the known proportion of Sines.
+
+  Why the incident and refracted Rays produce one another reciprocally.
+
+  Why Reflexion within a triangular glass prism is suddenly augmented
+  when the Light can no longer penetrate.
+
+  That bodies which cause greater Refraction also cause stronger
+  Reflexion.
+
+  Demonstration of the Theorem of Mr. Fermat.
+
+CHAPTER IV.
+On the Refraction of the Air.
+
+  That the emanations of Light in the air are not spherical.
+
+  How consequently some objects appear higher than they are.
+
+  How the Sun may appear on the Horizon before he has risen.
+
+  That the rays of light become curved in the Air of the Atmosphere,
+  and what effects this produces.
+
+CHAPTER V.
+On the Strange Refraction of Iceland Crystal.
+
+  That this Crystal grows also in other countries.
+
+  Who first-wrote about it.
+
+  Description of Iceland Crystal; its substance, shape, and properties.
+
+  That it has two different Refractions.
+
+  That the ray perpendicular to the surface suffers refraction, and
+  that some rays inclined to the surface pass without suffering
+  refraction.
+
+  Observation of the refractions in this Crystal.
+
+  That there is a Regular and an Irregular Refraction.
+
+  The way of measuring the two Refractions of Iceland Crystal.
+
+  Remarkable properties of the Irregular Refraction.
+
+  Hypothesis to explain the double Refraction.
+
+  That Rock Crystal has also a double Refraction.
+
+  Hypothesis of emanations of Light, within Iceland Crystal, of
+  spheroidal form, for the Irregular Refraction.
+
+  How a perpendicular ray can suffer Refraction.
+
+  How the position and form of the spheroidal emanations in this
+  Crystal can be defined.
+
+  Explanation of the Irregular Refraction by these spheroidal
+  emanations.
+
+  Easy way to find the Irregular Refraction of each incident ray.
+
+  Demonstration of the oblique ray which traverses the Crystal without
+  being refracted.
+
+  Other irregularities of Refraction explained.
+
+  That an object placed beneath the Crystal appears double, in two
+  images of different heights.
+
+  Why the apparent heights of one of the images change on changing the
+  position of the eyes above the Crystal.
+
+  Of the different sections of this Crystal which produce yet other
+  refractions, and confirm all this Theory.
+
+  Particular way of polishing the surfaces after it has been cut.
+
+  Surprising phenomenon touching the rays which pass through two
+  separated pieces; the cause of which is not explained.
+
+  Probable conjecture on the internal composition of Iceland Crystal,
+  and of what figure its particles are.
+
+  Tests to confirm this conjecture.
+
+  Calculations which have been supposed in this Chapter.
+
+CHAPTER VI.
+On the Figures of transparent bodies which serve for Refraction and
+for Reflexion.
+
+  General and easy rule to find these Figures.
+
+  Invention of the Ovals of Mr. Des Cartes for Dioptrics.
+
+  How he was able to find these Lines.
+
+  Way of finding the surface of a glass for perfect refraction, when
+  the other surface is given.
+
+  Remark on what happens to rays refracted at a spherical surface.
+
+  Remark on the curved line which is formed by reflexion in a spherical
+  concave mirror.
+
+
+
+
+
+CHAPTER I
+
+ON RAYS PROPAGATED IN STRAIGHT LINES
+
+
+As happens in all the sciences in which Geometry is applied to matter,
+the demonstrations concerning Optics are founded on truths drawn from
+experience. Such are that the rays of light are propagated in straight
+lines; that the angles of reflexion and of incidence are equal; and
+that in refraction the ray is bent according to the law of sines, now
+so well known, and which is no less certain than the preceding laws.
+
+The majority of those who have written touching the various parts of
+Optics have contented themselves with presuming these truths. But
+some, more inquiring, have desired to investigate the origin and the
+causes, considering these to be in themselves wonderful effects of
+Nature. In which they advanced some ingenious things, but not however
+such that the most intelligent folk do not wish for better and more
+satisfactory explanations. Wherefore I here desire to propound what I
+have meditated on the subject, so as to contribute as much as I can
+to the explanation of this department of Natural Science, which, not
+without reason, is reputed to be one of its most difficult parts. I
+recognize myself to be much indebted to those who were the first to
+begin to dissipate the strange obscurity in which these things were
+enveloped, and to give us hope that they might be explained by
+intelligible reasoning. But, on the other hand I am astonished also
+that even here these have often been willing to offer, as assured and
+demonstrative, reasonings which were far from conclusive. For I do not
+find that any one has yet given a probable explanation of the first
+and most notable phenomena of light, namely why it is not propagated
+except in straight lines, and how visible rays, coming from an
+infinitude of diverse places, cross one another without hindering one
+another in any way.
+
+I shall therefore essay in this book, to give, in accordance with the
+principles accepted in the Philosophy of the present day, some clearer
+and more probable reasons, firstly of these properties of light
+propagated rectilinearly; secondly of light which is reflected on
+meeting other bodies. Then I shall explain the phenomena of those rays
+which are said to suffer refraction on passing through transparent
+bodies of different sorts; and in this part I shall also explain the
+effects of the refraction of the air by the different densities of the
+Atmosphere.
+
+Thereafter I shall examine the causes of the strange refraction of a
+certain kind of Crystal which is brought from Iceland. And finally I
+shall treat of the various shapes of transparent and reflecting bodies
+by which rays are collected at a point or are turned aside in various
+ways. From this it will be seen with what facility, following our new
+Theory, we find not only the Ellipses, Hyperbolas, and other curves
+which Mr. Des Cartes has ingeniously invented for this purpose; but
+also those which the surface of a glass lens ought to possess when its
+other surface is given as spherical or plane, or of any other figure
+that may be.
+
+It is inconceivable to doubt that light consists in the motion of some
+sort of matter. For whether one considers its production, one sees
+that here upon the Earth it is chiefly engendered by fire and flame
+which contain without doubt bodies that are in rapid motion, since
+they dissolve and melt many other bodies, even the most solid; or
+whether one considers its effects, one sees that when light is
+collected, as by concave mirrors, it has the property of burning as a
+fire does, that is to say it disunites the particles of bodies. This
+is assuredly the mark of motion, at least in the true Philosophy, in
+which one conceives the causes of all natural effects in terms of
+mechanical motions. This, in my opinion, we must necessarily do, or
+else renounce all hopes of ever comprehending anything in Physics.
+
+And as, according to this Philosophy, one holds as certain that the
+sensation of sight is excited only by the impression of some movement
+of a kind of matter which acts on the nerves at the back of our eyes,
+there is here yet one reason more for believing that light consists in
+a movement of the matter which exists between us and the luminous
+body.
+
+Further, when one considers the extreme speed with which light spreads
+on every side, and how, when it comes from different regions, even
+from those directly opposite, the rays traverse one another without
+hindrance, one may well understand that when we see a luminous object,
+it cannot be by any transport of matter coming to us from this object,
+in the way in which a shot or an arrow traverses the air; for
+assuredly that would too greatly impugn these two properties of light,
+especially the second of them. It is then in some other way that light
+spreads; and that which can lead us to comprehend it is the knowledge
+which we have of the spreading of Sound in the air.
+
+We know that by means of the air, which is an invisible and impalpable
+body, Sound spreads around the spot where it has been produced, by a
+movement which is passed on successively from one part of the air to
+another; and that the spreading of this movement, taking place equally
+rapidly on all sides, ought to form spherical surfaces ever enlarging
+and which strike our ears. Now there is no doubt at all that light
+also comes from the luminous body to our eyes by some movement
+impressed on the matter which is between the two; since, as we have
+already seen, it cannot be by the transport of a body which passes
+from one to the other. If, in addition, light takes time for its
+passage--which we are now going to examine--it will follow that this
+movement, impressed on the intervening matter, is successive; and
+consequently it spreads, as Sound does, by spherical surfaces and
+waves: for I call them waves from their resemblance to those which are
+seen to be formed in water when a stone is thrown into it, and which
+present a successive spreading as circles, though these arise from
+another cause, and are only in a flat surface.
+
+To see then whether the spreading of light takes time, let us consider
+first whether there are any facts of experience which can convince us
+to the contrary. As to those which can be made here on the Earth, by
+striking lights at great distances, although they prove that light
+takes no sensible time to pass over these distances, one may say with
+good reason that they are too small, and that the only conclusion to
+be drawn from them is that the passage of light is extremely rapid.
+Mr. Des Cartes, who was of opinion that it is instantaneous, founded
+his views, not without reason, upon a better basis of experience,
+drawn from the Eclipses of the Moon; which, nevertheless, as I shall
+show, is not at all convincing. I will set it forth, in a way a little
+different from his, in order to make the conclusion more
+comprehensible.
+
+[Illustration]
+
+Let A be the place of the sun, BD a part of the orbit or annual path
+of the Earth: ABC a straight line which I suppose to meet the orbit of
+the Moon, which is represented by the circle CD, at C.
+
+Now if light requires time, for example one hour, to traverse the
+space which is between the Earth and the Moon, it will follow that the
+Earth having arrived at B, the shadow which it casts, or the
+interruption of the light, will not yet have arrived at the point C,
+but will only arrive there an hour after. It will then be one hour
+after, reckoning from the moment when the Earth was at B, that the
+Moon, arriving at C, will be obscured: but this obscuration or
+interruption of the light will not reach the Earth till after another
+hour. Let us suppose that the Earth in these two hours will have
+arrived at E. The Earth then, being at E, will see the Eclipsed Moon
+at C, which it left an hour before, and at the same time will see the
+sun at A. For it being immovable, as I suppose with Copernicus, and
+the light moving always in straight lines, it must always appear where
+it is. But one has always observed, we are told, that the eclipsed
+Moon appears at the point of the Ecliptic opposite to the Sun; and yet
+here it would appear in arrear of that point by an amount equal to the
+angle GEC, the supplement of AEC. This, however, is contrary to
+experience, since the angle GEC would be very sensible, and about 33
+degrees. Now according to our computation, which is given in the
+Treatise on the causes of the phenomena of Saturn, the distance BA
+between the Earth and the Sun is about twelve thousand diameters of
+the Earth, and hence four hundred times greater than BC the distance
+of the Moon, which is 30 diameters. Then the angle ECB will be nearly
+four hundred times greater than BAE, which is five minutes; namely,
+the path which the earth travels in two hours along its orbit; and
+thus the angle BCE will be nearly 33 degrees; and likewise the angle
+CEG, which is greater by five minutes.
+
+But it must be noted that the speed of light in this argument has been
+assumed such that it takes a time of one hour to make the passage from
+here to the Moon. If one supposes that for this it requires only one
+minute of time, then it is manifest that the angle CEG will only be 33
+minutes; and if it requires only ten seconds of time, the angle will
+be less than six minutes. And then it will not be easy to perceive
+anything of it in observations of the Eclipse; nor, consequently, will
+it be permissible to deduce from it that the movement of light is
+instantaneous.
+
+It is true that we are here supposing a strange velocity that would be
+a hundred thousand times greater than that of Sound. For Sound,
+according to what I have observed, travels about 180 Toises in the
+time of one Second, or in about one beat of the pulse. But this
+supposition ought not to seem to be an impossibility; since it is not
+a question of the transport of a body with so great a speed, but of a
+successive movement which is passed on from some bodies to others. I
+have then made no difficulty, in meditating on these things, in
+supposing that the emanation of light is accomplished with time,
+seeing that in this way all its phenomena can be explained, and that
+in following the contrary opinion everything is incomprehensible. For
+it has always seemed tome that even Mr. Des Cartes, whose aim has been
+to treat all the subjects of Physics intelligibly, and who assuredly
+has succeeded in this better than any one before him, has said nothing
+that is not full of difficulties, or even inconceivable, in dealing
+with Light and its properties.
+
+But that which I employed only as a hypothesis, has recently received
+great seemingness as an established truth by the ingenious proof of
+Mr. Römer which I am going here to relate, expecting him himself to
+give all that is needed for its confirmation. It is founded as is the
+preceding argument upon celestial observations, and proves not only
+that Light takes time for its passage, but also demonstrates how much
+time it takes, and that its velocity is even at least six times
+greater than that which I have just stated.
+
+For this he makes use of the Eclipses suffered by the little planets
+which revolve around Jupiter, and which often enter his shadow: and
+see what is his reasoning. Let A be the Sun, BCDE the annual orbit of
+the Earth, F Jupiter, GN the orbit of the nearest of his Satellites,
+for it is this one which is more apt for this investigation than any
+of the other three, because of the quickness of its revolution. Let G
+be this Satellite entering into the shadow of Jupiter, H the same
+Satellite emerging from the shadow.
+
+[Illustration]
+
+Let it be then supposed, the Earth being at B some time before the
+last quadrature, that one has seen the said Satellite emerge from the
+shadow; it must needs be, if the Earth remains at the same place,
+that, after 42-1/2 hours, one would again see a similar emergence,
+because that is the time in which it makes the round of its orbit, and
+when it would come again into opposition to the Sun. And if the Earth,
+for instance, were to remain always at B during 30 revolutions of this
+Satellite, one would see it again emerge from the shadow after 30
+times 42-1/2 hours. But the Earth having been carried along during
+this time to C, increasing thus its distance from Jupiter, it follows
+that if Light requires time for its passage the illumination of the
+little planet will be perceived later at C than it would have been at
+B, and that there must be added to this time of 30 times 42-1/2 hours
+that which the Light has required to traverse the space MC, the
+difference of the spaces CH, BH. Similarly at the other quadrature
+when the earth has come to E from D while approaching toward Jupiter,
+the immersions of the Satellite ought to be observed at E earlier than
+they would have been seen if the Earth had remained at D.
+
+Now in quantities of observations of these Eclipses, made during ten
+consecutive years, these differences have been found to be very
+considerable, such as ten minutes and more; and from them it has been
+concluded that in order to traverse the whole diameter of the annual
+orbit KL, which is double the distance from here to the sun, Light
+requires about 22 minutes of time.
+
+The movement of Jupiter in his orbit while the Earth passed from B to
+C, or from D to E, is included in this calculation; and this makes it
+evident that one cannot attribute the retardation of these
+illuminations or the anticipation of the eclipses, either to any
+irregularity occurring in the movement of the little planet or to its
+eccentricity.
+
+If one considers the vast size of the diameter KL, which according to
+me is some 24 thousand diameters of the Earth, one will acknowledge
+the extreme velocity of Light. For, supposing that KL is no more than
+22 thousand of these diameters, it appears that being traversed in 22
+minutes this makes the speed a thousand diameters in one minute, that
+is 16-2/3 diameters in one second or in one beat of the pulse, which
+makes more than 11 hundred times a hundred thousand toises; since the
+diameter of the Earth contains 2,865 leagues, reckoned at 25 to the
+degree, and each each league is 2,282 Toises, according to the exact
+measurement which Mr. Picard made by order of the King in 1669. But
+Sound, as I have said above, only travels 180 toises in the same time
+of one second: hence the velocity of Light is more than six hundred
+thousand times greater than that of Sound. This, however, is quite
+another thing from being instantaneous, since there is all the
+difference between a finite thing and an infinite. Now the successive
+movement of Light being confirmed in this way, it follows, as I have
+said, that it spreads by spherical waves, like the movement of Sound.
+
+But if the one resembles the other in this respect, they differ in
+many other things; to wit, in the first production of the movement
+which causes them; in the matter in which the movement spreads; and in
+the manner in which it is propagated. As to that which occurs in the
+production of Sound, one knows that it is occasioned by the agitation
+undergone by an entire body, or by a considerable part of one, which
+shakes all the contiguous air. But the movement of the Light must
+originate as from each point of the luminous object, else we should
+not be able to perceive all the different parts of that object, as
+will be more evident in that which follows. And I do not believe that
+this movement can be better explained than by supposing that all those
+of the luminous bodies which are liquid, such as flames, and
+apparently the sun and the stars, are composed of particles which
+float in a much more subtle medium which agitates them with great
+rapidity, and makes them strike against the particles of the ether
+which surrounds them, and which are much smaller than they. But I hold
+also that in luminous solids such as charcoal or metal made red hot in
+the fire, this same movement is caused by the violent agitation of
+the particles of the metal or of the wood; those of them which are on
+the surface striking similarly against the ethereal matter. The
+agitation, moreover, of the particles which engender the light ought
+to be much more prompt and more rapid than is that of the bodies which
+cause sound, since we do not see that the tremors of a body which is
+giving out a sound are capable of giving rise to Light, even as the
+movement of the hand in the air is not capable of producing Sound.
+
+Now if one examines what this matter may be in which the movement
+coming from the luminous body is propagated, which I call Ethereal
+matter, one will see that it is not the same that serves for the
+propagation of Sound. For one finds that the latter is really that
+which we feel and which we breathe, and which being removed from any
+place still leaves there the other kind of matter that serves to
+convey Light. This may be proved by shutting up a sounding body in a
+glass vessel from which the air is withdrawn by the machine which Mr.
+Boyle has given us, and with which he has performed so many beautiful
+experiments. But in doing this of which I speak, care must be taken to
+place the sounding body on cotton or on feathers, in such a way that
+it cannot communicate its tremors either to the glass vessel which
+encloses it, or to the machine; a precaution which has hitherto been
+neglected. For then after having exhausted all the air one hears no
+Sound from the metal, though it is struck.
+
+One sees here not only that our air, which does not penetrate through
+glass, is the matter by which Sound spreads; but also that it is not
+the same air but another kind of matter in which Light spreads; since
+if the air is removed from the vessel the Light does not cease to
+traverse it as before.
+
+And this last point is demonstrated even more clearly by the
+celebrated experiment of Torricelli, in which the tube of glass from
+which the quicksilver has withdrawn itself, remaining void of air,
+transmits Light just the same as when air is in it. For this proves
+that a matter different from air exists in this tube, and that this
+matter must have penetrated the glass or the quicksilver, either one
+or the other, though they are both impenetrable to the air. And when,
+in the same experiment, one makes the vacuum after putting a little
+water above the quicksilver, one concludes equally that the said
+matter passes through glass or water, or through both.
+
+As regards the different modes in which I have said the movements of
+Sound and of Light are communicated, one may sufficiently comprehend
+how this occurs in the case of Sound if one considers that the air is
+of such a nature that it can be compressed and reduced to a much
+smaller space than that which it ordinarily occupies. And in
+proportion as it is compressed the more does it exert an effort to
+regain its volume; for this property along with its penetrability,
+which remains notwithstanding its compression, seems to prove that it
+is made up of small bodies which float about and which are agitated
+very rapidly in the ethereal matter composed of much smaller parts. So
+that the cause of the spreading of Sound is the effort which these
+little bodies make in collisions with one another, to regain freedom
+when they are a little more squeezed together in the circuit of these
+waves than elsewhere.
+
+But the extreme velocity of Light, and other properties which it has,
+cannot admit of such a propagation of motion, and I am about to show
+here the way in which I conceive it must occur. For this, it is
+needful to explain the property which hard bodies must possess to
+transmit movement from one to another.
+
+When one takes a number of spheres of equal size, made of some very
+hard substance, and arranges them in a straight line, so that they
+touch one another, one finds, on striking with a similar sphere
+against the first of these spheres, that the motion passes as in an
+instant to the last of them, which separates itself from the row,
+without one's being able to perceive that the others have been
+stirred. And even that one which was used to strike remains motionless
+with them. Whence one sees that the movement passes with an extreme
+velocity which is the greater, the greater the hardness of the
+substance of the spheres.
+
+But it is still certain that this progression of motion is not
+instantaneous, but successive, and therefore must take time. For if
+the movement, or the disposition to movement, if you will have it so,
+did not pass successively through all these spheres, they would all
+acquire the movement at the same time, and hence would all advance
+together; which does not happen. For the last one leaves the whole row
+and acquires the speed of the one which was pushed. Moreover there are
+experiments which demonstrate that all the bodies which we reckon of
+the hardest kind, such as quenched steel, glass, and agate, act as
+springs and bend somehow, not only when extended as rods but also when
+they are in the form of spheres or of other shapes. That is to say
+they yield a little in themselves at the place where they are struck,
+and immediately regain their former figure. For I have found that on
+striking with a ball of glass or of agate against a large and quite
+thick thick piece of the same substance which had a flat surface,
+slightly soiled with breath or in some other way, there remained round
+marks, of smaller or larger size according as the blow had been weak
+or strong. This makes it evident that these substances yield where
+they meet, and spring back: and for this time must be required.
+
+Now in applying this kind of movement to that which produces Light
+there is nothing to hinder us from estimating the particles of the
+ether to be of a substance as nearly approaching to perfect hardness
+and possessing a springiness as prompt as we choose. It is not
+necessary to examine here the causes of this hardness, or of that
+springiness, the consideration of which would lead us too far from our
+subject. I will say, however, in passing that we may conceive that the
+particles of the ether, notwithstanding their smallness, are in turn
+composed of other parts and that their springiness consists in the
+very rapid movement of a subtle matter which penetrates them from
+every side and constrains their structure to assume such a disposition
+as to give to this fluid matter the most overt and easy passage
+possible. This accords with the explanation which Mr. Des Cartes gives
+for the spring, though I do not, like him, suppose the pores to be in
+the form of round hollow canals. And it must not be thought that in
+this there is anything absurd or impossible, it being on the contrary
+quite credible that it is this infinite series of different sizes of
+corpuscles, having different degrees of velocity, of which Nature
+makes use to produce so many marvellous effects.
+
+But though we shall ignore the true cause of springiness we still see
+that there are many bodies which possess this property; and thus there
+is nothing strange in supposing that it exists also in little
+invisible bodies like the particles of the Ether. Also if one wishes
+to seek for any other way in which the movement of Light is
+successively communicated, one will find none which agrees better,
+with uniform progression, as seems to be necessary, than the property
+of springiness; because if this movement should grow slower in
+proportion as it is shared over a greater quantity of matter, in
+moving away from the source of the light, it could not conserve this
+great velocity over great distances. But by supposing springiness in
+the ethereal matter, its particles will have the property of equally
+rapid restitution whether they are pushed strongly or feebly; and thus
+the propagation of Light will always go on with an equal velocity.
+
+[Illustration]
+
+And it must be known that although the particles of the ether are not
+ranged thus in straight lines, as in our row of spheres, but
+confusedly, so that one of them touches several others, this does not
+hinder them from transmitting their movement and from spreading it
+always forward. As to this it is to be remarked that there is a law of
+motion serving for this propagation, and verifiable by experiment. It
+is that when a sphere, such as A here, touches several other similar
+spheres CCC, if it is struck by another sphere B in such a way as to
+exert an impulse against all the spheres CCC which touch it, it
+transmits to them the whole of its movement, and remains after that
+motionless like the sphere B. And without supposing that the ethereal
+particles are of spherical form (for I see indeed no need to suppose
+them so) one may well understand that this property of communicating
+an impulse does not fail to contribute to the aforesaid propagation
+of movement.
+
+Equality of size seems to be more necessary, because otherwise there
+ought to be some reflexion of movement backwards when it passes from a
+smaller particle to a larger one, according to the Laws of Percussion
+which I published some years ago.
+
+However, one will see hereafter that we have to suppose such an
+equality not so much as a necessity for the propagation of light as
+for rendering that propagation easier and more powerful; for it is not
+beyond the limits of probability that the particles of the ether have
+been made equal for a purpose so important as that of light, at least
+in that vast space which is beyond the region of atmosphere and which
+seems to serve only to transmit the light of the Sun and the Stars.
+
+I have then shown in what manner one may conceive Light to spread
+successively, by spherical waves, and how it is possible that this
+spreading is accomplished with as great a velocity as that which
+experiments and celestial observations demand. Whence it may be
+further remarked that although the particles are supposed to be in
+continual movement (for there are many reasons for this) the
+successive propagation of the waves cannot be hindered by this;
+because the propagation consists nowise in the transport of those
+particles but merely in a small agitation which they cannot help
+communicating to those surrounding, notwithstanding any movement which
+may act on them causing them to be changing positions amongst
+themselves.
+
+But we must consider still more particularly the origin of these
+waves, and the manner in which they spread. And, first, it follows
+from what has been said on the production of Light, that each little
+region of a luminous body, such as the Sun, a candle, or a burning
+coal, generates its own waves of which that region is the centre. Thus
+in the flame of a candle, having distinguished the points A, B, C,
+concentric circles described about each of these points represent the
+waves which come from them. And one must imagine the same about every
+point of the surface and of the part within the flame.
+
+[Illustration]
+
+But as the percussions at the centres of these waves possess no
+regular succession, it must not be supposed that the waves themselves
+follow one another at equal distances: and if the distances marked in
+the figure appear to be such, it is rather to mark the progression of
+one and the same wave at equal intervals of time than to represent
+several of them issuing from one and the same centre.
+
+After all, this prodigious quantity of waves which traverse one
+another without confusion and without effacing one another must not be
+deemed inconceivable; it being certain that one and the same particle
+of matter can serve for many waves coming from different sides or even
+from contrary directions, not only if it is struck by blows which
+follow one another closely but even for those which act on it at the
+same instant. It can do so because the spreading of the movement is
+successive. This may be proved by the row of equal spheres of hard
+matter, spoken of above. If against this row there are pushed from two
+opposite sides at the same time two similar spheres A and D, one will
+see each of them rebound with the same velocity which it had in
+striking, yet the whole row will remain in its place, although the
+movement has passed along its whole length twice over. And if these
+contrary movements happen to meet one another at the middle sphere, B,
+or at some other such as C, that sphere will yield and act as a spring
+at both sides, and so will serve at the same instant to transmit these
+two movements.
+
+[Illustration]
+
+But what may at first appear full strange and even incredible is that
+the undulations produced by such small movements and corpuscles,
+should spread to such immense distances; as for example from the Sun
+or from the Stars to us. For the force of these waves must grow feeble
+in proportion as they move away from their origin, so that the action
+of each one in particular will without doubt become incapable of
+making itself felt to our sight. But one will cease to be astonished
+by considering how at a great distance from the luminous body an
+infinitude of waves, though they have issued from different points of
+this body, unite together in such a way that they sensibly compose one
+single wave only, which, consequently, ought to have enough force to
+make itself felt. Thus this infinite number of waves which originate
+at the same instant from all points of a fixed star, big it may be as
+the Sun, make practically only one single wave which may well have
+force enough to produce an impression on our eyes. Moreover from each
+luminous point there may come many thousands of waves in the smallest
+imaginable time, by the frequent percussion of the corpuscles which
+strike the Ether at these points: which further contributes to
+rendering their action more sensible.
+
+[Illustration]
+
+There is the further consideration in the emanation of these waves,
+that each particle of matter in which a wave spreads, ought not to
+communicate its motion only to the next particle which is in the
+straight line drawn from the luminous point, but that it also imparts
+some of it necessarily to all the others which touch it and which
+oppose themselves to its movement. So it arises that around each
+particle there is made a wave of which that particle is the centre.
+Thus if DCF is a wave emanating from the luminous point A, which is
+its centre, the particle B, one of those comprised within the sphere
+DCF, will have made its particular or partial wave KCL, which will
+touch the wave DCF at C at the same moment that the principal wave
+emanating from the point A has arrived at DCF; and it is clear that it
+will be only the region C of the wave KCL which will touch the wave
+DCF, to wit, that which is in the straight line drawn through AB.
+Similarly the other particles of the sphere DCF, such as _bb_, _dd_,
+etc., will each make its own wave. But each of these waves can be
+infinitely feeble only as compared with the wave DCF, to the
+composition of which all the others contribute by the part of their
+surface which is most distant from the centre A.
+
+One sees, in addition, that the wave DCF is determined by the
+distance attained in a certain space of time by the movement which
+started from the point A; there being no movement beyond this wave,
+though there will be in the space which it encloses, namely in parts
+of the particular waves, those parts which do not touch the sphere
+DCF. And all this ought not to seem fraught with too much minuteness
+or subtlety, since we shall see in the sequel that all the properties
+of Light, and everything pertaining to its reflexion and its
+refraction, can be explained in principle by this means. This is a
+matter which has been quite unknown to those who hitherto have begun
+to consider the waves of light, amongst whom are Mr. Hooke in his
+_Micrographia_, and Father Pardies, who, in a treatise of which he let
+me see a portion, and which he was unable to complete as he died
+shortly afterward, had undertaken to prove by these waves the effects
+of reflexion and refraction. But the chief foundation, which consists
+in the remark I have just made, was lacking in his demonstrations; and
+for the rest he had opinions very different from mine, as may be will
+appear some day if his writing has been preserved.
+
+To come to the properties of Light. We remark first that each portion
+of a wave ought to spread in such a way that its extremities lie
+always between the same straight lines drawn from the luminous point.
+Thus the portion BG of the wave, having the luminous point A as its
+centre, will spread into the arc CE bounded by the straight lines ABC,
+AGE. For although the particular waves produced by the particles
+comprised within the space CAE spread also outside this space, they
+yet do not concur at the same instant to compose a wave which
+terminates the movement, as they do precisely at the circumference
+CE, which is their common tangent.
+
+And hence one sees the reason why light, at least if its rays are not
+reflected or broken, spreads only by straight lines, so that it
+illuminates no object except when the path from its source to that
+object is open along such lines.
+
+For if, for example, there were an opening BG, limited by opaque
+bodies BH, GI, the wave of light which issues from the point A will
+always be terminated by the straight lines AC, AE, as has just been
+shown; the parts of the partial waves which spread outside the space
+ACE being too feeble to produce light there.
+
+Now, however small we make the opening BG, there is always the same
+reason causing the light there to pass between straight lines; since
+this opening is always large enough to contain a great number of
+particles of the ethereal matter, which are of an inconceivable
+smallness; so that it appears that each little portion of the wave
+necessarily advances following the straight line which comes from the
+luminous point. Thus then we may take the rays of light as if they
+were straight lines.
+
+It appears, moreover, by what has been remarked touching the
+feebleness of the particular waves, that it is not needful that all
+the particles of the Ether should be equal amongst themselves, though
+equality is more apt for the propagation of the movement. For it is
+true that inequality will cause a particle by pushing against another
+larger one to strive to recoil with a part of its movement; but it
+will thereby merely generate backwards towards the luminous point some
+partial waves incapable of causing light, and not a wave compounded of
+many as CE was.
+
+Another property of waves of light, and one of the most marvellous,
+is that when some of them come from different or even from opposing
+sides, they produce their effect across one another without any
+hindrance. Whence also it comes about that a number of spectators may
+view different objects at the same time through the same opening, and
+that two persons can at the same time see one another's eyes. Now
+according to the explanation which has been given of the action of
+light, how the waves do not destroy nor interrupt one another when
+they cross one another, these effects which I have just mentioned are
+easily conceived. But in my judgement they are not at all easy to
+explain according to the views of Mr. Des Cartes, who makes Light to
+consist in a continuous pressure merely tending to movement. For this
+pressure not being able to act from two opposite sides at the same
+time, against bodies which have no inclination to approach one
+another, it is impossible so to understand what I have been saying
+about two persons mutually seeing one another's eyes, or how two
+torches can illuminate one another.
+
+
+
+
+CHAPTER II
+
+ON REFLEXION
+
+
+Having explained the effects of waves of light which spread in a
+homogeneous matter, we will examine next that which happens to them on
+encountering other bodies. We will first make evident how the
+Reflexion of light is explained by these same waves, and why it
+preserves equality of angles.
+
+Let there be a surface AB; plane and polished, of some metal, glass,
+or other body, which at first I will consider as perfectly uniform
+(reserving to myself to deal at the end of this demonstration with the
+inequalities from which it cannot be exempt), and let a line AC,
+inclined to AD, represent a portion of a wave of light, the centre of
+which is so distant that this portion AC may be considered as a
+straight line; for I consider all this as in one plane, imagining to
+myself that the plane in which this figure is, cuts the sphere of the
+wave through its centre and intersects the plane AB at right angles.
+This explanation will suffice once for all.
+
+[Illustration]
+
+The piece C of the wave AC, will in a certain space of time advance as
+far as the plane AB at B, following the straight line CB, which may be
+supposed to come from the luminous centre, and which in consequence is
+perpendicular to AC. Now in this same space of time the portion A of
+the same wave, which has been hindered from communicating its movement
+beyond the plane AB, or at least partly so, ought to have continued
+its movement in the matter which is above this plane, and this along a
+distance equal to CB, making its own partial spherical wave,
+according to what has been said above. Which wave is here represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter AN equal to CB.
+
+If one considers further the other pieces H of the wave AC, it appears
+that they will not only have reached the surface AB by straight lines
+HK parallel to CB, but that in addition they will have generated in
+the transparent air, from the centres K, K, K, particular spherical
+waves, represented here by circumferences the semi-diameters of which
+are equal to KM, that is to say to the continuations of HK as far as
+the line BG parallel to AC. But all these circumferences have as a
+common tangent the straight line BN, namely the same which is drawn
+from B as a tangent to the first of the circles, of which A is the
+centre, and AN the semi-diameter equal to BC, as is easy to see.
+
+It is then the line BN (comprised between B and the point N where the
+perpendicular from the point A falls) which is as it were formed by
+all these circumferences, and which terminates the movement which is
+made by the reflexion of the wave AC; and it is also the place where
+the movement occurs in much greater quantity than anywhere else.
+Wherefore, according to that which has been explained, BN is the
+propagation of the wave AC at the moment when the piece C of it has
+arrived at B. For there is no other line which like BN is a common
+tangent to all the aforesaid circles, except BG below the plane AB;
+which line BG would be the propagation of the wave if the movement
+could have spread in a medium homogeneous with that which is above the
+plane. And if one wishes to see how the wave AC has come successively
+to BN, one has only to draw in the same figure the straight lines KO
+parallel to BN, and the straight lines KL parallel to AC. Thus one
+will see that the straight wave AC has become broken up into all the
+OKL parts successively, and that it has become straight again at NB.
+
+Now it is apparent here that the angle of reflexion is made equal to
+the angle of incidence. For the triangles ACB, BNA being rectangular
+and having the side AB common, and the side CB equal to NA, it follows
+that the angles opposite to these sides will be equal, and therefore
+also the angles CBA, NAB. But as CB, perpendicular to CA, marks the
+direction of the incident ray, so AN, perpendicular to the wave BN,
+marks the direction of the reflected ray; hence these rays are equally
+inclined to the plane AB.
+
+But in considering the preceding demonstration, one might aver that it
+is indeed true that BN is the common tangent of the circular waves in
+the plane of this figure, but that these waves, being in truth
+spherical, have still an infinitude of similar tangents, namely all
+the straight lines which are drawn from the point B in the surface
+generated by the straight line BN about the axis BA. It remains,
+therefore, to demonstrate that there is no difficulty herein: and by
+the same argument one will see why the incident ray and the reflected
+ray are always in one and the same plane perpendicular to the
+reflecting plane. I say then that the wave AC, being regarded only as
+a line, produces no light. For a visible ray of light, however narrow
+it may be, has always some width, and consequently it is necessary, in
+representing the wave whose progression constitutes the ray, to put
+instead of a line AC some plane figure such as the circle HC in the
+following figure, by supposing, as we have done, the luminous point to
+be infinitely distant. Now it is easy to see, following the preceding
+demonstration, that each small piece of this wave HC having arrived at
+the plane AB, and there generating each one its particular wave, these
+will all have, when C arrives at B, a common plane which will touch
+them, namely a circle BN similar to CH; and this will be intersected
+at its middle and at right angles by the same plane which likewise
+intersects the circle CH and the ellipse AB.
+
+[Illustration]
+
+One sees also that the said spheres of the partial waves cannot have
+any common tangent plane other than the circle BN; so that it will be
+this plane where there will be more reflected movement than anywhere
+else, and which will therefore carry on the light in continuance from
+the wave CH.
+
+I have also stated in the preceding demonstration that the movement of
+the piece A of the incident wave is not able to communicate itself
+beyond the plane AB, or at least not wholly. Whence it is to be
+remarked that though the movement of the ethereal matter might
+communicate itself partly to that of the reflecting body, this could
+in nothing alter the velocity of progression of the waves, on which
+the angle of reflexion depends. For a slight percussion ought to
+generate waves as rapid as strong percussion in the same matter. This
+comes about from the property of bodies which act as springs, of which
+we have spoken above; namely that whether compressed little or much
+they recoil in equal times. Equally so in every reflexion of the
+light, against whatever body it may be, the angles of reflexion and
+incidence ought to be equal notwithstanding that the body might be of
+such a nature that it takes away a portion of the movement made by the
+incident light. And experiment shows that in fact there is no polished
+body the reflexion of which does not follow this rule.
+
+
+But the thing to be above all remarked in our demonstration is that it
+does not require that the reflecting surface should be considered as a
+uniform plane, as has been supposed by all those who have tried to
+explain the effects of reflexion; but only an evenness such as may be
+attained by the particles of the matter of the reflecting body being
+set near to one another; which particles are larger than those of the
+ethereal matter, as will appear by what we shall say in treating of
+the transparency and opacity of bodies. For the surface consisting
+thus of particles put together, and the ethereal particles being
+above, and smaller, it is evident that one could not demonstrate the
+equality of the angles of incidence and reflexion by similitude to
+that which happens to a ball thrown against a wall, of which writers
+have always made use. In our way, on the other hand, the thing is
+explained without difficulty. For the smallness of the particles of
+quicksilver, for example, being such that one must conceive millions
+of them, in the smallest visible surface proposed, arranged like a
+heap of grains of sand which has been flattened as much as it is
+capable of being, this surface then becomes for our purpose as even
+as a polished glass is: and, although it always remains rough with
+respect to the particles of the Ether it is evident that the centres
+of all the particular spheres of reflexion, of which we have spoken,
+are almost in one uniform plane, and that thus the common tangent can
+fit to them as perfectly as is requisite for the production of light.
+And this alone is requisite, in our method of demonstration, to cause
+equality of the said angles without the remainder of the movement
+reflected from all parts being able to produce any contrary effect.
+
+
+
+
+CHAPTER III
+
+ON REFRACTION
+
+
+In the same way as the effects of Reflexion have been explained by
+waves of light reflected at the surface of polished bodies, we will
+explain transparency and the phenomena of refraction by waves which
+spread within and across diaphanous bodies, both solids, such as
+glass, and liquids, such as water, oils, etc. But in order that it may
+not seem strange to suppose this passage of waves in the interior of
+these bodies, I will first show that one may conceive it possible in
+more than one mode.
+
+First, then, if the ethereal matter cannot penetrate transparent
+bodies at all, their own particles would be able to communicate
+successively the movement of the waves, the same as do those of the
+Ether, supposing that, like those, they are of a nature to act as a
+spring. And this is easy to conceive as regards water and other
+transparent liquids, they being composed of detached particles. But it
+may seem more difficult as regards glass and other transparent and
+hard bodies, because their solidity does not seem to permit them to
+receive movement except in their whole mass at the same time. This,
+however, is not necessary because this solidity is not such as it
+appears to us, it being probable rather that these bodies are composed
+of particles merely placed close to one another and held together by
+some pressure from without of some other matter, and by the
+irregularity of their shapes. For primarily their rarity is shown by
+the facility with which there passes through them the matter of the
+vortices of the magnet, and that which causes gravity. Further, one
+cannot say that these bodies are of a texture similar to that of a
+sponge or of light bread, because the heat of the fire makes them flow
+and thereby changes the situation of the particles amongst themselves.
+It remains then that they are, as has been said, assemblages of
+particles which touch one another without constituting a continuous
+solid. This being so, the movement which these particles receive to
+carry on the waves of light, being merely communicated from some of
+them to others, without their going for that purpose out of their
+places or without derangement, it may very well produce its effect
+without prejudicing in any way the apparent solidity of the compound.
+
+By pressure from without, of which I have spoken, must not be
+understood that of the air, which would not be sufficient, but that of
+some other more subtle matter, a pressure which I chanced upon by
+experiment long ago, namely in the case of water freed from air, which
+remains suspended in a tube open at its lower end, notwithstanding
+that the air has been removed from the vessel in which this tube is
+enclosed.
+
+One can then in this way conceive of transparency in a solid without
+any necessity that the ethereal matter which serves for light should
+pass through it, or that it should find pores in which to insinuate
+itself. But the truth is that this matter not only passes through
+solids, but does so even with great facility; of which the experiment
+of Torricelli, above cited, is already a proof. Because on the
+quicksilver and the water quitting the upper part of the glass tube,
+it appears that it is immediately filled with ethereal matter, since
+light passes across it. But here is another argument which proves this
+ready penetrability, not only in transparent bodies but also in all
+others.
+
+When light passes across a hollow sphere of glass, closed on all
+sides, it is certain that it is full of ethereal matter, as much as
+the spaces outside the sphere. And this ethereal matter, as has been
+shown above, consists of particles which just touch one another. If
+then it were enclosed in the sphere in such a way that it could not
+get out through the pores of the glass, it would be obliged to follow
+the movement of the sphere when one changes its place: and it would
+require consequently almost the same force to impress a certain
+velocity on this sphere, when placed on a horizontal plane, as if it
+were full of water or perhaps of quicksilver: because every body
+resists the velocity of the motion which one would give to it, in
+proportion to the quantity of matter which it contains, and which is
+obliged to follow this motion. But on the contrary one finds that the
+sphere resists the impress of movement only in proportion to the
+quantity of matter of the glass of which it is made. Then it must be
+that the ethereal matter which is inside is not shut up, but flows
+through it with very great freedom. We shall demonstrate hereafter
+that by this process the same penetrability may be inferred also as
+relating to opaque bodies.
+
+The second mode then of explaining transparency, and one which appears
+more probably true, is by saying that the waves of light are carried
+on in the ethereal matter, which continuously occupies the interstices
+or pores of transparent bodies. For since it passes through them
+continuously and freely, it follows that they are always full of it.
+And one may even show that these interstices occupy much more space
+than the coherent particles which constitute the bodies. For if what
+we have just said is true: that force is required to impress a certain
+horizontal velocity on bodies in proportion as they contain coherent
+matter; and if the proportion of this force follows the law of
+weights, as is confirmed by experiment, then the quantity of the
+constituent matter of bodies also follows the proportion of their
+weights. Now we see that water weighs only one fourteenth part as much
+as an equal portion of quicksilver: therefore the matter of the water
+does not occupy the fourteenth part of the space which its mass
+obtains. It must even occupy much less of it, since quicksilver is
+less heavy than gold, and the matter of gold is by no means dense, as
+follows from the fact that the matter of the vortices of the magnet
+and of that which is the cause of gravity pass very freely through it.
+
+But it may be objected here that if water is a body of so great
+rarity, and if its particles occupy so small a portion of the space of
+its apparent bulk, it is very strange how it yet resists Compression
+so strongly without permitting itself to be condensed by any force
+which one has hitherto essayed to employ, preserving even its entire
+liquidity while subjected to this pressure.
+
+This is no small difficulty. It may, however, be resolved by saying
+that the very violent and rapid motion of the subtle matter which
+renders water liquid, by agitating the particles of which it is
+composed, maintains this liquidity in spite of the pressure which
+hitherto any one has been minded to apply to it.
+
+The rarity of transparent bodies being then such as we have said, one
+easily conceives that the waves might be carried on in the ethereal
+matter which fills the interstices of the particles. And, moreover,
+one may believe that the progression of these waves ought to be a
+little slower in the interior of bodies, by reason of the small
+detours which the same particles cause. In which different velocity of
+light I shall show the cause of refraction to consist.
+
+Before doing so, I will indicate the third and last mode in which
+transparency may be conceived; which is by supposing that the movement
+of the waves of light is transmitted indifferently both in the
+particles of the ethereal matter which occupy the interstices of
+bodies, and in the particles which compose them, so that the movement
+passes from one to the other. And it will be seen hereafter that this
+hypothesis serves excellently to explain the double refraction of
+certain transparent bodies.
+
+Should it be objected that if the particles of the ether are smaller
+than those of transparent bodies (since they pass through their
+intervals), it would follow that they can communicate to them but
+little of their movement, it may be replied that the particles of
+these bodies are in turn composed of still smaller particles, and so
+it will be these secondary particles which will receive the movement
+from those of the ether.
+
+Furthermore, if the particles of transparent bodies have a recoil a
+little less prompt than that of the ethereal particles, which nothing
+hinders us from supposing, it will again follow that the progression
+of the waves of light will be slower in the interior of such bodies
+than it is outside in the ethereal matter.
+
+All this I have found as most probable for the mode in which the waves
+of light pass across transparent bodies. To which it must further be
+added in what respect these bodies differ from those which are opaque;
+and the more so since it might seem because of the easy penetration of
+bodies by the ethereal matter, of which mention has been made, that
+there would not be any body that was not transparent. For by the same
+reasoning about the hollow sphere which I have employed to prove the
+smallness of the density of glass and its easy penetrability by the
+ethereal matter, one might also prove that the same penetrability
+obtains for metals and for every other sort of body. For this sphere
+being for example of silver, it is certain that it contains some of
+the ethereal matter which serves for light, since this was there as
+well as in the air when the opening of the sphere was closed. Yet,
+being closed and placed upon a horizontal plane, it resists the
+movement which one wishes to give to it, merely according to the
+quantity of silver of which it is made; so that one must conclude, as
+above, that the ethereal matter which is enclosed does not follow the
+movement of the sphere; and that therefore silver, as well as glass,
+is very easily penetrated by this matter. Some of it is therefore
+present continuously and in quantities between the particles of silver
+and of all other opaque bodies: and since it serves for the
+propagation of light it would seem that these bodies ought also to be
+transparent, which however is not the case.
+
+Whence then, one will say, does their opacity come? Is it because the
+particles which compose them are soft; that is to say, these particles
+being composed of others that are smaller, are they capable of
+changing their figure on receiving the pressure of the ethereal
+particles, the motion of which they thereby damp, and so hinder the
+continuance of the waves of light? That cannot be: for if the
+particles of the metals are soft, how is it that polished silver and
+mercury reflect light so strongly? What I find to be most probable
+herein, is to say that metallic bodies, which are almost the only
+really opaque ones, have mixed amongst their hard particles some soft
+ones; so that some serve to cause reflexion and the others to hinder
+transparency; while, on the other hand, transparent bodies contain
+only hard particles which have the faculty of recoil, and serve
+together with those of the ethereal matter for the propagation of the
+waves of light, as has been said.
+
+[Illustration]
+
+Let us pass now to the explanation of the effects of Refraction,
+assuming, as we have done, the passage of waves of light through
+transparent bodies, and the diminution of velocity which these same
+waves suffer in them.
+
+The chief property of Refraction is that a ray of light, such as AB,
+being in the air, and falling obliquely upon the polished surface of a
+transparent body, such as FG, is broken at the point of incidence B,
+in such a way that with the straight line DBE which cuts the surface
+perpendicularly it makes an angle CBE less than ABD which it made with
+the same perpendicular when in the air. And the measure of these
+angles is found by describing, about the point B, a circle which cuts
+the radii AB, BC. For the perpendiculars AD, CE, let fall from the
+points of intersection upon the straight line DE, which are called the
+Sines of the angles ABD, CBE, have a certain ratio between themselves;
+which ratio is always the same for all inclinations of the incident
+ray, at least for a given transparent body. This ratio is, in glass,
+very nearly as 3 to 2; and in water very nearly as 4 to 3; and is
+likewise different in other diaphanous bodies.
+
+Another property, similar to this, is that the refractions are
+reciprocal between the rays entering into a transparent body and those
+which are leaving it. That is to say that if the ray AB in entering
+the transparent body is refracted into BC, then likewise CB being
+taken as a ray in the interior of this body will be refracted, on
+passing out, into BA.
+
+[Illustration]
+
+To explain then the reasons of these phenomena according to our
+principles, let AB be the straight line which represents a plane
+surface bounding the transparent substances which lie towards C and
+towards N. When I say plane, that does not signify a perfect evenness,
+but such as has been understood in treating of reflexion, and for the
+same reason. Let the line AC represent a portion of a wave of light,
+the centre of which is supposed so distant that this portion may be
+considered as a straight line. The piece C, then, of the wave AC, will
+in a certain space of time have advanced as far as the plane AB
+following the straight line CB, which may be imagined as coming from
+the luminous centre, and which consequently will cut AC at right
+angles. Now in the same time the piece A would have come to G along
+the straight line AG, equal and parallel to CB; and all the portion of
+wave AC would be at GB if the matter of the transparent body
+transmitted the movement of the wave as quickly as the matter of the
+Ether. But let us suppose that it transmits this movement less
+quickly, by one-third, for instance. Movement will then be spread from
+the point A, in the matter of the transparent body through a distance
+equal to two-thirds of CB, making its own particular spherical wave
+according to what has been said before. This wave is then represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter equal to two-thirds of CB. Then if one considers in
+order the other pieces H of the wave AC, it appears that in the same
+time that the piece C reaches B they will not only have arrived at the
+surface AB along the straight lines HK parallel to CB, but that, in
+addition, they will have generated in the diaphanous substance from
+the centres K, partial waves, represented here by circumferences the
+semi-diameters of which are equal to two-thirds of the lines KM, that
+is to say, to two-thirds of the prolongations of HK down to the
+straight line BG; for these semi-diameters would have been equal to
+entire lengths of KM if the two transparent substances had been of the
+same penetrability.
+
+Now all these circumferences have for a common tangent the straight
+line BN; namely the same line which is drawn as a tangent from the
+point B to the circumference SNR which we considered first. For it is
+easy to see that all the other circumferences will touch the same BN,
+from B up to the point of contact N, which is the same point where AN
+falls perpendicularly on BN.
+
+It is then BN, which is formed by small arcs of these circumferences,
+which terminates the movement that the wave AC has communicated within
+the transparent body, and where this movement occurs in much greater
+amount than anywhere else. And for that reason this line, in
+accordance with what has been said more than once, is the propagation
+of the wave AC at the moment when its piece C has reached B. For there
+is no other line below the plane AB which is, like BN, a common
+tangent to all these partial waves. And if one would know how the wave
+AC has come progressively to BN, it is necessary only to draw in the
+same figure the straight lines KO parallel to BN, and all the lines KL
+parallel to AC. Thus one will see that the wave CA, from being a
+straight line, has become broken in all the positions LKO
+successively, and that it has again become a straight line at BN. This
+being evident by what has already been demonstrated, there is no need
+to explain it further.
+
+Now, in the same figure, if one draws EAF, which cuts the plane AB at
+right angles at the point A, since AD is perpendicular to the wave AC,
+it will be DA which will mark the ray of incident light, and AN which
+was perpendicular to BN, the refracted ray: since the rays are nothing
+else than the straight lines along which the portions of the waves
+advance.
+
+Whence it is easy to recognize this chief property of refraction,
+namely that the Sine of the angle DAE has always the same ratio to the
+Sine of the angle NAF, whatever be the inclination of the ray DA: and
+that this ratio is the same as that of the velocity of the waves in
+the transparent substance which is towards AE to their velocity in the
+transparent substance towards AF. For, considering AB as the radius of
+a circle, the Sine of the angle BAC is BC, and the Sine of the angle
+ABN is AN. But the angle BAC is equal to DAE, since each of them added
+to CAE makes a right angle. And the angle ABN is equal to NAF, since
+each of them with BAN makes a right angle. Then also the Sine of the
+angle DAE is to the Sine of NAF as BC is to AN. But the ratio of BC to
+AN was the same as that of the velocities of light in the substance
+which is towards AE and in that which is towards AF; therefore also
+the Sine of the angle DAE will be to the Sine of the angle NAF the
+same as the said velocities of light.
+
+To see, consequently, what the refraction will be when the waves of
+light pass into a substance in which the movement travels more quickly
+than in that from which they emerge (let us again assume the ratio of
+3 to 2), it is only necessary to repeat all the same construction and
+demonstration which we have just used, merely substituting everywhere
+3/2 instead of 2/3. And it will be found by the same reasoning, in
+this other figure, that when the piece C of the wave AC shall have
+reached the surface AB at B, all the portions of the wave AC will
+have advanced as far as BN, so that BC the perpendicular on AC is to
+AN the perpendicular on BN as 2 to 3. And there will finally be this
+same ratio of 2 to 3 between the Sine of the angle BAD and the Sine of
+the angle FAN.
+
+Hence one sees the reciprocal relation of the refractions of the ray
+on entering and on leaving one and the same transparent body: namely
+that if NA falling on the external surface AB is refracted into the
+direction AD, so the ray AD will be refracted on leaving the
+transparent body into the direction AN.
+
+[Illustration]
+
+One sees also the reason for a noteworthy accident which happens in
+this refraction: which is this, that after a certain obliquity of the
+incident ray DA, it begins to be quite unable to penetrate into the
+other transparent substance. For if the angle DAQ or CBA is such that
+in the triangle ACB, CB is equal to 2/3 of AB, or is greater, then AN
+cannot form one side of the triangle ANB, since it becomes equal to or
+greater than AB: so that the portion of wave BN cannot be found
+anywhere, neither consequently can AN, which ought to be perpendicular
+to it. And thus the incident ray DA does not then pierce the surface
+AB.
+
+When the ratio of the velocities of the waves is as two to three, as
+in our example, which is that which obtains for glass and air, the
+angle DAQ must be more than 48 degrees 11 minutes in order that the
+ray DA may be able to pass by refraction. And when the ratio of the
+velocities is as 3 to 4, as it is very nearly in water and air, this
+angle DAQ must exceed 41 degrees 24 minutes. And this accords
+perfectly with experiment.
+
+But it might here be asked: since the meeting of the wave AC against
+the surface AB ought to produce movement in the matter which is on the
+other side, why does no light pass there? To which the reply is easy
+if one remembers what has been said before. For although it generates
+an infinitude of partial waves in the matter which is at the other
+side of AB, these waves never have a common tangent line (either
+straight or curved) at the same moment; and so there is no line
+terminating the propagation of the wave AC beyond the plane AB, nor
+any place where the movement is gathered together in sufficiently
+great quantity to produce light. And one will easily see the truth of
+this, namely that CB being larger than 2/3 of AB, the waves excited
+beyond the plane AB will have no common tangent if about the centres K
+one then draws circles having radii equal to 3/2 of the lengths LB to
+which they correspond. For all these circles will be enclosed in one
+another and will all pass beyond the point B.
+
+Now it is to be remarked that from the moment when the angle DAQ is
+smaller than is requisite to permit the refracted ray DA to pass into
+the other transparent substance, one finds that the interior reflexion
+which occurs at the surface AB is much augmented in brightness, as is
+easy to realize by experiment with a triangular prism; and for this
+our theory can afford this reason. When the angle DAQ is still large
+enough to enable the ray DA to pass, it is evident that the light from
+the portion AC of the wave is collected in a minimum space when it
+reaches BN. It appears also that the wave BN becomes so much the
+smaller as the angle CBA or DAQ is made less; until when the latter is
+diminished to the limit indicated a little previously, this wave BN is
+collected together always at one point. That is to say, that when the
+piece C of the wave AC has then arrived at B, the wave BN which is the
+propagation of AC is entirely reduced to the same point B. Similarly
+when the piece H has reached K, the part AH is entirely reduced to the
+same point K. This makes it evident that in proportion as the wave CA
+comes to meet the surface AB, there occurs a great quantity of
+movement along that surface; which movement ought also to spread
+within the transparent body and ought to have much re-enforced the
+partial waves which produce the interior reflexion against the surface
+AB, according to the laws of reflexion previously explained.
+
+And because a slight diminution of the angle of incidence DAQ causes
+the wave BN, however great it was, to be reduced to zero, (for this
+angle being 49 degrees 11 minutes in the glass, the angle BAN is still
+11 degrees 21 minutes, and the same angle being reduced by one degree
+only the angle BAN is reduced to zero, and so the wave BN reduced to a
+point) thence it comes about that the interior reflexion from being
+obscure becomes suddenly bright, so soon as the angle of incidence is
+such that it no longer gives passage to the refraction.
+
+Now as concerns ordinary external reflexion, that is to say which
+occurs when the angle of incidence DAQ is still large enough to enable
+the refracted ray to penetrate beyond the surface AB, this reflexion
+should occur against the particles of the substance which touches the
+transparent body on its outside. And it apparently occurs against the
+particles of the air or others mingled with the ethereal particles and
+larger than they. So on the other hand the external reflexion of these
+bodies occurs against the particles which compose them, and which are
+also larger than those of the ethereal matter, since the latter flows
+in their interstices. It is true that there remains here some
+difficulty in those experiments in which this interior reflexion
+occurs without the particles of air being able to contribute to it, as
+in vessels or tubes from which the air has been extracted.
+
+Experience, moreover, teaches us that these two reflexions are of
+nearly equal force, and that in different transparent bodies they are
+so much the stronger as the refraction of these bodies is the greater.
+Thus one sees manifestly that the reflexion of glass is stronger than
+that of water, and that of diamond stronger than that of glass.
+
+I will finish this theory of refraction by demonstrating a remarkable
+proposition which depends on it; namely, that a ray of light in order
+to go from one point to another, when these points are in different
+media, is refracted in such wise at the plane surface which joins
+these two media that it employs the least possible time: and exactly
+the same happens in the case of reflexion against a plane surface. Mr.
+Fermat was the first to propound this property of refraction, holding
+with us, and directly counter to the opinion of Mr. Des Cartes, that
+light passes more slowly through glass and water than through air.
+But he assumed besides this a constant ratio of Sines, which we have
+just proved by these different degrees of velocity alone: or rather,
+what is equivalent, he assumed not only that the velocities were
+different but that the light took the least time possible for its
+passage, and thence deduced the constant ratio of the Sines. His
+demonstration, which may be seen in his printed works, and in the
+volume of letters of Mr. Des Cartes, is very long; wherefore I give
+here another which is simpler and easier.
+
+[Illustration]
+
+Let KF be the plane surface; A the point in the medium which the light
+traverses more easily, as the air; C the point in the other which is
+more difficult to penetrate, as water. And suppose that a ray has come
+from A, by B, to C, having been refracted at B according to the law
+demonstrated a little before; that is to say that, having drawn PBQ,
+which cuts the plane at right angles, let the sine of the angle ABP
+have to the sine of the angle CBQ the same ratio as the velocity of
+light in the medium where A is to the velocity of light in the medium
+where C is. It is to be shown that the time of passage of light along
+AB and BC taken together, is the shortest that can be. Let us assume
+that it may have come by other lines, and, in the first place, along
+AF, FC, so that the point of refraction F may be further from B than
+the point A; and let AO be a line perpendicular to AB, and FO parallel
+to AB; BH perpendicular to FO, and FG to BC.
+
+Since then the angle HBF is equal to PBA, and the angle BFG equal to
+QBC, it follows that the sine of the angle HBF will also have the same
+ratio to the sine of BFG, as the velocity of light in the medium A is
+to its velocity in the medium C. But these sines are the straight
+lines HF, BG, if we take BF as the semi-diameter of a circle. Then
+these lines HF, BG, will bear to one another the said ratio of the
+velocities. And, therefore, the time of the light along HF, supposing
+that the ray had been OF, would be equal to the time along BG in the
+interior of the medium C. But the time along AB is equal to the time
+along OH; therefore the time along OF is equal to the time along AB,
+BG. Again the time along FC is greater than that along GC; then the
+time along OFC will be longer than that along ABC. But AF is longer
+than OF, then the time along AFC will by just so much more exceed the
+time along ABC.
+
+Now let us assume that the ray has come from A to C along AK, KC; the
+point of refraction K being nearer to A than the point B is; and let
+CN be the perpendicular upon BC, KN parallel to BC: BM perpendicular
+upon KN, and KL upon BA.
+
+Here BL and KM are the sines of angles BKL, KBM; that is to say, of
+the angles PBA, QBC; and therefore they are to one another as the
+velocity of light in the medium A is to the velocity in the medium C.
+Then the time along LB is equal to the time along KM; and since the
+time along BC is equal to the time along MN, the time along LBC will
+be equal to the time along KMN. But the time along AK is longer than
+that along AL: hence the time along AKN is longer than that along ABC.
+And KC being longer than KN, the time along AKC will exceed, by as
+much more, the time along ABC. Hence it appears that the time along
+ABC is the shortest possible; which was to be proven.
+
+
+
+
+CHAPTER IV
+
+ON THE REFRACTION OF THE AIR
+
+
+We have shown how the movement which constitutes light spreads by
+spherical waves in any homogeneous matter. And it is evident that when
+the matter is not homogeneous, but of such a constitution that the
+movement is communicated in it more rapidly toward one side than
+toward another, these waves cannot be spherical: but that they must
+acquire their figure according to the different distances over which
+the successive movement passes in equal times.
+
+It is thus that we shall in the first place explain the refractions
+which occur in the air, which extends from here to the clouds and
+beyond. The effects of which refractions are very remarkable; for by
+them we often see objects which the rotundity of the Earth ought
+otherwise to hide; such as Islands, and the tops of mountains when one
+is at sea. Because also of them the Sun and the Moon appear as risen
+before in fact they have, and appear to set later: so that at times
+the Moon has been seen eclipsed while the Sun appeared still above the
+horizon. And so also the heights of the Sun and of the Moon, and those
+of all the Stars always appear a little greater than they are in
+reality, because of these same refractions, as Astronomers know. But
+there is one experiment which renders this refraction very evident;
+which is that of fixing a telescope on some spot so that it views an
+object, such as a steeple or a house, at a distance of half a league
+or more. If then you look through it at different hours of the day,
+leaving it always fixed in the same way, you will see that the same
+spots of the object will not always appear at the middle of the
+aperture of the telescope, but that generally in the morning and in
+the evening, when there are more vapours near the Earth, these objects
+seem to rise higher, so that the half or more of them will no longer
+be visible; and so that they seem lower toward mid-day when these
+vapours are dissipated.
+
+Those who consider refraction to occur only in the surfaces which
+separate transparent bodies of different nature, would find it
+difficult to give a reason for all that I have just related; but
+according to our Theory the thing is quite easy. It is known that the
+air which surrounds us, besides the particles which are proper to it
+and which float in the ethereal matter as has been explained, is full
+also of particles of water which are raised by the action of heat; and
+it has been ascertained further by some very definite experiments that
+as one mounts up higher the density of air diminishes in proportion.
+Now whether the particles of water and those of air take part, by
+means of the particles of ethereal matter, in the movement which
+constitutes light, but have a less prompt recoil than these, or
+whether the encounter and hindrance which these particles of air and
+water offer to the propagation of movement of the ethereal progress,
+retard the progression, it follows that both kinds of particles flying
+amidst the ethereal particles, must render the air, from a great
+height down to the Earth, gradually less easy for the spreading of the
+waves of light.
+
+[Illustration]
+
+Whence the configuration of the waves ought to become nearly such as
+this figure represents: namely, if A is a light, or the visible point
+of a steeple, the waves which start from it ought to spread more
+widely upwards and less widely downwards, but in other directions more
+or less as they approximate to these two extremes. This being so, it
+necessarily follows that every line intersecting one of these waves at
+right angles will pass above the point A, always excepting the one
+line which is perpendicular to the horizon.
+
+[Illustration]
+
+Let BC be the wave which brings the light to the spectator who is at
+B, and let BD be the straight line which intersects this wave at right
+angles. Now because the ray or straight line by which we judge the
+spot where the object appears to us is nothing else than the
+perpendicular to the wave that reaches our eye, as will be understood
+by what was said above, it is manifest that the point A will be
+perceived as being in the line BD, and therefore higher than in fact it
+is.
+
+Similarly if the Earth be AB, and the top of the Atmosphere CD, which
+probably is not a well defined spherical surface (since we know that
+the air becomes rare in proportion as one ascends, for above there is
+so much less of it to press down upon it), the waves of light from the
+sun coming, for instance, in such a way that so long as they have not
+reached the Atmosphere CD the straight line AE intersects them
+perpendicularly, they ought, when they enter the Atmosphere, to
+advance more quickly in elevated regions than in regions nearer to the
+Earth. So that if CA is the wave which brings the light to the
+spectator at A, its region C will be the furthest advanced; and the
+straight line AF, which intersects this wave at right angles, and
+which determines the apparent place of the Sun, will pass above the
+real Sun, which will be seen along the line AE. And so it may occur
+that when it ought not to be visible in the absence of vapours,
+because the line AE encounters the rotundity of the Earth, it will be
+perceived in the line AF by refraction. But this angle EAF is scarcely
+ever more than half a degree because the attenuation of the vapours
+alters the waves of light but little. Furthermore these refractions
+are not altogether constant in all weathers, particularly at small
+elevations of 2 or 3 degrees; which results from the different
+quantity of aqueous vapours rising above the Earth.
+
+And this same thing is the cause why at certain times a distant object
+will be hidden behind another less distant one, and yet may at another
+time be able to be seen, although the spot from which it is viewed is
+always the same. But the reason for this effect will be still more
+evident from what we are going to remark touching the curvature of
+rays. It appears from the things explained above that the progression
+or propagation of a small part of a wave of light is properly what one
+calls a ray. Now these rays, instead of being straight as they are in
+homogeneous media, ought to be curved in an atmosphere of unequal
+penetrability. For they necessarily follow from the object to the eye
+the line which intersects at right angles all the progressions of the
+waves, as in the first figure the line AEB does, as will be shown
+hereafter; and it is this line which determines what interposed bodies
+would or would not hinder us from seeing the object. For although the
+point of the steeple A appears raised to D, it would yet not appear to
+the eye B if the tower H was between the two, because it crosses the
+curve AEB. But the tower E, which is beneath this curve, does not
+hinder the point A from being seen. Now according as the air near the
+Earth exceeds in density that which is higher, the curvature of the
+ray AEB becomes greater: so that at certain times it passes above the
+summit E, which allows the point A to be perceived by the eye at B;
+and at other times it is intercepted by the same tower E which hides A
+from this same eye.
+
+[Illustration]
+
+But to demonstrate this curvature of the rays conformably to all our
+preceding Theory, let us imagine that AB is a small portion of a wave
+of light coming from the side C, which we may consider as a straight
+line. Let us also suppose that it is perpendicular to the Horizon, the
+portion B being nearer to the Earth than the portion A; and that
+because the vapours are less hindering at A than at B, the particular
+wave which comes from the point A spreads through a certain space AD
+while the particular wave which starts from the point B spreads
+through a shorter space BE; AD and BE being parallel to the Horizon.
+Further, supposing the straight lines FG, HI, etc., to be drawn from
+an infinitude of points in the straight line AB and to terminate on
+the line DE (which is straight or may be considered as such), let the
+different penetrabilities at the different heights in the air between
+A and B be represented by all these lines; so that the particular
+wave, originating from the point F, will spread across the space FG,
+and that from the point H across the space HI, while that from the
+point A spreads across the space AD.
+
+Now if about the centres A, B, one describes the circles DK, EL, which
+represent the spreading of the waves which originate from these two
+points, and if one draws the straight line KL which touches these two
+circles, it is easy to see that this same line will be the common
+tangent to all the other circles drawn about the centres F, H, etc.;
+and that all the points of contact will fall within that part of this
+line which is comprised between the perpendiculars AK, BL. Then it
+will be the line KL which will terminate the movement of the
+particular waves originating from the points of the wave AB; and this
+movement will be stronger between the points KL, than anywhere else at
+the same instant, since an infinitude of circumferences concur to form
+this straight line; and consequently KL will be the propagation of the
+portion of wave AB, as has been said in explaining reflexion and
+ordinary refraction. Now it appears that AK and BL dip down toward the
+side where the air is less easy to penetrate: for AK being longer than
+BL, and parallel to it, it follows that the lines AB and KL, being
+prolonged, would meet at the side L. But the angle K is a right angle:
+hence KAB is necessarily acute, and consequently less than DAB. If one
+investigates in the same way the progression of the portion of the
+wave KL, one will find that after a further time it has arrived at MN
+in such a manner that the perpendiculars KM, LN, dip down even more
+than do AK, BL. And this suffices to show that the ray will continue
+along the curved line which intersects all the waves at right angles,
+as has been said.
+
+
+
+
+CHAPTER V
+
+ON THE STRANGE REFRACTION OF ICELAND CRYSTAL
+
+
+1.
+
+There is brought from Iceland, which is an Island in the North Sea, in
+the latitude of 66 degrees, a kind of Crystal or transparent stone,
+very remarkable for its figure and other qualities, but above all for
+its strange refractions. The causes of this have seemed to me to be
+worthy of being carefully investigated, the more so because amongst
+transparent bodies this one alone does not follow the ordinary rules
+with respect to rays of light. I have even been under some necessity
+to make this research, because the refractions of this Crystal seemed
+to overturn our preceding explanation of regular refraction; which
+explanation, on the contrary, they strongly confirm, as will be seen
+after they have been brought under the same principle. In Iceland are
+found great lumps of this Crystal, some of which I have seen of 4 or 5
+pounds. But it occurs also in other countries, for I have had some of
+the same sort which had been found in France near the town of Troyes
+in Champagne, and some others which came from the Island of Corsica,
+though both were less clear and only in little bits, scarcely capable
+of letting any effect of refraction be observed.
+
+2. The first knowledge which the public has had about it is due to Mr.
+Erasmus Bartholinus, who has given a description of Iceland Crystal
+and of its chief phenomena. But here I shall not desist from giving my
+own, both for the instruction of those who may not have seen his book,
+and because as respects some of these phenomena there is a slight
+difference between his observations and those which I have made: for I
+have applied myself with great exactitude to examine these properties
+of refraction, in order to be quite sure before undertaking to explain
+the causes of them.
+
+3. As regards the hardness of this stone, and the property which it
+has of being easily split, it must be considered rather as a species
+of Talc than of Crystal. For an iron spike effects an entrance into it
+as easily as into any other Talc or Alabaster, to which it is equal in
+gravity.
+
+[Illustration]
+
+4. The pieces of it which are found have the figure of an oblique
+parallelepiped; each of the six faces being a parallelogram; and it
+admits of being split in three directions parallel to two of these
+opposed faces. Even in such wise, if you will, that all the six faces
+are equal and similar rhombuses. The figure here added represents a
+piece of this Crystal. The obtuse angles of all the parallelograms, as
+C, D, here, are angles of 101 degrees 52 minutes, and consequently
+the acute angles, such as A and B, are of 78 degrees 8 minutes.
+
+5. Of the solid angles there are two opposite to one another, such as
+C and E, which are each composed of three equal obtuse plane angles.
+The other six are composed of two acute angles and one obtuse. All
+that I have just said has been likewise remarked by Mr. Bartholinus in
+the aforesaid treatise; if we differ it is only slightly about the
+values of the angles. He recounts moreover some other properties of
+this Crystal; to wit, that when rubbed against cloth it attracts
+straws and other light things as do amber, diamond, glass, and Spanish
+wax. Let a piece be covered with water for a day or more, the surface
+loses its natural polish. When aquafortis is poured on it it produces
+ebullition, especially, as I have found, if the Crystal has been
+pulverized. I have also found by experiment that it may be heated to
+redness in the fire without being in anywise altered or rendered less
+transparent; but a very violent fire calcines it nevertheless. Its
+transparency is scarcely less than that of water or of Rock Crystal,
+and devoid of colour. But rays of light pass through it in another
+fashion and produce those marvellous refractions the causes of which I
+am now going to try to explain; reserving for the end of this Treatise
+the statement of my conjectures touching the formation and
+extraordinary configuration of this Crystal.
+
+6. In all other transparent bodies that we know there is but one sole
+and simple refraction; but in this substance there are two different
+ones. The effect is that objects seen through it, especially such as
+are placed right against it, appear double; and that a ray of
+sunlight, falling on one of its surfaces, parts itself into two rays
+and traverses the Crystal thus.
+
+7. It is again a general law in all other transparent bodies that the
+ray which falls perpendicularly on their surface passes straight on
+without suffering refraction, and that an oblique ray is always
+refracted. But in this Crystal the perpendicular ray suffers
+refraction, and there are oblique rays which pass through it quite
+straight.
+
+[Illustration]
+
+8. But in order to explain these phenomena more particularly, let
+there be, in the first place, a piece ABFE of the same Crystal, and
+let the obtuse angle ACB, one of the three which constitute the
+equilateral solid angle C, be divided into two equal parts by the
+straight line CG, and let it be conceived that the Crystal is
+intersected by a plane which passes through this line and through the
+side CF, which plane will necessarily be perpendicular to the surface
+AB; and its section in the Crystal will form a parallelogram GCFH. We
+will call this section the principal section of the Crystal.
+
+9. Now if one covers the surface AB, leaving there only a small
+aperture at the point K, situated in the straight line CG, and if one
+exposes it to the sun, so that his rays face it perpendicularly above,
+then the ray IK will divide itself at the point K into two, one of
+which will continue to go on straight by KL, and the other will
+separate itself along the straight line KM, which is in the plane
+GCFH, and which makes with KL an angle of about 6 degrees 40 minutes,
+tending from the side of the solid angle C; and on emerging from the
+other side of the Crystal it will turn again parallel to JK, along MZ.
+And as, in this extraordinary refraction, the point M is seen by the
+refracted ray MKI, which I consider as going to the eye at I, it
+necessarily follows that the point L, by virtue of the same
+refraction, will be seen by the refracted ray LRI, so that LR will be
+parallel to MK if the distance from the eye KI is supposed very great.
+The point L appears then as being in the straight line IRS; but the
+same point appears also, by ordinary refraction, to be in the straight
+line IK, hence it is necessarily judged to be double. And similarly if
+L be a small hole in a sheet of paper or other substance which is laid
+against the Crystal, it will appear when turned towards daylight as if
+there were two holes, which will seem the wider apart from one another
+the greater the thickness of the Crystal.
+
+10. Again, if one turns the Crystal in such wise that an incident ray
+NO, of sunlight, which I suppose to be in the plane continued from
+GCFH, makes with GC an angle of 73 degrees and 20 minutes, and is
+consequently nearly parallel to the edge CF, which makes with FH an
+angle of 70 degrees 57 minutes, according to the calculation which I
+shall put at the end, it will divide itself at the point O into two
+rays, one of which will continue along OP in a straight line with NO,
+and will similarly pass out of the other side of the crystal without
+any refraction; but the other will be refracted and will go along OQ.
+And it must be noted that it is special to the plane through GCF and
+to those which are parallel to it, that all incident rays which are in
+one of these planes continue to be in it after they have entered the
+Crystal and have become double; for it is quite otherwise for rays in
+all other planes which intersect the Crystal, as we shall see
+afterwards.
+
+11. I recognized at first by these experiments and by some others that
+of the two refractions which the ray suffers in this Crystal, there is
+one which follows the ordinary rules; and it is this to which the rays
+KL and OQ belong. This is why I have distinguished this ordinary
+refraction from the other; and having measured it by exact
+observation, I found that its proportion, considered as to the Sines
+of the angles which the incident and refracted rays make with the
+perpendicular, was very precisely that of 5 to 3, as was found also by
+Mr. Bartholinus, and consequently much greater than that of Rock
+Crystal, or of glass, which is nearly 3 to 2.
+
+[Illustration]
+
+12. The mode of making these observations exactly is as follows. Upon
+a leaf of paper fixed on a thoroughly flat table there is traced a
+black line AB, and two others, CED and KML, which cut it at right
+angles and are more or less distant from one another according as it
+is desired to examine a ray that is more or less oblique. Then place
+the Crystal upon the intersection E so that the line AB concurs with
+that which bisects the obtuse angle of the lower surface, or with some
+line parallel to it. Then by placing the eye directly above the line
+AB it will appear single only; and one will see that the portion
+viewed through the Crystal and the portions which appear outside it,
+meet together in a straight line: but the line CD will appear double,
+and one can distinguish the image which is due to regular refraction
+by the circumstance that when one views it with both eyes it seems
+raised up more than the other, or again by the circumstance that, when
+the Crystal is turned around on the paper, this image remains
+stationary, whereas the other image shifts and moves entirely around.
+Afterwards let the eye be placed at I (remaining always in the plane
+perpendicular through AB) so that it views the image which is formed
+by regular refraction of the line CD making a straight line with the
+remainder of that line which is outside the Crystal. And then, marking
+on the surface of the Crystal the point H where the intersection E
+appears, this point will be directly above E. Then draw back the eye
+towards O, keeping always in the plane perpendicular through AB, so
+that the image of the line CD, which is formed by ordinary refraction,
+may appear in a straight line with the line KL viewed without
+refraction; and then mark on the Crystal the point N where the point
+of intersection E appears.
+
+13. Then one will know the length and position of the lines NH, EM,
+and of HE, which is the thickness of the Crystal: which lines being
+traced separately upon a plan, and then joining NE and NM which cuts
+HE at P, the proportion of the refraction will be that of EN to NP,
+because these lines are to one another as the sines of the angles NPH,
+NEP, which are equal to those which the incident ray ON and its
+refraction NE make with the perpendicular to the surface. This
+proportion, as I have said, is sufficiently precisely as 5 to 3, and
+is always the same for all inclinations of the incident ray.
+
+14. The same mode of observation has also served me for examining the
+extraordinary or irregular refraction of this Crystal. For, the point
+H having been found and marked, as aforesaid, directly above the point
+E, I observed the appearance of the line CD, which is made by the
+extraordinary refraction; and having placed the eye at Q, so that this
+appearance made a straight line with the line KL viewed without
+refraction, I ascertained the triangles REH, RES, and consequently the
+angles RSH, RES, which the incident and the refracted ray make with
+the perpendicular.
+
+15. But I found in this refraction that the ratio of FR to RS was not
+constant, like the ordinary refraction, but that it varied with the
+varying obliquity of the incident ray.
+
+16. I found also that when QRE made a straight line, that is, when the
+incident ray entered the Crystal without being refracted (as I
+ascertained by the circumstance that then the point E viewed by the
+extraordinary refraction appeared in the line CD, as seen without
+refraction) I found, I say, then that the angle QRG was 73 degrees 20
+minutes, as has been already remarked; and so it is not the ray
+parallel to the edge of the Crystal, which crosses it in a straight
+line without being refracted, as Mr. Bartholinus believed, since that
+inclination is only 70 degrees 57 minutes, as was stated above. And
+this is to be noted, in order that no one may search in vain for the
+cause of the singular property of this ray in its parallelism to the
+edges mentioned.
+
+[Illustration]
+
+17. Finally, continuing my observations to discover the nature of
+this refraction, I learned that it obeyed the following remarkable
+rule. Let the parallelogram GCFH, made by the principal section of the
+Crystal, as previously determined, be traced separately. I found then
+that always, when the inclinations of two rays which come from
+opposite sides, as VK, SK here, are equal, their refractions KX and KT
+meet the bottom line HF in such wise that points X and T are equally
+distant from the point M, where the refraction of the perpendicular
+ray IK falls; and this occurs also for refractions in other sections
+of this Crystal. But before speaking of those, which have also other
+particular properties, we will investigate the causes of the phenomena
+which I have already reported.
+
+It was after having explained the refraction of ordinary transparent
+bodies by means of the spherical emanations of light, as above, that I
+resumed my examination of the nature of this Crystal, wherein I had
+previously been unable to discover anything.
+
+18. As there were two different refractions, I conceived that there
+were also two different emanations of waves of light, and that one
+could occur in the ethereal matter extending through the body of the
+Crystal. Which matter, being present in much larger quantity than is
+that of the particles which compose it, was alone capable of causing
+transparency, according to what has been explained heretofore. I
+attributed to this emanation of waves the regular refraction which is
+observed in this stone, by supposing these waves to be ordinarily of
+spherical form, and having a slower progression within the Crystal
+than they have outside it; whence proceeds refraction as I have
+demonstrated.
+
+19. As to the other emanation which should produce the irregular
+refraction, I wished to try what Elliptical waves, or rather
+spheroidal waves, would do; and these I supposed would spread
+indifferently both in the ethereal matter diffused throughout the
+crystal and in the particles of which it is composed, according to the
+last mode in which I have explained transparency. It seemed to me that
+the disposition or regular arrangement of these particles could
+contribute to form spheroidal waves (nothing more being required for
+this than that the successive movement of light should spread a little
+more quickly in one direction than in the other) and I scarcely
+doubted that there were in this crystal such an arrangement of equal
+and similar particles, because of its figure and of its angles with
+their determinate and invariable measure. Touching which particles,
+and their form and disposition, I shall, at the end of this Treatise,
+propound my conjectures and some experiments which confirm them.
+
+20. The double emission of waves of light, which I had imagined,
+became more probable to me after I had observed a certain phenomenon
+in the ordinary [Rock] Crystal, which occurs in hexagonal form, and
+which, because of this regularity, seems also to be composed of
+particles, of definite figure, and ranged in order. This was, that
+this crystal, as well as that from Iceland, has a double refraction,
+though less evident. For having had cut from it some well polished
+Prisms of different sections, I remarked in all, in viewing through
+them the flame of a candle or the lead of window panes, that
+everything appeared double, though with images not very distant from
+one another. Whence I understood the reason why this substance, though
+so transparent, is useless for Telescopes, when they have ever so
+little length.
+
+21. Now this double refraction, according to my Theory hereinbefore
+established, seemed to demand a double emission of waves of light,
+both of them spherical (for both the refractions are regular) and
+those of one series a little slower only than the others. For thus the
+phenomenon is quite naturally explained, by postulating substances
+which serve as vehicle for these waves, as I have done in the case of
+Iceland Crystal. I had then less trouble after that in admitting two
+emissions of waves in one and the same body. And since it might have
+been objected that in composing these two kinds of crystal of equal
+particles of a certain figure, regularly piled, the interstices which
+these particles leave and which contain the ethereal matter would
+scarcely suffice to transmit the waves of light which I have localized
+there, I removed this difficulty by regarding these particles as being
+of a very rare texture, or rather as composed of other much smaller
+particles, between which the ethereal matter passes quite freely.
+This, moreover, necessarily follows from that which has been already
+demonstrated touching the small quantity of matter of which the bodies
+are built up.
+
+22. Supposing then these spheroidal waves besides the spherical ones,
+I began to examine whether they could serve to explain the phenomena
+of the irregular refraction, and how by these same phenomena I could
+determine the figure and position of the spheroids: as to which I
+obtained at last the desired success, by proceeding as follows.
+
+[Illustration]
+
+23. I considered first the effect of waves so formed, as respects the
+ray which falls perpendicularly on the flat surface of a transparent
+body in which they should spread in this manner. I took AB for the
+exposed region of the surface. And, since a ray perpendicular to a
+plane, and coming from a very distant source of light, is nothing
+else, according to the precedent Theory, than the incidence of a
+portion of the wave parallel to that plane, I supposed the straight
+line RC, parallel and equal to AB, to be a portion of a wave of light,
+in which an infinitude of points such as RH_h_C come to meet the
+surface AB at the points AK_k_B. Then instead of the hemispherical
+partial waves which in a body of ordinary refraction would spread from
+each of these last points, as we have above explained in treating of
+refraction, these must here be hemi-spheroids. The axes (or rather the
+major diameters) of these I supposed to be oblique to the plane AB, as
+is AV the semi-axis or semi-major diameter of the spheroid SVT, which
+represents the partial wave coming from the point A, after the wave RC
+has reached AB. I say axis or major diameter, because the same ellipse
+SVT may be considered as the section of a spheroid of which the axis
+is AZ perpendicular to AV. But, for the present, without yet deciding
+one or other, we will consider these spheroids only in those sections
+of them which make ellipses in the plane of this figure. Now taking a
+certain space of time during which the wave SVT has spread from A, it
+would needs be that from all the other points K_k_B there should
+proceed, in the same time, waves similar to SVT and similarly
+situated. And the common tangent NQ of all these semi-ellipses would
+be the propagation of the wave RC which fell on AB, and would be the
+place where this movement occurs in much greater amount than anywhere
+else, being made up of arcs of an infinity of ellipses, the centres of
+which are along the line AB.
+
+24. Now it appeared that this common tangent NQ was parallel to AB,
+and of the same length, but that it was not directly opposite to it,
+since it was comprised between the lines AN, BQ, which are diameters
+of ellipses having A and B for centres, conjugate with respect to
+diameters which are not in the straight line AB. And in this way I
+comprehended, a matter which had seemed to me very difficult, how a
+ray perpendicular to a surface could suffer refraction on entering a
+transparent body; seeing that the wave RC, having come to the aperture
+AB, went on forward thence, spreading between the parallel lines AN,
+BQ, yet itself remaining always parallel to AB, so that here the light
+does not spread along lines perpendicular to its waves, as in ordinary
+refraction, but along lines cutting the waves obliquely.
+
+[Illustration]
+
+25. Inquiring subsequently what might be the position and form of
+these spheroids in the crystal, I considered that all the six faces
+produced precisely the same refractions. Taking, then, the
+parallelopiped AFB, of which the obtuse solid angle C is contained
+between the three equal plane angles, and imagining in it the three
+principal sections, one of which is perpendicular to the face DC and
+passes through the edge CF, another perpendicular to the face BF
+passing through the edge CA, and the third perpendicular to the face
+AF passing through the edge BC; I knew that the refractions of the
+incident rays belonging to these three planes were all similar. But
+there could be no position of the spheroid which would have the same
+relation to these three sections except that in which the axis was
+also the axis of the solid angle C. Consequently I saw that the axis
+of this angle, that is to say the straight line which traversed the
+crystal from the point C with equal inclination to the edges CF, CA,
+CB was the line which determined the position of the axis of all the
+spheroidal waves which one imagined to originate from some point,
+taken within or on the surface of the crystal, since all these
+spheroids ought to be alike, and have their axes parallel to one
+another.
+
+26. Considering after this the plane of one of these three sections,
+namely that through GCF, the angle of which is 109 degrees 3 minutes,
+since the angle F was shown above to be 70 degrees 57 minutes; and,
+imagining a spheroidal wave about the centre C, I knew, because I have
+just explained it, that its axis must be in the same plane, the half
+of which axis I have marked CS in the next figure: and seeking by
+calculation (which will be given with others at the end of this
+discourse) the value of the angle CGS, I found it 45 degrees 20
+minutes.
+
+[Illustration]
+
+27. To know from this the form of this spheroid, that is to say the
+proportion of the semi-diameters CS, CP, of its elliptical section,
+which are perpendicular to one another, I considered that the point M
+where the ellipse is touched by the straight line FH, parallel to CG,
+ought to be so situated that CM makes with the perpendicular CL an
+angle of 6 degrees 40 minutes; since, this being so, this ellipse
+satisfies what has been said about the refraction of the ray
+perpendicular to the surface CG, which is inclined to the
+perpendicular CL by the same angle. This, then, being thus disposed,
+and taking CM at 100,000 parts, I found by the calculation which will
+be given at the end, the semi-major diameter CP to be 105,032, and the
+semi-axis CS to be 93,410, the ratio of which numbers is very nearly 9
+to 8; so that the spheroid was of the kind which resembles a
+compressed sphere, being generated by the revolution of an ellipse
+about its smaller diameter. I found also the value of CG the
+semi-diameter parallel to the tangent ML to be 98,779.
+
+[Illustration]
+
+28. Now passing to the investigation of the refractions which
+obliquely incident rays must undergo, according to our hypothesis of
+spheroidal waves, I saw that these refractions depended on the ratio
+between the velocity of movement of the light outside the crystal in
+the ether, and that within the crystal. For supposing, for example,
+this proportion to be such that while the light in the crystal forms
+the spheroid GSP, as I have just said, it forms outside a sphere the
+semi-diameter of which is equal to the line N which will be determined
+hereafter, the following is the way of finding the refraction of the
+incident rays. Let there be such a ray RC falling upon the surface
+CK. Make CO perpendicular to RC, and across the angle KCO adjust OK,
+equal to N and perpendicular to CO; then draw KI, which touches the
+Ellipse GSP, and from the point of contact I join IC, which will be
+the required refraction of the ray RC. The demonstration of this is,
+it will be seen, entirely similar to that of which we made use in
+explaining ordinary refraction. For the refraction of the ray RC is
+nothing else than the progression of the portion C of the wave CO,
+continued in the crystal. Now the portions H of this wave, during the
+time that O came to K, will have arrived at the surface CK along the
+straight lines H_x_, and will moreover have produced in the crystal
+around the centres _x_ some hemi-spheroidal partial waves similar to
+the hemi-spheroidal GSP_g_, and similarly disposed, and of which the
+major and minor diameters will bear the same proportions to the lines
+_xv_ (the continuations of the lines H_x_ up to KB parallel to CO)
+that the diameters of the spheroid GSP_g_ bear to the line CB, or N.
+And it is quite easy to see that the common tangent of all these
+spheroids, which are here represented by Ellipses, will be the
+straight line IK, which consequently will be the propagation of the
+wave CO; and the point I will be that of the point C, conformably with
+that which has been demonstrated in ordinary refraction.
+
+Now as to finding the point of contact I, it is known that one must
+find CD a third proportional to the lines CK, CG, and draw DI parallel
+to CM, previously determined, which is the conjugate diameter to CG;
+for then, by drawing KI it touches the Ellipse at I.
+
+29. Now as we have found CI the refraction of the ray RC, similarly
+one will find C_i_ the refraction of the ray _r_C, which comes from
+the opposite side, by making C_o_ perpendicular to _r_C and following
+out the rest of the construction as before. Whence one sees that if
+the ray _r_C is inclined equally with RC, the line C_d_ will
+necessarily be equal to CD, because C_k_ is equal to CK, and C_g_ to
+CG. And in consequence I_i_ will be cut at E into equal parts by the
+line CM, to which DI and _di_ are parallel. And because CM is the
+conjugate diameter to CG, it follows that _i_I will be parallel to
+_g_G. Therefore if one prolongs the refracted rays CI, C_i_, until
+they meet the tangent ML at T and _t_, the distances MT, M_t_, will
+also be equal. And so, by our hypothesis, we explain perfectly the
+phenomenon mentioned above; to wit, that when there are two rays
+equally inclined, but coming from opposite sides, as here the rays RC,
+_rc_, their refractions diverge equally from the line followed by the
+refraction of the ray perpendicular to the surface, by considering
+these divergences in the direction parallel to the surface of the
+crystal.
+
+30. To find the length of the line N, in proportion to CP, CS, CG, it
+must be determined by observations of the irregular refraction which
+occurs in this section of the crystal; and I find thus that the ratio
+of N to GC is just a little less than 8 to 5. And having regard to
+some other observations and phenomena of which I shall speak
+afterwards, I put N at 156,962 parts, of which the semi-diameter CG is
+found to contain 98,779, making this ratio 8 to 5-1/29. Now this
+proportion, which there is between the line N and CG, may be called
+the Proportion of the Refraction; similarly as in glass that of 3 to
+2, as will be manifest when I shall have explained a short process in
+the preceding way to find the irregular refractions.
+
+31. Supposing then, in the next figure, as previously, the surface of
+the crystal _g_G, the Ellipse GP_g_, and the line N; and CM the
+refraction of the perpendicular ray FC, from which it diverges by 6
+degrees 40 minutes. Now let there be some other ray RC, the refraction
+of which must be found.
+
+About the centre C, with semi-diameter CG, let the circumference _g_RG
+be described, cutting the ray RC at R; and let RV be the perpendicular
+on CG. Then as the line N is to CG let CV be to CD, and let DI be
+drawn parallel to CM, cutting the Ellipse _g_MG at I; then joining CI,
+this will be the required refraction of the ray RC. Which is
+demonstrated thus.
+
+[Illustration]
+
+Let CO be perpendicular to CR, and across the angle OCG let OK be
+adjusted, equal to N and perpendicular to CO, and let there be drawn
+the straight line KI, which if it is demonstrated to be a tangent to
+the Ellipse at I, it will be evident by the things heretofore
+explained that CI is the refraction of the ray RC. Now since the angle
+RCO is a right angle, it is easy to see that the right-angled
+triangles RCV, KCO, are similar. As then, CK is to KO, so also is RC
+to CV. But KO is equal to N, and RC to CG: then as CK is to N so will
+CG be to CV. But as N is to CG, so, by construction, is CV to CD. Then
+as CK is to CG so is CG to CD. And because DI is parallel to CM, the
+conjugate diameter to CG, it follows that KI touches the Ellipse at I;
+which remained to be shown.
+
+32. One sees then that as there is in the refraction of ordinary
+media a certain constant proportion between the sines of the angles
+which the incident ray and the refracted ray make with the
+perpendicular, so here there is such a proportion between CV and CD or
+IE; that is to say between the Sine of the angle which the incident
+ray makes with the perpendicular, and the horizontal intercept, in the
+Ellipse, between the refraction of this ray and the diameter CM. For
+the ratio of CV to CD is, as has been said, the same as that of N to
+the semi-diameter CG.
+
+33. I will add here, before passing away, that in comparing together
+the regular and irregular refraction of this crystal, there is this
+remarkable fact, that if ABPS be the spheroid by which light spreads
+in the Crystal in a certain space of time (which spreading, as has
+been said, serves for the irregular refraction), then the inscribed
+sphere BVST is the extension in the same space of time of the light
+which serves for the regular refraction.
+
+[Illustration]
+
+For we have stated before this, that the line N being the radius of a
+spherical wave of light in air, while in the crystal it spread through
+the spheroid ABPS, the ratio of N to CS will be 156,962 to 93,410. But
+it has also been stated that the proportion of the regular refraction
+was 5 to 3; that is to say, that N being the radius of a spherical
+wave of light in air, its extension in the crystal would, in the same
+space of time, form a sphere the radius of which would be to N as 3 to
+5. Now 156,962 is to 93,410 as 5 to 3 less 1/41. So that it is
+sufficiently nearly, and may be exactly, the sphere BVST, which the
+light describes for the regular refraction in the crystal, while it
+describes the spheroid BPSA for the irregular refraction, and while it
+describes the sphere of radius N in air outside the crystal.
+
+Although then there are, according to what we have supposed, two
+different propagations of light within the crystal, it appears that it
+is only in directions perpendicular to the axis BS of the spheroid
+that one of these propagations occurs more rapidly than the other; but
+that they have an equal velocity in the other direction, namely, in
+that parallel to the same axis BS, which is also the axis of the
+obtuse angle of the crystal.
+
+[Illustration]
+
+34. The proportion of the refraction being what we have just seen, I
+will now show that there necessarily follows thence that notable
+property of the ray which falling obliquely on the surface of the
+crystal enters it without suffering refraction. For supposing the same
+things as before, and that the ray makes with the same surface _g_G
+the angle RCG of 73 degrees 20 minutes, inclining to the same side as
+the crystal (of which ray mention has been made above); if one
+investigates, by the process above explained, the refraction CI, one
+will find that it makes exactly a straight line with RC, and that thus
+this ray is not deviated at all, conformably with experiment. This is
+proved as follows by calculation.
+
+CG or CR being, as precedently, 98,779; CM being 100,000; and the
+angle RCV 73 degrees 20 minutes, CV will be 28,330. But because CI is
+the refraction of the ray RC, the proportion of CV to CD is 156,962 to
+98,779, namely, that of N to CG; then CD is 17,828.
+
+Now the rectangle _g_DC is to the square of DI as the square of CG is
+to the square of CM; hence DI or CE will be 98,353. But as CE is to
+EI, so will CM be to MT, which will then be 18,127. And being added to
+ML, which is 11,609 (namely the sine of the angle LCM, which is 6
+degrees 40 minutes, taking CM 100,000 as radius) we get LT 27,936; and
+this is to LC 99,324 as CV to VR, that is to say, as 29,938, the
+tangent of the complement of the angle RCV, which is 73 degrees 20
+minutes, is to the radius of the Tables. Whence it appears that RCIT
+is a straight line; which was to be proved.
+
+35. Further it will be seen that the ray CI in emerging through the
+opposite surface of the crystal, ought to pass out quite straight,
+according to the following demonstration, which proves that the
+reciprocal relation of refraction obtains in this crystal the same as
+in other transparent bodies; that is to say, that if a ray RC in
+meeting the surface of the crystal CG is refracted as CI, the ray CI
+emerging through the opposite parallel surface of the crystal, which
+I suppose to be IB, will have its refraction IA parallel to the ray
+RC.
+
+[Illustration]
+
+Let the same things be supposed as before; that is to say, let CO,
+perpendicular to CR, represent a portion of a wave the continuation of
+which in the crystal is IK, so that the piece C will be continued on
+along the straight line CI, while O comes to K. Now if one takes a
+second period of time equal to the first, the piece K of the wave IK
+will, in this second period, have advanced along the straight line KB,
+equal and parallel to CI, because every piece of the wave CO, on
+arriving at the surface CK, ought to go on in the crystal the same as
+the piece C; and in this same time there will be formed in the air
+from the point I a partial spherical wave having a semi-diameter IA
+equal to KO, since KO has been traversed in an equal time. Similarly,
+if one considers some other point of the wave IK, such as _h_, it will
+go along _hm_, parallel to CI, to meet the surface IB, while the point
+K traverses K_l_ equal to _hm_; and while this accomplishes the
+remainder _l_B, there will start from the point _m_ a partial wave the
+semi-diameter of which, _mn_, will have the same ratio to _l_B as IA
+to KB. Whence it is evident that this wave of semi-diameter _mn_, and
+the other of semi-diameter IA will have the same tangent BA. And
+similarly for all the partial spherical waves which will be formed
+outside the crystal by the impact of all the points of the wave IK
+against the surface of the Ether IB. It is then precisely the tangent
+BA which will be the continuation of the wave IK, outside the crystal,
+when the piece K has reached B. And in consequence IA, which is
+perpendicular to BA, will be the refraction of the ray CI on emerging
+from the crystal. Now it is clear that IA is parallel to the incident
+ray RC, since IB is equal to CK, and IA equal to KO, and the angles A
+and O are right angles.
+
+It is seen then that, according to our hypothesis, the reciprocal
+relation of refraction holds good in this crystal as well as in
+ordinary transparent bodies; as is thus in fact found by observation.
+
+36. I pass now to the consideration of other sections of the crystal,
+and of the refractions there produced, on which, as will be seen, some
+other very remarkable phenomena depend.
+
+Let ABH be a parallelepiped of crystal, and let the top surface AEHF
+be a perfect rhombus, the obtuse angles of which are equally divided
+by the straight line EF, and the acute angles by the straight line AH
+perpendicular to FE.
+
+The section which we have hitherto considered is that which passes
+through the lines EF, EB, and which at the same time cuts the plane
+AEHF at right angles. Refractions in this section have this in common
+with the refractions in ordinary media that the plane which is drawn
+through the incident ray and which also intersects the surface of the
+crystal at right angles, is that in which the refracted ray also is
+found. But the refractions which appertain to every other section of
+this crystal have this strange property that the refracted ray always
+quits the plane of the incident ray perpendicular to the surface, and
+turns away towards the side of the slope of the crystal. For which
+fact we shall show the reason, in the first place, for the section
+through AH; and we shall show at the same time how one can determine
+the refraction, according to our hypothesis. Let there be, then, in
+the plane which passes through AH, and which is perpendicular to the
+plane AFHE, the incident ray RC; it is required to find its refraction
+in the crystal.
+
+[Illustration]
+
+37. About the centre C, which I suppose to be in the intersection of
+AH and FE, let there be imagined a hemi-spheroid QG_qg_M, such as the
+light would form in spreading in the crystal, and let its section by
+the plane AEHF form the Ellipse QG_qg_, the major diameter of which
+Q_q_, which is in the line AH, will necessarily be one of the major
+diameters of the spheroid; because the axis of the spheroid being in
+the plane through FEB, to which QC is perpendicular, it follows that
+QC is also perpendicular to the axis of the spheroid, and consequently
+QC_q_ one of its major diameters. But the minor diameter of this
+Ellipse, G_g_, will bear to Q_q_ the proportion which has been defined
+previously, Article 27, between CG and the major semi-diameter of the
+spheroid, CP, namely, that of 98,779 to 105,032.
+
+Let the line N be the length of the travel of light in air during the
+time in which, within the crystal, it makes, from the centre C, the
+spheroid QC_qg_M. Then having drawn CO perpendicular to the ray CR and
+situate in the plane through CR and AH, let there be adjusted, across
+the angle ACO, the straight line OK equal to N and perpendicular to
+CO, and let it meet the straight line AH at K. Supposing consequently
+that CL is perpendicular to the surface of the crystal AEHF, and that
+CM is the refraction of the ray which falls perpendicularly on this
+same surface, let there be drawn a plane through the line CM and
+through KCH, making in the spheroid the semi-ellipse QM_q_, which will
+be given, since the angle MCL is given of value 6 degrees 40 minutes.
+And it is certain, according to what has been explained above, Article
+27, that a plane which would touch the spheroid at the point M, where
+I suppose the straight line CM to meet the surface, would be parallel
+to the plane QG_q_. If then through the point K one now draws KS
+parallel to G_g_, which will be parallel also to QX, the tangent to
+the Ellipse QG_q_ at Q; and if one conceives a plane passing through
+KS and touching the spheroid, the point of contact will necessarily be
+in the Ellipse QM_q_, because this plane through KS, as well as the
+plane which touches the spheroid at the point M, are parallel to QX,
+the tangent of the spheroid: for this consequence will be demonstrated
+at the end of this Treatise. Let this point of contact be at I, then
+making KC, QC, DC proportionals, draw DI parallel to CM; also join CI.
+I say that CI will be the required refraction of the ray RC. This will
+be manifest if, in considering CO, which is perpendicular to the ray
+RC, as a portion of the wave of light, we can demonstrate that the
+continuation of its piece C will be found in the crystal at I, when O
+has arrived at K.
+
+38. Now as in the Chapter on Reflexion, in demonstrating that the
+incident and reflected rays are always in the same plane perpendicular
+to the reflecting surface, we considered the breadth of the wave of
+light, so, similarly, we must here consider the breadth of the wave CO
+in the diameter G_g_. Taking then the breadth C_c_ on the side toward
+the angle E, let the parallelogram CO_oc_ be taken as a portion of a
+wave, and let us complete the parallelograms CK_kc_, CI_ic_, Kl_ik_,
+OK_ko_. In the time then that the line O_o_ arrives at the surface of
+the crystal at K_k_, all the points of the wave CO_oc_ will have
+arrived at the rectangle K_c_ along lines parallel to OK; and from the
+points of their incidences there will originate, beyond that, in the
+crystal partial hemi-spheroids, similar to the hemi-spheroid QM_q_,
+and similarly disposed. These hemi-spheroids will necessarily all
+touch the plane of the parallelogram KI_ik_ at the same instant that
+O_o_ has reached K_k_. Which is easy to comprehend, since, of these
+hemi-spheroids, all those which have their centres along the line CK,
+touch this plane in the line KI (for this is to be shown in the same
+way as we have demonstrated the refraction of the oblique ray in the
+principal section through EF) and all those which have their centres
+in the line C_c_ will touch the same plane KI in the line I_i_; all
+these being similar to the hemi-spheroid QM_q_. Since then the
+parallelogram K_i_ is that which touches all these spheroids, this
+same parallelogram will be precisely the continuation of the wave
+CO_oc_ in the crystal, when O_o_ has arrived at K_k_, because it forms
+the termination of the movement and because of the quantity of
+movement which occurs more there than anywhere else: and thus it
+appears that the piece C of the wave CO_oc_ has its continuation at I;
+that is to say, that the ray RC is refracted as CI.
+
+From this it is to be noted that the proportion of the refraction for
+this section of the crystal is that of the line N to the semi-diameter
+CQ; by which one will easily find the refractions of all incident
+rays, in the same way as we have shown previously for the case of the
+section through FE; and the demonstration will be the same. But it
+appears that the said proportion of the refraction is less here than
+in the section through FEB; for it was there the same as the ratio of
+N to CG, that is to say, as 156,962 to 98,779, very nearly as 8 to 5;
+and here it is the ratio of N to CQ the major semi-diameter of the
+spheroid, that is to say, as 156,962 to 105,032, very nearly as 3 to
+2, but just a little less. Which still agrees perfectly with what one
+finds by observation.
+
+39. For the rest, this diversity of proportion of refraction produces
+a very singular effect in this Crystal; which is that when it is
+placed upon a sheet of paper on which there are letters or anything
+else marked, if one views it from above with the two eyes situated in
+the plane of the section through EF, one sees the letters raised up by
+this irregular refraction more than when one puts one's eyes in the
+plane of section through AH: and the difference of these elevations
+appears by comparison with the other ordinary refraction of the
+crystal, the proportion of which is as 5 to 3, and which always raises
+the letters equally, and higher than the irregular refraction does.
+For one sees the letters and the paper on which they are written, as
+on two different stages at the same time; and in the first position of
+the eyes, namely, when they are in the plane through AH these two
+stages are four times more distant from one another than when the eyes
+are in the plane through EF.
+
+We will show that this effect follows from the refractions; and it
+will enable us at the same time to ascertain the apparent place of a
+point of an object placed immediately under the crystal, according to
+the different situation of the eyes.
+
+40. Let us see first by how much the irregular refraction of the plane
+through AH ought to lift the bottom of the crystal. Let the plane of
+this figure represent separately the section through Q_q_ and CL, in
+which section there is also the ray RC, and let the semi-elliptic
+plane through Q_q_ and CM be inclined to the former, as previously, by
+an angle of 6 degrees 40 minutes; and in this plane CI is then the
+refraction of the ray RC.
+
+[Illustration]
+
+If now one considers the point I as at the bottom of the crystal, and
+that it is viewed by the rays ICR, _Icr_, refracted equally at the
+points C_c_, which should be equally distant from D, and that these
+rays meet the two eyes at R_r_; it is certain that the point I will
+appear raised to S where the straight lines RC, _rc_, meet; which
+point S is in DP, perpendicular to Q_q_. And if upon DP there is drawn
+the perpendicular IP, which will lie at the bottom of the crystal, the
+length SP will be the apparent elevation of the point I above the
+bottom.
+
+Let there be described on Q_q_ a semicircle cutting the ray CR at B,
+from which BV is drawn perpendicular to Q_q_; and let the proportion
+of the refraction for this section be, as before, that of the line N
+to the semi-diameter CQ.
+
+Then as N is to CQ so is VC to CD, as appears by the method of finding
+the refraction which we have shown above, Article 31; but as VC is to
+CD, so is VB to DS. Then as N is to CQ, so is VB to DS. Let ML be
+perpendicular to CL. And because I suppose the eyes R_r_ to be distant
+about a foot or so from the crystal, and consequently the angle RS_r_
+very small, VB may be considered as equal to the semi-diameter CQ, and
+DP as equal to CL; then as N is to CQ so is CQ to DS. But N is valued
+at 156,962 parts, of which CM contains 100,000 and CQ 105,032. Then DS
+will have 70,283. But CL is 99,324, being the sine of the complement
+of the angle MCL which is 6 degrees 40 minutes; CM being supposed as
+radius. Then DP, considered as equal to CL, will be to DS as 99,324 to
+70,283. And so the elevation of the point I by the refraction of this
+section is known.
+
+[Illustration]
+
+41. Now let there be represented the other section through EF in the
+figure before the preceding one; and let CM_g_ be the semi-ellipse,
+considered in Articles 27 and 28, which is made by cutting a
+spheroidal wave having centre C. Let the point I, taken in this
+ellipse, be imagined again at the bottom of the Crystal; and let it be
+viewed by the refracted rays ICR, I_cr_, which go to the two eyes; CR
+and _cr_ being equally inclined to the surface of the crystal G_g_.
+This being so, if one draws ID parallel to CM, which I suppose to be
+the refraction of the perpendicular ray incident at the point C, the
+distances DC, D_c_, will be equal, as is easy to see by that which has
+been demonstrated in Article 28. Now it is certain that the point I
+should appear at S where the straight lines RC, _rc_, meet when
+prolonged; and that this point will fall in the line DP perpendicular
+to G_g_. If one draws IP perpendicular to this DP, it will be the
+distance PS which will mark the apparent elevation of the point I. Let
+there be described on G_g_ a semicircle cutting CR at B, from which
+let BV be drawn perpendicular to G_g_; and let N to GC be the
+proportion of the refraction in this section, as in Article 28. Since
+then CI is the refraction of the radius BC, and DI is parallel to CM,
+VC must be to CD as N to GC, according to what has been demonstrated
+in Article 31. But as VC is to CD so is BV to DS. Let ML be drawn
+perpendicular to CL. And because I consider, again, the eyes to be
+distant above the crystal, BV is deemed equal to the semi-diameter CG;
+and hence DS will be a third proportional to the lines N and CG: also
+DP will be deemed equal to CL. Now CG consisting of 98,778 parts, of
+which CM contains 100,000, N is taken as 156,962. Then DS will be
+62,163. But CL is also determined, and contains 99,324 parts, as has
+been said in Articles 34 and 40. Then the ratio of PD to DS will be as
+99,324 to 62,163. And thus one knows the elevation of the point at the
+bottom I by the refraction of this section; and it appears that this
+elevation is greater than that by the refraction of the preceding
+section, since the ratio of PD to DS was there as 99,324 to 70,283.
+
+[Illustration]
+
+But by the regular refraction of the crystal, of which we have above
+said that the proportion is 5 to 3, the elevation of the point I, or
+P, from the bottom, will be 2/5 of the height DP; as appears by this
+figure, where the point P being viewed by the rays PCR, P_cr_,
+refracted equally at the surface C_c_, this point must needs appear
+to be at S, in the perpendicular PD where the lines RC, _rc_, meet
+when prolonged: and one knows that the line PC is to CS as 5 to 3,
+since they are to one another as the sine of the angle CSP or DSC is
+to the sine of the angle SPC. And because the ratio of PD to DS is
+deemed the same as that of PC to CS, the two eyes Rr being supposed
+very far above the crystal, the elevation PS will thus be 2/5 of PD.
+
+[Illustration]
+
+42. If one takes a straight line AB for the thickness of the crystal,
+its point B being at the bottom, and if one divides it at the points
+C, D, E, according to the proportions of the elevations found, making
+AE 3/5 of AB, AB to AC as 99,324 to 70,283, and AB to AD as 99,324 to
+62,163, these points will divide AB as in this figure. And it will be
+found that this agrees perfectly with experiment; that is to say by
+placing the eyes above in the plane which cuts the crystal according
+to the shorter diameter of the rhombus, the regular refraction will
+lift up the letters to E; and one will see the bottom, and the letters
+over which it is placed, lifted up to D by the irregular refraction.
+But by placing the eyes above in the plane which cuts the crystal
+according to the longer diameter of the rhombus, the regular
+refraction will lift the letters to E as before; but the irregular
+refraction will make them, at the same time, appear lifted up only to
+C; and in such a way that the interval CE will be quadruple the
+interval ED, which one previously saw.
+
+
+43. I have only to make the remark here that in both the positions of
+the eyes the images caused by the irregular refraction do not appear
+directly below those which proceed from the regular refraction, but
+they are separated from them by being more distant from the
+equilateral solid angle of the Crystal. That follows, indeed, from all
+that has been hitherto demonstrated about the irregular refraction;
+and it is particularly shown by these last demonstrations, from which
+one sees that the point I appears by irregular refraction at S in the
+perpendicular line DP, in which line also the image of the point P
+ought to appear by regular refraction, but not the image of the point
+I, which will be almost directly above the same point, and higher than
+S.
+
+But as to the apparent elevation of the point I in other positions of
+the eyes above the crystal, besides the two positions which we have
+just examined, the image of that point by the irregular refraction
+will always appear between the two heights of D and C, passing from
+one to the other as one turns one's self around about the immovable
+crystal, while looking down from above. And all this is still found
+conformable to our hypothesis, as any one can assure himself after I
+shall have shown here the way of finding the irregular refractions
+which appear in all other sections of the crystal, besides the two
+which we have considered. Let us suppose one of the faces of the
+crystal, in which let there be the Ellipse HDE, the centre C of which
+is also the centre of the spheroid HME in which the light spreads, and
+of which the said Ellipse is the section. And let the incident ray be
+RC, the refraction of which it is required to find.
+
+Let there be taken a plane passing through the ray RC and which is
+perpendicular to the plane of the ellipse HDE, cutting it along the
+straight line BCK; and having in the same plane through RC made CO
+perpendicular to CR, let OK be adjusted across the angle OCK, so as
+to be perpendicular to OC and equal to the line N, which I suppose to
+measure the travel of the light in air during the time that it spreads
+in the crystal through the spheroid HDEM. Then in the plane of the
+Ellipse HDE let KT be drawn, through the point K, perpendicular to
+BCK. Now if one conceives a plane drawn through the straight line KT
+and touching the spheroid HME at I, the straight line CI will be the
+refraction of the ray RC, as is easy to deduce from that which has
+been demonstrated in Article 36.
+
+[Illustration]
+
+But it must be shown how one can determine the point of contact I. Let
+there be drawn parallel to the line KT a line HF which touches the
+Ellipse HDE, and let this point of contact be at H. And having drawn a
+straight line along CH to meet KT at T, let there be imagined a plane
+passing through the same CH and through CM (which I suppose to be the
+refraction of the perpendicular ray), which makes in the spheroid the
+elliptical section HME. It is certain that the plane which will pass
+through the straight line KT, and which will touch the spheroid, will
+touch it at a point in the Ellipse HME, according to the Lemma which
+will be demonstrated at the end of the Chapter. Now this point is
+necessarily the point I which is sought, since the plane drawn through
+TK can touch the spheroid at one point only. And this point I is easy
+to determine, since it is needful only to draw from the point T, which
+is in the plane of this Ellipse, the tangent TI, in the way shown
+previously. For the Ellipse HME is given, and its conjugate
+semi-diameters are CH and CM; because a straight line drawn through M,
+parallel to HE, touches the Ellipse HME, as follows from the fact that
+a plane taken through M, and parallel to the plane HDE, touches the
+spheroid at that point M, as is seen from Articles 27 and 23. For the
+rest, the position of this ellipse, with respect to the plane through
+the ray RC and through CK, is also given; from which it will be easy
+to find the position of CI, the refraction corresponding to the ray
+RC.
+
+Now it must be noted that the same ellipse HME serves to find the
+refractions of any other ray which may be in the plane through RC and
+CK. Because every plane, parallel to the straight line HF, or TK,
+which will touch the spheroid, will touch it in this ellipse,
+according to the Lemma quoted a little before.
+
+I have investigated thus, in minute detail, the properties of the
+irregular refraction of this Crystal, in order to see whether each
+phenomenon that is deduced from our hypothesis accords with that which
+is observed in fact. And this being so it affords no slight proof of
+the truth of our suppositions and principles. But what I am going to
+add here confirms them again marvellously. It is this: that there are
+different sections of this Crystal, the surfaces of which, thereby
+produced, give rise to refractions precisely such as they ought to be,
+and as I had foreseen them, according to the preceding Theory.
+
+In order to explain what these sections are, let ABKF _be_ the
+principal section through the axis of the crystal ACK, in which there
+will also be the axis SS of a spheroidal wave of light spreading in
+the crystal from the centre C; and the straight line which cuts SS
+through the middle and at right angles, namely PP, will be one of the
+major diameters.
+
+[Illustration: {Section ABKF}]
+
+Now as in the natural section of the crystal, made by a plane parallel
+to two opposite faces, which plane is here represented by the line GG,
+the refraction of the surfaces which are produced by it will be
+governed by the hemi-spheroids GNG, according to what has been
+explained in the preceding Theory. Similarly, cutting the Crystal
+through NN, by a plane perpendicular to the parallelogram ABKF, the
+refraction of the surfaces will be governed by the hemi-spheroids NGN.
+And if one cuts it through PP, perpendicularly to the said
+parallelogram, the refraction of the surfaces ought to be governed by
+the hemi-spheroids PSP, and so for others. But I saw that if the plane
+NN was almost perpendicular to the plane GG, making the angle NCG,
+which is on the side A, an angle of 90 degrees 40 minutes, the
+hemi-spheroids NGN would become similar to the hemi-spheroids GNG,
+since the planes NN and GG were equally inclined by an angle of 45
+degrees 20 minutes to the axis SS. In consequence it must needs be, if
+our theory is true, that the surfaces which the section through NN
+produces should effect the same refractions as the surfaces of the
+section through GG. And not only the surfaces of the section NN but
+all other sections produced by planes which might be inclined to the
+axis at an angle equal to 45 degrees 20 minutes. So that there are an
+infinitude of planes which ought to produce precisely the same
+refractions as the natural surfaces of the crystal, or as the section
+parallel to any one of those surfaces which are made by cleavage.
+
+I saw also that by cutting it by a plane taken through PP, and
+perpendicular to the axis SS, the refraction of the surfaces ought to
+be such that the perpendicular ray should suffer thereby no deviation;
+and that for oblique rays there would always be an irregular
+refraction, differing from the regular, and by which objects placed
+beneath the crystal would be less elevated than by that other
+refraction.
+
+That, similarly, by cutting the crystal by any plane through the axis
+SS, such as the plane of the figure is, the perpendicular ray ought to
+suffer no refraction; and that for oblique rays there were different
+measures for the irregular refraction according to the situation of
+the plane in which the incident ray was.
+
+Now these things were found in fact so; and, after that, I could not
+doubt that a similar success could be met with everywhere. Whence I
+concluded that one might form from this crystal solids similar to
+those which are its natural forms, which should produce, at all their
+surfaces, the same regular and irregular refractions as the natural
+surfaces, and which nevertheless would cleave in quite other ways, and
+not in directions parallel to any of their faces. That out of it one
+would be able to fashion pyramids, having their base square,
+pentagonal, hexagonal, or with as many sides as one desired, all the
+surfaces of which should have the same refractions as the natural
+surfaces of the crystal, except the base, which will not refract the
+perpendicular ray. These surfaces will each make an angle of 45
+degrees 20 minutes with the axis of the crystal, and the base will be
+the section perpendicular to the axis.
+
+That, finally, one could also fashion out of it triangular prisms, or
+prisms with as many sides as one would, of which neither the sides nor
+the bases would refract the perpendicular ray, although they would yet
+all cause double refraction for oblique rays. The cube is included
+amongst these prisms, the bases of which are sections perpendicular to
+the axis of the crystal, and the sides are sections parallel to the
+same axis.
+
+From all this it further appears that it is not at all in the
+disposition of the layers of which this crystal seems to be composed,
+and according to which it splits in three different senses, that the
+cause resides of its irregular refraction; and that it would be in
+vain to wish to seek it there.
+
+But in order that any one who has some of this stone may be able to
+find, by his own experience, the truth of what I have just advanced, I
+will state here the process of which I have made use to cut it, and to
+polish it. Cutting is easy by the slicing wheels of lapidaries, or in
+the way in which marble is sawn: but polishing is very difficult, and
+by employing the ordinary means one more often depolishes the surfaces
+than makes them lucent.
+
+After many trials, I have at last found that for this service no plate
+of metal must be used, but a piece of mirror glass made matt and
+depolished. Upon this, with fine sand and water, one smoothes the
+crystal little by little, in the same way as spectacle glasses, and
+polishes it simply by continuing the work, but ever reducing the
+material. I have not, however, been able to give it perfect clarity
+and transparency; but the evenness which the surfaces acquire enables
+one to observe in them the effects of refraction better than in those
+made by cleaving the stone, which always have some inequality.
+
+Even when the surface is only moderately smoothed, if one rubs it over
+with a little oil or white of egg, it becomes quite transparent, so
+that the refraction is discerned in it quite distinctly. And this aid
+is specially necessary when it is wished to polish the natural
+surfaces to remove the inequalities; because one cannot render them
+lucent equally with the surfaces of other sections, which take a
+polish so much the better the less nearly they approximate to these
+natural planes.
+
+Before finishing the treatise on this Crystal, I will add one more
+marvellous phenomenon which I discovered after having written all the
+foregoing. For though I have not been able till now to find its cause,
+I do not for that reason wish to desist from describing it, in order
+to give opportunity to others to investigate it. It seems that it will
+be necessary to make still further suppositions besides those which I
+have made; but these will not for all that cease to keep their
+probability after having been confirmed by so many tests.
+
+[Illustration]
+
+The phenomenon is, that by taking two pieces of this crystal and
+applying them one over the other, or rather holding them with a space
+between the two, if all the sides of one are parallel to those of the
+other, then a ray of light, such as AB, is divided into two in the
+first piece, namely into BD and BC, following the two refractions,
+regular and irregular. On penetrating thence into the other piece
+each ray will pass there without further dividing itself in two; but
+that one which underwent the regular refraction, as here DG, will
+undergo again only a regular refraction at GH; and the other, CE, an
+irregular refraction at EF. And the same thing occurs not only in this
+disposition, but also in all those cases in which the principal
+section of each of the pieces is situated in one and the same plane,
+without it being needful for the two neighbouring surfaces to be
+parallel. Now it is marvellous why the rays CE and DG, incident from
+the air on the lower crystal, do not divide themselves the same as the
+first ray AB. One would say that it must be that the ray DG in passing
+through the upper piece has lost something which is necessary to move
+the matter which serves for the irregular refraction; and that
+likewise CE has lost that which was necessary to move the matter
+which serves for regular refraction: but there is yet another thing
+which upsets this reasoning. It is that when one disposes the two
+crystals in such a way that the planes which constitute the principal
+sections intersect one another at right angles, whether the
+neighbouring surfaces are parallel or not, then the ray which has come
+by the regular refraction, as DG, undergoes only an irregular
+refraction in the lower piece; and on the contrary the ray which has
+come by the irregular refraction, as CE, undergoes only a regular
+refraction.
+
+But in all the infinite other positions, besides those which I have
+just stated, the rays DG, CE, divide themselves anew each one into
+two, by refraction in the lower crystal so that from the single ray AB
+there are four, sometimes of equal brightness, sometimes some much
+less bright than others, according to the varying agreement in the
+positions of the crystals: but they do not appear to have all together
+more light than the single ray AB.
+
+When one considers here how, while the rays CE, DG, remain the same,
+it depends on the position that one gives to the lower piece, whether
+it divides them both in two, or whether it does not divide them, and
+yet how the ray AB above is always divided, it seems that one is
+obliged to conclude that the waves of light, after having passed
+through the first crystal, acquire a certain form or disposition in
+virtue of which, when meeting the texture of the second crystal, in
+certain positions, they can move the two different kinds of matter
+which serve for the two species of refraction; and when meeting the
+second crystal in another position are able to move only one of these
+kinds of matter. But to tell how this occurs, I have hitherto found
+nothing which satisfies me.
+
+Leaving then to others this research, I pass to what I have to say
+touching the cause of the extraordinary figure of this crystal, and
+why it cleaves easily in three different senses, parallel to any one
+of its surfaces.
+
+There are many bodies, vegetable, mineral, and congealed salts, which
+are formed with certain regular angles and figures. Thus among flowers
+there are many which have their leaves disposed in ordered polygons,
+to the number of 3, 4, 5, or 6 sides, but not more. This well deserves
+to be investigated, both as to the polygonal figure, and as to why it
+does not exceed the number 6.
+
+Rock Crystal grows ordinarily in hexagonal bars, and diamonds are
+found which occur with a square point and polished surfaces. There is
+a species of small flat stones, piled up directly upon one another,
+which are all of pentagonal figure with rounded angles, and the sides
+a little folded inwards. The grains of gray salt which are formed from
+sea water affect the figure, or at least the angle, of the cube; and
+in the congelations of other salts, and in that of sugar, there are
+found other solid angles with perfectly flat faces. Small snowflakes
+almost always fall in little stars with 6 points, and sometimes in
+hexagons with straight sides. And I have often observed, in water
+which is beginning to freeze, a kind of flat and thin foliage of ice,
+the middle ray of which throws out branches inclined at an angle of 60
+degrees. All these things are worthy of being carefully investigated
+to ascertain how and by what artifice nature there operates. But it is
+not now my intention to treat fully of this matter. It seems that in
+general the regularity which occurs in these productions comes from
+the arrangement of the small invisible equal particles of which they
+are composed. And, coming to our Iceland Crystal, I say that if there
+were a pyramid such as ABCD, composed of small rounded corpuscles, not
+spherical but flattened spheroids, such as would be made by the
+rotation of the ellipse GH around its lesser diameter EF (of which the
+ratio to the greater diameter is very nearly that of 1 to the square
+root of 8)--I say that then the solid angle of the point D would be
+equal to the obtuse and equilateral angle of this Crystal. I say,
+further, that if these corpuscles were lightly stuck together, on
+breaking this pyramid it would break along faces parallel to those
+that make its point: and by this means, as it is easy to see, it would
+produce prisms similar to those of the same crystal as this other
+figure represents. The reason is that when broken in this fashion a
+whole layer separates easily from its neighbouring layer since each
+spheroid has to be detached only from the three spheroids of the next
+layer; of which three there is but one which touches it on its
+flattened surface, and the other two at the edges. And the reason why
+the surfaces separate sharp and polished is that if any spheroid of
+the neighbouring surface would come out by attaching itself to the
+surface which is being separated, it would be needful for it to detach
+itself from six other spheroids which hold it locked, and four of
+which press it by these flattened surfaces. Since then not only the
+angles of our crystal but also the manner in which it splits agree
+precisely with what is observed in the assemblage composed of such
+spheroids, there is great reason to believe that the particles are
+shaped and ranged in the same way.
+
+[Illustration: {Pyramid and section of spheroids}]
+
+There is even probability enough that the prisms of this crystal are
+produced by the breaking up of pyramids, since Mr. Bartholinus relates
+that he occasionally found some pieces of triangularly pyramidal
+figure. But when a mass is composed interiorly only of these little
+spheroids thus piled up, whatever form it may have exteriorly, it is
+certain, by the same reasoning which I have just explained, that if
+broken it would produce similar prisms. It remains to be seen whether
+there are other reasons which confirm our conjecture, and whether
+there are none which are repugnant to it.
+
+[Illustration: {paralleloid arrangement of spheroids with planes of
+potential cleavage}]
+
+It may be objected that this crystal, being so composed, might be
+capable of cleavage in yet two more fashions; one of which would be
+along planes parallel to the base of the pyramid, that is to say to
+the triangle ABC; the other would be parallel to a plane the trace of
+which is marked by the lines GH, HK, KL. To which I say that both the
+one and the other, though practicable, are more difficult than those
+which were parallel to any one of the three planes of the pyramid; and
+that therefore, when striking on the crystal in order to break it, it
+ought always to split rather along these three planes than along the
+two others. When one has a number of spheroids of the form above
+described, and ranges them in a pyramid, one sees why the two methods
+of division are more difficult. For in the case of that division which
+would be parallel to the base, each spheroid would be obliged to
+detach itself from three others which it touches upon their flattened
+surfaces, which hold more strongly than the contacts at the edges. And
+besides that, this division will not occur along entire layers,
+because each of the spheroids of a layer is scarcely held at all by
+the 6 of the same layer that surround it, since they only touch it at
+the edges; so that it adheres readily to the neighbouring layer, and
+the others to it, for the same reason; and this causes uneven
+surfaces. Also one sees by experiment that when grinding down the
+crystal on a rather rough stone, directly on the equilateral solid
+angle, one verily finds much facility in reducing it in this
+direction, but much difficulty afterwards in polishing the surface
+which has been flattened in this manner.
+
+As for the other method of division along the plane GHKL, it will be
+seen that each spheroid would have to detach itself from four of the
+neighbouring layer, two of which touch it on the flattened surfaces,
+and two at the edges. So that this division is likewise more difficult
+than that which is made parallel to one of the surfaces of the
+crystal; where, as we have said, each spheroid is detached from only
+three of the neighbouring layer: of which three there is one only
+which touches it on the flattened surface, and the other two at the
+edges only.
+
+However, that which has made me know that in the crystal there are
+layers in this last fashion, is that in a piece weighing half a pound
+which I possess, one sees that it is split along its length, as is the
+above-mentioned prism by the plane GHKL; as appears by colours of the
+Iris extending throughout this whole plane although the two pieces
+still hold together. All this proves then that the composition of the
+crystal is such as we have stated. To which I again add this
+experiment; that if one passes a knife scraping along any one of the
+natural surfaces, and downwards as it were from the equilateral obtuse
+angle, that is to say from the apex of the pyramid, one finds it quite
+hard; but by scraping in the opposite sense an incision is easily
+made. This follows manifestly from the situation of the small
+spheroids; over which, in the first manner, the knife glides; but in
+the other manner it seizes them from beneath almost as if they were
+the scales of a fish.
+
+I will not undertake to say anything touching the way in which so many
+corpuscles all equal and similar are generated, nor how they are set
+in such beautiful order; whether they are formed first and then
+assembled, or whether they arrange themselves thus in coming into
+being and as fast as they are produced, which seems to me more
+probable. To develop truths so recondite there would be needed a
+knowledge of nature much greater than that which we have. I will add
+only that these little spheroids could well contribute to form the
+spheroids of the waves of light, here above supposed, these as well as
+those being similarly situated, and with their axes parallel.
+
+
+_Calculations which have been supposed in this Chapter_.
+
+Mr. Bartholinus, in his treatise of this Crystal, puts at 101 degrees
+the obtuse angles of the faces, which I have stated to be 101 degrees
+52 minutes. He states that he measured these angles directly on the
+crystal, which is difficult to do with ultimate exactitude, because
+the edges such as CA, CB, in this figure, are generally worn, and not
+quite straight. For more certainty, therefore, I preferred to measure
+actually the obtuse angle by which the faces CBDA, CBVF, are inclined
+to one another, namely the angle OCN formed by drawing CN
+perpendicular to FV, and CO perpendicular to DA. This angle OCN I
+found to be 105 degrees; and its supplement CNP, to be 75 degrees, as
+it should be.
+
+[Illustration]
+
+To find from this the obtuse angle BCA, I imagined a sphere having its
+centre at C, and on its surface a spherical triangle, formed by the
+intersection of three planes which enclose the solid angle C. In this
+equilateral triangle, which is ABF in this other figure, I see that
+each of the angles should be 105 degrees, namely equal to the angle
+OCN; and that each of the sides should be of as many degrees as the
+angle ACB, or ACF, or BCF. Having then drawn the arc FQ perpendicular
+to the side AB, which it divides equally at Q, the triangle FQA has a
+right angle at Q, the angle A 105 degrees, and F half as much, namely
+52 degrees 30 minutes; whence the hypotenuse AF is found to be 101
+degrees 52 minutes. And this arc AF is the measure of the angle ACF in
+the figure of the crystal.
+
+[Illustration]
+
+In the same figure, if the plane CGHF cuts the crystal so that it
+divides the obtuse angles ACB, MHV, in the middle, it is stated, in
+Article 10, that the angle CFH is 70 degrees 57 minutes. This again is
+easily shown in the same spherical triangle ABF, in which it appears
+that the arc FQ is as many degrees as the angle GCF in the crystal,
+the supplement of which is the angle CFH. Now the arc FQ is found to
+be 109 degrees 3 minutes. Then its supplement, 70 degrees 57 minutes,
+is the angle CFH.
+
+It was stated, in Article 26, that the straight line CS, which in the
+preceding figure is CH, being the axis of the crystal, that is to say
+being equally inclined to the three sides CA, CB, CF, the angle GCH is
+45 degrees 20 minutes. This is also easily calculated by the same
+spherical triangle. For by drawing the other arc AD which cuts BF
+equally, and intersects FQ at S, this point will be the centre of the
+triangle. And it is easy to see that the arc SQ is the measure of the
+angle GCH in the figure which represents the crystal. Now in the
+triangle QAS, which is right-angled, one knows also the angle A, which
+is 52 degrees 30 minutes, and the side AQ 50 degrees 56 minutes;
+whence the side SQ is found to be 45 degrees 20 minutes.
+
+In Article 27 it was required to show that PMS being an ellipse the
+centre of which is C, and which touches the straight line MD at M so
+that the angle MCL which CM makes with CL, perpendicular on DM, is 6
+degrees 40 minutes, and its semi-minor axis CS making with CG (which
+is parallel to MD) an angle GCS of 45 degrees 20 minutes, it was
+required to show, I say, that, CM being 100,000 parts, PC the
+semi-major diameter of this ellipse is 105,032 parts, and CS, the
+semi-minor diameter, 93,410.
+
+Let CP and CS be prolonged and meet the tangent DM at D and Z; and
+from the point of contact M let MN and MO be drawn as perpendiculars
+to CP and CS. Now because the angles SCP, GCL, are right angles, the
+angle PCL will be equal to GCS which was 45 degrees 20 minutes. And
+deducting the angle LCM, which is 6 degrees 40 minutes, from LCP,
+which is 45 degrees 20 minutes, there remains MCP, 38 degrees 40
+minutes. Considering then CM as a radius of 100,000 parts, MN, the
+sine of 38 degrees 40 minutes, will be 62,479. And in the right-angled
+triangle MND, MN will be to ND as the radius of the Tables is to the
+tangent of 45 degrees 20 minutes (because the angle NMD is equal to
+DCL, or GCS); that is to say as 100,000 to 101,170: whence results ND
+63,210. But NC is 78,079 of the same parts, CM being 100,000, because
+NC is the sine of the complement of the angle MCP, which was 38
+degrees 40 minutes. Then the whole line DC is 141,289; and CP, which
+is a mean proportional between DC and CN, since MD touches the
+Ellipse, will be 105,032.
+
+[Illustration]
+
+Similarly, because the angle OMZ is equal to CDZ, or LCZ, which is 44
+degrees 40 minutes, being the complement of GCS, it follows that, as
+the radius of the Tables is to the tangent of 44 degrees 40 minutes,
+so will OM 78,079 be to OZ 77,176. But OC is 62,479 of these same
+parts of which CM is 100,000, because it is equal to MN, the sine of
+the angle MCP, which is 38 degrees 40 minutes. Then the whole line CZ
+is 139,655; and CS, which is a mean proportional between CZ and CO
+will be 93,410.
+
+At the same place it was stated that GC was found to be 98,779 parts.
+To prove this, let PE be drawn in the same figure parallel to DM, and
+meeting CM at E. In the right-angled triangle CLD the side CL is
+99,324 (CM being 100,000), because CL is the sine of the complement of
+the angle LCM, which is 6 degrees 40 minutes. And since the angle LCD
+is 45 degrees 20 minutes, being equal to GCS, the side LD is found to
+be 100,486: whence deducting ML 11,609 there will remain MD 88,877.
+Now as CD (which was 141,289) is to DM 88,877, so will CP 105,032 be
+to PE 66,070. But as the rectangle MEH (or rather the difference of
+the squares on CM and CE) is to the square on MC, so is the square on
+PE to the square on C_g_; then also as the difference of the squares
+on DC and CP to the square on CD, so also is the square on PE to the
+square on _g_C. But DP, CP, and PE are known; hence also one knows GC,
+which is 98,779.
+
+
+_Lemma which has been supposed_.
+
+If a spheroid is touched by a straight line, and also by two or more
+planes which are parallel to this line, though not parallel to one
+another, all the points of contact of the line, as well as of the
+planes, will be in one and the same ellipse made by a plane which
+passes through the centre of the spheroid.
+
+Let LED be the spheroid touched by the line BM at the point B, and
+also by the planes parallel to this line at the points O and A. It is
+required to demonstrate that the points B, O, and A are in one and the
+same Ellipse made in the spheroid by a plane which passes through its
+centre.
+
+[Illustration]
+
+Through the line BM, and through the points O and A, let there be
+drawn planes parallel to one another, which, in cutting the spheroid
+make the ellipses LBD, POP, QAQ; which will all be similar and
+similarly disposed, and will have their centres K, N, R, in one and
+the same diameter of the spheroid, which will also be the diameter of
+the ellipse made by the section of the plane that passes through the
+centre of the spheroid, and which cuts the planes of the three said
+Ellipses at right angles: for all this is manifest by proposition 15
+of the book of Conoids and Spheroids of Archimedes. Further, the two
+latter planes, which are drawn through the points O and A, will also,
+by cutting the planes which touch the spheroid in these same points,
+generate straight lines, as OH and AS, which will, as is easy to see,
+be parallel to BM; and all three, BM, OH, AS, will touch the Ellipses
+LBD, POP, QAQ in these points, B, O, A; since they are in the planes
+of these ellipses, and at the same time in the planes which touch the
+spheroid. If now from these points B, O, A, there are drawn the
+straight lines BK, ON, AR, through the centres of the same ellipses,
+and if through these centres there are drawn also the diameters LD,
+PP, QQ, parallel to the tangents BM, OH, AS; these will be conjugate
+to the aforesaid BK, ON, AR. And because the three ellipses are
+similar and similarly disposed, and have their diameters LD, PP, QQ
+parallel, it is certain that their conjugate diameters BK, ON, AR,
+will also be parallel. And the centres K, N, R being, as has been
+stated, in one and the same diameter of the spheroid, these parallels
+BK, ON, AR will necessarily be in one and the same plane, which passes
+through this diameter of the spheroid, and, in consequence, the points
+R, O, A are in one and the same ellipse made by the intersection of
+this plane. Which was to be proved. And it is manifest that the
+demonstration would be the same if, besides the points O, A, there had
+been others in which the spheroid had been touched by planes parallel
+to the straight line BM.
+
+
+
+
+CHAPTER VI
+
+ON THE FIGURES OF THE TRANSPARENT BODIES
+
+Which serve for Refraction and for Reflexion
+
+
+After having explained how the properties of reflexion and refraction
+follow from what we have supposed concerning the nature of light, and
+of opaque bodies, and of transparent media, I will here set forth a
+very easy and natural way of deducing, from the same principles, the
+true figures which serve, either by reflexion or by refraction, to
+collect or disperse the rays of light, as may be desired. For though I
+do not see yet that there are means of making use of these figures, so
+far as relates to Refraction, not only because of the difficulty of
+shaping the glasses of Telescopes with the requisite exactitude
+according to these figures, but also because there exists in
+refraction itself a property which hinders the perfect concurrence of
+the rays, as Mr. Newton has very well proved by experiment, I will yet
+not desist from relating the invention, since it offers itself, so to
+speak, of itself, and because it further confirms our Theory of
+refraction, by the agreement which here is found between the refracted
+ray and the reflected ray. Besides, it may occur that some one in the
+future will discover in it utilities which at present are not seen.
+
+[Illustration]
+
+To proceed then to these figures, let us suppose first that it is
+desired to find a surface CDE which shall reassemble at a point B rays
+coming from another point A; and that the summit of the surface shall
+be the given point D in the straight line AB. I say that, whether by
+reflexion or by refraction, it is only necessary to make this surface
+such that the path of the light from the point A to all points of the
+curved line CDE, and from these to the point of concurrence (as here
+the path along the straight lines AC, CB, along AL, LB, and along AD,
+DB), shall be everywhere traversed in equal times: by which principle
+the finding of these curves becomes very easy.
+
+[Illustration]
+
+So far as relates to the reflecting surface, since the sum of the
+lines AC, CB ought to be equal to that of AD, DB, it appears that DCE
+ought to be an ellipse; and for refraction, the ratio of the
+velocities of waves of light in the media A and B being supposed to be
+known, for example that of 3 to 2 (which is the same, as we have
+shown, as the ratio of the Sines in the refraction), it is only
+necessary to make DH equal to 3/2 of DB; and having after that
+described from the centre A some arc FC, cutting DB at F, then
+describe another from centre B with its semi-diameter BX equal to 2/3
+of FH; and the point of intersection of the two arcs will be one of
+the points required, through which the curve should pass. For this
+point, having been found in this fashion, it is easy forthwith to
+demonstrate that the time along AC, CB, will be equal to the time
+along AD, DB.
+
+For assuming that the line AD represents the time which the light
+takes to traverse this same distance AD in air, it is evident that DH,
+equal to 3/2 of DB, will represent the time of the light along DB in
+the medium, because it needs here more time in proportion as its speed
+is slower. Therefore the whole line AH will represent the time along
+AD, DB. Similarly the line AC or AF will represent the time along AC;
+and FH being by construction equal to 3/2 of CB, it will represent the
+time along CB in the medium; and in consequence the whole line AH will
+represent also the time along AC, CB. Whence it appears that the time
+along AC, CB, is equal to the time along AD, DB. And similarly it can
+be shown if L and K are other points in the curve CDE, that the times
+along AL, LB, and along AK, KB, are always represented by the line AH,
+and therefore equal to the said time along AD, DB.
+
+In order to show further that the surfaces, which these curves will
+generate by revolution, will direct all the rays which reach them from
+the point A in such wise that they tend towards B, let there be
+supposed a point K in the curve, farther from D than C is, but such
+that the straight line AK falls from outside upon the curve which
+serves for the refraction; and from the centre B let the arc KS be
+described, cutting BD at S, and the straight line CB at R; and from
+the centre A describe the arc DN meeting AK at N.
+
+Since the sums of the times along AK, KB, and along AC, CB are equal,
+if from the former sum one deducts the time along KB, and if from the
+other one deducts the time along RB, there will remain the time along
+AK as equal to the time along the two parts AC, CR. Consequently in
+the time that the light has come along AK it will also have come along
+AC and will in addition have made, in the medium from the centre C, a
+partial spherical wave, having a semi-diameter equal to CR. And this
+wave will necessarily touch the circumference KS at R, since CB cuts
+this circumference at right angles. Similarly, having taken any other
+point L in the curve, one can show that in the same time as the light
+passes along AL it will also have come along AL and in addition will
+have made a partial wave, from the centre L, which will touch the same
+circumference KS. And so with all other points of the curve CDE. Then
+at the moment that the light reaches K the arc KRS will be the
+termination of the movement, which has spread from A through DCK. And
+thus this same arc will constitute in the medium the propagation of
+the wave emanating from A; which wave may be represented by the arc
+DN, or by any other nearer the centre A. But all the pieces of the arc
+KRS are propagated successively along straight lines which are
+perpendicular to them, that is to say, which tend to the centre B (for
+that can be demonstrated in the same way as we have proved above that
+the pieces of spherical waves are propagated along the straight lines
+coming from their centre), and these progressions of the pieces of the
+waves constitute the rays themselves of light. It appears then that
+all these rays tend here towards the point B.
+
+One might also determine the point C, and all the others, in this
+curve which serves for the refraction, by dividing DA at G in such a
+way that DG is 2/3 of DA, and describing from the centre B any arc CX
+which cuts BD at N, and another from the centre A with its
+semi-diameter AF equal to 3/2 of GX; or rather, having described, as
+before, the arc CX, it is only necessary to make DF equal to 3/2 of
+DX, and from-the centre A to strike the arc FC; for these two
+constructions, as may be easily known, come back to the first one
+which was shown before. And it is manifest by the last method that
+this curve is the same that Mr. Des Cartes has given in his Geometry,
+and which he calls the first of his Ovals.
+
+It is only a part of this oval which serves for the refraction,
+namely, the part DK, ending at K, if AK is the tangent. As to the,
+other part, Des Cartes has remarked that it could serve for
+reflexions, if there were some material of a mirror of such a nature
+that by its means the force of the rays (or, as we should say, the
+velocity of the light, which he could not say, since he held that the
+movement of light was instantaneous) could be augmented in the
+proportion of 3 to 2. But we have shown that in our way of explaining
+reflexion, such a thing could not arise from the matter of the mirror,
+and it is entirely impossible.
+
+[Illustration]
+
+[Illustration]
+
+From what has been demonstrated about this oval, it will be easy to
+find the figure which serves to collect to a point incident parallel
+rays. For by supposing just the same construction, but the point A
+infinitely distant, giving parallel rays, our oval becomes a true
+Ellipse, the construction of which differs in no way from that of the
+oval, except that FC, which previously was an arc of a circle, is here
+a straight line, perpendicular to DB. For the wave of light DN, being
+likewise represented by a straight line, it will be seen that all the
+points of this wave, travelling as far as the surface KD along lines
+parallel to DB, will advance subsequently towards the point B, and
+will arrive there at the same time. As for the Ellipse which served
+for reflexion, it is evident that it will here become a parabola,
+since its focus A may be regarded as infinitely distant from the
+other, B, which is here the focus of the parabola, towards which all
+the reflexions of rays parallel to AB tend. And the demonstration of
+these effects is just the same as the preceding.
+
+But that this curved line CDE which serves for refraction is an
+Ellipse, and is such that its major diameter is to the distance
+between its foci as 3 to 2, which is the proportion of the refraction,
+can be easily found by the calculus of Algebra. For DB, which is
+given, being called _a_; its undetermined perpendicular DT being
+called _x_; and TC _y_; FB will be _a - y_; CB will be sqrt(_xx + aa
+-2ay + yy_). But the nature of the curve is such that 2/3 of TC
+together with CB is equal to DB, as was stated in the last
+construction: then the equation will be between _(2/3)y + sqrt(xx + aa
+- 2ay + yy)_ and _a_; which being reduced, gives _(6/5)ay - yy_ equal
+to _(9/5)xx_; that is to say that having made DO equal to 6/5 of DB,
+the rectangle DFO is equal to 9/5 of the square on FC. Whence it is
+seen that DC is an ellipse, of which the axis DO is to the parameter
+as 9 to 5; and therefore the square on DO is to the square of the
+distance between the foci as 9 to 9 - 5, that is to say 4; and finally
+the line DO will be to this distance as 3 to 2.
+
+[Illustration]
+
+Again, if one supposes the point B to be infinitely distant, in lieu
+of our first oval we shall find that CDE is a true Hyperbola; which
+will make those rays become parallel which come from the point A. And
+in consequence also those which are parallel within the transparent
+body will be collected outside at the point A. Now it must be remarked
+that CX and KS become straight lines perpendicular to BA, because they
+represent arcs of circles the centre of which is infinitely distant.
+And the intersection of the perpendicular CX with the arc FC will give
+the point C, one of those through which the curve ought to pass. And
+this operates so that all the parts of the wave of light DN, coming to
+meet the surface KDE, will advance thence along parallels to KS and
+will arrive at this straight line at the same time; of which the proof
+is again the same as that which served for the first oval. Besides one
+finds by a calculation as easy as the preceding one, that CDE is here
+a hyperbola of which the axis DO is 4/5 of AD, and the parameter
+equal to AD. Whence it is easily proved that DO is to the distance
+between the foci as 3 to 2.
+
+[Illustration]
+
+These are the two cases in which Conic sections serve for refraction,
+and are the same which are explained, in his _Dioptrique_, by Des
+Cartes, who first found out the use of these lines in relation to
+refraction, as also that of the Ovals the first of which we have
+already set forth. The second oval is that which serves for rays that
+tend to a given point; in which oval, if the apex of the surface which
+receives the rays is D, it will happen that the other apex will be
+situated between B and A, or beyond A, according as the ratio of AD to
+DB is given of greater or lesser value. And in this latter case it is
+the same as that which Des Cartes calls his 3rd oval.
+
+Now the finding and construction of this second oval is the same as
+that of the first, and the demonstration of its effect likewise. But
+it is worthy of remark that in one case this oval becomes a perfect
+circle, namely when the ratio of AD to DB is the same as the ratio of
+the refractions, here as 3 to 2, as I observed a long time ago. The
+4th oval, serving only for impossible reflexions, there is no need to
+set it forth.
+
+[Illustration]
+
+As for the manner in which Mr. Des Cartes discovered these lines,
+since he has given no explanation of it, nor any one else since that I
+know of, I will say here, in passing, what it seems to me it must have
+been. Let it be proposed to find the surface generated by the
+revolution of the curve KDE, which, receiving the incident rays coming
+to it from the point A, shall deviate them toward the point B. Then
+considering this other curve as already known, and that its apex D is
+in the straight line AB, let us divide it up into an infinitude of
+small pieces by the points G, C, F; and having drawn from each of
+these points, straight lines towards A to represent the incident rays,
+and other straight lines towards B, let there also be described with
+centre A the arcs GL, CM, FN, DO, cutting the rays that come from A at
+L, M, N, O; and from the points K, G, C, F, let there be described
+the arcs KQ, GR, CS, FT cutting the rays towards B at Q, R, S, T; and
+let us suppose that the straight line HKZ cuts the curve at K at
+right-angles.
+
+[Illustration]
+
+Then AK being an incident ray, and KB its refraction within the
+medium, it needs must be, according to the law of refraction which was
+known to Mr. Des Cartes, that the sine of the angle ZKA should be to
+the sine of the angle HKB as 3 to 2, supposing that this is the
+proportion of the refraction of glass; or rather, that the sine of the
+angle KGL should have this same ratio to the sine of the angle GKQ,
+considering KG, GL, KQ as straight lines because of their smallness.
+But these sines are the lines KL and GQ, if GK is taken as the radius
+of the circle. Then LK ought to be to GQ as 3 to 2; and in the same
+ratio MG to CR, NC to FS, OF to DT. Then also the sum of all the
+antecedents to all the consequents would be as 3 to 2. Now by
+prolonging the arc DO until it meets AK at X, KX is the sum of the
+antecedents. And by prolonging the arc KQ till it meets AD at Y, the
+sum of the consequents is DY. Then KX ought to be to DY as 3 to 2.
+Whence it would appear that the curve KDE was of such a nature that
+having drawn from some point which had been assumed, such as K, the
+straight lines KA, KB, the excess by which AK surpasses AD should be
+to the excess of DB over KB, as 3 to 2. For it can similarly be
+demonstrated, by taking any other point in the curve, such as G, that
+the excess of AG over AD, namely VG, is to the excess of BD over DG,
+namely DP, in this same ratio of 3 to 2. And following this principle
+Mr. Des Cartes constructed these curves in his _Geometric_; and he
+easily recognized that in the case of parallel rays, these curves
+became Hyperbolas and Ellipses.
+
+Let us now return to our method and let us see how it leads without
+difficulty to the finding of the curves which one side of the glass
+requires when the other side is of a given figure; a figure not only
+plane or spherical, or made by one of the conic sections (which is the
+restriction with which Des Cartes proposed this problem, leaving the
+solution to those who should come after him) but generally any figure
+whatever: that is to say, one made by the revolution of any given
+curved line to which one must merely know how to draw straight lines
+as tangents.
+
+Let the given figure be that made by the revolution of some curve such
+as AK about the axis AV, and that this side of the glass receives rays
+coming from the point L. Furthermore, let the thickness AB of the
+middle of the glass be given, and the point F at which one desires the
+rays to be all perfectly reunited, whatever be the first refraction
+occurring at the surface AK.
+
+I say that for this the sole requirement is that the outline BDK which
+constitutes the other surface shall be such that the path of the
+light from the point L to the surface AK, and from thence to the
+surface BDK, and from thence to the point F, shall be traversed
+everywhere in equal times, and in each case in a time equal to that
+which the light employs, to pass along the straight line LF of which
+the part AB is within the glass.
+
+[Illustration]
+
+Let LG be a ray falling on the arc AK. Its refraction GV will be given
+by means of the tangent which will be drawn at the point G. Now in GV
+the point D must be found such that FD together with 3/2 of DG and the
+straight line GL, may be equal to FB together with 3/2 of BA and the
+straight line AL; which, as is clear, make up a given length. Or
+rather, by deducting from each the length of LG, which is also given,
+it will merely be needful to adjust FD up to the straight line VG in
+such a way that FD together with 3/2 of DG is equal to a given
+straight line, which is a quite easy plane problem: and the point D
+will be one of those through which the curve BDK ought to pass. And
+similarly, having drawn another ray LM, and found its refraction MO,
+the point N will be found in this line, and so on as many times as one
+desires.
+
+To demonstrate the effect of the curve, let there be described about
+the centre L the circular arc AH, cutting LG at H; and about the
+centre F the arc BP; and in AB let AS be taken equal to 2/3 of HG; and
+SE equal to GD. Then considering AH as a wave of light emanating from
+the point L, it is certain that during the time in which its piece H
+arrives at G the piece A will have advanced within the transparent
+body only along AS; for I suppose, as above, the proportion of the
+refraction to be as 3 to 2. Now we know that the piece of wave which
+is incident on G, advances thence along the line GD, since GV is the
+refraction of the ray LG. Then during the time that this piece of wave
+has taken from G to D, the other piece which was at S has reached E,
+since GD, SE are equal. But while the latter will advance from E to B,
+the piece of wave which was at D will have spread into the air its
+partial wave, the semi-diameter of which, DC (supposing this wave to
+cut the line DF at C), will be 3/2 of EB, since the velocity of light
+outside the medium is to that inside as 3 to 2. Now it is easy to show
+that this wave will touch the arc BP at this point C. For since, by
+construction, FD + 3/2 DG + GL are equal to FB + 3/2 BA + AL; on
+deducting the equals LH, LA, there will remain FD + 3/2 DG + GH equal
+to FB + 3/2 BA. And, again, deducting from one side GH, and from the
+other side 3/2 of AS, which are equal, there will remain FD with 3/2
+DG equal to FB with 3/2 of BS. But 3/2 of DG are equal to 3/2 of ES;
+then FD is equal to FB with 3/2 of BE. But DC was equal to 3/2 of EB;
+then deducting these equal lengths from one side and from the other,
+there will remain CF equal to FB. And thus it appears that the wave,
+the semi-diameter of which is DC, touches the arc BP at the moment
+when the light coming from the point L has arrived at B along the line
+LB. It can be demonstrated similarly that at this same moment the
+light that has come along any other ray, such as LM, MN, will have
+propagated the movement which is terminated at the arc BP. Whence it
+follows, as has been often said, that the propagation of the wave AH,
+after it has passed through the thickness of the glass, will be the
+spherical wave BP, all the pieces of which ought to advance along
+straight lines, which are the rays of light, to the centre F. Which
+was to be proved. Similarly these curved lines can be found in all the
+cases which can be proposed, as will be sufficiently shown by one or
+two examples which I will add.
+
+Let there be given the surface of the glass AK, made by the revolution
+about the axis BA of the line AK, which may be straight or curved. Let
+there be also given in the axis the point L and the thickness BA of
+the glass; and let it be required to find the other surface KDB, which
+receiving rays that are parallel to AB will direct them in such wise
+that after being again refracted at the given surface AK they will all
+be reassembled at the point L.
+
+[Illustration]
+
+From the point L let there be drawn to some point of the given line
+AK the straight line LG, which, being considered as a ray of light,
+its refraction GD will then be found. And this line being then
+prolonged at one side or the other will meet the straight line BL, as
+here at V. Let there then be erected on AB the perpendicular BC, which
+will represent a wave of light coming from the infinitely distant
+point F, since we have supposed the rays to be parallel. Then all the
+parts of this wave BC must arrive at the same time at the point L; or
+rather all the parts of a wave emanating from the point L must arrive
+at the same time at the straight line BC. And for that, it is
+necessary to find in the line VGD the point D such that having drawn
+DC parallel to AB, the sum of CD, plus 3/2 of DG, plus GL may be equal
+to 3/2 of AB, plus AL: or rather, on deducting from both sides GL,
+which is given, CD plus 3/2 of DG must be equal to a given length;
+which is a still easier problem than the preceding construction. The
+point D thus found will be one of those through which the curve ought
+to pass; and the proof will be the same as before. And by this it will
+be proved that the waves which come from the point L, after having
+passed through the glass KAKB, will take the form of straight lines,
+as BC; which is the same thing as saying that the rays will become
+parallel. Whence it follows reciprocally that parallel rays falling on
+the surface KDB will be reassembled at the point L.
+
+[Illustration]
+
+Again, let there be given the surface AK, of any desired form,
+generated by revolution about the axis AB, and let the thickness of
+the glass at the middle be AB. Also let the point L be given in the
+axis behind the glass; and let it be supposed that the rays which fall
+on the surface AK tend to this point, and that it is required to find
+the surface BD, which on their emergence from the glass turns them as
+if they came from the point F in front of the glass.
+
+Having taken any point G in the line AK, and drawing the straight line
+IGL, its part GI will represent one of the incident rays, the
+refraction of which, GV, will then be found: and it is in this line
+that we must find the point D, one of those through which the curve DG
+ought to pass. Let us suppose that it has been found: and about L as
+centre let there be described GT, the arc of a circle cutting the
+straight line AB at T, in case the distance LG is greater than LA; for
+otherwise the arc AH must be described about the same centre, cutting
+the straight line LG at H. This arc GT (or AH, in the other case) will
+represent an incident wave of light, the rays of which tend towards
+L. Similarly, about the centre F let there be described the circular
+arc DQ, which will represent a wave emanating from the point F.
+
+Then the wave TG, after having passed through the glass, must form the
+wave QD; and for this I observe that the time taken by the light along
+GD in the glass must be equal to that taken along the three, TA, AB,
+and BQ, of which AB alone is within the glass. Or rather, having taken
+AS equal to 2/3 of AT, I observe that 3/2 of GD ought to be equal to
+3/2 of SB, plus BQ; and, deducting both of them from FD or FQ, that FD
+less 3/2 of GD ought to be equal to FB less 3/2 of SB. And this last
+difference is a given length: and all that is required is to draw the
+straight line FD from the given point F to meet VG so that it may be
+thus. Which is a problem quite similar to that which served for the
+first of these constructions, where FD plus 3/2 of GD had to be equal
+to a given length.
+
+In the demonstration it is to be observed that, since the arc BC falls
+within the glass, there must be conceived an arc RX, concentric with
+it and on the other side of QD. Then after it shall have been shown
+that the piece G of the wave GT arrives at D at the same time that the
+piece T arrives at Q, which is easily deduced from the construction,
+it will be evident as a consequence that the partial wave generated at
+the point D will touch the arc RX at the moment when the piece Q shall
+have come to R, and that thus this arc will at the same moment be the
+termination of the movement that comes from the wave TG; whence all
+the rest may be concluded.
+
+Having shown the method of finding these curved lines which serve for
+the perfect concurrence of the rays, there remains to be explained a
+notable thing touching the uncoordinated refraction of spherical,
+plane, and other surfaces: an effect which if ignored might cause some
+doubt concerning what we have several times said, that rays of light
+are straight lines which intersect at right angles the waves which
+travel along them.
+
+[Illustration]
+
+For in the case of rays which, for example, fall parallel upon a
+spherical surface AFE, intersecting one another, after refraction, at
+different points, as this figure represents; what can the waves of
+light be, in this transparent body, which are cut at right angles by
+the converging rays? For they can not be spherical. And what will
+these waves become after the said rays begin to intersect one another?
+It will be seen in the solution of this difficulty that something very
+remarkable comes to pass herein, and that the waves do not cease to
+persist though they do not continue entire, as when they cross the
+glasses designed according to the construction we have seen.
+
+According to what has been shown above, the straight line AD, which
+has been drawn at the summit of the sphere, at right angles to the
+axis parallel to which the rays come, represents the wave of light;
+and in the time taken by its piece D to reach the spherical surface
+AGE at E, its other parts will have met the same surface at F, G, H,
+etc., and will have also formed spherical partial waves of which these
+points are the centres. And the surface EK which all those waves will
+touch, will be the continuation of the wave AD in the sphere at the
+moment when the piece D has reached E. Now the line EK is not an arc
+of a circle, but is a curved line formed as the evolute of another
+curve ENC, which touches all the rays HL, GM, FO, etc., that are the
+refractions of the parallel rays, if we imagine laid over the
+convexity ENC a thread which in unwinding describes at its end E the
+said curve EK. For, supposing that this curve has been thus described,
+we will show that the said waves formed from the centres F, G, H,
+etc., will all touch it.
+
+It is certain that the curve EK and all the others described by the
+evolution of the curve ENC, with different lengths of thread, will cut
+all the rays HL, GM, FO, etc., at right angles, and in such wise that
+the parts of them intercepted between two such curves will all be
+equal; for this follows from what has been demonstrated in our
+treatise _de Motu Pendulorum_. Now imagining the incident rays as
+being infinitely near to one another, if we consider two of them, as
+RG, TF, and draw GQ perpendicular to RG, and if we suppose the curve
+FS which intersects GM at P to have been described by evolution from
+the curve NC, beginning at F, as far as which the thread is supposed
+to extend, we may assume the small piece FP as a straight line
+perpendicular to the ray GM, and similarly the arc GF as a straight
+line. But GM being the refraction of the ray RG, and FP being
+perpendicular to it, QF must be to GP as 3 to 2, that is to say in the
+proportion of the refraction; as was shown above in explaining the
+discovery of Des Cartes. And the same thing occurs in all the small
+arcs GH, HA, etc., namely that in the quadrilaterals which enclose
+them the side parallel to the axis is to the opposite side as 3 to 2.
+Then also as 3 to 2 will the sum of the one set be to the sum of the
+other; that is to say, TF to AS, and DE to AK, and BE to SK or DV,
+supposing V to be the intersection of the curve EK and the ray FO.
+But, making FB perpendicular to DE, the ratio of 3 to 2 is also that
+of BE to the semi-diameter of the spherical wave which emanated from
+the point F while the light outside the transparent body traversed the
+space BE. Then it appears that this wave will intersect the ray FM at
+the same point V where it is intersected at right angles by the curve
+EK, and consequently that the wave will touch this curve. In the same
+way it can be proved that the same will apply to all the other waves
+above mentioned, originating at the points G, H, etc.; to wit, that
+they will touch the curve EK at the moment when the piece D of the
+wave ED shall have reached E.
+
+Now to say what these waves become after the rays have begun to cross
+one another: it is that from thence they fold back and are composed of
+two contiguous parts, one being a curve formed as evolute of the curve
+ENC in one sense, and the other as evolute of the same curve in the
+opposite sense. Thus the wave KE, while advancing toward the meeting
+place becomes _abc_, whereof the part _ab_ is made by the evolute
+_b_C, a portion of the curve ENC, while the end C remains attached;
+and the part _bc_ by the evolute of the portion _b_E while the end E
+remains attached. Consequently the same wave becomes _def_, then
+_ghk_, and finally CY, from whence it subsequently spreads without any
+fold, but always along curved lines which are evolutes of the curve
+ENC, increased by some straight line at the end C.
+
+There is even, in this curve, a part EN which is straight, N being the
+point where the perpendicular from the centre X of the sphere falls
+upon the refraction of the ray DE, which I now suppose to touch the
+sphere. The folding of the waves of light begins from the point N up
+to the end of the curve C, which point is formed by taking AC to CX in
+the proportion of the refraction, as here 3 to 2.
+
+As many other points as may be desired in the curve NC are found by a
+Theorem which Mr. Barrow has demonstrated in section 12 of his
+_Lectiones Opticae_, though for another purpose. And it is to be noted
+that a straight line equal in length to this curve can be given. For
+since it together with the line NE is equal to the line CK, which is
+known, since DE is to AK in the proportion of the refraction, it
+appears that by deducting EN from CK the remainder will be equal to
+the curve NC.
+
+Similarly the waves that are folded back in reflexion by a concave
+spherical mirror can be found. Let ABC be the section, through the
+axis, of a hollow hemisphere, the centre of which is D, its axis being
+DB, parallel to which I suppose the rays of light to come. All the
+reflexions of those rays which fall upon the quarter-circle AB will
+touch a curved line AFE, of which line the end E is at the focus of
+the hemisphere, that is to say, at the point which divides the
+semi-diameter BD into two equal parts. The points through which this
+curve ought to pass are found by taking, beyond A, some arc AO, and
+making the arc OP double the length of it; then dividing the chord OP
+at F in such wise that the part FP is three times the part FO; for
+then F is one of the required points.
+
+[Illustration]
+
+And as the parallel rays are merely perpendiculars to the waves which
+fall on the concave surface, which waves are parallel to AD, it will
+be found that as they come successively to encounter the surface AB,
+they form on reflexion folded waves composed of two curves which
+originate from two opposite evolutions of the parts of the curve AFE.
+So, taking AD as an incident wave, when the part AG shall have met the
+surface AI, that is to say when the piece G shall have reached I, it
+will be the curves HF, FI, generated as evolutes of the curves FA, FE,
+both beginning at F, which together constitute the propagation of the
+part AG. And a little afterwards, when the part AK has met the surface
+AM, the piece K having come to M, then the curves LN, NM, will
+together constitute the propagation of that part. And thus this folded
+wave will continue to advance until the point N has reached the focus
+E. The curve AFE can be seen in smoke, or in flying dust, when a
+concave mirror is held opposite the sun. And it should be known that
+it is none other than that curve which is described by the point E on
+the circumference of the circle EB, when that circle is made to roll
+within another whose semi-diameter is ED and whose centre is D. So
+that it is a kind of Cycloid, of which, however, the points can be
+found geometrically.
+
+Its length is exactly equal to 3/4 of the diameter of the sphere, as
+can be found and demonstrated by means of these waves, nearly in the
+same way as the mensuration of the preceding curve; though it may also
+be demonstrated in other ways, which I omit as outside the subject.
+The area AOBEFA, comprised between the arc of the quarter-circle, the
+straight line BE, and the curve EFA, is equal to the fourth part of
+the quadrant DAB.
+
+
+
+
+
+INDEX
+
+Archimedes, 104.
+
+Atmospheric refraction, 45.
+
+Barrow, Isaac, 126.
+
+Bartholinus, Erasmus, 53, 54, 57, 60, 97, 99.
+
+Boyle, Hon. Robert, 11.
+
+Cassini, Jacques, iii.
+
+Caustic Curves, 123.
+
+Crystals, see Iceland Crystal, Rock Crystal.
+
+Crystals, configuration of, 95.
+
+Descartes, RĂ©nĂª, 3, 5, 7, 14, 22, 42, 43, 109, 113.
+
+Double Refraction, discovery of, 54, 81, 93.
+
+Elasticity, 12, 14.
+
+Ether, the, or Ethereal matter, 11, 14, 16, 28.
+
+Extraordinary refraction, 55, 56.
+
+Fermat, principle of, 42.
+
+Figures of transparent bodies, 105.
+
+Hooke, Robert, 20.
+
+Iceland Crystal, 2, 52 sqq.
+
+Iceland Crystal, Cutting and Polishing of, 91, 92, 98.
+
+Leibnitz, G.W., vi.
+
+Light, nature of, 3.
+
+Light, velocity of, 4, 15.
+
+Molecular texture of bodies, 27, 95.
+
+Newton, Sir Isaac, vi, 106.
+
+Opacity, 34.
+
+Ovals, Cartesian, 107, 113.
+
+Pardies, Rev. Father, 20.
+
+Rays, definition of, 38, 49.
+
+Reflexion, 22.
+
+Refraction, 28, 34.
+
+Rock Crystal, 54, 57, 62, 95.
+
+Römer, Olaf, v, 7.
+
+Roughness of surfaces, 27.
+
+Sines, law of, 1, 35, 38, 43.
+
+Spheres, elasticity of, 15.
+
+Spheroidal waves in crystals, 63.
+
+Spheroids, lemma about, 103.
+
+Sound, speed of, 7, 10, 12.
+
+Telescopes, lenses for, 62, 105.
+
+Torricelli's experiment, 12, 30.
+
+Transparency, explanation of, 28, 31, 32.
+
+Waves, no regular succession of, 17.
+
+Waves, principle of wave envelopes, 19, 24.
+
+Waves, principle of elementary wave fronts, 19.
+
+Waves, propagation of light as, 16, 63.
+
+*** END OF THE PROJECT GUTENBERG EBOOK 14725 ***
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+<div>*** START OF THE PROJECT GUTENBERG EBOOK 14725 ***</div>
+<h1>The Project Gutenberg eBook, Treatise on Light, by Christiaan Huygens,
+Translated by Silvanus P. Thompson</h1>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+<hr class="full" />
+<p>&nbsp;</p>
+
+<h1><a name="Page_iii" id="Page_iii" /><b>TREATISE ON LIGHT</b></h1>
+
+
+<p class="center">In which are explained<br />
+The causes of that which occurs<br />
+<b>In REFLEXION, &amp; in REFRACTION</b></p>
+
+<p class="center">And particularly<br />
+<b>In the strange REFRACTION</b><br />
+<b>OF ICELAND CRYSTAL</b></p>
+
+
+<h3>By</h3>
+
+<h2><b>CHRISTIAAN HUYGENS</b></h2>
+
+
+<p class="center">Rendered into English</p>
+
+<p class="center">By</p>
+
+<p class="center"><b>SILVANUS P. THOMPSON</b></p>
+
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+<h6>University of Chicago Press</h6>
+<p><a name="Page_iv" id="Page_iv" /></p>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+
+
+<div class="pagenum">[Pg v]<a name="Page_v" id="Page_v" /></div>
+<div class="figcenter" style="width: 600px;">
+<img src="images/prefhead.png" width="600" height="150" alt="" title="" />
+</div>
+<h2>PREFACE</h2>
+
+
+<div style="width: 147px; float: left; margin-right: .2em;">
+<img src="images/pref.png" width="147" height="150" alt="I" title="I" />
+</div><p> wrote this Treatise during my sojourn in France twelve years ago,
+and I communicated it in the year 1678 to the learned persons who then
+composed the Royal Academy of Science, to the membership of which the
+King had done me the honour of calling, me. Several of that body who
+are still alive will remember having been present when I read it, and
+above the rest those amongst them who applied themselves particularly
+to the study of Mathematics; of whom I cannot cite more than the
+celebrated gentlemen Cassini, R&ouml;mer, and De la Hire. And, although I
+have since corrected and changed some parts, the copies which I had
+made of it at that time may serve for proof that I have yet added
+nothing to it save some conjectures touching the formation of Iceland
+Crystal, and a novel observation on the refraction of Rock Crystal. I
+have desired to relate these particulars to make known how long I have
+meditated the things which now I publish, and not for the purpose of
+detracting from the merit of those who, without having seen anything
+that I have written, may be found to have treated <span class="pagenum">[Pg vi]</span><a name="Page_vi" id="Page_vi" />of like matters: as
+has in fact occurred to two eminent Geometricians, Messieurs Newton
+and Leibnitz, with respect to the Problem of the figure of glasses for
+collecting rays when one of the surfaces is given.</p>
+
+<p>One may ask why I have so long delayed to bring this work to the
+light. The reason is that I wrote it rather carelessly in the Language
+in which it appears, with the intention of translating it into Latin,
+so doing in order to obtain greater attention to the thing. After
+which I proposed to myself to give it out along with another Treatise
+on Dioptrics, in which I explain the effects of Telescopes and those
+things which belong more to that Science. But the pleasure of novelty
+being past, I have put off from time to time the execution of this
+design, and I know not when I shall ever come to an end if it, being
+often turned aside either by business or by some new study.
+Considering which I have finally judged that it was better worth while
+to publish this writing, such as it is, than to let it run the risk,
+by waiting longer, of remaining lost.</p>
+
+<p>There will be seen in it demonstrations of those kinds which do not
+produce as great a certitude as those of Geometry, and which even
+differ much therefrom, since whereas the Geometers prove their
+Propositions by fixed and incontestable Principles, here the
+Principles are verified by the conclusions to be drawn from them; the
+nature of these things not allowing of this being done otherwise.</p>
+
+<p>It is always possible to attain thereby to a degree of probability
+which very often is scarcely less than complete proof. To wit, when
+things which have been demonstrated by the Principles that have been
+assumed correspond perfectly to the phenomena which experiment has
+brought under observation; especially when there are a great number of
+<span class="pagenum">[Pg vii]</span><a name="Page_vii" id="Page_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 fact corresponds to our prevision.
+But if all these proofs of probability are met with in that which I
+propose to discuss, as it seems to me they are, this ought to be a
+very strong confirmation of the success of my inquiry; and it must be
+ill if the facts are not pretty much as I represent them. I would
+believe then that those who love to know the Causes of things and who
+are able to admire the marvels of Light, will find some satisfaction
+in these various speculations regarding it, and in the new explanation
+of its famous property which is the main foundation of the
+construction of our eyes and of those great inventions which extend so
+vastly the use of them.</p>
+
+<p>I hope also that there will be some who by following these beginnings
+will penetrate much further into this question than I have been able
+to do, since the subject must be far from being exhausted. This
+appears from the passages which I have indicated where I leave certain
+difficulties without having resolved them, and still more from matters
+which I have not touched at all, such as Luminous Bodies of several
+sorts, and all that concerns Colours; in which no one until now can
+boast of having succeeded. Finally, there remains much more to be
+investigated touching the nature of Light which I do not pretend to
+have disclosed, and I shall owe much in return to him who shall be
+able to supplement that which is here lacking to me in knowledge. The
+Hague. The 8 January 1690.<span class="pagenum">[Pg viii]</span><a name="Page_viii" id="Page_viii" /></p>
+
+
+<div class="pagenum">[Pg ix]<a name="Page_ix" id="Page_ix" /></div>
+<div class="figcenter" style="width: 600px;">
+<img src="images/tranhead.png" width="600" height="151" alt="." title="" />
+</div>
+<h2><a name="NOTE_BY_THE_TRANSLATOR" id="NOTE_BY_THE_TRANSLATOR" />NOTE BY THE TRANSLATOR</h2>
+
+
+<div style="width: 150px; float: left; margin-right: .2em;">
+<img src="images/trans.png" width="150" height="150" alt="C" title="C" />
+</div><p>onsidering the great influence which this Treatise has exercised in
+the development of the Science of Optics, it seems strange that two
+centuries should have passed before an English edition of the work
+appeared. Perhaps the circumstance is due to the mistaken zeal with
+which formerly everything that conflicted with the cherished ideas of
+Newton was denounced by his followers. The Treatise on Light of
+Huygens has, however, withstood the test of time: and even now the
+exquisite skill with which he applied his conception of the
+propagation of waves of light to unravel the intricacies of the
+phenomena of the double refraction of crystals, and of the refraction
+of the atmosphere, will excite the admiration of the student of
+Optics. It is true that his wave theory was far from the complete
+doctrine as subsequently developed by Thomas Young and Augustin
+Fresnel, and belonged rather to geometrical than to physical Optics.
+If Huygens had no conception of transverse vibrations, of the
+principle of interference, or of the existence of the ordered sequence
+of waves in trains, he nevertheless attained to a remarkably clear
+understanding of the prin<span class="pagenum">[Pg x]</span><a name="Page_x" id="Page_x" />ciples of wave-propagation; and his
+exposition of the subject marks an epoch in the treatment of Optical
+problems. It has been needful in preparing this translation to
+exercise care lest one should import into the author's text ideas of
+subsequent date, by using words that have come to imply modern
+conceptions. Hence the adoption of as literal a rendering as possible.
+A few of the author's terms need explanation. He uses the word
+&quot;refraction,&quot; for example, both for the phenomenon or process usually
+so denoted, and for the result of that process: thus the refracted ray
+he habitually terms &quot;the refraction&quot; of the incident ray. When a
+wave-front, or, as he terms it, a &quot;wave,&quot; has passed from some initial
+position to a subsequent one, he terms the wave-front in its
+subsequent position &quot;the continuation&quot; of the wave. He also speaks of
+the envelope of a set of elementary waves, formed by coalescence of
+those elementary wave-fronts, as &quot;the termination&quot; of the wave; and
+the elementary wave-fronts he terms &quot;particular&quot; waves. Owing to the
+circumstance that the French word <i>rayon</i> possesses the double
+signification of ray of light and radius of a circle, he avoids its
+use in the latter sense and speaks always of the semi-diameter, not of
+the radius. His speculations as to the ether, his suggestive views of
+the structure of crystalline bodies, and his explanation of opacity,
+slight as they are, will possibly surprise the reader by their seeming
+modernness. And none can read his investigation of the phenomena found
+in Iceland spar without marvelling at his insight and sagacity.</p>
+
+<div style="margin-left: 80%;"><p>S.P.T.</p>
+
+<p><i>June</i>, 1912.</p></div>
+
+
+
+<hr style="width: 65%;" />
+<div class="pagenum">[Pg xi]<a name="Page_xi" id="Page_xi" /></div>
+
+<h2><a name="TABLE_OF_MATTERS" id="TABLE_OF_MATTERS" />TABLE OF MATTERS</h2>
+
+<h3><i>Contained in this Treatise</i></h3>
+
+
+<table border="0" cellpadding="4" cellspacing="0" summary="" width="500">
+<tr><td><a href="#CHAPTER_I"><b>CHAP. I. On Rays Propagated in Straight Lines.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>That Light is produced by a certain movement.</i></td><td align='left'><a href="#Page_3">p.&nbsp;3</a></td></tr>
+<tr><td align='left'><i>That no substance passes from the luminous object to the eyes.</i></td><td align='left'><a href="#Page_3">p.&nbsp;3</a></td></tr>
+<tr><td align='left'><i>That Light spreads spherically, almost as Sound does.</i></td><td align='left'><a href="#Page_4">p.&nbsp;4</a></td></tr>
+<tr><td align='left'><i>Whether Light takes time to spread.</i></td><td align='left'><a href="#Page_4">p.&nbsp;4</a></td></tr>
+<tr><td align='left'><i>Experience seeming to prove that it passes instantaneously.</i></td><td align='left'><a href="#Page_5">p.&nbsp;5</a></td></tr>
+<tr><td align='left'><i>Experience proving that it takes time.</i></td><td align='left'><a href="#Page_8">p.&nbsp;8</a></td></tr>
+<tr><td align='left'><i>How much its speed is greater than that of Sound.</i></td><td align='left'><a href="#Page_10">p.&nbsp;10</a></td></tr>
+<tr><td align='left'><i>In what the emission of Light differs from that of Sound.</i></td><td align='left'><a href="#Page_10">p.&nbsp;10</a></td></tr>
+<tr><td align='left'><i>That it is not the same medium which serves for Light and Sound.</i></td><td align='left'><a href="#Page_11">p.&nbsp;11</a></td></tr>
+<tr><td align='left'><i>How Sound is propagated.</i></td><td align='left'><a href="#Page_12">p.&nbsp;12</a></td></tr>
+<tr><td align='left'><i>How Light is propagated.</i></td><td align='left'><a href="#Page_14">p.&nbsp;14</a></td></tr>
+<tr><td align='left'><i>Detailed Remarks on the propagation of Light.</i></td><td align='left'><a href="#Page_15">p.&nbsp;15</a></td></tr>
+<tr><td align='left'><i>Why Rays are propagated only in straight lines.</i></td><td align='left'><a href="#Page_20">p.&nbsp;20</a></td></tr>
+<tr><td align='left'><i>How Light coming in different directions can cross itself.</i></td><td align='left'><a href="#Page_22">p.&nbsp;22</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_II"><b>CHAP. II. On Reflexion.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>Demonstration of equality of angles of incidence and reflexion.</i></td><td align='left'><a href="#Page_23">p.&nbsp;23</a> </td></tr>
+<tr><td align='left'><i>Why the incident and reflected rays are in the same plane perpendicular to the reflecting surface.</i></td><td align='left'><a href="#Page_25">p.&nbsp;25</a></td></tr>
+<tr><td align='left'><i>That it is not needful for the reflecting surface to be perfectly flat to attain equality of the angles of incidence and reflexion.</i></td><td align='left'><a href="#Page_27">p.&nbsp;27</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_III"><b>CHAP. III. On Refraction.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>That bodies may be transparent without any substance passing through them.</i></td><td align='left'><a href="#Page_29">p.&nbsp;29</a></td></tr>
+<tr><td align='left'><i>Proof that the ethereal matter passes through transparent bodies.</i></td><td align='left'><a href="#Page_30">p.&nbsp;30</a></td></tr>
+<tr><td align='left'><i>How this matter passing through can render them transparent.</i></td><td align='left'><a href="#Page_31">p.&nbsp;31</a></td></tr>
+<tr><td align='left'><i>That the most solid bodies in appearance are of a very loose texture.</i></td><td align='left'><a href="#Page_31">p.&nbsp;31</a></td></tr>
+<tr><td align='left'><i>That Light spreads more slowly in water and in glass than in air.</i></td><td align='left'><a href="#Page_32">p.&nbsp;32</a></td></tr>
+<tr><td align='left'><i>Third hypothesis to explain transparency, and the retardation which Light suffers.</i></td><td align='left'><a href="#Page_32">p.&nbsp;32</a></td></tr>
+<tr><td align='left'><i>On that which makes bodies opaque.</i></td><td align='left'><a href="#Page_34">p.&nbsp;34</a></td></tr>
+<tr><td align='left'><i>Demonstration why Refraction obeys the known proportion of Sines.</i></td><td align='left'><a href="#Page_35">p.&nbsp;35</a></td></tr>
+<tr><td align='left'><i>Why the incident and refracted Rays produce one another reciprocally.</i></td><td align='left'><a href="#Page_39">p.&nbsp;39</a></td></tr>
+<tr><td align='left'><i>Why Reflexion within a triangular glass prism is suddenly augmented when the Light can no longer penetrate.</i></td><td align='left'><a href="#Page_40">p.&nbsp;40</a></td></tr>
+<tr><td align='left'><i>That bodies which cause greater Refraction also cause stronger Reflexion.</i></td><td align='left'><a href="#Page_42">p.&nbsp;42</a></td></tr>
+<tr><td align='left'><i>Demonstration of the Theorem of Mr. Fermat.</i></td><td align='left'><a href="#Page_43">p.&nbsp;43</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_IV"><b>CHAP. IV. On the Refraction of the Air.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>That the emanations of Light in the air are not spherical.</i></td><td align='left'><a href="#Page_45">p.&nbsp;45</a></td></tr>
+<tr><td align='left'><i>How consequently some objects appear higher than they are.</i></td><td align='left'><a href="#Page_47">p.&nbsp;47</a></td></tr>
+<tr><td align='left'><i>How the Sun may appear on the Horizon before he has risen.</i></td><td align='left'><a href="#Page_49">p.&nbsp;49</a></td></tr>
+<tr><td align='left'><i>That the rays of light become curved in the Air of the Atmosphere, and what effects this produces.</i></td><td align='left'><a href="#Page_50">p.&nbsp;50</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_V"><b>CHAP. V. On the Strange Refraction of Iceland Crystal.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>That this Crystal grows also in other countries.</i></td><td align='left'><a href="#Page_52">p.&nbsp;52</a></td></tr>
+<tr><td align='left'><i>Who first-wrote about it.</i></td><td align='left'><a href="#Page_53">p.&nbsp;53</a></td></tr>
+<tr><td align='left'><i>Description of Iceland Crystal; its substance, shape, and properties.</i></td><td align='left'><a href="#Page_53">p.&nbsp;53</a></td></tr>
+<tr><td align='left'><i>That it has two different Refractions.</i></td><td align='left'><a href="#Page_54">p.&nbsp;54</a></td></tr>
+<tr><td align='left'><i>That the ray perpendicular to the surface suffers refraction, and that some rays inclined to the surface pass without suffering refraction.</i></td><td align='left'><a href="#Page_55">p.&nbsp;55</a></td></tr>
+<tr><td align='left'><i>Observation of the refractions in this Crystal.</i></td><td align='left'><a href="#Page_56">p.&nbsp;56</a></td></tr>
+<tr><td align='left'><i>That there is a Regular and an Irregular Refraction.</i></td><td align='left'><a href="#Page_57">p.&nbsp;57</a></td></tr>
+<tr><td align='left'><i>The way of measuring the two Refractions of Iceland Crystal.</i></td><td align='left'><a href="#Page_57">p.&nbsp;57</a></td></tr>
+<tr><td align='left'><i>Remarkable properties of the Irregular Refraction.</i></td><td align='left'><a href="#Page_60">p.&nbsp;60</a></td></tr>
+<tr><td align='left'><i>Hypothesis to explain the double Refraction.</i></td><td align='left'><a href="#Page_61">p.&nbsp;61</a></td></tr>
+<tr><td align='left'><i>That Rock Crystal has also a double Refraction.</i></td><td align='left'><a href="#Page_62">p.&nbsp;62</a></td></tr>
+<tr><td align='left'><i>Hypothesis of emanations of Light, within Iceland Crystal, of spheroidal form, for the Irregular Refraction.</i></td><td align='left'><a href="#Page_63">p.&nbsp;63</a></td></tr>
+<tr><td align='left'><i>How a perpendicular ray can suffer Refraction.</i></td><td align='left'><a href="#Page_64">p.&nbsp;64</a></td></tr>
+<tr><td align='left'><i>How the position and form of the spheroidal emanations in this Crystal can be defined.</i></td><td align='left'><a href="#Page_65">p.&nbsp;65</a></td></tr>
+<tr><td align='left'><i>Explanation of the Irregular Refraction by these spheroidal emanations.</i></td><td align='left'><a href="#Page_67">p.&nbsp;67</a></td></tr>
+<tr><td align='left'><i>Easy way to find the Irregular Refraction of each incident ray.</i></td><td align='left'><a href="#Page_70">p.&nbsp;70</a></td></tr>
+<tr><td align='left'><i>Demonstration of the oblique ray which traverses the Crystal without being refracted.</i></td><td align='left'><a href="#Page_73">p.&nbsp;73</a></td></tr>
+<tr><td align='left'><i>Other irregularities of Refraction explained.</i></td><td align='left'><a href="#Page_76">p.&nbsp;76</a></td></tr>
+<tr><td align='left'><i>That an object placed beneath the Crystal appears double, in two images of different heights.</i></td><td align='left'><a href="#Page_81">p.&nbsp;81</a></td></tr>
+<tr><td align='left'><i>Why the apparent heights of one of the images change on changing the position of the eyes above the Crystal.</i></td><td align='left'><a href="#Page_85">p.&nbsp;85</a></td></tr>
+<tr><td align='left'><i>Of the different sections of this Crystal which produce yet other refractions, and confirm all this Theory.</i></td><td align='left'><a href="#Page_88">p.&nbsp;88</a></td></tr>
+<tr><td align='left'><i>Particular way of polishing the surfaces after it has been cut.</i></td><td align='left'><a href="#Page_91">p.&nbsp;91</a></td></tr>
+<tr><td align='left'><i>Surprising phenomenon touching the rays which pass through two separated pieces; the cause of which is not explained.</i></td><td align='left'><a href="#Page_92">p.&nbsp;92</a></td></tr>
+<tr><td align='left'><i>Probable conjecture on the internal composition of Iceland Crystal, and of what figure its particles are.</i></td><td align='left'><a href="#Page_95">p.&nbsp;95</a></td></tr>
+<tr><td align='left'><i>Tests to confirm this conjecture.</i></td><td align='left'><a href="#Page_97">p.&nbsp;97</a></td></tr>
+<tr><td align='left'><i>Calculations which have been supposed in this Chapter.</i></td><td align='left'><a href="#Page_99">p.&nbsp;99</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_VI"><b>CHAP. VI. On the Figures of transparent bodies which serve for Refraction and for Reflexion.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>General and easy rule to find these Figures.</i></td><td align='left'><a href="#Page_106">p.&nbsp;106</a></td></tr>
+<tr><td align='left'><i>Invention of the Ovals of Mr. Des Cartes for Dioptrics.</i></td><td align='left'><a href="#Page_109">p.&nbsp;109</a></td></tr>
+<tr><td align='left'><i>How he was able to find these Lines.</i></td><td align='left'><a href="#Page_114">p.&nbsp;114</a></td></tr>
+<tr><td align='left'><i>Way of finding the surface of a glass for perfect refraction, when the other surface is given.</i></td><td align='left'><a href="#Page_116">p.&nbsp;116</a></td></tr>
+<tr><td align='left'><i>Remark on what happens to rays refracted at a spherical surface.</i></td><td align='left'><a href="#Page_123">p.&nbsp;123</a></td></tr>
+<tr><td align='left'><i>Remark on the curved line which is formed by reflexion in a spherical concave mirror.</i></td><td align='left'><a href="#Page_126">p.&nbsp;126</a></td></tr>
+</table>
+</td></tr>
+</table>
+
+<hr style="width: 65%;" />
+<div><span class="pagenum">[Pg 1]</span><a name="Page_1" id="Page_1" /></div>
+<div class="figcenter" style="width: 600px;">
+<img src="images/ch01head.png" width="600" height="137" alt="" title="" />
+</div>
+<h1>TREATISE ON LIGHT</h1>
+
+
+<h2><a name="CHAPTER_I" id="CHAPTER_I" />CHAPTER I</h2>
+
+<h3>ON RAYS PROPAGATED IN STRAIGHT LINES</h3>
+
+
+<div style="width: 154px; float: left; margin-right: .2em;">
+<img src="images/ch01.png" width="154" height="150" alt="A" title="A" />
+</div><p>s happens in all the sciences in which Geometry is applied to matter,
+the demonstrations concerning Optics are founded on truths drawn from
+experience. Such are that the rays of light are propagated in straight
+lines; that the angles of reflexion and of incidence are equal; and
+that in refraction the ray is bent according to the law of sines, now
+so well known, and which is no less certain than the preceding laws.</p>
+
+<p>The majority of those who have written touching the various parts of
+Optics have contented themselves with presuming these truths. But
+some, more inquiring, have desired to investigate the origin and the
+causes, considering these to be in themselves wonderful effects of
+Nature. In which they advanced some ingenious things, but not however
+such that the most intelligent folk do not wish for better and more
+satisfactory explanations. Wherefore I here desire to propound what I
+have meditated on the sub<span class="pagenum">[Pg 2]</span><a name="Page_2" id="Page_2" />ject, 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.</p>
+
+<p>I shall therefore essay in this book, to give, in accordance with the
+principles accepted in the Philosophy of the present day, some clearer
+and more probable reasons, firstly of these properties of light
+propagated rectilinearly; secondly of light which is reflected on
+meeting other bodies. Then I shall explain the phenomena of those rays
+which are said to suffer refraction on passing through transparent
+bodies of different sorts; and in this part I shall also explain the
+effects of the refraction of the air by the different densities of the
+Atmosphere.</p>
+
+<p>Thereafter I shall examine the causes of the strange refraction of a
+certain kind of Crystal which is brought from Iceland. And finally I
+shall treat of the various shapes of transparent and reflecting bodies
+by which rays are collected at a point or are turned aside in various
+ways. From this it will be seen with what facility, following our new
+Theory, we find not only the Ellipses, Hyperbolas, and <span class="pagenum">[Pg 3]</span><a name="Page_3" id="Page_3" />other curves
+which Mr. Des Cartes has ingeniously invented for this purpose; but
+also those which the surface of a glass lens ought to possess when its
+other surface is given as spherical or plane, or of any other figure
+that may be.</p>
+
+<p>It is inconceivable to doubt that light consists in the motion of some
+sort of matter. For whether one considers its production, one sees
+that here upon the Earth it is chiefly engendered by fire and flame
+which contain without doubt bodies that are in rapid motion, since
+they dissolve and melt many other bodies, even the most solid; or
+whether one considers its effects, one sees that when light is
+collected, as by concave mirrors, it has the property of burning as a
+fire does, that is to say it disunites the particles of bodies. This
+is assuredly the mark of motion, at least in the true Philosophy, in
+which one conceives the causes of all natural effects in terms of
+mechanical motions. This, in my opinion, we must necessarily do, or
+else renounce all hopes of ever comprehending anything in Physics.</p>
+
+<p>And as, according to this Philosophy, one holds as certain that the
+sensation of sight is excited only by the impression of some movement
+of a kind of matter which acts on the nerves at the back of our eyes,
+there is here yet one reason more for believing that light consists in
+a movement of the matter which exists between us and the luminous
+body.</p>
+
+<p>Further, when one considers the extreme speed with which light spreads
+on every side, and how, when it comes from different regions, even
+from those directly opposite, the rays traverse one another without
+hindrance, one may well understand that when we see a luminous object,
+it cannot be by any transport of matter coming to us from this object,
+<span class="pagenum">[Pg 4]</span><a name="Page_4" id="Page_4" />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.</p>
+
+<p>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&mdash;which we are now going to examine&mdash;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.</p>
+
+<p>To see then whether the spreading of light takes time, let us consider
+first whether there are any facts of experience which can convince us
+to the contrary. As to those which can be made here on the Earth, by
+striking lights at great distances, although they prove that light
+takes no sensible time to pass over these distances, one may say with
+good <span class="pagenum">[Pg 5]</span><a name="Page_5" id="Page_5" />reason that they are too small, and that the only conclusion to
+be drawn from them is that the passage of light is extremely rapid.
+Mr. Des Cartes, who was of opinion that it is instantaneous, founded
+his views, not without reason, upon a better basis of experience,
+drawn from the Eclipses of the Moon; which, nevertheless, as I shall
+show, is not at all convincing. I will set it forth, in a way a little
+different from his, in order to make the conclusion more
+comprehensible.</p>
+
+<div class="figcenter" style="width: 400px;">
+<img src="images/pg005.png" width="400" height="196" alt="" title="" />
+</div>
+
+<p>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.</p>
+
+<p>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, <span class="pagenum">[Pg 6]</span><a name="Page_6" id="Page_6" />that the
+Moon, arriving at C, will be obscured: but this obscuration or
+interruption of the light will not reach the Earth till after another
+hour. Let us suppose that the Earth in these two hours will have
+arrived at E. The Earth then, being at E, will see the Eclipsed Moon
+at C, which it left an hour before, and at the same time will see the
+sun at A. For it being immovable, as I suppose with Copernicus, and
+the light moving always in straight lines, it must always appear where
+it is. But one has always observed, we are told, that the eclipsed
+Moon appears at the point of the Ecliptic opposite to the Sun; and yet
+here it would appear in arrear of that point by an amount equal to the
+angle GEC, the supplement of AEC. This, however, is contrary to
+experience, since the angle GEC would be very sensible, and about 33
+degrees. Now according to our computation, which is given in the
+Treatise on the causes of the phenomena of Saturn, the distance BA
+between the Earth and the Sun is about twelve thousand diameters of
+the Earth, and hence four hundred times greater than BC the distance
+of the Moon, which is 30 diameters. Then the angle ECB will be nearly
+four hundred times greater than BAE, which is five minutes; namely,
+the path which the earth travels in two hours along its orbit; and
+thus the angle BCE will be nearly 33 degrees; and likewise the angle
+CEG, which is greater by five minutes.</p>
+
+<p>But it must be noted that the speed of light in this argument has been
+assumed such that it takes a time of one hour to make the passage from
+here to the Moon. If one supposes that for this it requires only one
+minute of time, then it is manifest that the angle CEG will only be 33
+minutes; and if it requires only ten seconds of time, <span class="pagenum">[Pg 7]</span><a name="Page_7" id="Page_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.</p>
+
+<p>It is true that we are here supposing a strange velocity that would be
+a hundred thousand times greater than that of Sound. For Sound,
+according to what I have observed, travels about 180 Toises in the
+time of one Second, or in about one beat of the pulse. But this
+supposition ought not to seem to be an impossibility; since it is not
+a question of the transport of a body with so great a speed, but of a
+successive movement which is passed on from some bodies to others. I
+have then made no difficulty, in meditating on these things, in
+supposing that the emanation of light is accomplished with time,
+seeing that in this way all its phenomena can be explained, and that
+in following the contrary opinion everything is incomprehensible. For
+it has always seemed tome that even Mr. Des Cartes, whose aim has been
+to treat all the subjects of Physics intelligibly, and who assuredly
+has succeeded in this better than any one before him, has said nothing
+that is not full of difficulties, or even inconceivable, in dealing
+with Light and its properties.</p>
+
+<p>But that which I employed only as a hypothesis, has recently received
+great seemingness as an established truth by the ingenious proof of
+Mr. R&ouml;mer 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.</p>
+
+<p><span class="pagenum">[Pg 8]</span><a name="Page_8" id="Page_8" />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.</p>
+
+<div class="figleft" style="width: 166px;">
+<img src="images/pg008.png" width="166" height="400" alt="" title="" />
+</div>
+
+<p>Let it be then supposed, the Earth being at B some time before the
+last quadrature, that one has seen the said Satellite emerge from the
+shadow; it must needs be, if the Earth remains at the same place,
+that, after 42-1/2 hours, one would again see a similar emergence,
+because that is the time in which it makes the round of its orbit, and
+when it would come again into opposition to the Sun. And if the Earth,
+for instance, were to remain always at B during 30 revolutions of this
+Satellite, one would see it again emerge from the shadow after 30
+times 42-1/2 hours. But the Earth having been carried along during
+this time to C, increasing thus its distance from Jupiter, it follows
+that if Light requires time for its passage the illumination of the
+little planet will be perceived later at <span class="pagenum">[Pg 9]</span><a name="Page_9" id="Page_9" />C than it would have been at
+B, and that there must be added to this time of 30 times 42-1/2 hours
+that which the Light has required to traverse the space MC, the
+difference of the spaces CH, BH. Similarly at the other quadrature
+when the earth has come to E from D while approaching toward Jupiter,
+the immersions of the Satellite ought to be observed at E earlier than
+they would have been seen if the Earth had remained at D.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>If one considers the vast size of the diameter KL, which according to
+me is some 24 thousand diameters of the Earth, one will acknowledge
+the extreme velocity of Light. For, supposing that KL is no more than
+22 thousand of these diameters, it appears that being traversed in 22
+minutes this makes the speed a thousand diameters in one minute, that
+is 16-2/3 diameters in one second or in one beat of the pulse, which
+makes more than 11 hundred times a hundred thousand toises; since the
+diameter of the Earth contains 2,865 leagues, reckoned at 25 to the
+degree, and each <span class="pagenum">[Pg 10]</span><a name="Page_10" id="Page_10" />each league is 2,282 Toises, according to the exact
+measurement which Mr. Picard made by order of the King in 1669. But
+Sound, as I have said above, only travels 180 toises in the same time
+of one second: hence the velocity of Light is more than six hundred
+thousand times greater than that of Sound. This, however, is quite
+another thing from being instantaneous, since there is all the
+difference between a finite thing and an infinite. Now the successive
+movement of Light being confirmed in this way, it follows, as I have
+said, that it spreads by spherical waves, like the movement of Sound.</p>
+
+<p>But if the one resembles the other in this respect, they differ in
+many other things; to wit, in the first production of the movement
+which causes them; in the matter in which the movement spreads; and in
+the manner in which it is propagated. As to that which occurs in the
+production of Sound, one knows that it is occasioned by the agitation
+undergone by an entire body, or by a considerable part of one, which
+shakes all the contiguous air. But the movement of the Light must
+originate as from each point of the luminous object, else we should
+not be able to perceive all the different parts of that object, as
+will be more evident in that which follows. And I do not believe that
+this movement can be better explained than by supposing that all those
+of the luminous bodies which are liquid, such as flames, and
+apparently the sun and the stars, are composed of particles which
+float in a much more subtle medium which agitates them with great
+rapidity, and makes them strike against the particles of the ether
+which surrounds them, and which are much smaller than they. But I hold
+also that in luminous solids such as charcoal or metal made red hot in
+the fire, this same movement is caused by the violent <span class="pagenum">[Pg 11]</span><a name="Page_11" id="Page_11" />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.</p>
+
+<p>Now if one examines what this matter may be in which the movement
+coming from the luminous body is propagated, which I call Ethereal
+matter, one will see that it is not the same that serves for the
+propagation of Sound. For one finds that the latter is really that
+which we feel and which we breathe, and which being removed from any
+place still leaves there the other kind of matter that serves to
+convey Light. This may be proved by shutting up a sounding body in a
+glass vessel from which the air is withdrawn by the machine which Mr.
+Boyle has given us, and with which he has performed so many beautiful
+experiments. But in doing this of which I speak, care must be taken to
+place the sounding body on cotton or on feathers, in such a way that
+it cannot communicate its tremors either to the glass vessel which
+encloses it, or to the machine; a precaution which has hitherto been
+neglected. For then after having exhausted all the air one hears no
+Sound from the metal, though it is struck.</p>
+
+<p>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 <span class="pagenum">[Pg 12]</span><a name="Page_12" id="Page_12" />removed from the vessel the Light does not cease to
+traverse it as before.</p>
+
+<p>And this last point is demonstrated even more clearly by the
+celebrated experiment of Torricelli, in which the tube of glass from
+which the quicksilver has withdrawn itself, remaining void of air,
+transmits Light just the same as when air is in it. For this proves
+that a matter different from air exists in this tube, and that this
+matter must have penetrated the glass or the quicksilver, either one
+or the other, though they are both impenetrable to the air. And when,
+in the same experiment, one makes the vacuum after putting a little
+water above the quicksilver, one concludes equally that the said
+matter passes through glass or water, or through both.</p>
+
+<p>As regards the different modes in which I have said the movements of
+Sound and of Light are communicated, one may sufficiently comprehend
+how this occurs in the case of Sound if one considers that the air is
+of such a nature that it can be compressed and reduced to a much
+smaller space than that which it ordinarily occupies. And in
+proportion as it is compressed the more does it exert an effort to
+regain its volume; for this property along with its penetrability,
+which remains notwithstanding its compression, seems to prove that it
+is made up of small bodies which float about and which are agitated
+very rapidly in the ethereal matter composed of much smaller parts. So
+that the cause of the spreading of Sound is the effort which these
+little bodies make in collisions with one another, to regain freedom
+when they are a little more squeezed together in the circuit of these
+waves than elsewhere.</p>
+
+<p>But the extreme velocity of Light, and other properties which it has,
+cannot admit of such a propagation of motion, <span class="pagenum">[Pg 13]</span><a name="Page_13" id="Page_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 property which hard bodies must possess to
+transmit movement from one to another.</p>
+
+<p>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.</p>
+
+<p>But it is still certain that this progression of motion is not
+instantaneous, but successive, and therefore must take time. For if
+the movement, or the disposition to movement, if you will have it so,
+did not pass successively through all these spheres, they would all
+acquire the movement at the same time, and hence would all advance
+together; which does not happen. For the last one leaves the whole row
+and acquires the speed of the one which was pushed. Moreover there are
+experiments which demonstrate that all the bodies which we reckon of
+the hardest kind, such as quenched steel, glass, and agate, act as
+springs and bend somehow, not only when extended as rods but also when
+they are in the form of spheres or of other shapes. That is to say
+they yield a little in themselves at the place where they are struck,
+and immediately regain their former figure. For I have found that on
+striking with a ball of glass or of agate against a large and quite
+thick <span class="pagenum">[Pg 14]</span><a name="Page_14" id="Page_14" />thick piece of the same substance which had a flat surface,
+slightly soiled with breath or in some other way, there remained round
+marks, of smaller or larger size according as the blow had been weak
+or strong. This makes it evident that these substances yield where
+they meet, and spring back: and for this time must be required.</p>
+
+<p>Now in applying this kind of movement to that which produces Light
+there is nothing to hinder us from estimating the particles of the
+ether to be of a substance as nearly approaching to perfect hardness
+and possessing a springiness as prompt as we choose. It is not
+necessary to examine here the causes of this hardness, or of that
+springiness, the consideration of which would lead us too far from our
+subject. I will say, however, in passing that we may conceive that the
+particles of the ether, notwithstanding their smallness, are in turn
+composed of other parts and that their springiness consists in the
+very rapid movement of a subtle matter which penetrates them from
+every side and constrains their structure to assume such a disposition
+as to give to this fluid matter the most overt and easy passage
+possible. This accords with the explanation which Mr. Des Cartes gives
+for the spring, though I do not, like him, suppose the pores to be in
+the form of round hollow canals. And it must not be thought that in
+this there is anything absurd or impossible, it being on the contrary
+quite credible that it is this infinite series of different sizes of
+corpuscles, having different degrees of velocity, of which Nature
+makes use to produce so many marvellous effects.</p>
+
+<p>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 <span class="pagenum">[Pg 15]</span><a name="Page_15" id="Page_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 communicated, one will find none which agrees better,
+with uniform progression, as seems to be necessary, than the property
+of springiness; because if this movement should grow slower in
+proportion as it is shared over a greater quantity of matter, in
+moving away from the source of the light, it could not conserve this
+great velocity over great distances. But by supposing springiness in
+the ethereal matter, its particles will have the property of equally
+rapid restitution whether they are pushed strongly or feebly; and thus
+the propagation of Light will always go on with an equal velocity.</p>
+
+<div class="figleft" style="width: 131px;">
+<img src="images/pg015.png" width="131" height="200" alt="" title="" />
+</div>
+
+<p>And it must be known that although the particles of the ether are not
+ranged thus in straight lines, as in our row of spheres, but
+confusedly, so that one of them touches several others, this does not
+hinder them from transmitting their movement and from spreading it
+always forward. As to this it is to be remarked that there is a law of
+motion serving for this propagation, and verifiable by experiment. It
+is that when a sphere, such as A here, touches several other similar
+spheres CCC, if it is struck by another sphere B in such a way as to
+exert an impulse against all the spheres CCC which touch it, it
+transmits to them the whole of its movement, and remains after that
+motionless like the sphere B. And without supposing that the ethereal
+particles are of spherical form (for I see indeed no need to suppose
+them so) one may well understand that this property of communicating
+an impulse <span class="pagenum">[Pg 16]</span><a name="Page_16" id="Page_16" />does not fail to contribute to the aforesaid propagation
+of movement.</p>
+
+<p>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.</p>
+
+<p>However, one will see hereafter that we have to suppose such an
+equality not so much as a necessity for the propagation of light as
+for rendering that propagation easier and more powerful; for it is not
+beyond the limits of probability that the particles of the ether have
+been made equal for a purpose so important as that of light, at least
+in that vast space which is beyond the region of atmosphere and which
+seems to serve only to transmit the light of the Sun and the Stars.</p>
+
+<div class="figright" style="width: 182px;">
+<img src="images/pg017.png" width="182" height="300" alt="" title="" />
+</div>
+
+<p>I have then shown in what manner one may conceive Light to spread
+successively, by spherical waves, and how it is possible that this
+spreading is accomplished with as great a velocity as that which
+experiments and celestial observations demand. Whence it may be
+further remarked that although the particles are supposed to be in
+continual movement (for there are many reasons for this) the
+successive propagation of the waves cannot be hindered by this;
+because the propagation consists nowise in the transport of those
+particles but merely in a small agitation which they cannot help
+communicating to those surrounding, notwithstanding any movement which
+may act on them causing them to be changing positions amongst
+themselves.</p>
+
+<p>But we must consider still more particularly the origin of these
+waves, and the manner in which they spread. And, first, it follows
+from what has been said on the production <span class="pagenum">[Pg 17]</span><a name="Page_17" id="Page_17" />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.</p>
+
+<p>But as the percussions at the centres of these waves possess no
+regular succession, it must not be supposed that the waves themselves
+follow one another at equal distances: and if the distances marked in
+the figure appear to be such, it is rather to mark the progression of
+one and the same wave at equal intervals of time than to represent
+several of them issuing from one and the same centre.</p>
+
+<p>After all, this prodigious quantity of waves which traverse one
+another without confusion and without effacing one another must not be
+deemed inconceivable; it being certain that one and the same particle
+of matter can serve for many waves coming from different sides or even
+from contrary directions, not only if it is struck by blows which
+follow one another closely but even for those which act on it at the
+same instant. It can do so because the spreading of the movement is
+successive. This may be proved by the row of equal spheres of hard
+matter, spoken of above. If against this row there are pushed from two
+opposite sides at the same time two similar spheres A and <span class="pagenum">[Pg 18]</span><a name="Page_18" id="Page_18" />D, one will
+see each of them rebound with the same velocity which it had in
+striking, yet the whole row will remain in its place, although the
+movement has passed along its whole length twice over. And if these
+contrary movements happen to meet one another at the middle sphere, B,
+or at some other such as C, that sphere will yield and act as a spring
+at both sides, and so will serve at the same instant to transmit these
+two movements.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg018.png" width="500" height="59" alt="" title="" />
+</div>
+
+<p>But what may at first appear full strange and even incredible is that
+the undulations produced by such small movements and corpuscles,
+should spread to such immense distances; as for example from the Sun
+or from the Stars to us. For the force of these waves must grow feeble
+in proportion as they move away from their origin, so that the action
+of each one in particular will without doubt become incapable of
+making itself felt to our sight. But one will cease to be astonished
+by considering how at a great distance from the luminous body an
+infinitude of waves, though they have issued from different points of
+this body, unite together in such a way that they sensibly compose one
+single wave only, which, consequently, ought to have enough force to
+make itself felt. Thus this infinite number of waves which originate
+at the same instant from all points of a fixed star, big it may be as
+the Sun, make practically only one single wave which may well have
+force enough to produce an impression on our eyes. Moreover from each
+luminous point there may come many thousands of waves in the smallest
+imaginable time, by the frequent percussion of the corpuscles which
+strike the <span class="pagenum">[Pg 19]</span><a name="Page_19" id="Page_19" />Ether at these points: which further contributes to
+rendering their action more sensible.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg019.png" width="300" height="284" alt="" title="" />
+</div>
+
+<p>There is the further consideration in the emanation of these waves,
+that each particle of matter in which a wave spreads, ought not to
+communicate its motion only to the next particle which is in the
+straight line drawn from the luminous point, but that it also imparts
+some of it necessarily to all the others which touch it and which
+oppose themselves to its movement. So it arises that around each
+particle there is made a wave of which that particle is the centre.
+Thus if DCF is a wave emanating from the luminous point A, which is
+its centre, the particle B, one of those comprised within the sphere
+DCF, will have made its particular or partial wave KCL, which will
+touch the wave DCF at C at the same moment that the principal wave
+emanating from the point A has arrived at DCF; and it is clear that it
+will be only the region C of the wave KCL which will touch the wave
+DCF, to wit, that which is in the straight line drawn through AB.
+Similarly the other particles of the sphere DCF, such as <i>bb</i>, <i>dd</i>,
+etc., will each make its own wave. But each of these waves can be
+infinitely feeble only as compared with the wave DCF, to the
+composition of which all the others contribute by the part of their
+surface which is most distant from the centre A.</p>
+
+<p><span class="pagenum">[Pg 20]</span><a name="Page_20" id="Page_20" />One sees, in addition, that the wave DCF is determined by the
+distance attained in a certain space of time by the movement which
+started from the point A; there being no movement beyond this wave,
+though there will be in the space which it encloses, namely in parts
+of the particular waves, those parts which do not touch the sphere
+DCF. And all this ought not to seem fraught with too much minuteness
+or subtlety, since we shall see in the sequel that all the properties
+of Light, and everything pertaining to its reflexion and its
+refraction, can be explained in principle by this means. This is a
+matter which has been quite unknown to those who hitherto have begun
+to consider the waves of light, amongst whom are Mr. Hooke in his
+<i>Micrographia</i>, and Father Pardies, who, in a treatise of which he let
+me see a portion, and which he was unable to complete as he died
+shortly afterward, had undertaken to prove by these waves the effects
+of reflexion and refraction. But the chief foundation, which consists
+in the remark I have just made, was lacking in his demonstrations; and
+for the rest he had opinions very different from mine, as may be will
+appear some day if his writing has been preserved.</p>
+
+<p>To come to the properties of Light. We remark first that each portion
+of a wave ought to spread in such a way that its extremities lie
+always between the same straight lines drawn from the luminous point.
+Thus the portion BG of the wave, having the luminous point A as its
+centre, will spread into the arc CE bounded by the straight lines ABC,
+AGE. For although the particular waves produced by the particles
+comprised within the space CAE spread also outside this space, they
+yet do not concur at the same instant to compose a wave which
+terminates the <span class="pagenum">[Pg 21]</span><a name="Page_21" id="Page_21" />movement, as they do precisely at the circumference
+CE, which is their common tangent.</p>
+
+<p>And hence one sees the reason why light, at least if its rays are not
+reflected or broken, spreads only by straight lines, so that it
+illuminates no object except when the path from its source to that
+object is open along such lines.</p>
+
+<p>For if, for example, there were an opening BG, limited by opaque
+bodies BH, GI, the wave of light which issues from the point A will
+always be terminated by the straight lines AC, AE, as has just been
+shown; the parts of the partial waves which spread outside the space
+ACE being too feeble to produce light there.</p>
+
+<p>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.</p>
+
+<p>It appears, moreover, by what has been remarked touching the
+feebleness of the particular waves, that it is not needful that all
+the particles of the Ether should be equal amongst themselves, though
+equality is more apt for the propagation of the movement. For it is
+true that inequality will cause a particle by pushing against another
+larger one to strive to recoil with a part of its movement; but it
+will thereby merely generate backwards towards the luminous point some
+partial waves incapable of causing light, and not a wave compounded of
+many as CE was.</p>
+
+<p>Another property of waves of light, and one of the most <span class="pagenum">[Pg 22]</span><a name="Page_22" id="Page_22" />marvellous,
+is that when some of them come from different or even from opposing
+sides, they produce their effect across one another without any
+hindrance. Whence also it comes about that a number of spectators may
+view different objects at the same time through the same opening, and
+that two persons can at the same time see one another's eyes. Now
+according to the explanation which has been given of the action of
+light, how the waves do not destroy nor interrupt one another when
+they cross one another, these effects which I have just mentioned are
+easily conceived. But in my judgement they are not at all easy to
+explain according to the views of Mr. Des Cartes, who makes Light to
+consist in a continuous pressure merely tending to movement. For this
+pressure not being able to act from two opposite sides at the same
+time, against bodies which have no inclination to approach one
+another, it is impossible so to understand what I have been saying
+about two persons mutually seeing one another's eyes, or how two
+torches can illuminate one another.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_II" id="CHAPTER_II" />CHAPTER II</h2>
+
+<h3>ON REFLEXION</h3>
+
+
+<div style="width: 158px; float: left; margin-right: .2em;">
+<img src="images/ch02.png" width="158" height="150" alt="H" title="H" />
+</div><p>aving explained the effects of waves of light which spread in a
+homogeneous matter, we will examine next that which happens to them on
+encountering other bodies. We will first make evident how the
+Reflexion of light is explained by these same waves, and why it
+preserves equality of angles.</p>
+
+<p><span class="pagenum">[Pg 23]</span><a name="Page_23" id="Page_23" />Let there be a surface AB; plane and polished, of some metal, glass,
+or other body, which at first I will consider as perfectly uniform
+(reserving to myself to deal at the end of this demonstration with the
+inequalities from which it cannot be exempt), and let a line AC,
+inclined to AD, represent a portion of a wave of light, the centre of
+which is so distant that this portion AC may be considered as a
+straight line; for I consider all this as in one plane, imagining to
+myself that the plane in which this figure is, cuts the sphere of the
+wave through its centre and intersects the plane AB at right angles.
+This explanation will suffice once for all.</p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg023.png" width="350" height="352" alt="" title="" />
+</div>
+
+<p>The piece C of the wave AC, will in a certain space of time advance as
+far as the plane AB at B, following the straight line CB, which may be
+supposed to come from the luminous centre, and which in consequence is
+perpendicular to AC. Now in this same space of time the portion A of
+the same wave, which has been hindered from communicating its movement
+beyond the plane AB, or at least partly so, ought to have continued
+its movement in the matter which is above this plane, and this along a
+distance equal to CB, making its <span class="pagenum">[Pg 24]</span><a name="Page_24" id="Page_24" />own partial spherical wave,
+according to what has been said above. Which wave is here represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter AN equal to CB.</p>
+
+<p>If one considers further the other pieces H of the wave AC, it appears
+that they will not only have reached the surface AB by straight lines
+HK parallel to CB, but that in addition they will have generated in
+the transparent air, from the centres K, K, K, particular spherical
+waves, represented here by circumferences the semi-diameters of which
+are equal to KM, that is to say to the continuations of HK as far as
+the line BG parallel to AC. But all these circumferences have as a
+common tangent the straight line BN, namely the same which is drawn
+from B as a tangent to the first of the circles, of which A is the
+centre, and AN the semi-diameter equal to BC, as is easy to see.</p>
+
+<p>It is then the line BN (comprised between B and the point N where the
+perpendicular from the point A falls) which is as it were formed by
+all these circumferences, and which terminates the movement which is
+made by the reflexion of the wave AC; and it is also the place where
+the movement occurs in much greater quantity than anywhere else.
+Wherefore, according to that which has been explained, BN is the
+propagation of the wave AC at the moment when the piece C of it has
+arrived at B. For there is no other line which like BN is a common
+tangent to all the aforesaid circles, except BG below the plane AB;
+which line BG would be the propagation of the wave if the movement
+could have spread in a medium homogeneous with that which is above the
+plane. And if one wishes to see how the wave AC has come successively
+to BN, one has only to draw in the same figure the straight lines KO
+<span class="pagenum">[Pg 25]</span><a name="Page_25" id="Page_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.</p>
+
+<p>Now it is apparent here that the angle of reflexion is made equal to
+the angle of incidence. For the triangles ACB, BNA being rectangular
+and having the side AB common, and the side CB equal to NA, it follows
+that the angles opposite to these sides will be equal, and therefore
+also the angles CBA, NAB. But as CB, perpendicular to CA, marks the
+direction of the incident ray, so AN, perpendicular to the wave BN,
+marks the direction of the reflected ray; hence these rays are equally
+inclined to the plane AB.</p>
+
+<p>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. <span class="pagenum">[Pg 26]</span><a name="Page_26" id="Page_26" />Now it is easy to see, following the preceding
+demonstration, that each small piece of this wave HC having arrived at
+the plane AB, and there generating each one its particular wave, these
+will all have, when C arrives at B, a common plane which will touch
+them, namely a circle BN similar to CH; and this will be intersected
+at its middle and at right angles by the same plane which likewise
+intersects the circle CH and the ellipse AB.</p>
+
+<div class="figcenter" style="width: 400px;">
+<img src="images/pg026.png" width="400" height="214" alt="" title="" />
+</div>
+
+<p>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.</p>
+
+<p>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
+<span class="pagenum">[Pg 27]</span><a name="Page_27" id="Page_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 act as springs, of which
+we have spoken above; namely that whether compressed little or much
+they recoil in equal times. Equally so in every reflexion of the
+light, against whatever body it may be, the angles of reflexion and
+incidence ought to be equal notwithstanding that the body might be of
+such a nature that it takes away a portion of the movement made by the
+incident light. And experiment shows that in fact there is no polished
+body the reflexion of which does not follow this rule.</p>
+
+
+<p>But the thing to be above all remarked in our demonstration is that it
+does not require that the reflecting surface should be considered as a
+uniform plane, as has been supposed by all those who have tried to
+explain the effects of reflexion; but only an evenness such as may be
+attained by the particles of the matter of the reflecting body being
+set near to one another; which particles are larger than those of the
+ethereal matter, as will appear by what we shall say in treating of
+the transparency and opacity of bodies. For the surface consisting
+thus of particles put together, and the ethereal particles being
+above, and smaller, it is evident that one could not demonstrate the
+equality of the angles of incidence and reflexion by similitude to
+that which happens to a ball thrown against a wall, of which writers
+have always made use. In our way, on the other hand, the thing is
+explained without difficulty. For the smallness of the particles of
+quicksilver, for example, being such that one must conceive millions
+of them, in the smallest visible surface proposed, arranged like a
+heap of grains of sand which has been flattened as much as it is
+capable of being, <span class="pagenum">[Pg 28]</span><a name="Page_28" id="Page_28" />this surface then becomes for our purpose as even
+as a polished glass is: and, although it always remains rough with
+respect to the particles of the Ether it is evident that the centres
+of all the particular spheres of reflexion, of which we have spoken,
+are almost in one uniform plane, and that thus the common tangent can
+fit to them as perfectly as is requisite for the production of light.
+And this alone is requisite, in our method of demonstration, to cause
+equality of the said angles without the remainder of the movement
+reflected from all parts being able to produce any contrary effect.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_III" id="CHAPTER_III" />CHAPTER III</h2>
+
+<h3>ON REFRACTION</h3>
+
+
+<div style="width: 145px; float: left; margin-right: .2em;">
+<img src="images/ch03.png" width="145" height="150" alt="I" title="I" />
+</div><p>n the same way as the effects of Reflexion have been explained by
+waves of light reflected at the surface of polished bodies, we will
+explain transparency and the phenomena of refraction by waves which
+spread within and across diaphanous bodies, both solids, such as
+glass, and liquids, such as water, oils, etc. But in order that it may
+not seem strange to suppose this passage of waves in the interior of
+these bodies, I will first show that one may conceive it possible in
+more than one mode.</p>
+
+<p>First, then, if the ethereal matter cannot penetrate transparent
+bodies at all, their own particles would be able to communicate
+successively the movement of the waves, the same as do those of the
+Ether, supposing that, like those, they are of a nature to act as a
+spring. And this is <span class="pagenum">[Pg 29]</span><a name="Page_29" id="Page_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 transparent and
+hard bodies, because their solidity does not seem to permit them to
+receive movement except in their whole mass at the same time. This,
+however, is not necessary because this solidity is not such as it
+appears to us, it being probable rather that these bodies are composed
+of particles merely placed close to one another and held together by
+some pressure from without of some other matter, and by the
+irregularity of their shapes. For primarily their rarity is shown by
+the facility with which there passes through them the matter of the
+vortices of the magnet, and that which causes gravity. Further, one
+cannot say that these bodies are of a texture similar to that of a
+sponge or of light bread, because the heat of the fire makes them flow
+and thereby changes the situation of the particles amongst themselves.
+It remains then that they are, as has been said, assemblages of
+particles which touch one another without constituting a continuous
+solid. This being so, the movement which these particles receive to
+carry on the waves of light, being merely communicated from some of
+them to others, without their going for that purpose out of their
+places or without derangement, it may very well produce its effect
+without prejudicing in any way the apparent solidity of the compound.</p>
+
+<p>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
+<span class="pagenum">[Pg 30]</span><a name="Page_30" id="Page_30" />that the air has been removed from the vessel in which this tube is
+enclosed.</p>
+
+<p>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.</p>
+
+<p>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 <span class="pagenum">[Pg 31]</span><a name="Page_31" id="Page_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.</p>
+
+<p>The second mode then of explaining transparency, and one which appears
+more probably true, is by saying that the waves of light are carried
+on in the ethereal matter, which continuously occupies the interstices
+or pores of transparent bodies. For since it passes through them
+continuously and freely, it follows that they are always full of it.
+And one may even show that these interstices occupy much more space
+than the coherent particles which constitute the bodies. For if what
+we have just said is true: that force is required to impress a certain
+horizontal velocity on bodies in proportion as they contain coherent
+matter; and if the proportion of this force follows the law of
+weights, as is confirmed by experiment, then the quantity of the
+constituent matter of bodies also follows the proportion of their
+weights. Now we see that water weighs only one fourteenth part as much
+as an equal portion of quicksilver: therefore the matter of the water
+does not occupy the fourteenth part of the space which its mass
+obtains. It must even occupy much less of it, since quicksilver is
+less heavy than gold, and the matter of gold is by no means dense, as
+follows from the fact that the matter of the vortices of the magnet
+and of that which is the cause of gravity pass very freely through it.</p>
+
+<p>But it may be objected here that if water is a body of so great
+rarity, and if its particles occupy so small a portion of the space of
+its apparent bulk, it is very strange how it yet resists Compression
+so strongly without permitting itself to be condensed by any force
+which one has <span class="pagenum">[Pg 32]</span><a name="Page_32" id="Page_32" />hitherto essayed to employ, preserving even its entire
+liquidity while subjected to this pressure.</p>
+
+<p>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.</p>
+
+<p>The rarity of transparent bodies being then such as we have said, one
+easily conceives that the waves might be carried on in the ethereal
+matter which fills the interstices of the particles. And, moreover,
+one may believe that the progression of these waves ought to be a
+little slower in the interior of bodies, by reason of the small
+detours which the same particles cause. In which different velocity of
+light I shall show the cause of refraction to consist.</p>
+
+<p>Before doing so, I will indicate the third and last mode in which
+transparency may be conceived; which is by supposing that the movement
+of the waves of light is transmitted indifferently both in the
+particles of the ethereal matter which occupy the interstices of
+bodies, and in the particles which compose them, so that the movement
+passes from one to the other. And it will be seen hereafter that this
+hypothesis serves excellently to explain the double refraction of
+certain transparent bodies.</p>
+
+<p>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 <span class="pagenum">[Pg 33]</span><a name="Page_33" id="Page_33" />these secondary particles which will receive the movement
+from those of the ether.</p>
+
+<p>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.</p>
+
+<p>All this I have found as most probable for the mode in which the waves
+of light pass across transparent bodies. To which it must further be
+added in what respect these bodies differ from those which are opaque;
+and the more so since it might seem because of the easy penetration of
+bodies by the ethereal matter, of which mention has been made, that
+there would not be any body that was not transparent. For by the same
+reasoning about the hollow sphere which I have employed to prove the
+smallness of the density of glass and its easy penetrability by the
+ethereal matter, one might also prove that the same penetrability
+obtains for metals and for every other sort of body. For this sphere
+being for example of silver, it is certain that it contains some of
+the ethereal matter which serves for light, since this was there as
+well as in the air when the opening of the sphere was closed. Yet,
+being closed and placed upon a horizontal plane, it resists the
+movement which one wishes to give to it, merely according to the
+quantity of silver of which it is made; so that one must conclude, as
+above, that the ethereal matter which is enclosed does not follow the
+movement of the sphere; and that therefore silver, as well as glass,
+is very easily penetrated by this matter. Some of it is therefore
+present continuously and in quantities between the particles of silver
+and of all other opaque <span class="pagenum">[Pg 34]</span><a name="Page_34" id="Page_34" />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.</p>
+
+<p>Whence then, one will say, does their opacity come? Is it because the
+particles which compose them are soft; that is to say, these particles
+being composed of others that are smaller, are they capable of
+changing their figure on receiving the pressure of the ethereal
+particles, the motion of which they thereby damp, and so hinder the
+continuance of the waves of light? That cannot be: for if the
+particles of the metals are soft, how is it that polished silver and
+mercury reflect light so strongly? What I find to be most probable
+herein, is to say that metallic bodies, which are almost the only
+really opaque ones, have mixed amongst their hard particles some soft
+ones; so that some serve to cause reflexion and the others to hinder
+transparency; while, on the other hand, transparent bodies contain
+only hard particles which have the faculty of recoil, and serve
+together with those of the ethereal matter for the propagation of the
+waves of light, as has been said.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg034.png" width="300" height="283" alt="" title="" />
+</div>
+
+<p>Let us pass now to the explanation of the effects of Refraction,
+assuming, as we have done, the passage of waves of light through
+transparent bodies, and the diminution of velocity which these same
+waves suffer in them.</p>
+
+<p>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 <span class="pagenum">[Pg 35]</span><a name="Page_35" id="Page_35" />broken at the point of incidence B,
+in such a way that with the straight line DBE which cuts the surface
+perpendicularly it makes an angle CBE less than ABD which it made with
+the same perpendicular when in the air. And the measure of these
+angles is found by describing, about the point B, a circle which cuts
+the radii AB, BC. For the perpendiculars AD, CE, let fall from the
+points of intersection upon the straight line DE, which are called the
+Sines of the angles ABD, CBE, have a certain ratio between themselves;
+which ratio is always the same for all inclinations of the incident
+ray, at least for a given transparent body. This ratio is, in glass,
+very nearly as 3 to 2; and in water very nearly as 4 to 3; and is
+likewise different in other diaphanous bodies.</p>
+
+<p>Another property, similar to this, is that the refractions are
+reciprocal between the rays entering into a transparent body and those
+which are leaving it. That is to say that if the ray AB in entering
+the transparent body is refracted into BC, then likewise CB being
+taken as a ray in the interior of this body will be refracted, on
+passing out, into BA.</p>
+
+<div class="figleft" style="width: 400px;">
+<img src="images/pg035.png" width="400" height="298" alt="" title="" />
+</div>
+
+<p>To explain then the reasons of these phenomena according to our
+principles, let AB be the straight line which <span class="pagenum">[Pg 36]</span><a name="Page_36" id="Page_36" />represents a plane
+surface bounding the transparent substances which lie towards C and
+towards N. When I say plane, that does not signify a perfect evenness,
+but such as has been understood in treating of reflexion, and for the
+same reason. Let the line AC represent a portion of a wave of light,
+the centre of which is supposed so distant that this portion may be
+considered as a straight line. The piece C, then, of the wave AC, will
+in a certain space of time have advanced as far as the plane AB
+following the straight line CB, which may be imagined as coming from
+the luminous centre, and which consequently will cut AC at right
+angles. Now in the same time the piece A would have come to G along
+the straight line AG, equal and parallel to CB; and all the portion of
+wave AC would be at GB if the matter of the transparent body
+transmitted the movement of the wave as quickly as the matter of the
+Ether. But let us suppose that it transmits this movement less
+quickly, by one-third, for instance. Movement will then be spread from
+the point A, in the matter of the transparent body through a distance
+equal to two-thirds of CB, making its own particular spherical wave
+according to what has been said before. This wave is then represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter equal to two-thirds of CB. Then if one considers in
+order the other pieces H of the wave AC, it appears that in the same
+time that the piece C reaches B they will not only have arrived at the
+surface AB along the straight lines HK parallel to CB, but that, in
+addition, they will have generated in the diaphanous substance from
+the centres K, partial waves, represented here by circumferences the
+semi-diameters of which are equal to two-thirds of the lines KM, that
+is to say, to <span class="pagenum">[Pg 37]</span><a name="Page_37" id="Page_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.</p>
+
+<p>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.</p>
+
+<p>It is then BN, which is formed by small arcs of these circumferences,
+which terminates the movement that the wave AC has communicated within
+the transparent body, and where this movement occurs in much greater
+amount than anywhere else. And for that reason this line, in
+accordance with what has been said more than once, is the propagation
+of the wave AC at the moment when its piece C has reached B. For there
+is no other line below the plane AB which is, like BN, a common
+tangent to all these partial waves. And if one would know how the wave
+AC has come progressively to BN, it is necessary only to draw in the
+same figure the straight lines KO parallel to BN, and all the lines KL
+parallel to AC. Thus one will see that the wave CA, from being a
+straight line, has become broken in all the positions LKO
+successively, and that it has again become a straight line at BN. This
+being evident by what has already been demonstrated, there is no need
+to explain it further.</p>
+
+<p>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 <span class="pagenum">[Pg 38]</span><a name="Page_38" id="Page_38" />mark the ray of incident light, and AN which
+was perpendicular to BN, the refracted ray: since the rays are nothing
+else than the straight lines along which the portions of the waves
+advance.</p>
+
+<p>Whence it is easy to recognize this chief property of refraction,
+namely that the Sine of the angle DAE has always the same ratio to the
+Sine of the angle NAF, whatever be the inclination of the ray DA: and
+that this ratio is the same as that of the velocity of the waves in
+the transparent substance which is towards AE to their velocity in the
+transparent substance towards AF. For, considering AB as the radius of
+a circle, the Sine of the angle BAC is BC, and the Sine of the angle
+ABN is AN. But the angle BAC is equal to DAE, since each of them added
+to CAE makes a right angle. And the angle ABN is equal to NAF, since
+each of them with BAN makes a right angle. Then also the Sine of the
+angle DAE is to the Sine of NAF as BC is to AN. But the ratio of BC to
+AN was the same as that of the velocities of light in the substance
+which is towards AE and in that which is towards AF; therefore also
+the Sine of the angle DAE will be to the Sine of the angle NAF the
+same as the said velocities of light.</p>
+
+<p>To see, consequently, what the refraction will be when the waves of
+light pass into a substance in which the movement travels more quickly
+than in that from which they emerge (let us again assume the ratio of
+3 to 2), it is only necessary to repeat all the same construction and
+demonstration which we have just used, merely substituting everywhere
+3/2 instead of 2/3. And it will be found by the same reasoning, in
+this other figure, that when the piece C of the wave AC shall have
+reached the surface AB at B, <span class="pagenum">[Pg 39]</span><a name="Page_39" id="Page_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.</p>
+
+<p>Hence one sees the reciprocal relation of the refractions of the ray
+on entering and on leaving one and the same transparent body: namely
+that if NA falling on the external surface AB is refracted into the
+direction AD, so the ray AD will be refracted on leaving the
+transparent body into the direction AN.</p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg039.png" width="350" height="311" alt="" title="" />
+</div>
+
+<p>One sees also the reason for a noteworthy accident which happens in
+this refraction: which is this, that after a certain obliquity of the
+incident ray DA, it begins to be quite unable to penetrate into the
+other transparent substance. For if the angle DAQ or CBA is such that
+in the triangle ACB, CB is equal to 2/3 of AB, or is greater, then AN
+cannot form one side of the triangle ANB, since it becomes equal to or
+greater than AB: so that the portion of wave BN cannot be found
+anywhere, neither consequently can AN, which ought to be perpendicular
+to it. And thus the incident ray DA does not then pierce the surface
+AB.</p>
+
+<p><span class="pagenum">[Pg 40]</span><a name="Page_40" id="Page_40" />When the ratio of the velocities of the waves is as two to three, as
+in our example, which is that which obtains for glass and air, the
+angle DAQ must be more than 48 degrees 11 minutes in order that the
+ray DA may be able to pass by refraction. And when the ratio of the
+velocities is as 3 to 4, as it is very nearly in water and air, this
+angle DAQ must exceed 41 degrees 24 minutes. And this accords
+perfectly with experiment.</p>
+
+<p>But it might here be asked: since the meeting of the wave AC against
+the surface AB ought to produce movement in the matter which is on the
+other side, why does no light pass there? To which the reply is easy
+if one remembers what has been said before. For although it generates
+an infinitude of partial waves in the matter which is at the other
+side of AB, these waves never have a common tangent line (either
+straight or curved) at the same moment; and so there is no line
+terminating the propagation of the wave AC beyond the plane AB, nor
+any place where the movement is gathered together in sufficiently
+great quantity to produce light. And one will easily see the truth of
+this, namely that CB being larger than 2/3 of AB, the waves excited
+beyond the plane AB will have no common tangent if about the centres K
+one then draws circles having radii equal to 3/2 of the lengths LB to
+which they correspond. For all these circles will be enclosed in one
+another and will all pass beyond the point B.</p>
+
+<p>Now it is to be remarked that from the moment when the angle DAQ is
+smaller than is requisite to permit the refracted ray DA to pass into
+the other transparent substance, one finds that the interior reflexion
+which occurs at the surface AB is much augmented in brightness, as <span class="pagenum">[Pg 41]</span><a name="Page_41" id="Page_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 collected in a minimum space when it
+reaches BN. It appears also that the wave BN becomes so much the
+smaller as the angle CBA or DAQ is made less; until when the latter is
+diminished to the limit indicated a little previously, this wave BN is
+collected together always at one point. That is to say, that when the
+piece C of the wave AC has then arrived at B, the wave BN which is the
+propagation of AC is entirely reduced to the same point B. Similarly
+when the piece H has reached K, the part AH is entirely reduced to the
+same point K. This makes it evident that in proportion as the wave CA
+comes to meet the surface AB, there occurs a great quantity of
+movement along that surface; which movement ought also to spread
+within the transparent body and ought to have much re-enforced the
+partial waves which produce the interior reflexion against the surface
+AB, according to the laws of reflexion previously explained.</p>
+
+<p>And because a slight diminution of the angle of incidence DAQ causes
+the wave BN, however great it was, to be reduced to zero, (for this
+angle being 49 degrees 11 minutes in the glass, the angle BAN is still
+11 degrees 21 minutes, and the same angle being reduced by one degree
+only the angle BAN is reduced to zero, and so the wave BN reduced to a
+point) thence it comes about that the interior reflexion from being
+obscure becomes suddenly bright, so soon as the angle of incidence is
+such that it no longer gives passage to the refraction.</p>
+
+<p><span class="pagenum">[Pg 42]</span><a name="Page_42" id="Page_42" />Now as concerns ordinary external reflexion, that is to say which
+occurs when the angle of incidence DAQ is still large enough to enable
+the refracted ray to penetrate beyond the surface AB, this reflexion
+should occur against the particles of the substance which touches the
+transparent body on its outside. And it apparently occurs against the
+particles of the air or others mingled with the ethereal particles and
+larger than they. So on the other hand the external reflexion of these
+bodies occurs against the particles which compose them, and which are
+also larger than those of the ethereal matter, since the latter flows
+in their interstices. It is true that there remains here some
+difficulty in those experiments in which this interior reflexion
+occurs without the particles of air being able to contribute to it, as
+in vessels or tubes from which the air has been extracted.</p>
+
+<p>Experience, moreover, teaches us that these two reflexions are of
+nearly equal force, and that in different transparent bodies they are
+so much the stronger as the refraction of these bodies is the greater.
+Thus one sees manifestly that the reflexion of glass is stronger than
+that of water, and that of diamond stronger than that of glass.</p>
+
+<p>I will finish this theory of refraction by demonstrating a remarkable
+proposition which depends on it; namely, that a ray of light in order
+to go from one point to another, when these points are in different
+media, is refracted in such wise at the plane surface which joins
+these two media that it employs the least possible time: and exactly
+the same happens in the case of reflexion against a plane surface. Mr.
+Fermat was the first to propound this property of refraction, holding
+with us, and directly counter to the opinion of Mr. Des Cartes, that
+light passes <span class="pagenum">[Pg 43]</span><a name="Page_43" id="Page_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
+demonstration, which may be seen in his printed works, and in the
+volume of letters of Mr. Des Cartes, is very long; wherefore I give
+here another which is simpler and easier.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg043.png" width="350" height="320" alt="" title="" />
+</div>
+
+<p>Let KF be the plane surface; A the point in the medium which the light
+traverses more easily, as the air; C the point in the other which is
+more difficult to penetrate, as water. And suppose that a ray has come
+from A, by B, to C, having been refracted at B according to the law
+demonstrated a little before; that is to say that, having drawn PBQ,
+which cuts the plane at right angles, let the sine of the angle ABP
+have to the sine of the angle CBQ the same ratio as the velocity of
+light in the medium where A is to the velocity of light in the medium
+where C is. It is to be shown that the time of passage of light along
+AB and BC taken together, is the shortest that can be. Let us assume
+that it may have come by other lines, and, in the first place, along
+AF, FC, so <span class="pagenum">[Pg 44]</span><a name="Page_44" id="Page_44" />that the point of refraction F may be further from B than
+the point A; and let AO be a line perpendicular to AB, and FO parallel
+to AB; BH perpendicular to FO, and FG to BC.</p>
+
+<p>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.</p>
+
+<p>Now let us assume that the ray has come from A to C along AK, KC; the
+point of refraction K being nearer to A than the point B is; and let
+CN be the perpendicular upon BC, KN parallel to BC: BM perpendicular
+upon KN, and KL upon BA.</p>
+
+<p>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 <span class="pagenum">[Pg 45]</span><a name="Page_45" id="Page_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.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_IV" id="CHAPTER_IV" />CHAPTER IV</h2>
+
+<h3>ON THE REFRACTION OF THE AIR</h3>
+
+
+<div style="width: 208px; float: left; margin-right: .2em;">
+<img src="images/ch04.png" width="208" height="150" alt="W" title="W" />
+</div><p>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
+movement is communicated in it more rapidly toward one side than
+toward another, these waves cannot be spherical: but that they must
+acquire their figure according to the different distances over which
+the successive movement passes in equal times.</p>
+
+<p>It is thus that we shall in the first place explain the refractions
+which occur in the air, which extends from here to the clouds and
+beyond. The effects of which refractions are very remarkable; for by
+them we often see objects which the rotundity of the Earth ought
+otherwise to hide; such as Islands, and the tops of mountains when one
+is at sea. Because also of them the Sun and the Moon appear as risen
+before in fact they have, and appear to set <span class="pagenum">[Pg 46]</span><a name="Page_46" id="Page_46" />later: so that at times
+the Moon has been seen eclipsed while the Sun appeared still above the
+horizon. And so also the heights of the Sun and of the Moon, and those
+of all the Stars always appear a little greater than they are in
+reality, because of these same refractions, as Astronomers know. But
+there is one experiment which renders this refraction very evident;
+which is that of fixing a telescope on some spot so that it views an
+object, such as a steeple or a house, at a distance of half a league
+or more. If then you look through it at different hours of the day,
+leaving it always fixed in the same way, you will see that the same
+spots of the object will not always appear at the middle of the
+aperture of the telescope, but that generally in the morning and in
+the evening, when there are more vapours near the Earth, these objects
+seem to rise higher, so that the half or more of them will no longer
+be visible; and so that they seem lower toward mid-day when these
+vapours are dissipated.</p>
+
+<p>Those who consider refraction to occur only in the surfaces which
+separate transparent bodies of different nature, would find it
+difficult to give a reason for all that I have just related; but
+according to our Theory the thing is quite easy. It is known that the
+air which surrounds us, besides the particles which are proper to it
+and which float in the ethereal matter as has been explained, is full
+also of particles of water which are raised by the action of heat; and
+it has been ascertained further by some very definite experiments that
+as one mounts up higher the density of air diminishes in proportion.
+Now whether the particles of water and those of air take part, by
+means of the particles of ethereal matter, in the movement which
+constitutes light, but have a less prompt recoil than these, <span class="pagenum">[Pg 47]</span><a name="Page_47" id="Page_47" />or
+whether the encounter and hindrance which these particles of air and
+water offer to the propagation of movement of the ethereal progress,
+retard the progression, it follows that both kinds of particles flying
+amidst the ethereal particles, must render the air, from a great
+height down to the Earth, gradually less easy for the spreading of the
+waves of light.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg047.png" width="500" height="302" alt="" title="" />
+</div>
+
+<p>Whence the configuration of the waves ought to become nearly such as
+this figure represents: namely, if A is a light, or the visible point
+of a steeple, the waves which start from it ought to spread more
+widely upwards and less widely downwards, but in other directions more
+or less as they approximate to these two extremes. This being so, it
+necessarily follows that every line intersecting one of these waves at
+right angles will pass above the point A, always excepting the one
+line which is perpendicular to the horizon.<span class="pagenum">[Pg 48]</span><a name="Page_48" id="Page_48" /></p>
+
+<div class="figcenter" style="width: 400px;">
+<img src="images/pg048.png" width="400" height="371" alt="" title="" />
+</div>
+
+<p>Let BC be the wave which brings the light to the spectator who is at
+B, and let BD be the straight line which intersects this wave at right
+angles. Now because the ray or straight line by which we judge the
+spot where the object appears to us is nothing else than the
+perpendicular to the wave that reaches our eye, as will be understood
+by what was said above, it is manifest that the point A will be
+perceived as being in the line BD, and therefore higher than in fact it
+is.</p>
+
+<p>Similarly if the Earth be AB, and the top of the Atmosphere CD, which
+probably is not a well defined spherical surface (since we know that
+the air becomes rare in proportion as one ascends, for above there is
+so much less of it to press down upon it), the waves of light from the
+sun coming, for instance, in such a way that so long as they have not
+reached the Atmosphere CD the straight line AE intersects them
+perpendicularly, they ought, when they enter the Atmosphere, to
+advance more quickly in elevated regions than in regions nearer to the
+Earth. So that if <span class="pagenum">[Pg 49]</span><a name="Page_49" id="Page_49" />CA is the wave which brings the light to the
+spectator at A, its region C will be the furthest advanced; and the
+straight line AF, which intersects this wave at right angles, and
+which determines the apparent place of the Sun, will pass above the
+real Sun, which will be seen along the line AE. And so it may occur
+that when it ought not to be visible in the absence of vapours,
+because the line AE encounters the rotundity of the Earth, it will be
+perceived in the line AF by refraction. But this angle EAF is scarcely
+ever more than half a degree because the attenuation of the vapours
+alters the waves of light but little. Furthermore these refractions
+are not altogether constant in all weathers, particularly at small
+elevations of 2 or 3 degrees; which results from the different
+quantity of aqueous vapours rising above the Earth.</p>
+
+<p>And this same thing is the cause why at certain times a distant object
+will be hidden behind another less distant one, and yet may at another
+time be able to be seen, although the spot from which it is viewed is
+always the same. But the reason for this effect will be still more
+evident from what we are going to remark touching the curvature of
+rays. It appears from the things explained above that the progression
+or propagation of a small part of a wave of light is properly what one
+calls a ray. Now these rays, instead of being straight as they are in
+homogeneous media, ought to be curved in an atmosphere of unequal
+penetrability. For they necessarily follow from the object to the eye
+the line which intersects at right angles all the progressions of the
+waves, as in the first figure the line AEB does, as will be shown
+hereafter; and it is this line which determines what interposed bodies
+would or would not hinder us from seeing the object. For <span class="pagenum">[Pg 50]</span><a name="Page_50" id="Page_50" />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.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg050.png" width="350" height="367" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 51]</span><a name="Page_51" id="Page_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 penetrabilities at the different heights in the air between
+A and B be represented by all these lines; so that the particular
+wave, originating from the point F, will spread across the space FG,
+and that from the point H across the space HI, while that from the
+point A spreads across the space AD.</p>
+
+<p>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 <span class="pagenum">[Pg 52]</span><a name="Page_52" id="Page_52" />arrived at MN
+in such a manner that the perpendiculars KM, LN, dip down even more
+than do AK, BL. And this suffices to show that the ray will continue
+along the curved line which intersects all the waves at right angles,
+as has been said.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_V" id="CHAPTER_V" />CHAPTER V</h2>
+
+<h3>ON THE STRANGE REFRACTION OF ICELAND CRYSTAL</h3>
+
+
+<p>1.</p>
+
+<div style="width: 156px; float: left; margin-right: .2em;">
+<img src="images/ch05.png" width="156" height="150" alt="T" title="T" />
+</div><p>here is brought from Iceland, which is an Island in the North Sea, in
+the latitude of 66 degrees, a kind of Crystal or transparent stone,
+very remarkable for its figure and other qualities, but above all for
+its strange refractions. The causes of this have seemed to me to be
+worthy of being carefully investigated, the more so because amongst
+transparent bodies this one alone does not follow the ordinary rules
+with respect to rays of light. I have even been under some necessity
+to make this research, because the refractions of this Crystal seemed
+to overturn our preceding explanation of regular refraction; which
+explanation, on the contrary, they strongly confirm, as will be seen
+after they have been brought under the same principle. In Iceland are
+found great lumps of this Crystal, some of which I have seen of 4 or 5
+pounds. But it occurs also in other countries, for I have had some of
+the same sort which had been found in France near the town of Troyes
+in Champagne, and some others which came from the Island of Corsica,
+though both were <span class="pagenum">[Pg 53]</span><a name="Page_53" id="Page_53" />less clear and only in little bits, scarcely capable
+of letting any effect of refraction be observed.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg053.png" width="300" height="255" alt="" title="" />
+</div>
+
+<p>4. The pieces of it which are found have the figure of an oblique
+parallelepiped; each of the six faces being a parallelogram; and it
+admits of being split in three directions parallel to two of these
+opposed faces. Even in such wise, if you will, that all the six faces
+are equal and similar rhombuses. The figure here added represents a
+piece of this Crystal. The obtuse angles of all the parallelograms, as
+C, D, here, are angles of 101 degrees 52 minutes, <span class="pagenum">[Pg 54]</span><a name="Page_54" id="Page_54" />and consequently
+the acute angles, such as A and B, are of 78 degrees 8 minutes.</p>
+
+<p>5. Of the solid angles there are two opposite to one another, such as
+C and E, which are each composed of three equal obtuse plane angles.
+The other six are composed of two acute angles and one obtuse. All
+that I have just said has been likewise remarked by Mr. Bartholinus in
+the aforesaid treatise; if we differ it is only slightly about the
+values of the angles. He recounts moreover some other properties of
+this Crystal; to wit, that when rubbed against cloth it attracts
+straws and other light things as do amber, diamond, glass, and Spanish
+wax. Let a piece be covered with water for a day or more, the surface
+loses its natural polish. When aquafortis is poured on it it produces
+ebullition, especially, as I have found, if the Crystal has been
+pulverized. I have also found by experiment that it may be heated to
+redness in the fire without being in anywise altered or rendered less
+transparent; but a very violent fire calcines it nevertheless. Its
+transparency is scarcely less than that of water or of Rock Crystal,
+and devoid of colour. But rays of light pass through it in another
+fashion and produce those marvellous refractions the causes of which I
+am now going to try to explain; reserving for the end of this Treatise
+the statement of my conjectures touching the formation and
+extraordinary configuration of this Crystal.</p>
+
+<p>6. In all other transparent bodies that we know there is but one sole
+and simple refraction; but in this substance there are two different
+ones. The effect is that objects seen through it, especially such as
+are placed right against it, appear double; and that a ray of
+sunlight, falling on one of its surfaces, parts itself into two rays
+and traverses the Crystal thus.</p>
+
+<p><span class="pagenum">[Pg 55]</span><a name="Page_55" id="Page_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
+refracted. But in this Crystal the perpendicular ray suffers
+refraction, and there are oblique rays which pass through it quite
+straight.</p>
+
+<div class="figcenter" style="width: 450px;">
+<img src="images/pg055.png" width="450" height="444" alt="" title="" />
+</div>
+
+<p>8. But in order to explain these phenomena more particularly, let
+there be, in the first place, a piece ABFE of the same Crystal, and
+let the obtuse angle ACB, one of the three which constitute the
+equilateral solid angle C, be divided into two equal parts by the
+straight line CG, and let it be conceived that the Crystal is
+intersected by a plane which passes through this line and through the
+side CF, which plane will necessarily be perpendicular to <span class="pagenum">[Pg 56]</span><a name="Page_56" id="Page_56" />the surface
+AB; and its section in the Crystal will form a parallelogram GCFH. We
+will call this section the principal section of the Crystal.</p>
+
+<p>9. Now if one covers the surface AB, leaving there only a small
+aperture at the point K, situated in the straight line CG, and if one
+exposes it to the sun, so that his rays face it perpendicularly above,
+then the ray IK will divide itself at the point K into two, one of
+which will continue to go on straight by KL, and the other will
+separate itself along the straight line KM, which is in the plane
+GCFH, and which makes with KL an angle of about 6 degrees 40 minutes,
+tending from the side of the solid angle C; and on emerging from the
+other side of the Crystal it will turn again parallel to JK, along MZ.
+And as, in this extraordinary refraction, the point M is seen by the
+refracted ray MKI, which I consider as going to the eye at I, it
+necessarily follows that the point L, by virtue of the same
+refraction, will be seen by the refracted ray LRI, so that LR will be
+parallel to MK if the distance from the eye KI is supposed very great.
+The point L appears then as being in the straight line IRS; but the
+same point appears also, by ordinary refraction, to be in the straight
+line IK, hence it is necessarily judged to be double. And similarly if
+L be a small hole in a sheet of paper or other substance which is laid
+against the Crystal, it will appear when turned towards daylight as if
+there were two holes, which will seem the wider apart from one another
+the greater the thickness of the Crystal.</p>
+
+<p>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 <span class="pagenum">[Pg 57]</span><a name="Page_57" id="Page_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 calculation which I
+shall put at the end, it will divide itself at the point O into two
+rays, one of which will continue along OP in a straight line with NO,
+and will similarly pass out of the other side of the crystal without
+any refraction; but the other will be refracted and will go along OQ.
+And it must be noted that it is special to the plane through GCF and
+to those which are parallel to it, that all incident rays which are in
+one of these planes continue to be in it after they have entered the
+Crystal and have become double; for it is quite otherwise for rays in
+all other planes which intersect the Crystal, as we shall see
+afterwards.</p>
+
+<p>11. I recognized at first by these experiments and by some others that
+of the two refractions which the ray suffers in this Crystal, there is
+one which follows the ordinary rules; and it is this to which the rays
+KL and OQ belong. This is why I have distinguished this ordinary
+refraction from the other; and having measured it by exact
+observation, I found that its proportion, considered as to the Sines
+of the angles which the incident and refracted rays make with the
+perpendicular, was very precisely that of 5 to 3, as was found also by
+Mr. Bartholinus, and consequently much greater than that of Rock
+Crystal, or of glass, which is nearly 3 to 2.</p>
+
+<div class="figright" style="width: 400px;">
+<img src="images/pg058.png" width="400" height="307" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 58]</span><a name="Page_58" id="Page_58" />as it
+is desired to examine a ray that is more or less oblique. Then place
+the Crystal upon the intersection E so that the line AB concurs with
+that which bisects the obtuse angle of the lower surface, or with some
+line parallel to it. Then by placing the eye directly above the line
+AB it will appear single only; and one will see that the portion
+viewed through the Crystal and the portions which appear outside it,
+meet together in a straight line: but the line CD will appear double,
+and one can distinguish the image which is due to regular refraction
+by the circumstance that when one views it with both eyes it seems
+raised up more than the other, or again by the circumstance that, when
+the Crystal is turned around on the paper, this image remains
+stationary, whereas the other image shifts and moves entirely around.
+Afterwards let the eye be placed at I (remaining <span class="pagenum">[Pg 59]</span><a name="Page_59" id="Page_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.</p>
+
+<p>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.</p>
+
+<p>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, <span class="pagenum">[Pg 60]</span><a name="Page_60" id="Page_60" />RES, which the incident and the refracted ray make with
+the perpendicular.</p>
+
+<p>15. But I found in this refraction that the ratio of FR to RS was not
+constant, like the ordinary refraction, but that it varied with the
+varying obliquity of the incident ray.</p>
+
+<p>16. I found also that when QRE made a straight line, that is, when the
+incident ray entered the Crystal without being refracted (as I
+ascertained by the circumstance that then the point E viewed by the
+extraordinary refraction appeared in the line CD, as seen without
+refraction) I found, I say, then that the angle QRG was 73 degrees 20
+minutes, as has been already remarked; and so it is not the ray
+parallel to the edge of the Crystal, which crosses it in a straight
+line without being refracted, as Mr. Bartholinus believed, since that
+inclination is only 70 degrees 57 minutes, as was stated above. And
+this is to be noted, in order that no one may search in vain for the
+cause of the singular property of this ray in its parallelism to the
+edges mentioned.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg060.png" width="350" height="347" alt="" title="" />
+</div>
+
+<p>17. Finally, continuing my observations to discover the <span class="pagenum">[Pg 61]</span><a name="Page_61" id="Page_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 refractions KX and KT
+meet the bottom line HF in such wise that points X and T are equally
+distant from the point M, where the refraction of the perpendicular
+ray IK falls; and this occurs also for refractions in other sections
+of this Crystal. But before speaking of those, which have also other
+particular properties, we will investigate the causes of the phenomena
+which I have already reported.</p>
+
+<p>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.</p>
+
+<p>18. As there were two different refractions, I conceived that there
+were also two different emanations of waves of light, and that one
+could occur in the ethereal matter extending through the body of the
+Crystal. Which matter, being present in much larger quantity than is
+that of the particles which compose it, was alone capable of causing
+transparency, according to what has been explained heretofore. I
+attributed to this emanation of waves the regular refraction which is
+observed in this stone, by supposing these waves to be ordinarily of
+spherical form, and having a slower progression within the Crystal
+than they have outside it; whence proceeds refraction as I have
+demonstrated.</p>
+
+<p>19. As to the other emanation which should produce <span class="pagenum">[Pg 62]</span><a name="Page_62" id="Page_62" />the irregular
+refraction, I wished to try what Elliptical waves, or rather
+spheroidal waves, would do; and these I supposed would spread
+indifferently both in the ethereal matter diffused throughout the
+crystal and in the particles of which it is composed, according to the
+last mode in which I have explained transparency. It seemed to me that
+the disposition or regular arrangement of these particles could
+contribute to form spheroidal waves (nothing more being required for
+this than that the successive movement of light should spread a little
+more quickly in one direction than in the other) and I scarcely
+doubted that there were in this crystal such an arrangement of equal
+and similar particles, because of its figure and of its angles with
+their determinate and invariable measure. Touching which particles,
+and their form and disposition, I shall, at the end of this Treatise,
+propound my conjectures and some experiments which confirm them.</p>
+
+<p>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.</p>
+
+<p><span class="pagenum">[Pg 63]</span><a name="Page_63" id="Page_63" />21. Now this double refraction, according to my Theory hereinbefore
+established, seemed to demand a double emission of waves of light,
+both of them spherical (for both the refractions are regular) and
+those of one series a little slower only than the others. For thus the
+phenomenon is quite naturally explained, by postulating substances
+which serve as vehicle for these waves, as I have done in the case of
+Iceland Crystal. I had then less trouble after that in admitting two
+emissions of waves in one and the same body. And since it might have
+been objected that in composing these two kinds of crystal of equal
+particles of a certain figure, regularly piled, the interstices which
+these particles leave and which contain the ethereal matter would
+scarcely suffice to transmit the waves of light which I have localized
+there, I removed this difficulty by regarding these particles as being
+of a very rare texture, or rather as composed of other much smaller
+particles, between which the ethereal matter passes quite freely.
+This, moreover, necessarily follows from that which has been already
+demonstrated touching the small quantity of matter of which the bodies
+are built up.</p>
+
+<p>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.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg064.png" width="300" height="253" alt="" title="" />
+</div>
+
+<p>23. I considered first the effect of waves so formed, as respects the
+ray which falls perpendicularly on the flat surface of a transparent
+body in which they should spread in this manner. I took AB for the
+exposed region of the surface. And, since a ray perpendicular to a
+plane, and <span class="pagenum">[Pg 64]</span><a name="Page_64" id="Page_64" />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<i>h</i>C come to meet the
+surface AB at the points AK<i>k</i>B. Then instead of the hemispherical
+partial waves which in a body of ordinary refraction would spread from
+each of these last points, as we have above explained in treating of
+refraction, these must here be hemi-spheroids. The axes (or rather the
+major diameters) of these I supposed to be oblique to the plane AB, as
+is AV the semi-axis or semi-major diameter of the spheroid SVT, which
+represents the partial wave coming from the point A, after the wave RC
+has reached AB. I say axis or major diameter, because the same ellipse
+SVT may be considered as the section of a spheroid of which the axis
+is AZ perpendicular to AV. But, for the present, without yet deciding
+one or other, we will consider these spheroids only in those sections
+of them which make ellipses in the plane of this figure. Now taking a
+certain space of time during which the wave SVT has spread from A, it
+would needs be that from all the other points K<i>k</i>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 <span class="pagenum">[Pg 65]</span><a name="Page_65" id="Page_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.</p>
+
+<p>24. Now it appeared that this common tangent NQ was parallel to AB,
+and of the same length, but that it was not directly opposite to it,
+since it was comprised between the lines AN, BQ, which are diameters
+of ellipses having A and B for centres, conjugate with respect to
+diameters which are not in the straight line AB. And in this way I
+comprehended, a matter which had seemed to me very difficult, how a
+ray perpendicular to a surface could suffer refraction on entering a
+transparent body; seeing that the wave RC, having come to the aperture
+AB, went on forward thence, spreading between the parallel lines AN,
+BQ, yet itself remaining always parallel to AB, so that here the light
+does not spread along lines perpendicular to its waves, as in ordinary
+refraction, but along lines cutting the waves obliquely.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg065.png" width="300" height="253" alt="" title="" />
+</div>
+
+<p>25. Inquiring subsequently what might be the position and form of
+these spheroids in the crystal, I considered that all the six faces
+produced precisely the same refractions. Taking, then, the
+parallelopiped AFB, of which the obtuse solid angle C is contained
+between the three equal plane angles, and imagining in it the three
+principal sections, one of which is perpendicular to the face DC and
+passes through the edge CF, another perpendicular to the face BF
+passing through the edge <span class="pagenum">[Pg 66]</span><a name="Page_66" id="Page_66" />CA, and the third perpendicular to the face
+AF passing through the edge BC; I knew that the refractions of the
+incident rays belonging to these three planes were all similar. But
+there could be no position of the spheroid which would have the same
+relation to these three sections except that in which the axis was
+also the axis of the solid angle C. Consequently I saw that the axis
+of this angle, that is to say the straight line which traversed the
+crystal from the point C with equal inclination to the edges CF, CA,
+CB was the line which determined the position of the axis of all the
+spheroidal waves which one imagined to originate from some point,
+taken within or on the surface of the crystal, since all these
+spheroids ought to be alike, and have their axes parallel to one
+another.</p>
+
+<p>26. Considering after this the plane of one of these three sections,
+namely that through GCF, the angle of which is 109 degrees 3 minutes,
+since the angle F was shown above to be 70 degrees 57 minutes; and,
+imagining a spheroidal wave about the centre C, I knew, because I have
+just explained it, that its axis must be in the same plane, the half
+of which axis I have marked CS in the next figure: and seeking by
+calculation (which will be given with others at the end of this
+discourse) the value of the angle CGS, I found it 45 degrees 20
+minutes.</p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg067.png" width="350" height="234" alt="" title="" />
+</div>
+
+<p>27. To know from this the form of this spheroid, that is to say the
+proportion of the semi-diameters CS, CP, of its elliptical section,
+which are perpendicular to one another, I considered that the point M
+where the ellipse is touched by the straight line FH, parallel to CG,
+ought to be so situated that CM makes with the perpendicular CL an
+angle of 6 degrees 40 minutes; since, this being so, this ellipse
+satisfies what has been said about the refraction of <span class="pagenum">[Pg 67]</span><a name="Page_67" id="Page_67" />the ray
+perpendicular to the surface CG, which is inclined to the
+perpendicular CL by the same angle. This, then, being thus disposed,
+and taking CM at 100,000 parts, I found by the calculation which will
+be given at the end, the semi-major diameter CP to be 105,032, and the
+semi-axis CS to be 93,410, the ratio of which numbers is very nearly 9
+to 8; so that the spheroid was of the kind which resembles a
+compressed sphere, being generated by the revolution of an ellipse
+about its smaller diameter. I found also the value of CG the
+semi-diameter parallel to the tangent ML to be 98,779.</p>
+
+<div class="figleft" style="width: 500px;">
+<img src="images/pg068.png" width="500" height="314" alt="" title="" />
+</div>
+
+<p>28. Now passing to the investigation of the refractions which
+obliquely incident rays must undergo, according to our hypothesis of
+spheroidal waves, I saw that these refractions depended on the ratio
+between the velocity of movement of the light outside the crystal in
+the ether, and that within the crystal. For supposing, for example,
+this proportion to be such that while the light in the crystal forms
+the spheroid GSP, as I have just said, it forms outside a sphere the
+semi-diameter of which is equal to the line N which will be determined
+hereafter, the following is the way of finding the refraction of the
+incident rays. Let there be such a ray RC falling upon the <span class="pagenum">[Pg 68]</span><a name="Page_68" id="Page_68" />surface
+CK. Make CO perpendicular to RC, and across the angle KCO adjust OK,
+equal to N and perpendicular to CO; then draw KI, which touches the
+Ellipse GSP, and from the point of contact I join IC, which will be
+the required refraction of the ray RC. The demonstration of this is,
+it will be seen, entirely similar to that of which we made use in
+explaining ordinary refraction. For the refraction of the ray RC is
+nothing else than the progression of the portion C of the wave CO,
+continued in the crystal. Now the portions H of this wave, during the
+time that O came to K, will have arrived at the surface CK along the
+straight lines H<i>x</i>, and will moreover have produced in the crystal
+around the centres <i>x</i> some hemi-spheroidal partial waves similar to
+the hemi-spheroidal GSP<i>g</i>, and similarly disposed, and of which the
+major <span class="pagenum">[Pg 69]</span><a name="Page_69" id="Page_69" />and minor diameters will bear the same proportions to the lines
+<i>xv</i> (the continuations of the lines H<i>x</i> up to KB parallel to CO)
+that the diameters of the spheroid GSP<i>g</i> 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.</p>
+
+<p>Now as to finding the point of contact I, it is known that one must
+find CD a third proportional to the lines CK, CG, and draw DI parallel
+to CM, previously determined, which is the conjugate diameter to CG;
+for then, by drawing KI it touches the Ellipse at I.</p>
+
+<p>29. Now as we have found CI the refraction of the ray RC, similarly
+one will find C<i>i</i> the refraction of the ray <i>r</i>C, which comes from
+the opposite side, by making C<i>o</i> perpendicular to <i>r</i>C and following
+out the rest of the construction as before. Whence one sees that if
+the ray <i>r</i>C is inclined equally with RC, the line C<i>d</i> will
+necessarily be equal to CD, because C<i>k</i> is equal to CK, and C<i>g</i> to
+CG. And in consequence I<i>i</i> will be cut at E into equal parts by the
+line CM, to which DI and <i>di</i> are parallel. And because CM is the
+conjugate diameter to CG, it follows that <i>i</i>I will be parallel to
+<i>g</i>G. Therefore if one prolongs the refracted rays CI, C<i>i</i>, until
+they meet the tangent ML at T and <i>t</i>, the distances MT, M<i>t</i>, will
+also be equal. And so, by our hypothesis, we explain perfectly the
+phenomenon mentioned above; to wit, that when there are two rays
+equally inclined, but coming from opposite sides, as here the rays RC,
+<i>rc</i>, their refractions diverge equally from the line <span class="pagenum">[Pg 70]</span><a name="Page_70" id="Page_70" />followed by the
+refraction of the ray perpendicular to the surface, by considering
+these divergences in the direction parallel to the surface of the
+crystal.</p>
+
+<p>30. To find the length of the line N, in proportion to CP, CS, CG, it
+must be determined by observations of the irregular refraction which
+occurs in this section of the crystal; and I find thus that the ratio
+of N to GC is just a little less than 8 to 5. And having regard to
+some other observations and phenomena of which I shall speak
+afterwards, I put N at 156,962 parts, of which the semi-diameter CG is
+found to contain 98,779, making this ratio 8 to 5-1/29. Now this
+proportion, which there is between the line N and CG, may be called
+the Proportion of the Refraction; similarly as in glass that of 3 to
+2, as will be manifest when I shall have explained a short process in
+the preceding way to find the irregular refractions.</p>
+
+<p>31. Supposing then, in the next figure, as previously, the surface of
+the crystal <i>g</i>G, the Ellipse GP<i>g</i>, and the line N; and CM the
+refraction of the perpendicular ray FC, from which it diverges by 6
+degrees 40 minutes. Now let there be some other ray RC, the refraction
+of which must be found.</p>
+
+<p>About the centre C, with semi-diameter CG, let the circumference <i>g</i>RG
+be described, cutting the ray RC at R; and let RV be the perpendicular
+on CG. Then as the line N is to CG let CV be to CD, and let DI be
+drawn parallel to CM, cutting the Ellipse <i>g</i>MG at I; then joining CI,
+this will be the required refraction of the ray RC. Which is
+demonstrated thus.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg071.png" width="500" height="364" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 71]</span><a name="Page_71" id="Page_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 refraction of the ray RC. Now since the angle
+RCO is a right angle, it is easy to see that the right-angled
+triangles RCV, KCO, are similar. As then, CK is to KO, so also is RC
+to CV. But KO is equal to N, and RC to CG: then as CK is to N so will
+CG be to CV. But as N is to CG, so, by construction, is CV to CD. Then
+as CK is to CG so is CG to CD. And because DI is parallel to CM, the
+conjugate diameter to CG, it follows that KI touches the Ellipse at I;
+which remained to be shown.</p>
+
+<p>32. One sees then that as there is in the refraction of <span class="pagenum">[Pg 72]</span><a name="Page_72" id="Page_72" />ordinary
+media a certain constant proportion between the sines of the angles
+which the incident ray and the refracted ray make with the
+perpendicular, so here there is such a proportion between CV and CD or
+IE; that is to say between the Sine of the angle which the incident
+ray makes with the perpendicular, and the horizontal intercept, in the
+Ellipse, between the refraction of this ray and the diameter CM. For
+the ratio of CV to CD is, as has been said, the same as that of N to
+the semi-diameter CG.</p>
+
+<p>33. I will add here, before passing away, that in comparing together
+the regular and irregular refraction of this crystal, there is this
+remarkable fact, that if ABPS be the spheroid by which light spreads
+in the Crystal in a certain space of time (which spreading, as has
+been said, serves for the irregular refraction), then the inscribed
+sphere BVST is the extension in the same space of time of the light
+which serves for the regular refraction.</p>
+
+<div class="figright" style="width: 250px;">
+<img src="images/pg072.png" width="250" height="314" alt="" title="" />
+</div>
+
+<p>For we have stated before this, that the line N being the radius of a
+spherical wave of light in air, while in the crystal it spread through
+the spheroid ABPS, the ratio of N to CS will be 156,962 to 93,410. But
+it has also been stated that the proportion of the regular refraction
+was 5 to 3; that is to say, that N being the radius of a spherical
+wave of light in air, its extension in the crystal would, in the same
+space of time, form a sphere the radius of which would be to N as 3 to
+5. Now 156,962 is to 93,410 as 5 to 3 less 1/41. So that it is
+sufficiently nearly, and may <span class="pagenum">[Pg 73]</span><a name="Page_73" id="Page_73" />be exactly, the sphere BVST, which the
+light describes for the regular refraction in the crystal, while it
+describes the spheroid BPSA for the irregular refraction, and while it
+describes the sphere of radius N in air outside the crystal.</p>
+
+<p>Although then there are, according to what we have supposed, two
+different propagations of light within the crystal, it appears that it
+is only in directions perpendicular to the axis BS of the spheroid
+that one of these propagations occurs more rapidly than the other; but
+that they have an equal velocity in the other direction, namely, in
+that parallel to the same axis BS, which is also the axis of the
+obtuse angle of the crystal.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg073.png" width="300" height="336" alt="" title="" />
+</div>
+
+<p>34. The proportion of the refraction being what we have just seen, I
+will now show that there necessarily follows thence that notable
+property of the ray which falling obliquely on the surface of the
+crystal enters it without suffering refraction. For supposing the same
+things as before, and that the ray makes with the same surface <i>g</i>G
+the angle RCG of <span class="pagenum">[Pg 74]</span><a name="Page_74" id="Page_74" />73 degrees 20 minutes, inclining to the same side as
+the crystal (of which ray mention has been made above); if one
+investigates, by the process above explained, the refraction CI, one
+will find that it makes exactly a straight line with RC, and that thus
+this ray is not deviated at all, conformably with experiment. This is
+proved as follows by calculation.</p>
+
+<p>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.</p>
+
+<p>Now the rectangle <i>g</i>DC is to the square of DI as the square of CG is
+to the square of CM; hence DI or CE will be 98,353. But as CE is to
+EI, so will CM be to MT, which will then be 18,127. And being added to
+ML, which is 11,609 (namely the sine of the angle LCM, which is 6
+degrees 40 minutes, taking CM 100,000 as radius) we get LT 27,936; and
+this is to LC 99,324 as CV to VR, that is to say, as 29,938, the
+tangent of the complement of the angle RCV, which is 73 degrees 20
+minutes, is to the radius of the Tables. Whence it appears that RCIT
+is a straight line; which was to be proved.</p>
+
+<p>35. Further it will be seen that the ray CI in emerging through the
+opposite surface of the crystal, ought to pass out quite straight,
+according to the following demonstration, which proves that the
+reciprocal relation of refraction obtains in this crystal the same as
+in other transparent bodies; that is to say, that if a ray RC in
+meeting the surface of the crystal CG is refracted as CI, the ray CI
+emerging through the opposite parallel surface of the <span class="pagenum">[Pg 75]</span><a name="Page_75" id="Page_75" />crystal, which
+I suppose to be IB, will have its refraction IA parallel to the ray
+RC.</p>
+
+<div class="figright" style="width: 400px;">
+<img src="images/pg075.png" width="400" height="337" alt="" title="" />
+</div>
+
+<p>Let the same things be supposed as before; that is to say, let CO,
+perpendicular to CR, represent a portion of a wave the continuation of
+which in the crystal is IK, so that the piece C will be continued on
+along the straight line CI, while O comes to K. Now if one takes a
+second period of time equal to the first, the piece K of the wave IK
+will, in this second period, have advanced along the straight line KB,
+equal and parallel to CI, because every piece of the wave CO, on
+arriving at the surface CK, ought to go on in the crystal the same as
+the piece C; and in this same time there will be formed in the air
+from the point I a partial spherical wave having a semi-diameter IA
+equal to KO, since KO has been traversed in an equal time. Similarly,
+if one considers some other point of the wave IK, such as <i>h</i>, it will
+go along <i>hm</i>, parallel to CI, to meet the surface IB, while the point
+K traverses K<i>l</i> equal to <i>hm</i>; and while this accomplishes the
+remainder <i>l</i>B, there will start from the point <i>m</i> a partial wave the
+semi-diameter of which, <i>mn</i>, will have the same ratio to <i>l</i>B as IA
+to <span class="pagenum">[Pg 76]</span><a name="Page_76" id="Page_76" />KB. Whence it is evident that this wave of semi-diameter <i>mn</i>, and
+the other of semi-diameter IA will have the same tangent BA. And
+similarly for all the partial spherical waves which will be formed
+outside the crystal by the impact of all the points of the wave IK
+against the surface of the Ether IB. It is then precisely the tangent
+BA which will be the continuation of the wave IK, outside the crystal,
+when the piece K has reached B. And in consequence IA, which is
+perpendicular to BA, will be the refraction of the ray CI on emerging
+from the crystal. Now it is clear that IA is parallel to the incident
+ray RC, since IB is equal to CK, and IA equal to KO, and the angles A
+and O are right angles.</p>
+
+<p>It is seen then that, according to our hypothesis, the reciprocal
+relation of refraction holds good in this crystal as well as in
+ordinary transparent bodies; as is thus in fact found by observation.</p>
+
+<p>36. I pass now to the consideration of other sections of the crystal,
+and of the refractions there produced, on which, as will be seen, some
+other very remarkable phenomena depend.</p>
+
+<p>Let ABH be a parallelepiped of crystal, and let the top surface AEHF
+be a perfect rhombus, the obtuse angles of which are equally divided
+by the straight line EF, and the acute angles by the straight line AH
+perpendicular to FE.</p>
+
+<p>The section which we have hitherto considered is that which passes
+through the lines EF, EB, and which at the same time cuts the plane
+AEHF at right angles. Refractions in this section have this in common
+with the refractions in ordinary media that the plane which is drawn
+through the incident ray and which also intersects the <span class="pagenum">[Pg 77]</span><a name="Page_77" id="Page_77" />surface of the
+crystal at right angles, is that in which the refracted ray also is
+found. But the refractions which appertain to every other section of
+this crystal have this strange property that the refracted ray always
+quits the plane of the incident ray perpendicular to the surface, and
+turns away towards the side of the slope of the crystal. For which
+fact we shall show the reason, in the first place, for the section
+through AH; and we shall show at the same time how one can determine
+the refraction, according to our hypothesis. Let there be, then, in
+the plane which passes through AH, and which is perpendicular to the
+plane AFHE, the incident ray RC; it is required to find its refraction
+in the crystal.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg077.png" width="500" height="432" alt="" title="" />
+</div>
+
+<p><span class="pagenum">[Pg 78]</span><a name="Page_78" id="Page_78" />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<i>qg</i>M, such as the
+light would form in spreading in the crystal, and let its section by
+the plane AEHF form the Ellipse QG<i>qg</i>, the major diameter of which
+Q<i>q</i>, which is in the line AH, will necessarily be one of the major
+diameters of the spheroid; because the axis of the spheroid being in
+the plane through FEB, to which QC is perpendicular, it follows that
+QC is also perpendicular to the axis of the spheroid, and consequently
+QC<i>q</i> one of its major diameters. But the minor diameter of this
+Ellipse, G<i>g</i>, will bear to Q<i>q</i> 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.</p>
+
+<p>Let the line N be the length of the travel of light in air during the
+time in which, within the crystal, it makes, from the centre C, the
+spheroid QC<i>qg</i>M. Then having drawn CO perpendicular to the ray CR and
+situate in the plane through CR and AH, let there be adjusted, across
+the angle ACO, the straight line OK equal to N and perpendicular to
+CO, and let it meet the straight line AH at K. Supposing consequently
+that CL is perpendicular to the surface of the crystal AEHF, and that
+CM is the refraction of the ray which falls perpendicularly on this
+same surface, let there be drawn a plane through the line CM and
+through KCH, making in the spheroid the semi-ellipse QM<i>q</i>, 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 <span class="pagenum">[Pg 79]</span><a name="Page_79" id="Page_79" />straight line CM to meet the surface, would be parallel
+to the plane QG<i>q</i>. If then through the point K one now draws KS
+parallel to G<i>g</i>, which will be parallel also to QX, the tangent to
+the Ellipse QG<i>q</i> 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<i>q</i>, because this plane through KS, as well as the
+plane which touches the spheroid at the point M, are parallel to QX,
+the tangent of the spheroid: for this consequence will be demonstrated
+at the end of this Treatise. Let this point of contact be at I, then
+making KC, QC, DC proportionals, draw DI parallel to CM; also join CI.
+I say that CI will be the required refraction of the ray RC. This will
+be manifest if, in considering CO, which is perpendicular to the ray
+RC, as a portion of the wave of light, we can demonstrate that the
+continuation of its piece C will be found in the crystal at I, when O
+has arrived at K.</p>
+
+<p>38. Now as in the Chapter on Reflexion, in demonstrating that the
+incident and reflected rays are always in the same plane perpendicular
+to the reflecting surface, we considered the breadth of the wave of
+light, so, similarly, we must here consider the breadth of the wave CO
+in the diameter G<i>g</i>. Taking then the breadth C<i>c</i> on the side toward
+the angle E, let the parallelogram CO<i>oc</i> be taken as a portion of a
+wave, and let us complete the parallelograms CK<i>kc</i>, CI<i>ic</i>, Kl<i>ik</i>,
+OK<i>ko</i>. In the time then that the line O<i>o</i> arrives at the surface of
+the crystal at K<i>k</i>, all the points of the wave CO<i>oc</i> will have
+arrived at the rectangle K<i>c</i> 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 <span class="pagenum">[Pg 80]</span><a name="Page_80" id="Page_80" />hemi-spheroid QM<i>q</i>,
+and similarly disposed. These hemi-spheroids will necessarily all
+touch the plane of the parallelogram KI<i>ik</i> at the same instant that
+O<i>o</i> has reached K<i>k</i>. Which is easy to comprehend, since, of these
+hemi-spheroids, all those which have their centres along the line CK,
+touch this plane in the line KI (for this is to be shown in the same
+way as we have demonstrated the refraction of the oblique ray in the
+principal section through EF) and all those which have their centres
+in the line C<i>c</i> will touch the same plane KI in the line I<i>i</i>; all
+these being similar to the hemi-spheroid QM<i>q</i>. Since then the
+parallelogram K<i>i</i> is that which touches all these spheroids, this
+same parallelogram will be precisely the continuation of the wave
+CO<i>oc</i> in the crystal, when O<i>o</i> has arrived at K<i>k</i>, because it forms
+the termination of the movement and because of the quantity of
+movement which occurs more there than anywhere else: and thus it
+appears that the piece C of the wave CO<i>oc</i> has its continuation at I;
+that is to say, that the ray RC is refracted as CI.</p>
+
+<p>From this it is to be noted that the proportion of the refraction for
+this section of the crystal is that of the line N to the semi-diameter
+CQ; by which one will easily find the refractions of all incident
+rays, in the same way as we have shown previously for the case of the
+section through FE; and the demonstration will be the same. But it
+appears that the said proportion of the refraction is less here than
+in the section through FEB; for it was there the same as the ratio of
+N to CG, that is to say, as 156,962 to 98,779, very nearly as 8 to 5;
+and here it is the ratio of N to CQ the major semi-diameter of the
+spheroid, that is to say, as 156,962 to 105,032, very nearly <span class="pagenum">[Pg 81]</span><a name="Page_81" id="Page_81" />as 3 to
+2, but just a little less. Which still agrees perfectly with what one
+finds by observation.</p>
+
+<p>39. For the rest, this diversity of proportion of refraction produces
+a very singular effect in this Crystal; which is that when it is
+placed upon a sheet of paper on which there are letters or anything
+else marked, if one views it from above with the two eyes situated in
+the plane of the section through EF, one sees the letters raised up by
+this irregular refraction more than when one puts one's eyes in the
+plane of section through AH: and the difference of these elevations
+appears by comparison with the other ordinary refraction of the
+crystal, the proportion of which is as 5 to 3, and which always raises
+the letters equally, and higher than the irregular refraction does.
+For one sees the letters and the paper on which they are written, as
+on two different stages at the same time; and in the first position of
+the eyes, namely, when they are in the plane through AH these two
+stages are four times more distant from one another than when the eyes
+are in the plane through EF.</p>
+
+<p>We will show that this effect follows from the refractions; and it
+will enable us at the same time to ascertain the apparent place of a
+point of an object placed immediately under the crystal, according to
+the different situation of the eyes.</p>
+
+<p>40. Let us see first by how much the irregular refraction of the plane
+through AH ought to lift the bottom of the crystal. Let the plane of
+this figure represent separately the section through Q<i>q</i> and CL, in
+which section there is also the ray RC, and let the semi-elliptic
+plane through Q<i>q</i> and CM be inclined to the former, as previously, by
+an angle of 6 degrees 40 minutes; and in this plane CI is then the
+refraction of the ray RC.<span class="pagenum">[Pg 82]</span><a name="Page_82" id="Page_82" /></p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg082.png" width="350" height="336" alt="" title="" />
+</div>
+
+<p>If now one considers the point I as at the bottom of the crystal, and
+that it is viewed by the rays ICR, <i>Icr</i>, refracted equally at the
+points C<i>c</i>, which should be equally distant from D, and that these
+rays meet the two eyes at R<i>r</i>; it is certain that the point I will
+appear raised to S where the straight lines RC, <i>rc</i>, meet; which
+point S is in DP, perpendicular to Q<i>q</i>. 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.</p>
+
+<p>Let there be described on Q<i>q</i> a semicircle cutting the ray CR at B,
+from which BV is drawn perpendicular to Q<i>q</i>; and let the proportion
+of the refraction for this section be, as before, that of the line N
+to the semi-diameter CQ.</p>
+
+<p>Then as N is to CQ so is VC to CD, as appears by the method of finding
+the refraction which we have shown above, Article 31; but as VC is to
+CD, so is VB to DS. Then as N is to CQ, so is VB to DS. Let ML be
+perpendicular to CL. And because I suppose the eyes R<i>r</i> to be distant
+about a foot or so from the crystal, and consequently the angle RS<i>r</i>
+very small, VB may be considered as equal to the semi-diameter CQ, and
+DP as equal to CL; then as N is to <span class="pagenum">[Pg 83]</span><a name="Page_83" id="Page_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, considered as equal to CL, will be to DS as 99,324 to
+70,283. And so the elevation of the point I by the refraction of this
+section is known.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg083.png" width="350" height="370" alt="" title="" />
+</div>
+
+<p>41. Now let there be represented the other section through EF in the
+figure before the preceding one; and let CM<i>g</i> be the semi-ellipse,
+considered in Articles 27 and 28, which is made by cutting a
+spheroidal wave having centre C. Let the point I, taken in this
+ellipse, be imagined again at the bottom of the Crystal; and let it be
+viewed by the refracted rays ICR, I<i>cr</i>, which go to the two eyes; CR
+and <i>cr</i> being equally inclined to the surface of the crystal G<i>g</i>.
+This being so, if one draws ID parallel to CM, which I suppose to be
+the refraction of the perpendicular ray incident at the point C, the
+distances DC, D<i>c</i>, will be equal, as is easy to see by that which has
+been demonstrated in Article 28. Now it is certain that the point I
+should appear at S where the straight lines RC, <i>rc</i>, meet when
+prolonged; and that this point will fall in the <span class="pagenum">[Pg 84]</span><a name="Page_84" id="Page_84" />line DP perpendicular
+to G<i>g</i>. If one draws IP perpendicular to this DP, it will be the
+distance PS which will mark the apparent elevation of the point I. Let
+there be described on G<i>g</i> a semicircle cutting CR at B, from which
+let BV be drawn perpendicular to G<i>g</i>; and let N to GC be the
+proportion of the refraction in this section, as in Article 28. Since
+then CI is the refraction of the radius BC, and DI is parallel to CM,
+VC must be to CD as N to GC, according to what has been demonstrated
+in Article 31. But as VC is to CD so is BV to DS. Let ML be drawn
+perpendicular to CL. And because I consider, again, the eyes to be
+distant above the crystal, BV is deemed equal to the semi-diameter CG;
+and hence DS will be a third proportional to the lines N and CG: also
+DP will be deemed equal to CL. Now CG consisting of 98,778 parts, of
+which CM contains 100,000, N is taken as 156,962. Then DS will be
+62,163. But CL is also determined, and contains 99,324 parts, as has
+been said in Articles 34 and 40. Then the ratio of PD to DS will be as
+99,324 to 62,163. And thus one knows the elevation of the point at the
+bottom I by the refraction of this section; and it appears that this
+elevation is greater than that by the refraction of the preceding
+section, since the ratio of PD to DS was there as 99,324 to 70,283.</p>
+
+<div class="figleft" style="width: 150px;">
+<img src="images/pg084.png" width="150" height="287" alt="" title="" />
+</div>
+
+<p>But by the regular refraction of the crystal, of which we have above
+said that the proportion is 5 to 3, the elevation of the point I, or
+P, from the bottom, will be 2/5 of the height DP; as appears by this
+figure, where the point P being viewed by the rays PCR, P<i>cr</i>,
+refracted equally <span class="pagenum">[Pg 85]</span><a name="Page_85" id="Page_85" />at the surface C<i>c</i>, this point must needs appear
+to be at S, in the perpendicular PD where the lines RC, <i>rc</i>, meet
+when prolonged: and one knows that the line PC is to CS as 5 to 3,
+since they are to one another as the sine of the angle CSP or DSC is
+to the sine of the angle SPC. And because the ratio of PD to DS is
+deemed the same as that of PC to CS, the two eyes Rr being supposed
+very far above the crystal, the elevation PS will thus be 2/5 of PD.</p>
+
+<div class="figright" style="width: 75px;">
+<img src="images/pg085.png" width="75" height="325" alt="" title="" />
+</div>
+
+<p>42. If one takes a straight line AB for the thickness of the crystal,
+its point B being at the bottom, and if one divides it at the points
+C, D, E, according to the proportions of the elevations found, making
+AE 3/5 of AB, AB to AC as 99,324 to 70,283, and AB to AD as 99,324 to
+62,163, these points will divide AB as in this figure. And it will be
+found that this agrees perfectly with experiment; that is to say by
+placing the eyes above in the plane which cuts the crystal according
+to the shorter diameter of the rhombus, the regular refraction will
+lift up the letters to E; and one will see the bottom, and the letters
+over which it is placed, lifted up to D by the irregular refraction.
+But by placing the eyes above in the plane which cuts the crystal
+according to the longer diameter of the rhombus, the regular
+refraction will lift the letters to E as before; but the irregular
+refraction will make them, at the same time, appear lifted up only to
+C; and in such a way that the interval CE will be quadruple the
+interval ED, which one previously saw.</p>
+
+
+<p>43. I have only to make the remark here that in both the positions of
+the eyes the images caused by the irregular refraction do not appear
+directly below those which proceed <span class="pagenum">[Pg 86]</span><a name="Page_86" id="Page_86" />from the regular refraction, but
+they are separated from them by being more distant from the
+equilateral solid angle of the Crystal. That follows, indeed, from all
+that has been hitherto demonstrated about the irregular refraction;
+and it is particularly shown by these last demonstrations, from which
+one sees that the point I appears by irregular refraction at S in the
+perpendicular line DP, in which line also the image of the point P
+ought to appear by regular refraction, but not the image of the point
+I, which will be almost directly above the same point, and higher than
+S.</p>
+
+<p>But as to the apparent elevation of the point I in other positions of
+the eyes above the crystal, besides the two positions which we have
+just examined, the image of that point by the irregular refraction
+will always appear between the two heights of D and C, passing from
+one to the other as one turns one's self around about the immovable
+crystal, while looking down from above. And all this is still found
+conformable to our hypothesis, as any one can assure himself after I
+shall have shown here the way of finding the irregular refractions
+which appear in all other sections of the crystal, besides the two
+which we have considered. Let us suppose one of the faces of the
+crystal, in which let there be the Ellipse HDE, the centre C of which
+is also the centre of the spheroid HME in which the light spreads, and
+of which the said Ellipse is the section. And let the incident ray be
+RC, the refraction of which it is required to find.</p>
+
+<p>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, <span class="pagenum">[Pg 87]</span><a name="Page_87" id="Page_87" />let OK be adjusted across the angle OCK, so as
+to be perpendicular to OC and equal to the line N, which I suppose to
+measure the travel of the light in air during the time that it spreads
+in the crystal through the spheroid HDEM. Then in the plane of the
+Ellipse HDE let KT be drawn, through the point K, perpendicular to
+BCK. Now if one conceives a plane drawn through the straight line KT
+and touching the spheroid HME at I, the straight line CI will be the
+refraction of the ray RC, as is easy to deduce from that which has
+been demonstrated in Article 36.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg087.png" width="350" height="268" alt="" title="" />
+</div>
+
+<p>But it must be shown how one can determine the point of contact I. Let
+there be drawn parallel to the line KT a line HF which touches the
+Ellipse HDE, and let this point of contact be at H. And having drawn a
+straight line along CH to meet KT at T, let there be imagined a plane
+passing through the same CH and through CM (which I suppose to be the
+refraction of the perpendicular ray), which makes in the spheroid the
+elliptical section HME. It is certain that the plane which will pass
+through the straight line KT, and which will touch the spheroid, will
+touch it at a point in the Ellipse HME, according to the Lemma which
+will be demonstrated at the end of the <span class="pagenum">[Pg 88]</span><a name="Page_88" id="Page_88" />Chapter. Now this point is
+necessarily the point I which is sought, since the plane drawn through
+TK can touch the spheroid at one point only. And this point I is easy
+to determine, since it is needful only to draw from the point T, which
+is in the plane of this Ellipse, the tangent TI, in the way shown
+previously. For the Ellipse HME is given, and its conjugate
+semi-diameters are CH and CM; because a straight line drawn through M,
+parallel to HE, touches the Ellipse HME, as follows from the fact that
+a plane taken through M, and parallel to the plane HDE, touches the
+spheroid at that point M, as is seen from Articles 27 and 23. For the
+rest, the position of this ellipse, with respect to the plane through
+the ray RC and through CK, is also given; from which it will be easy
+to find the position of CI, the refraction corresponding to the ray
+RC.</p>
+
+<p>Now it must be noted that the same ellipse HME serves to find the
+refractions of any other ray which may be in the plane through RC and
+CK. Because every plane, parallel to the straight line HF, or TK,
+which will touch the spheroid, will touch it in this ellipse,
+according to the Lemma quoted a little before.</p>
+
+<p>I have investigated thus, in minute detail, the properties of the
+irregular refraction of this Crystal, in order to see whether each
+phenomenon that is deduced from our hypothesis accords with that which
+is observed in fact. And this being so it affords no slight proof of
+the truth of our suppositions and principles. But what I am going to
+add here confirms them again marvellously. It is this: that there are
+different sections of this Crystal, the surfaces of which, thereby
+produced, give rise to refractions precisely such as they ought to be,
+and as I had foreseen them, according to the preceding Theory.</p>
+
+<p><span class="pagenum">[Pg 89]</span><a name="Page_89" id="Page_89" />In order to explain what these sections are, let ABKF <i>be</i> the
+principal section through the axis of the crystal ACK, in which there
+will also be the axis SS of a spheroidal wave of light spreading in
+the crystal from the centre C; and the straight line which cuts SS
+through the middle and at right angles, namely PP, will be one of the
+major diameters.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg089.png" width="300" height="213" alt="{Section ABKF}" title="" />
+</div>
+
+<p>Now as in the natural section of the crystal, made by a plane parallel
+to two opposite faces, which plane is here represented by the line GG,
+the refraction of the surfaces which are produced by it will be
+governed by the hemi-spheroids GNG, according to what has been
+explained in the preceding Theory. Similarly, cutting the Crystal
+through NN, by a plane perpendicular to the parallelogram ABKF, the
+refraction of the surfaces will be governed by the hemi-spheroids NGN.
+And if one cuts it through PP, perpendicularly to the said
+parallelogram, the refraction of the surfaces ought to be governed by
+the hemi-spheroids PSP, and so for others. But I saw that if the plane
+NN was almost perpendicular to the plane GG, making the angle NCG,
+which is on the side A, an angle of 90 degrees 40 minutes, the
+hemi-spheroids NGN would become similar to the hemi-spheroids GNG,
+since the planes NN and GG were equally inclined by an angle of 45
+degrees 20 minutes to the axis SS. In consequence it must needs be, if
+our theory is true, that the surfaces which the section through <span class="pagenum">[Pg 90]</span><a name="Page_90" id="Page_90" />NN
+produces should effect the same refractions as the surfaces of the
+section through GG. And not only the surfaces of the section NN but
+all other sections produced by planes which might be inclined to the
+axis at an angle equal to 45 degrees 20 minutes. So that there are an
+infinitude of planes which ought to produce precisely the same
+refractions as the natural surfaces of the crystal, or as the section
+parallel to any one of those surfaces which are made by cleavage.</p>
+
+<p>I saw also that by cutting it by a plane taken through PP, and
+perpendicular to the axis SS, the refraction of the surfaces ought to
+be such that the perpendicular ray should suffer thereby no deviation;
+and that for oblique rays there would always be an irregular
+refraction, differing from the regular, and by which objects placed
+beneath the crystal would be less elevated than by that other
+refraction.</p>
+
+<p>That, similarly, by cutting the crystal by any plane through the axis
+SS, such as the plane of the figure is, the perpendicular ray ought to
+suffer no refraction; and that for oblique rays there were different
+measures for the irregular refraction according to the situation of
+the plane in which the incident ray was.</p>
+
+<p>Now these things were found in fact so; and, after that, I could not
+doubt that a similar success could be met with everywhere. Whence I
+concluded that one might form from this crystal solids similar to
+those which are its natural forms, which should produce, at all their
+surfaces, the same regular and irregular refractions as the natural
+surfaces, and which nevertheless would cleave in quite other ways, and
+not in directions parallel to any of their faces. That out of it one
+would be able to fashion pyramids, having their base square,
+pentagonal, hexagonal, or with as many sides <span class="pagenum">[Pg 91]</span><a name="Page_91" id="Page_91" />as one desired, all the
+surfaces of which should have the same refractions as the natural
+surfaces of the crystal, except the base, which will not refract the
+perpendicular ray. These surfaces will each make an angle of 45
+degrees 20 minutes with the axis of the crystal, and the base will be
+the section perpendicular to the axis.</p>
+
+<p>That, finally, one could also fashion out of it triangular prisms, or
+prisms with as many sides as one would, of which neither the sides nor
+the bases would refract the perpendicular ray, although they would yet
+all cause double refraction for oblique rays. The cube is included
+amongst these prisms, the bases of which are sections perpendicular to
+the axis of the crystal, and the sides are sections parallel to the
+same axis.</p>
+
+<p>From all this it further appears that it is not at all in the
+disposition of the layers of which this crystal seems to be composed,
+and according to which it splits in three different senses, that the
+cause resides of its irregular refraction; and that it would be in
+vain to wish to seek it there.</p>
+
+<p>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.</p>
+
+<p>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 <span class="pagenum">[Pg 92]</span><a name="Page_92" id="Page_92" />way as spectacle glasses, and
+polishes it simply by continuing the work, but ever reducing the
+material. I have not, however, been able to give it perfect clarity
+and transparency; but the evenness which the surfaces acquire enables
+one to observe in them the effects of refraction better than in those
+made by cleaving the stone, which always have some inequality.</p>
+
+<p>Even when the surface is only moderately smoothed, if one rubs it over
+with a little oil or white of egg, it becomes quite transparent, so
+that the refraction is discerned in it quite distinctly. And this aid
+is specially necessary when it is wished to polish the natural
+surfaces to remove the inequalities; because one cannot render them
+lucent equally with the surfaces of other sections, which take a
+polish so much the better the less nearly they approximate to these
+natural planes.</p>
+
+<p>Before finishing the treatise on this Crystal, I will add one more
+marvellous phenomenon which I discovered after having written all the
+foregoing. For though I have not been able till now to find its cause,
+I do not for that reason wish to desist from describing it, in order
+to give opportunity to others to investigate it. It seems that it will
+be necessary to make still further suppositions besides those which I
+have made; but these will not for all that cease to keep their
+probability after having been confirmed by so many tests.</p>
+
+<div class="figcenter" style="width: 450px;">
+<img src="images/pg093.png" width="450" height="360" alt="" title="" />
+
+</div>
+
+<p>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,
+<span class="pagenum">[Pg 93]</span><a name="Page_93" id="Page_93" />regular and irregular. On penetrating thence into the other piece
+each ray will pass there without further dividing itself in two; but
+that one which underwent the regular refraction, as here DG, will
+undergo again only a regular refraction at GH; and the other, CE, an
+irregular refraction at EF. And the same thing occurs not only in this
+disposition, but also in all those cases in which the principal
+section of each of the pieces is situated in one and the same plane,
+without it being needful for the two neighbouring surfaces to be
+parallel. Now it is marvellous why the rays CE and DG, incident from
+the air on the lower crystal, do not divide themselves the same as the
+first ray AB. One would say that it must be that the ray DG in passing
+through the upper piece has lost something which is necessary to move
+the matter which serves for the irregular refraction; and that
+likewise CE has lost that which <span class="pagenum">[Pg 94]</span><a name="Page_94" id="Page_94" />was necessary to move the matter
+which serves for regular refraction: but there is yet another thing
+which upsets this reasoning. It is that when one disposes the two
+crystals in such a way that the planes which constitute the principal
+sections intersect one another at right angles, whether the
+neighbouring surfaces are parallel or not, then the ray which has come
+by the regular refraction, as DG, undergoes only an irregular
+refraction in the lower piece; and on the contrary the ray which has
+come by the irregular refraction, as CE, undergoes only a regular
+refraction.</p>
+
+<p>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.</p>
+
+<p>When one considers here how, while the rays CE, DG, remain the same,
+it depends on the position that one gives to the lower piece, whether
+it divides them both in two, or whether it does not divide them, and
+yet how the ray AB above is always divided, it seems that one is
+obliged to conclude that the waves of light, after having passed
+through the first crystal, acquire a certain form or disposition in
+virtue of which, when meeting the texture of the second crystal, in
+certain positions, they can move the two different kinds of matter
+which serve for the two species of refraction; and when meeting the
+second crystal in another position are able to move only one of these
+kinds of matter. But to tell how this occurs, I have hitherto found
+nothing which satisfies me.</p>
+
+<p><span class="pagenum">[Pg 95]</span><a name="Page_95" id="Page_95" />Leaving then to others this research, I pass to what I have to say
+touching the cause of the extraordinary figure of this crystal, and
+why it cleaves easily in three different senses, parallel to any one
+of its surfaces.</p>
+
+<p>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.</p>
+
+<p>Rock Crystal grows ordinarily in hexagonal bars, and diamonds are
+found which occur with a square point and polished surfaces. There is
+a species of small flat stones, piled up directly upon one another,
+which are all of pentagonal figure with rounded angles, and the sides
+a little folded inwards. The grains of gray salt which are formed from
+sea water affect the figure, or at least the angle, of the cube; and
+in the congelations of other salts, and in that of sugar, there are
+found other solid angles with perfectly flat faces. Small snowflakes
+almost always fall in little stars with 6 points, and sometimes in
+hexagons with straight sides. And I have often observed, in water
+which is beginning to freeze, a kind of flat and thin foliage of ice,
+the middle ray of which throws out branches inclined at an angle of 60
+degrees. All these things are worthy of being carefully investigated
+to ascertain how and by what artifice nature there operates. But it is
+not now my intention to treat fully of this matter. It seems that in
+general the regularity which occurs in these productions comes from
+the arrangement of the small invisible equal particles of which they
+are composed. And, coming to our Iceland Crystal, I say <span class="pagenum">[Pg 96]</span><a name="Page_96" id="Page_96" />that if there
+were a pyramid such as ABCD, composed of small rounded corpuscles, not
+spherical but flattened spheroids, such as would be made by the
+rotation of the ellipse GH around its lesser diameter EF (of which the
+ratio to the greater diameter is very nearly that of 1 to the square
+root of 8)&mdash;I say that then the solid angle of the point D would be
+equal to the obtuse and equilateral angle of this Crystal. I say,
+further, that if these corpuscles were lightly stuck together, on
+breaking this pyramid it would break along faces parallel to those
+that make its point: and by this means, as it is easy to see, it would
+produce prisms similar to those of the same crystal as this other
+figure represents. The reason is that when broken in this fashion a
+whole layer separates easily from its neighbouring layer since each
+spheroid has to be detached only from the three spheroids of the next
+layer; of which three there is but one which touches it on its
+flattened surface, and the other two at the edges. And the reason why
+the surfaces separate sharp and polished is that if any spheroid of
+the neighbouring surface would come out by attaching itself to the
+surface which is being separated, it would be needful for it to detach
+itself from six other spheroids which hold it locked, and four of
+which press it by these flattened surfaces. Since then not only the
+angles of our crystal but also the manner in which it splits agree
+precisely with what is observed in the assemblage composed of such
+spheroids, there is great reason to believe that the particles are
+shaped and ranged in the same way.</p>
+
+<div class="figright" style="width: 200px;">
+<img src="images/pg096.png" width="200" height="310" alt="{Pyramid and section of spheroids}" title="" />
+</div>
+
+<p><span class="pagenum">[Pg 97]</span><a name="Page_97" id="Page_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.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg097.png" width="300" height="228" alt="{paralleloid arrangement of spheroids with planes of
+potential cleavage}" title="" />
+</div>
+
+<p>It may be objected that this crystal, being so composed, might be
+capable of cleavage in yet two more fashions; one of which would be
+along planes parallel to the base of the pyramid, that is to say to
+the triangle ABC; the other would be parallel to a plane the trace of
+which is marked by the lines GH, HK, KL. To which I say that both the
+one and the other, though practicable, are more difficult than those
+which were parallel to any one of the three planes of the pyramid; and
+that therefore, when striking on the crystal in order to break it, it
+ought always to split rather along these three planes than along the
+two others. When one has a number of spheroids of the form above
+described, and ranges them in a pyramid, one sees why the two methods
+of division are more difficult. For in the case of that division which
+would be parallel to the base, <span class="pagenum">[Pg 98]</span><a name="Page_98" id="Page_98" />each spheroid would be obliged to
+detach itself from three others which it touches upon their flattened
+surfaces, which hold more strongly than the contacts at the edges. And
+besides that, this division will not occur along entire layers,
+because each of the spheroids of a layer is scarcely held at all by
+the 6 of the same layer that surround it, since they only touch it at
+the edges; so that it adheres readily to the neighbouring layer, and
+the others to it, for the same reason; and this causes uneven
+surfaces. Also one sees by experiment that when grinding down the
+crystal on a rather rough stone, directly on the equilateral solid
+angle, one verily finds much facility in reducing it in this
+direction, but much difficulty afterwards in polishing the surface
+which has been flattened in this manner.</p>
+
+<p>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.</p>
+
+<p>However, that which has made me know that in the crystal there are
+layers in this last fashion, is that in a piece weighing half a pound
+which I possess, one sees that it is split along its length, as is the
+above-mentioned prism by the plane GHKL; as appears by colours of the
+Iris extending throughout this whole plane although the two pieces
+still hold together. All this proves then that the composition of the
+crystal is such as we have stated. To <span class="pagenum">[Pg 99]</span><a name="Page_99" id="Page_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 incision is easily
+made. This follows manifestly from the situation of the small
+spheroids; over which, in the first manner, the knife glides; but in
+the other manner it seizes them from beneath almost as if they were
+the scales of a fish.</p>
+
+<p>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.</p>
+
+
+<p><i>Calculations which have been supposed in this Chapter</i>.</p>
+
+<p>Mr. Bartholinus, in his treatise of this Crystal, puts at 101 degrees
+the obtuse angles of the faces, which I have stated to be 101 degrees
+52 minutes. He states that he measured these angles directly on the
+crystal, which is difficult to do with ultimate exactitude, because
+the edges such as CA, CB, in this figure, are generally worn, and not
+quite straight. For more certainty, therefore, I preferred to measure
+actually the obtuse angle by which the faces <span class="pagenum">[Pg 100]</span><a name="Page_100" id="Page_100" />CBDA, CBVF, are inclined
+to one another, namely the angle OCN formed by drawing CN
+perpendicular to FV, and CO perpendicular to DA. This angle OCN I
+found to be 105 degrees; and its supplement CNP, to be 75 degrees, as
+it should be.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg100.png" width="350" height="235" alt="" title="" />
+</div>
+
+<p>To find from this the obtuse angle BCA, I imagined a sphere having its
+centre at C, and on its surface a spherical triangle, formed by the
+intersection of three planes which enclose the solid angle C. In this
+equilateral triangle, which is ABF in this other figure, I see that
+each of the angles should be 105 degrees, namely equal to the angle
+OCN; and that each of the sides should be of as many degrees as the
+angle ACB, or ACF, or BCF. Having then drawn the arc FQ perpendicular
+to the side AB, which it divides equally at Q, the triangle FQA has a
+right angle at Q, the angle A 105 degrees, and F half as much, namely
+52 degrees 30 minutes; whence the hypotenuse AF is found to be 101
+degrees 52 minutes. And this arc AF is the measure of the angle ACF in
+the figure of the crystal.</p>
+
+<div class="figright" style="width: 150px;">
+<img src="images/pg100a.png" width="150" height="165" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 101]</span><a name="Page_101" id="Page_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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Let CP and CS be prolonged and meet the tangent DM at D and Z; and
+from the point of contact M let MN and MO be drawn as perpendiculars
+to CP and CS. Now because the angles SCP, GCL, are right angles, the
+<span class="pagenum">[Pg 102]</span><a name="Page_102" id="Page_102" />angle PCL will be equal to GCS which was 45 degrees 20 minutes. And
+deducting the angle LCM, which is 6 degrees 40 minutes, from LCP,
+which is 45 degrees 20 minutes, there remains MCP, 38 degrees 40
+minutes. Considering then CM as a radius of 100,000 parts, MN, the
+sine of 38 degrees 40 minutes, will be 62,479. And in the right-angled
+triangle MND, MN will be to ND as the radius of the Tables is to the
+tangent of 45 degrees 20 minutes (because the angle NMD is equal to
+DCL, or GCS); that is to say as 100,000 to 101,170: whence results ND
+63,210. But NC is 78,079 of the same parts, CM being 100,000, because
+NC is the sine of the complement of the angle MCP, which was 38
+degrees 40 minutes. Then the whole line DC is 141,289; and CP, which
+is a mean proportional between DC and CN, since MD touches the
+Ellipse, will be 105,032.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg102.png" width="300" height="300" alt="" title="" />
+</div>
+
+<p>Similarly, because the angle OMZ is equal to CDZ, or LCZ, which is 44
+degrees 40 minutes, being the complement of GCS, it follows that, as
+the radius of the Tables is to the tangent of 44 degrees 40 minutes,
+so will OM 78,079 be to OZ 77,176. But OC is 62,479 of these same
+parts of which CM is 100,000, because it is equal to MN, the sine of
+the angle MCP, which is 38 degrees 40 minutes. Then the whole line CZ
+is 139,655; and CS, which is a mean proportional between CZ and CO
+will be 93,410.</p>
+
+<p><span class="pagenum">[Pg 103]</span><a name="Page_103" id="Page_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 complement of
+the angle LCM, which is 6 degrees 40 minutes. And since the angle LCD
+is 45 degrees 20 minutes, being equal to GCS, the side LD is found to
+be 100,486: whence deducting ML 11,609 there will remain MD 88,877.
+Now as CD (which was 141,289) is to DM 88,877, so will CP 105,032 be
+to PE 66,070. But as the rectangle MEH (or rather the difference of
+the squares on CM and CE) is to the square on MC, so is the square on
+PE to the square on C<i>g</i>; 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 <i>g</i>C. But DP, CP, and PE are known; hence also one knows GC,
+which is 98,779.</p>
+
+
+<p><i>Lemma which has been supposed</i>.</p>
+
+<p>If a spheroid is touched by a straight line, and also by two or more
+planes which are parallel to this line, though not parallel to one
+another, all the points of contact of the line, as well as of the
+planes, will be in one and the same ellipse made by a plane which
+passes through the centre of the spheroid.</p>
+
+<p>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.<span class="pagenum">[Pg 104]</span><a name="Page_104" id="Page_104" /></p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg104.png" width="350" height="400" alt="" title="" />
+</div>
+
+<p>Through the line BM, and through the points O and A, let there be
+drawn planes parallel to one another, which, in cutting the spheroid
+make the ellipses LBD, POP, QAQ; which will all be similar and
+similarly disposed, and will have their centres K, N, R, in one and
+the same diameter of the spheroid, which will also be the diameter of
+the ellipse made by the section of the plane that passes through the
+centre of the spheroid, and which cuts the planes of the three said
+Ellipses at right angles: for all this is manifest by proposition 15
+of the book of Conoids and Spheroids of Archimedes. Further, the two
+latter planes, which are drawn through the points O and A, will also,
+by cutting the planes which touch the spheroid in these same points,
+generate straight lines, as OH and AS, which will, as is easy to see,
+be parallel to BM; and all three, BM, OH, AS, will touch the Ellipses
+LBD, POP, QAQ in these points, B, O, A; since they are in the planes
+of these ellipses, and at the same time in the planes which touch the
+spheroid. If now from these points B, O, A, there are drawn the
+straight lines BK, ON, AR, through the centres of the same ellipses,
+and if through these centres there are drawn also the diameters LD,
+PP, QQ, parallel to the tangents BM, OH, AS; these will be conjugate
+to the aforesaid BK, ON, AR. And because the three ellipses are
+similar and similarly <span class="pagenum">[Pg 105]</span><a name="Page_105" id="Page_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 intersection of
+this plane. Which was to be proved. And it is manifest that the
+demonstration would be the same if, besides the points O, A, there had
+been others in which the spheroid had been touched by planes parallel
+to the straight line BM.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_VI" id="CHAPTER_VI" />CHAPTER VI</h2>
+
+<h3>ON THE FIGURES OF THE TRANSPARENT BODIES</h3>
+
+<h4><i>Which serve for Refraction and for Reflexion</i>.</h4>
+
+<div style="width: 154px; float: left; margin-right: .2em;">
+<img src="images/ch06.png" width="154" height="150" alt="A" title="A" />
+</div><p>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 refraction, to
+collect or disperse the rays of light, as may be desired. For though I
+do not see yet that there are means of making use of these figures, so
+far as relates to Refraction, not only because of the difficulty of
+shaping the glasses of Telescopes with the requisite<span class="pagenum">[Pg 106]</span><a name="Page_106" id="Page_106" /> exactitude
+according to these figures, but also because there exists in
+refraction itself a property which hinders the perfect concurrence of
+the rays, as Mr. Newton has very well proved by experiment, I will yet
+not desist from relating the invention, since it offers itself, so to
+speak, of itself, and because it further confirms our Theory of
+refraction, by the agreement which here is found between the refracted
+ray and the reflected ray. Besides, it may occur that some one in the
+future will discover in it utilities which at present are not seen.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg106.png" width="500" height="191" alt="" title="" />
+</div>
+
+<p>To proceed then to these figures, let us suppose first that it is
+desired to find a surface CDE which shall reassemble at a point B rays
+coming from another point A; and that the summit of the surface shall
+be the given point D in the straight line AB. I say that, whether by
+reflexion or by refraction, it is only necessary to make this surface
+such that the path of the light from the point A to all points of the
+curved line CDE, and from these to the point of concurrence (as here
+the path along the straight lines AC, CB, along AL, LB, and along AD,
+DB), shall be everywhere traversed in equal times: by which principle
+the finding of these curves becomes very easy.<span class="pagenum">[Pg 107]</span><a name="Page_107" id="Page_107" /></p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg107.png" width="300" height="236" alt="" title="" />
+</div>
+
+<p>So far as relates to the reflecting surface, since the sum of the
+lines AC, CB ought to be equal to that of AD, DB, it appears that DCE
+ought to be an ellipse; and for refraction, the ratio of the
+velocities of waves of light in the media A and B being supposed to be
+known, for example that of 3 to 2 (which is the same, as we have
+shown, as the ratio of the Sines in the refraction), it is only
+necessary to make DH equal to 3/2 of DB; and having after that
+described from the centre A some arc FC, cutting DB at F, then
+describe another from centre B with its semi-diameter BX equal to 2/3
+of FH; and the point of intersection of the two arcs will be one of
+the points required, through which the curve should pass. For this
+point, having been found in this fashion, it is easy forthwith to
+demonstrate that the time along AC, CB, will be equal to the time
+along AD, DB.</p>
+
+<p>For assuming that the line AD represents the time which the light
+takes to traverse this same distance AD in air, it is evident that DH,
+equal to 3/2 of DB, will represent the time of the light along DB in
+the medium, because it needs here more time in proportion as its speed
+is slower. Therefore the whole line AH will represent the time along
+AD, DB. Similarly the line AC or AF will represent the time along AC;
+and FH being by construction equal to 3/2 of CB, it will represent the
+time along CB in the medium; and in consequence the whole line AH will
+represent also the time along AC, CB. Whence it appears that the <span class="pagenum">[Pg 108]</span><a name="Page_108" id="Page_108" />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.</p>
+
+<p>In order to show further that the surfaces, which these curves will
+generate by revolution, will direct all the rays which reach them from
+the point A in such wise that they tend towards B, let there be
+supposed a point K in the curve, farther from D than C is, but such
+that the straight line AK falls from outside upon the curve which
+serves for the refraction; and from the centre B let the arc KS be
+described, cutting BD at S, and the straight line CB at R; and from
+the centre A describe the arc DN meeting AK at N.</p>
+
+<p>Since the sums of the times along AK, KB, and along AC, CB are equal,
+if from the former sum one deducts the time along KB, and if from the
+other one deducts the time along RB, there will remain the time along
+AK as equal to the time along the two parts AC, CR. Consequently in
+the time that the light has come along AK it will also have come along
+AC and will in addition have made, in the medium from the centre C, a
+partial spherical wave, having a semi-diameter equal to CR. And this
+wave will necessarily touch the circumference KS at R, since CB cuts
+this circumference at right angles. Similarly, having taken any other
+point L in the curve, one can show that in the same time as the light
+passes along AL it will also have come along AL and in addition will
+have made a partial wave, from the centre L, which will touch the same
+circumference KS. And so with all other points of the curve CDE. Then
+at the moment that the light reaches K the arc KRS will be the
+termination <span class="pagenum">[Pg 109]</span><a name="Page_109" id="Page_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.</p>
+
+<p>One might also determine the point C, and all the others, in this
+curve which serves for the refraction, by dividing DA at G in such a
+way that DG is 2/3 of DA, and describing from the centre B any arc CX
+which cuts BD at N, and another from the centre A with its
+semi-diameter AF equal to 3/2 of GX; or rather, having described, as
+before, the arc CX, it is only necessary to make DF equal to 3/2 of
+DX, and from-the centre A to strike the arc FC; for these two
+constructions, as may be easily known, come back to the first one
+which was shown before. And it is manifest by the last method that
+this curve is the same that Mr. Des Cartes has given in his Geometry,
+and which he calls the first of his Ovals.</p>
+
+<p>It is only a part of this oval which serves for the refraction,
+namely, the part DK, ending at K, if AK is the tangent. As to the,
+other part, Des Cartes has remarked that it could serve for
+reflexions, if there were some material of a mirror of such a nature
+that by its <span class="pagenum">[Pg 110]</span><a name="Page_110" id="Page_110" />means the force of the rays (or, as we should say, the
+velocity of the light, which he could not say, since he held that the
+movement of light was instantaneous) could be augmented in the
+proportion of 3 to 2. But we have shown that in our way of explaining
+reflexion, such a thing could not arise from the matter of the mirror,
+and it is entirely impossible.</p>
+
+<div class="figcenter" style="width: 400px;">
+<img src="images/pg110.png" width="400" height="439" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 111]</span><a name="Page_111" id="Page_111" />construction of which differs in no way from that of the
+oval, except that FC, which previously was an arc of a circle, is here
+a straight line, perpendicular to DB. For the wave of light DN, being
+likewise represented by a straight line, it will be seen that all the
+points of this wave, travelling as far as the surface KD along lines
+parallel to DB, will advance subsequently towards the point B, and
+will arrive there at the same time. As for the Ellipse which served
+for reflexion, it is evident that it will here become a parabola,
+since its focus A may be regarded as infinitely distant from the
+other, B, which is here the focus of the parabola, towards which all
+the reflexions of rays parallel to AB tend. And the demonstration of
+these effects is just the same as the preceding.</p>
+
+<p>But that this curved line CDE which serves for refraction is an
+Ellipse, and is such that its major diameter is to the distance
+between its foci as 3 to 2, which is the proportion of the refraction,
+can be easily found by the calculus of Algebra. For DB, which is
+given, being called <i>a</i>; its undetermined perpendicular DT being
+called <i>x</i>; and TC <i>y</i>; FB will be <i>a - y</i>; CB will be
+sqrt(<i>xx + aa - 2ay + yy</i>). But the nature of the curve is such that
+2/3 of TC together with CB is equal to DB, as was stated in the last
+construction: then the equation will be between <i>(2/3)y +
+sqrt(xx + aa - 2ay + yy)</i> and <i>a</i>; which being reduced, gives
+<i>(6/5)ay - yy</i> equal to <i>(9/5)xx</i>; that is to say that
+having made DO equal to 6/5 of DB, the rectangle DFO is equal to 9/5
+of the square on FC. Whence it is seen that DC is an ellipse, of which
+the axis DO is to the parameter as 9 to 5; and therefore the square on
+DO is to the square of the distance between the foci as 9 to 9 - 5,
+that is to say 4; and finally the line DO will be to this distance as
+3 to 2.<span class="pagenum">[Pg 112]</span><a name="Page_112" id="Page_112" /></p>
+
+<div class="figleft" style="width: 400px;">
+<img src="images/pg112.png" width="400" height="307" alt="" title="" />
+</div>
+
+<p>Again, if one supposes the point B to be infinitely distant, in lieu
+of our first oval we shall find that CDE is a true Hyperbola; which
+will make those rays become parallel which come from the point A. And
+in consequence also those which are parallel within the transparent
+body will be collected outside at the point A. Now it must be remarked
+that CX and KS become straight lines perpendicular to BA, because they
+represent arcs of circles the centre of which is infinitely distant.
+And the intersection of the perpendicular CX with the arc FC will give
+the point C, one of those through which the curve ought to pass. And
+this operates so that all the parts of the wave of light DN, coming to
+meet the surface KDE, will advance thence along parallels to KS and
+will arrive at this straight line at the same time; of which the proof
+is again the same as that which served for the first oval. Besides one
+finds by a calculation as easy as the preceding one, that CDE is here
+a hyperbola of which the axis DO <span class="pagenum">[Pg 113]</span><a name="Page_113" id="Page_113" />is 4/5 of AD, and the parameter
+equal to AD. Whence it is easily proved that DO is to the distance
+between the foci as 3 to 2.</p>
+
+<div class="figright" style="width: 400px;">
+<img src="images/pg113.png" width="400" height="316" alt="" title="" />
+</div>
+
+<p>These are the two cases in which Conic sections serve for refraction,
+and are the same which are explained, in his <i>Dioptrique</i>, by Des
+Cartes, who first found out the use of these lines in relation to
+refraction, as also that of the Ovals the first of which we have
+already set forth. The second oval is that which serves for rays that
+tend to a given point; in which oval, if the apex of the surface which
+receives the rays is D, it will happen that the other apex will be
+situated between B and A, or beyond A, according as the ratio of AD to
+DB is given of greater or lesser value. And in this latter case it is
+the same as that which Des Cartes calls his 3rd oval.</p>
+
+<p>Now the finding and construction of this second oval is <span class="pagenum">[Pg 114]</span><a name="Page_114" id="Page_114" />the same as
+that of the first, and the demonstration of its effect likewise. But
+it is worthy of remark that in one case this oval becomes a perfect
+circle, namely when the ratio of AD to DB is the same as the ratio of
+the refractions, here as 3 to 2, as I observed a long time ago. The
+4th oval, serving only for impossible reflexions, there is no need to
+set it forth.</p>
+
+<div class="figleft" style="width: 400px;">
+<img src="images/pg114.png" width="400" height="316" alt="" title="" />
+</div>
+
+<p>As for the manner in which Mr. Des Cartes discovered these lines,
+since he has given no explanation of it, nor any one else since that I
+know of, I will say here, in passing, what it seems to me it must have
+been. Let it be proposed to find the surface generated by the
+revolution of the curve KDE, which, receiving the incident rays coming
+to it from the point A, shall deviate them toward the point B. Then
+considering this other curve as already known, and that its apex D is
+in the straight line AB, let us divide it up into an infinitude of
+small pieces by the points G, C, F; and having drawn from each of
+these points, straight lines towards A to represent the incident rays,
+and other straight lines towards B, let there also be described with
+centre A the arcs GL, CM, FN, DO, cutting the rays that come from A at
+L, M, N, O; and from the points K, G, C, F, <span class="pagenum">[Pg 115]</span><a name="Page_115" id="Page_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.</p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/pg115.png" width="600" height="274" alt="" title="" />
+</div>
+
+<p>Then AK being an incident ray, and KB its refraction within the
+medium, it needs must be, according to the law of refraction which was
+known to Mr. Des Cartes, that the sine of the angle ZKA should be to
+the sine of the angle HKB as 3 to 2, supposing that this is the
+proportion of the refraction of glass; or rather, that the sine of the
+angle KGL should have this same ratio to the sine of the angle GKQ,
+considering KG, GL, KQ as straight lines because of their smallness.
+But these sines are the lines KL and GQ, if GK is taken as the radius
+of the circle. Then LK ought to be to GQ as 3 to 2; and in the same
+ratio MG to CR, NC to FS, OF to DT. Then also the sum of all the
+antecedents to all the consequents would be as 3 to 2. Now by
+prolonging the arc DO until it meets AK at X, KX is the sum of the
+antecedents. And by prolonging the arc KQ till it meets AD at Y, the
+sum of <span class="pagenum">[Pg 116]</span><a name="Page_116" id="Page_116" />the consequents is DY. Then KX ought to be to DY as 3 to 2.
+Whence it would appear that the curve KDE was of such a nature that
+having drawn from some point which had been assumed, such as K, the
+straight lines KA, KB, the excess by which AK surpasses AD should be
+to the excess of DB over KB, as 3 to 2. For it can similarly be
+demonstrated, by taking any other point in the curve, such as G, that
+the excess of AG over AD, namely VG, is to the excess of BD over DG,
+namely DP, in this same ratio of 3 to 2. And following this principle
+Mr. Des Cartes constructed these curves in his <i>Geometric</i>; and he
+easily recognized that in the case of parallel rays, these curves
+became Hyperbolas and Ellipses.</p>
+
+<p>Let us now return to our method and let us see how it leads without
+difficulty to the finding of the curves which one side of the glass
+requires when the other side is of a given figure; a figure not only
+plane or spherical, or made by one of the conic sections (which is the
+restriction with which Des Cartes proposed this problem, leaving the
+solution to those who should come after him) but generally any figure
+whatever: that is to say, one made by the revolution of any given
+curved line to which one must merely know how to draw straight lines
+as tangents.</p>
+
+<p>Let the given figure be that made by the revolution of some curve such
+as AK about the axis AV, and that this side of the glass receives rays
+coming from the point L. Furthermore, let the thickness AB of the
+middle of the glass be given, and the point F at which one desires the
+rays to be all perfectly reunited, whatever be the first refraction
+occurring at the surface AK.</p>
+
+<p>I say that for this the sole requirement is that the outline BDK which
+constitutes the other surface shall be <span class="pagenum">[Pg 117]</span><a name="Page_117" id="Page_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
+everywhere in equal times, and in each case in a time equal to that
+which the light employs, to pass along the straight line LF of which
+the part AB is within the glass.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg117.png" width="300" height="677" alt="" title="" />
+</div>
+
+<p>Let LG be a ray falling on the arc AK. Its refraction GV will be given
+by means of the tangent which will be drawn at the point G. Now in GV
+the point D must be found such that FD together with 3/2 of DG and the
+straight line <span class="pagenum">[Pg 118]</span><a name="Page_118" id="Page_118" />GL, may be equal to FB together with 3/2 of BA and the
+straight line AL; which, as is clear, make up a given length. Or
+rather, by deducting from each the length of LG, which is also given,
+it will merely be needful to adjust FD up to the straight line VG in
+such a way that FD together with 3/2 of DG is equal to a given
+straight line, which is a quite easy plane problem: and the point D
+will be one of those through which the curve BDK ought to pass. And
+similarly, having drawn another ray LM, and found its refraction MO,
+the point N will be found in this line, and so on as many times as one
+desires.</p>
+
+<p>To demonstrate the effect of the curve, let there be described about
+the centre L the circular arc AH, cutting LG at H; and about the
+centre F the arc BP; and in AB let AS be taken equal to 2/3 of HG; and
+SE equal to GD. Then considering AH as a wave of light emanating from
+the point L, it is certain that during the time in which its piece H
+arrives at G the piece A will have advanced within the transparent
+body only along AS; for I suppose, as above, the proportion of the
+refraction to be as 3 to 2. Now we know that the piece of wave which
+is incident on G, advances thence along the line GD, since GV is the
+refraction of the ray LG. Then during the time that this piece of wave
+has taken from G to D, the other piece which was at S has reached E,
+since GD, SE are equal. But while the latter will advance from E to B,
+the piece of wave which was at D will have spread into the air its
+partial wave, the semi-diameter of which, DC (supposing this wave to
+cut the line DF at C), will be 3/2 of EB, since the velocity of light
+outside the medium is to that inside as 3 to 2. Now it is easy to show
+that this wave will touch the arc BP at this point C. For since, by
+construction, FD + <span class="pagenum">[Pg 119]</span><a name="Page_119" id="Page_119" />3/2 DG + GL are equal to FB + 3/2 BA + AL; on
+deducting the equals LH, LA, there will remain FD + 3/2 DG + GH equal
+to FB + 3/2 BA. And, again, deducting from one side GH, and from the
+other side 3/2 of AS, which are equal, there will remain FD with 3/2
+DG equal to FB with 3/2 of BS. But 3/2 of DG are equal to 3/2 of ES;
+then FD is equal to FB with 3/2 of BE. But DC was equal to 3/2 of EB;
+then deducting these equal lengths from one side and from the other,
+there will remain CF equal to FB. And thus it appears that the wave,
+the semi-diameter of which is DC, touches the arc BP at the moment
+when the light coming from the point L has arrived at B along the line
+LB. It can be demonstrated similarly that at this same moment the
+light that has come along any other ray, such as LM, MN, will have
+propagated the movement which is terminated at the arc BP. Whence it
+follows, as has been often said, that the propagation of the wave AH,
+after it has passed through the thickness of the glass, will be the
+spherical wave BP, all the pieces of which ought to advance along
+straight lines, which are the rays of light, to the centre F. Which
+was to be proved. Similarly these curved lines can be found in all the
+cases which can be proposed, as will be sufficiently shown by one or
+two examples which I will add.</p>
+
+<p>Let there be given the surface of the glass AK, made by the revolution
+about the axis BA of the line AK, which may be straight or curved. Let
+there be also given in the axis the point L and the thickness BA of
+the glass; and let it be required to find the other surface KDB, which
+receiving rays that are parallel to AB will direct them in such wise
+that after being again refracted at the given surface AK they will all
+be reassembled at the point L.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg120.png" width="300" height="416" alt="" title="" />
+</div>
+
+<p>From the point L let there be drawn to some point of <span class="pagenum">[Pg 120]</span><a name="Page_120" id="Page_120" />the given line
+AK the straight line LG, which, being considered as a ray of light,
+its refraction GD will then be found. And this line being then
+prolonged at one side or the other will meet the straight line BL, as
+here at V. Let there then be erected on AB the perpendicular BC, which
+will represent a wave of light coming from the infinitely distant
+point F, since we have supposed the rays to be parallel. Then all the
+parts of this wave BC must arrive at the same time at the point L; or
+rather all the parts of a wave emanating from the point L must arrive
+at the same time at the straight line BC. And for that, it is
+necessary to find in the line VGD the point D such that having drawn
+DC parallel to AB, the sum of CD, plus 3/2 of DG, plus GL may be equal
+to 3/2 of AB, plus AL: or rather, on deducting from both sides GL,
+which is given, CD plus 3/2 of DG must be equal to a given length;
+which is a still easier problem than the preceding construction. The
+point D thus found will be one of those through which the curve ought
+to pass; and the proof will be the same as before. And by this it will
+be proved that the waves which come from the point L, after having
+passed through the glass KAKB, will take <span class="pagenum">[Pg 121]</span><a name="Page_121" id="Page_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 parallel rays falling on
+the surface KDB will be reassembled at the point L.</p>
+
+<div class="figright" style="width: 250px;">
+<img src="images/pg121.png" width="250" height="383" alt="" title="" />
+</div>
+
+<p>Again, let there be given the surface AK, of any desired form,
+generated by revolution about the axis AB, and let the thickness of
+the glass at the middle be AB. Also let the point L be given in the
+axis behind the glass; and let it be supposed that the rays which fall
+on the surface AK tend to this point, and that it is required to find
+the surface BD, which on their emergence from the glass turns them as
+if they came from the point F in front of the glass.</p>
+
+<p>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 <span class="pagenum">[Pg 122]</span><a name="Page_122" id="Page_122" />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.</p>
+
+<p>Then the wave TG, after having passed through the glass, must form the
+wave QD; and for this I observe that the time taken by the light along
+GD in the glass must be equal to that taken along the three, TA, AB,
+and BQ, of which AB alone is within the glass. Or rather, having taken
+AS equal to 2/3 of AT, I observe that 3/2 of GD ought to be equal to
+3/2 of SB, plus BQ; and, deducting both of them from FD or FQ, that FD
+less 3/2 of GD ought to be equal to FB less 3/2 of SB. And this last
+difference is a given length: and all that is required is to draw the
+straight line FD from the given point F to meet VG so that it may be
+thus. Which is a problem quite similar to that which served for the
+first of these constructions, where FD plus 3/2 of GD had to be equal
+to a given length.</p>
+
+<p>In the demonstration it is to be observed that, since the arc BC falls
+within the glass, there must be conceived an arc RX, concentric with
+it and on the other side of QD. Then after it shall have been shown
+that the piece G of the wave GT arrives at D at the same time that the
+piece T arrives at Q, which is easily deduced from the construction,
+it will be evident as a consequence that the partial wave generated at
+the point D will touch the arc RX at the moment when the piece Q shall
+have come to R, and that thus this arc will at the same moment be the
+termination of the movement that comes from the wave TG; whence all
+the rest may be concluded.</p>
+
+<p>Having shown the method of finding these curved lines which serve for
+the perfect concurrence of the rays, <span class="pagenum">[Pg 123]</span><a name="Page_123" id="Page_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 angles the waves which
+travel along them.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg123.png" width="300" height="464" alt="" title="" />
+</div>
+
+<p>For in the case of rays which, for example, fall parallel upon a
+spherical surface AFE, intersecting one another, after refraction, at
+different points, as this figure represents; what can the waves of
+light be, in this transparent body, which are cut at right angles by
+the converging rays? For they can not be spherical. And what will
+these waves become after the said rays begin to intersect one another?
+It will be seen in the solution of this difficulty that something very
+remarkable comes to pass herein, and that the waves do not cease to
+persist though they do not continue entire, as when they cross the
+glasses designed according to the construction we have seen.</p>
+
+<p><span class="pagenum">[Pg 124]</span><a name="Page_124" id="Page_124" />According to what has been shown above, the straight line AD, which
+has been drawn at the summit of the sphere, at right angles to the
+axis parallel to which the rays come, represents the wave of light;
+and in the time taken by its piece D to reach the spherical surface
+AGE at E, its other parts will have met the same surface at F, G, H,
+etc., and will have also formed spherical partial waves of which these
+points are the centres. And the surface EK which all those waves will
+touch, will be the continuation of the wave AD in the sphere at the
+moment when the piece D has reached E. Now the line EK is not an arc
+of a circle, but is a curved line formed as the evolute of another
+curve ENC, which touches all the rays HL, GM, FO, etc., that are the
+refractions of the parallel rays, if we imagine laid over the
+convexity ENC a thread which in unwinding describes at its end E the
+said curve EK. For, supposing that this curve has been thus described,
+we will show that the said waves formed from the centres F, G, H,
+etc., will all touch it.</p>
+
+<p>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 <i>de Motu Pendulorum</i>. 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 <span class="pagenum">[Pg 125]</span><a name="Page_125" id="Page_125" />to the ray GM, and similarly the arc GF as a straight
+line. But GM being the refraction of the ray RG, and FP being
+perpendicular to it, QF must be to GP as 3 to 2, that is to say in the
+proportion of the refraction; as was shown above in explaining the
+discovery of Des Cartes. And the same thing occurs in all the small
+arcs GH, HA, etc., namely that in the quadrilaterals which enclose
+them the side parallel to the axis is to the opposite side as 3 to 2.
+Then also as 3 to 2 will the sum of the one set be to the sum of the
+other; that is to say, TF to AS, and DE to AK, and BE to SK or DV,
+supposing V to be the intersection of the curve EK and the ray FO.
+But, making FB perpendicular to DE, the ratio of 3 to 2 is also that
+of BE to the semi-diameter of the spherical wave which emanated from
+the point F while the light outside the transparent body traversed the
+space BE. Then it appears that this wave will intersect the ray FM at
+the same point V where it is intersected at right angles by the curve
+EK, and consequently that the wave will touch this curve. In the same
+way it can be proved that the same will apply to all the other waves
+above mentioned, originating at the points G, H, etc.; to wit, that
+they will touch the curve EK at the moment when the piece D of the
+wave ED shall have reached E.</p>
+
+<p>Now to say what these waves become after the rays have begun to cross
+one another: it is that from thence they fold back and are composed of
+two contiguous parts, one being a curve formed as evolute of the curve
+ENC in one sense, and the other as evolute of the same curve in the
+opposite sense. Thus the wave KE, while advancing toward the meeting
+place becomes <i>abc</i>, whereof the part <i>ab</i> is made by the evolute
+<i>b</i>C, a portion of the curve <span class="pagenum">[Pg 126]</span><a name="Page_126" id="Page_126" />ENC, while the end C remains attached;
+and the part <i>bc</i> by the evolute of the portion <i>b</i>E while the end E
+remains attached. Consequently the same wave becomes <i>def</i>, then
+<i>ghk</i>, 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.</p>
+
+<p>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.</p>
+
+<p>As many other points as may be desired in the curve NC are found by a
+Theorem which Mr. Barrow has demonstrated in section 12 of his
+<i>Lectiones Opticae</i>, 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.</p>
+
+<p>Similarly the waves that are folded back in reflexion by a concave
+spherical mirror can be found. Let ABC be the section, through the
+axis, of a hollow hemisphere, the centre of which is D, its axis being
+DB, parallel to which I suppose the rays of light to come. All the
+reflexions of those rays which fall upon the quarter-circle AB will
+touch a curved line AFE, of which line the end E is at the focus of
+the hemisphere, that is to say, at the point which divides the
+semi-diameter BD into two equal parts. <span class="pagenum">[Pg 127]</span><a name="Page_127" id="Page_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 points.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg127.png" width="300" height="232" alt="" title="" />
+</div>
+
+<p>And as the parallel rays are merely perpendiculars to the waves which
+fall on the concave surface, which waves are parallel to AD, it will
+be found that as they come successively to encounter the surface AB,
+they form on reflexion folded waves composed of two curves which
+originate from two opposite evolutions of the parts of the curve AFE.
+So, taking AD as an incident wave, when the part AG shall have met the
+surface AI, that is to say when the piece G shall have reached I, it
+will be the curves HF, FI, generated as evolutes of the curves FA, FE,
+both beginning at F, which together constitute the propagation of the
+part AG. And a little afterwards, when the part AK has met the surface
+AM, the piece K having come to M, then the curves LN, NM, will
+together constitute the propagation of that part. And thus this folded
+wave will continue to advance until the point N has reached the focus
+E. The curve AFE can be seen in smoke, or in flying dust, when a
+concave mirror is held opposite the sun. And it should be known that
+it is none other than that curve which is described <span class="pagenum">[Pg 128]</span><a name="Page_128" id="Page_128" />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.</p>
+
+<p>Its length is exactly equal to 3/4 of the diameter of the sphere, as
+can be found and demonstrated by means of these waves, nearly in the
+same way as the mensuration of the preceding curve; though it may also
+be demonstrated in other ways, which I omit as outside the subject.
+The area AOBEFA, comprised between the arc of the quarter-circle, the
+straight line BE, and the curve EFA, is equal to the fourth part of
+the quadrant DAB.</p>
+
+<h2>END.</h2>
+
+
+
+<hr style="width: 65%;" />
+<div class="pagenum">[Pg 129]<a name="Page_129" id="Page_129" /><a name="INDEX" id="INDEX" /></div>
+<h2>INDEX</h2>
+
+<p>
+<i>Archimedes</i>, <a href="#Page_104">104</a>.<br />
+<i>Atmospheric refraction</i>, <a href="#Page_45">45</a>.<br />
+<br />
+<i>Barrow, Isaac</i>, <a href="#Page_126">126</a>.<br />
+<i>Bartholinus, Erasmus</i>, <a href="#Page_53">53</a>, <a href="#Page_54">54</a>, <a href="#Page_57">57</a>, <a href="#Page_60">60</a>, <a href="#Page_97">97</a>, <a href="#Page_99">99</a>.<br />
+<i>Boyle, Hon. Robert,</i> <a href="#Page_11">11</a>.<br />
+<br />
+<i>Cassini, Jacques</i>, <a href="#Page_iii">iii</a>.<br />
+<i>Caustic Curves</i>, <a href="#Page_123">123</a>.<br />
+<i>Crystals</i>, see <i>Iceland Crystal, Rock Crystal</i>.<br />
+<i>Crystals, configuration of</i>, <a href="#Page_95">95</a>.<br />
+<br />
+<i>Descartes, R&eacute;n&ecirc;</i>, <a href="#Page_3">3</a>, <a href="#Page_5">5</a>, <a href="#Page_7">7</a>, <a href="#Page_14">14</a>, <a href="#Page_22">22</a>, <a href="#Page_42">42</a>, <a href="#Page_43">43</a>, <a href="#Page_109">109</a>, <a href="#Page_113">113</a>.<br />
+<i>Double Refraction, discovery of</i>, <a href="#Page_54">54</a>, <a href="#Page_81">81</a>, <a href="#Page_93">93</a>.<br />
+<br />
+<i>Elasticity</i>, <a href="#Page_12">12</a>, <a href="#Page_14">14</a>.<br />
+<i>Ether, the, or Ethereal matter</i>, <a href="#Page_11">11</a>, <a href="#Page_14">14</a>, <a href="#Page_16">16</a>, <a href="#Page_28">28</a>.<br />
+<i>Extraordinary refraction</i>, <a href="#Page_55">55</a>, <a href="#Page_56">56</a>.<br />
+<br />
+<i>Fermat, principle of</i>, <a href="#Page_42">42</a>.<br />
+<i>Figures of transparent bodies</i>, <a href="#Page_105">105</a>.<br />
+<br />
+<i>Hooke, Robert</i>, <a href="#Page_20">20</a>.<br />
+<br />
+<i>Iceland Crystal</i>, <a href="#Page_2">2</a>, <a href="#Page_52">52</a> sqq.<br />
+<i>Iceland Crystal, Cutting and Polishing of</i>, <a href="#Page_91">91</a>, <a href="#Page_92">92</a>, <a href="#Page_98">98</a>.<br />
+<br />
+<i>Leibnitz, G.W.</i>, <a href="#Page_vi">vi</a>.<br />
+<i>Light, nature of</i>, <a href="#Page_3">3</a>.<br />
+<i>Light, velocity of</i>, <a href="#Page_4">4</a>, <a href="#Page_15">15</a>.<br />
+<br />
+<i>Molecular texture of bodies</i>, <a href="#Page_27">27</a>, <a href="#Page_95">95</a>.<br />
+<br />
+<i>Newton, Sir Isaac</i>, <a href="#Page_vi">vi</a>, <a href="#Page_106">106</a>.<br />
+<br />
+<i>Opacity</i>, <a href="#Page_34">34</a>.<br />
+<i>Ovals, Cartesian</i>, <a href="#Page_107">107</a>, <a href="#Page_113">113</a>.<br />
+<br />
+<i>Pardies, Rev. Father</i>, <a href="#Page_20">20</a>.<br />
+<i>Rays, definition of</i>, <a href="#Page_38">38</a>, <a href="#Page_49">49</a>.<br />
+<i>Reflexion</i>, <a href="#Page_22">22</a>.<br />
+<i>Refraction,</i> <a href="#Page_28">28</a>, <a href="#Page_34">34</a>.<br />
+<i>Rock Crystal</i>, <a href="#Page_54">54</a>, <a href="#Page_57">57</a>, <a href="#Page_62">62</a>, <a href="#Page_95">95</a>.<br />
+<i>R&ouml;mer, Olaf</i>, <a href="#Page_v">v</a>, <a href="#Page_7">7</a>.<br />
+<i>Roughness of surfaces</i>, <a href="#Page_27">27</a>.<br />
+<br />
+<i>Sines, law of</i>, <a href="#Page_1">1</a>, <a href="#Page_35">35</a>, <a href="#Page_38">38</a>, <a href="#Page_43">43</a>.<br />
+<i>Spheres, elasticity of</i>, <a href="#Page_15">15</a>.<br />
+<i>Spheroidal waves in crystals</i>, <a href="#Page_63">63</a>.<br />
+<i>Spheroids, lemma about</i>, <a href="#Page_103">103</a>.<br />
+<i>Sound, speed of</i>, <a href="#Page_7">7</a>, <a href="#Page_10">10</a>, <a href="#Page_12">12</a>.<br />
+<br />
+<i>Telescopes, lenses for</i>, <a href="#Page_62">62</a>, <a href="#Page_105">105</a>.<br />
+<i>Torricelli's experiment</i>, <a href="#Page_12">12</a>, <a href="#Page_30">30</a>.<br />
+<i>Transparency, explanation of</i>, <a href="#Page_28">28</a>, <a href="#Page_31">31</a>, <a href="#Page_32">32</a>.<br />
+<br />
+<i>Waves, no regular succession of</i>, <a href="#Page_17">17</a>.<br />
+<i>Waves, principle of wave envelopes</i>, <a href="#Page_19">19</a>, <a href="#Page_24">24</a>.<br />
+<i>Waves, principle of elementary wave fronts</i>, <a href="#Page_19">19</a>.<br />
+<i>Waves, propagation of light as</i>, <a href="#Page_16">16</a>, <a href="#Page_63">63</a>.<br />
+</p>
+
+<p>&nbsp;</p>
+<div>*** END OF THE PROJECT GUTENBERG EBOOK 14725 ***</div>
+</body>
+</html>
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+This eBook, including all associated images, markup, improvements,
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+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #14725 (https://www.gutenberg.org/ebooks/14725)
diff --git a/old/14725-8.txt b/old/14725-8.txt
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+The Project Gutenberg eBook, Treatise on Light, by Christiaan Huygens,
+Translated by Silvanus P. Thompson
+
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+
+
+
+Title: Treatise on Light
+
+Author: Christiaan Huygens
+
+Release Date: January 18, 2005  [eBook #14725]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+
+***START OF THE PROJECT GUTENBERG EBOOK TREATISE ON LIGHT***
+
+
+E-text prepared by Clare Boothby, Stephen Schulze, and the Project
+Gutenberg Online Distributed Proofreading Team
+
+
+
+Note: Project Gutenberg also has an HTML version of this
+      file which includes the original illustrations.
+      See 14725-h.htm or 14725-h.zip:
+      (https://www.gutenberg.org/dirs/1/4/7/2/14725/14725-h/14725-h.htm)
+      or
+      (https://www.gutenberg.org/dirs/1/4/7/2/14725/14725-h.zip)
+
+
+
+
+
+TREATISE ON LIGHT
+
+In which are explained
+The causes of that which occurs
+In REFLEXION, & in REFRACTION
+
+And particularly
+In the strange REFRACTION
+OF ICELAND CRYSTAL
+
+by
+
+CHRISTIAAN HUYGENS
+
+Rendered into English by
+
+SILVANUS P. THOMPSON
+
+University of Chicago Press
+
+
+
+
+
+
+
+PREFACE
+
+
+I wrote this Treatise during my sojourn in France twelve years ago,
+and I communicated it in the year 1678 to the learned persons who then
+composed the Royal Academy of Science, to the membership of which the
+King had done me the honour of calling, me. Several of that body who
+are still alive will remember having been present when I read it, and
+above the rest those amongst them who applied themselves particularly
+to the study of Mathematics; of whom I cannot cite more than the
+celebrated gentlemen Cassini, Römer, and De la Hire. And, although I
+have since corrected and changed some parts, the copies which I had
+made of it at that time may serve for proof that I have yet added
+nothing to it save some conjectures touching the formation of Iceland
+Crystal, and a novel observation on the refraction of Rock Crystal. I
+have desired to relate these particulars to make known how long I have
+meditated the things which now I publish, and not for the purpose of
+detracting from the merit of those who, without having seen anything
+that I have written, may be found to have treated of like matters: as
+has in fact occurred to two eminent Geometricians, Messieurs Newton
+and Leibnitz, with respect to the Problem of the figure of glasses for
+collecting rays when one of the surfaces is given.
+
+One may ask why I have so long delayed to bring this work to the
+light. The reason is that I wrote it rather carelessly in the Language
+in which it appears, with the intention of translating it into Latin,
+so doing in order to obtain greater attention to the thing. After
+which I proposed to myself to give it out along with another Treatise
+on Dioptrics, in which I explain the effects of Telescopes and those
+things which belong more to that Science. But the pleasure of novelty
+being past, I have put off from time to time the execution of this
+design, and I know not when I shall ever come to an end if it, being
+often turned aside either by business or by some new study.
+Considering which I have finally judged that it was better worth while
+to publish this writing, such as it is, than to let it run the risk,
+by waiting longer, of remaining lost.
+
+There will be seen in it demonstrations of those kinds which do not
+produce as great a certitude as those of Geometry, and which even
+differ much therefrom, since whereas the Geometers prove their
+Propositions by fixed and incontestable Principles, here the
+Principles are verified by the conclusions to be drawn from them; the
+nature of these things not allowing of this being done otherwise.
+
+It is always possible to attain thereby to a degree of probability
+which very often is scarcely less than complete proof. To wit, when
+things which have been demonstrated by the Principles that have been
+assumed correspond perfectly to the phenomena which experiment has
+brought under observation; especially when there are a great number of
+them, and further, principally, when one can imagine and foresee new
+phenomena which ought to follow from the hypotheses which one employs,
+and when one finds that therein the fact corresponds to our prevision.
+But if all these proofs of probability are met with in that which I
+propose to discuss, as it seems to me they are, this ought to be a
+very strong confirmation of the success of my inquiry; and it must be
+ill if the facts are not pretty much as I represent them. I would
+believe then that those who love to know the Causes of things and who
+are able to admire the marvels of Light, will find some satisfaction
+in these various speculations regarding it, and in the new explanation
+of its famous property which is the main foundation of the
+construction of our eyes and of those great inventions which extend so
+vastly the use of them.
+
+I hope also that there will be some who by following these beginnings
+will penetrate much further into this question than I have been able
+to do, since the subject must be far from being exhausted. This
+appears from the passages which I have indicated where I leave certain
+difficulties without having resolved them, and still more from matters
+which I have not touched at all, such as Luminous Bodies of several
+sorts, and all that concerns Colours; in which no one until now can
+boast of having succeeded. Finally, there remains much more to be
+investigated touching the nature of Light which I do not pretend to
+have disclosed, and I shall owe much in return to him who shall be
+able to supplement that which is here lacking to me in knowledge. The
+Hague. The 8 January 1690.
+
+
+
+
+NOTE BY THE TRANSLATOR
+
+
+Considering the great influence which this Treatise has exercised in
+the development of the Science of Optics, it seems strange that two
+centuries should have passed before an English edition of the work
+appeared. Perhaps the circumstance is due to the mistaken zeal with
+which formerly everything that conflicted with the cherished ideas of
+Newton was denounced by his followers. The Treatise on Light of
+Huygens has, however, withstood the test of time: and even now the
+exquisite skill with which he applied his conception of the
+propagation of waves of light to unravel the intricacies of the
+phenomena of the double refraction of crystals, and of the refraction
+of the atmosphere, will excite the admiration of the student of
+Optics. It is true that his wave theory was far from the complete
+doctrine as subsequently developed by Thomas Young and Augustin
+Fresnel, and belonged rather to geometrical than to physical Optics.
+If Huygens had no conception of transverse vibrations, of the
+principle of interference, or of the existence of the ordered sequence
+of waves in trains, he nevertheless attained to a remarkably clear
+understanding of the principles of wave-propagation; and his
+exposition of the subject marks an epoch in the treatment of Optical
+problems. It has been needful in preparing this translation to
+exercise care lest one should import into the author's text ideas of
+subsequent date, by using words that have come to imply modern
+conceptions. Hence the adoption of as literal a rendering as possible.
+A few of the author's terms need explanation. He uses the word
+"refraction," for example, both for the phenomenon or process usually
+so denoted, and for the result of that process: thus the refracted ray
+he habitually terms "the refraction" of the incident ray. When a
+wave-front, or, as he terms it, a "wave," has passed from some initial
+position to a subsequent one, he terms the wave-front in its
+subsequent position "the continuation" of the wave. He also speaks of
+the envelope of a set of elementary waves, formed by coalescence of
+those elementary wave-fronts, as "the termination" of the wave; and
+the elementary wave-fronts he terms "particular" waves. Owing to the
+circumstance that the French word _rayon_ possesses the double
+signification of ray of light and radius of a circle, he avoids its
+use in the latter sense and speaks always of the semi-diameter, not of
+the radius. His speculations as to the ether, his suggestive views of
+the structure of crystalline bodies, and his explanation of opacity,
+slight as they are, will possibly surprise the reader by their seeming
+modernness. And none can read his investigation of the phenomena found
+in Iceland spar without marvelling at his insight and sagacity.
+
+S.P.T.
+
+June, 1912.
+
+
+
+
+TABLE OF MATTERS
+
+Contained in this Treatise
+
+
+CHAPTER I.
+On Rays Propagated in Straight Lines.
+
+  That Light is produced by a certain movement.
+
+  That no substance passes from the luminous object to the eyes.
+
+  That Light spreads spherically, almost as Sound does.
+
+  Whether Light takes time to spread.
+
+  Experience seeming to prove that it passes instantaneously.
+
+  Experience proving that it takes time.
+
+  How much its speed is greater than that of Sound.
+
+  In what the emission of Light differs from that of Sound.
+
+  That it is not the same medium which serves for Light and Sound.
+
+  How Sound is propagated.
+
+  How Light is propagated.
+
+  Detailed Remarks on the propagation of Light.
+
+  Why Rays are propagated only in straight lines.
+
+  How Light coming in different directions can cross itself.
+
+CHAPTER II.
+On Reflexion.
+
+  Demonstration of equality of angles of incidence and reflexion.
+
+  Why the incident and reflected rays are in the same plane
+  perpendicular to the reflecting surface.
+
+  That it is not needful for the reflecting surface to be perfectly
+  flat to attain equality of the angles of incidence and reflexion.
+
+CHAPTER III.
+On Refraction.
+
+  That bodies may be transparent without any substance passing through
+  them.
+
+  Proof that the ethereal matter passes through transparent bodies.
+
+  How this matter passing through can render them transparent.
+
+  That the most solid bodies in appearance are of a very loose texture.
+
+  That Light spreads more slowly in water and in glass than in air.
+
+  Third hypothesis to explain transparency, and the retardation which
+  Light suffers.
+
+  On that which makes bodies opaque.
+
+  Demonstration why Refraction obeys the known proportion of Sines.
+
+  Why the incident and refracted Rays produce one another reciprocally.
+
+  Why Reflexion within a triangular glass prism is suddenly augmented
+  when the Light can no longer penetrate.
+
+  That bodies which cause greater Refraction also cause stronger
+  Reflexion.
+
+  Demonstration of the Theorem of Mr. Fermat.
+
+CHAPTER IV.
+On the Refraction of the Air.
+
+  That the emanations of Light in the air are not spherical.
+
+  How consequently some objects appear higher than they are.
+
+  How the Sun may appear on the Horizon before he has risen.
+
+  That the rays of light become curved in the Air of the Atmosphere,
+  and what effects this produces.
+
+CHAPTER V.
+On the Strange Refraction of Iceland Crystal.
+
+  That this Crystal grows also in other countries.
+
+  Who first-wrote about it.
+
+  Description of Iceland Crystal; its substance, shape, and properties.
+
+  That it has two different Refractions.
+
+  That the ray perpendicular to the surface suffers refraction, and
+  that some rays inclined to the surface pass without suffering
+  refraction.
+
+  Observation of the refractions in this Crystal.
+
+  That there is a Regular and an Irregular Refraction.
+
+  The way of measuring the two Refractions of Iceland Crystal.
+
+  Remarkable properties of the Irregular Refraction.
+
+  Hypothesis to explain the double Refraction.
+
+  That Rock Crystal has also a double Refraction.
+
+  Hypothesis of emanations of Light, within Iceland Crystal, of
+  spheroidal form, for the Irregular Refraction.
+
+  How a perpendicular ray can suffer Refraction.
+
+  How the position and form of the spheroidal emanations in this
+  Crystal can be defined.
+
+  Explanation of the Irregular Refraction by these spheroidal
+  emanations.
+
+  Easy way to find the Irregular Refraction of each incident ray.
+
+  Demonstration of the oblique ray which traverses the Crystal without
+  being refracted.
+
+  Other irregularities of Refraction explained.
+
+  That an object placed beneath the Crystal appears double, in two
+  images of different heights.
+
+  Why the apparent heights of one of the images change on changing the
+  position of the eyes above the Crystal.
+
+  Of the different sections of this Crystal which produce yet other
+  refractions, and confirm all this Theory.
+
+  Particular way of polishing the surfaces after it has been cut.
+
+  Surprising phenomenon touching the rays which pass through two
+  separated pieces; the cause of which is not explained.
+
+  Probable conjecture on the internal composition of Iceland Crystal,
+  and of what figure its particles are.
+
+  Tests to confirm this conjecture.
+
+  Calculations which have been supposed in this Chapter.
+
+CHAPTER VI.
+On the Figures of transparent bodies which serve for Refraction and
+for Reflexion.
+
+  General and easy rule to find these Figures.
+
+  Invention of the Ovals of Mr. Des Cartes for Dioptrics.
+
+  How he was able to find these Lines.
+
+  Way of finding the surface of a glass for perfect refraction, when
+  the other surface is given.
+
+  Remark on what happens to rays refracted at a spherical surface.
+
+  Remark on the curved line which is formed by reflexion in a spherical
+  concave mirror.
+
+
+
+
+
+CHAPTER I
+
+ON RAYS PROPAGATED IN STRAIGHT LINES
+
+
+As happens in all the sciences in which Geometry is applied to matter,
+the demonstrations concerning Optics are founded on truths drawn from
+experience. Such are that the rays of light are propagated in straight
+lines; that the angles of reflexion and of incidence are equal; and
+that in refraction the ray is bent according to the law of sines, now
+so well known, and which is no less certain than the preceding laws.
+
+The majority of those who have written touching the various parts of
+Optics have contented themselves with presuming these truths. But
+some, more inquiring, have desired to investigate the origin and the
+causes, considering these to be in themselves wonderful effects of
+Nature. In which they advanced some ingenious things, but not however
+such that the most intelligent folk do not wish for better and more
+satisfactory explanations. Wherefore I here desire to propound what I
+have meditated on the subject, so as to contribute as much as I can
+to the explanation of this department of Natural Science, which, not
+without reason, is reputed to be one of its most difficult parts. I
+recognize myself to be much indebted to those who were the first to
+begin to dissipate the strange obscurity in which these things were
+enveloped, and to give us hope that they might be explained by
+intelligible reasoning. But, on the other hand I am astonished also
+that even here these have often been willing to offer, as assured and
+demonstrative, reasonings which were far from conclusive. For I do not
+find that any one has yet given a probable explanation of the first
+and most notable phenomena of light, namely why it is not propagated
+except in straight lines, and how visible rays, coming from an
+infinitude of diverse places, cross one another without hindering one
+another in any way.
+
+I shall therefore essay in this book, to give, in accordance with the
+principles accepted in the Philosophy of the present day, some clearer
+and more probable reasons, firstly of these properties of light
+propagated rectilinearly; secondly of light which is reflected on
+meeting other bodies. Then I shall explain the phenomena of those rays
+which are said to suffer refraction on passing through transparent
+bodies of different sorts; and in this part I shall also explain the
+effects of the refraction of the air by the different densities of the
+Atmosphere.
+
+Thereafter I shall examine the causes of the strange refraction of a
+certain kind of Crystal which is brought from Iceland. And finally I
+shall treat of the various shapes of transparent and reflecting bodies
+by which rays are collected at a point or are turned aside in various
+ways. From this it will be seen with what facility, following our new
+Theory, we find not only the Ellipses, Hyperbolas, and other curves
+which Mr. Des Cartes has ingeniously invented for this purpose; but
+also those which the surface of a glass lens ought to possess when its
+other surface is given as spherical or plane, or of any other figure
+that may be.
+
+It is inconceivable to doubt that light consists in the motion of some
+sort of matter. For whether one considers its production, one sees
+that here upon the Earth it is chiefly engendered by fire and flame
+which contain without doubt bodies that are in rapid motion, since
+they dissolve and melt many other bodies, even the most solid; or
+whether one considers its effects, one sees that when light is
+collected, as by concave mirrors, it has the property of burning as a
+fire does, that is to say it disunites the particles of bodies. This
+is assuredly the mark of motion, at least in the true Philosophy, in
+which one conceives the causes of all natural effects in terms of
+mechanical motions. This, in my opinion, we must necessarily do, or
+else renounce all hopes of ever comprehending anything in Physics.
+
+And as, according to this Philosophy, one holds as certain that the
+sensation of sight is excited only by the impression of some movement
+of a kind of matter which acts on the nerves at the back of our eyes,
+there is here yet one reason more for believing that light consists in
+a movement of the matter which exists between us and the luminous
+body.
+
+Further, when one considers the extreme speed with which light spreads
+on every side, and how, when it comes from different regions, even
+from those directly opposite, the rays traverse one another without
+hindrance, one may well understand that when we see a luminous object,
+it cannot be by any transport of matter coming to us from this object,
+in the way in which a shot or an arrow traverses the air; for
+assuredly that would too greatly impugn these two properties of light,
+especially the second of them. It is then in some other way that light
+spreads; and that which can lead us to comprehend it is the knowledge
+which we have of the spreading of Sound in the air.
+
+We know that by means of the air, which is an invisible and impalpable
+body, Sound spreads around the spot where it has been produced, by a
+movement which is passed on successively from one part of the air to
+another; and that the spreading of this movement, taking place equally
+rapidly on all sides, ought to form spherical surfaces ever enlarging
+and which strike our ears. Now there is no doubt at all that light
+also comes from the luminous body to our eyes by some movement
+impressed on the matter which is between the two; since, as we have
+already seen, it cannot be by the transport of a body which passes
+from one to the other. If, in addition, light takes time for its
+passage--which we are now going to examine--it will follow that this
+movement, impressed on the intervening matter, is successive; and
+consequently it spreads, as Sound does, by spherical surfaces and
+waves: for I call them waves from their resemblance to those which are
+seen to be formed in water when a stone is thrown into it, and which
+present a successive spreading as circles, though these arise from
+another cause, and are only in a flat surface.
+
+To see then whether the spreading of light takes time, let us consider
+first whether there are any facts of experience which can convince us
+to the contrary. As to those which can be made here on the Earth, by
+striking lights at great distances, although they prove that light
+takes no sensible time to pass over these distances, one may say with
+good reason that they are too small, and that the only conclusion to
+be drawn from them is that the passage of light is extremely rapid.
+Mr. Des Cartes, who was of opinion that it is instantaneous, founded
+his views, not without reason, upon a better basis of experience,
+drawn from the Eclipses of the Moon; which, nevertheless, as I shall
+show, is not at all convincing. I will set it forth, in a way a little
+different from his, in order to make the conclusion more
+comprehensible.
+
+[Illustration]
+
+Let A be the place of the sun, BD a part of the orbit or annual path
+of the Earth: ABC a straight line which I suppose to meet the orbit of
+the Moon, which is represented by the circle CD, at C.
+
+Now if light requires time, for example one hour, to traverse the
+space which is between the Earth and the Moon, it will follow that the
+Earth having arrived at B, the shadow which it casts, or the
+interruption of the light, will not yet have arrived at the point C,
+but will only arrive there an hour after. It will then be one hour
+after, reckoning from the moment when the Earth was at B, that the
+Moon, arriving at C, will be obscured: but this obscuration or
+interruption of the light will not reach the Earth till after another
+hour. Let us suppose that the Earth in these two hours will have
+arrived at E. The Earth then, being at E, will see the Eclipsed Moon
+at C, which it left an hour before, and at the same time will see the
+sun at A. For it being immovable, as I suppose with Copernicus, and
+the light moving always in straight lines, it must always appear where
+it is. But one has always observed, we are told, that the eclipsed
+Moon appears at the point of the Ecliptic opposite to the Sun; and yet
+here it would appear in arrear of that point by an amount equal to the
+angle GEC, the supplement of AEC. This, however, is contrary to
+experience, since the angle GEC would be very sensible, and about 33
+degrees. Now according to our computation, which is given in the
+Treatise on the causes of the phenomena of Saturn, the distance BA
+between the Earth and the Sun is about twelve thousand diameters of
+the Earth, and hence four hundred times greater than BC the distance
+of the Moon, which is 30 diameters. Then the angle ECB will be nearly
+four hundred times greater than BAE, which is five minutes; namely,
+the path which the earth travels in two hours along its orbit; and
+thus the angle BCE will be nearly 33 degrees; and likewise the angle
+CEG, which is greater by five minutes.
+
+But it must be noted that the speed of light in this argument has been
+assumed such that it takes a time of one hour to make the passage from
+here to the Moon. If one supposes that for this it requires only one
+minute of time, then it is manifest that the angle CEG will only be 33
+minutes; and if it requires only ten seconds of time, the angle will
+be less than six minutes. And then it will not be easy to perceive
+anything of it in observations of the Eclipse; nor, consequently, will
+it be permissible to deduce from it that the movement of light is
+instantaneous.
+
+It is true that we are here supposing a strange velocity that would be
+a hundred thousand times greater than that of Sound. For Sound,
+according to what I have observed, travels about 180 Toises in the
+time of one Second, or in about one beat of the pulse. But this
+supposition ought not to seem to be an impossibility; since it is not
+a question of the transport of a body with so great a speed, but of a
+successive movement which is passed on from some bodies to others. I
+have then made no difficulty, in meditating on these things, in
+supposing that the emanation of light is accomplished with time,
+seeing that in this way all its phenomena can be explained, and that
+in following the contrary opinion everything is incomprehensible. For
+it has always seemed tome that even Mr. Des Cartes, whose aim has been
+to treat all the subjects of Physics intelligibly, and who assuredly
+has succeeded in this better than any one before him, has said nothing
+that is not full of difficulties, or even inconceivable, in dealing
+with Light and its properties.
+
+But that which I employed only as a hypothesis, has recently received
+great seemingness as an established truth by the ingenious proof of
+Mr. Römer which I am going here to relate, expecting him himself to
+give all that is needed for its confirmation. It is founded as is the
+preceding argument upon celestial observations, and proves not only
+that Light takes time for its passage, but also demonstrates how much
+time it takes, and that its velocity is even at least six times
+greater than that which I have just stated.
+
+For this he makes use of the Eclipses suffered by the little planets
+which revolve around Jupiter, and which often enter his shadow: and
+see what is his reasoning. Let A be the Sun, BCDE the annual orbit of
+the Earth, F Jupiter, GN the orbit of the nearest of his Satellites,
+for it is this one which is more apt for this investigation than any
+of the other three, because of the quickness of its revolution. Let G
+be this Satellite entering into the shadow of Jupiter, H the same
+Satellite emerging from the shadow.
+
+[Illustration]
+
+Let it be then supposed, the Earth being at B some time before the
+last quadrature, that one has seen the said Satellite emerge from the
+shadow; it must needs be, if the Earth remains at the same place,
+that, after 42-1/2 hours, one would again see a similar emergence,
+because that is the time in which it makes the round of its orbit, and
+when it would come again into opposition to the Sun. And if the Earth,
+for instance, were to remain always at B during 30 revolutions of this
+Satellite, one would see it again emerge from the shadow after 30
+times 42-1/2 hours. But the Earth having been carried along during
+this time to C, increasing thus its distance from Jupiter, it follows
+that if Light requires time for its passage the illumination of the
+little planet will be perceived later at C than it would have been at
+B, and that there must be added to this time of 30 times 42-1/2 hours
+that which the Light has required to traverse the space MC, the
+difference of the spaces CH, BH. Similarly at the other quadrature
+when the earth has come to E from D while approaching toward Jupiter,
+the immersions of the Satellite ought to be observed at E earlier than
+they would have been seen if the Earth had remained at D.
+
+Now in quantities of observations of these Eclipses, made during ten
+consecutive years, these differences have been found to be very
+considerable, such as ten minutes and more; and from them it has been
+concluded that in order to traverse the whole diameter of the annual
+orbit KL, which is double the distance from here to the sun, Light
+requires about 22 minutes of time.
+
+The movement of Jupiter in his orbit while the Earth passed from B to
+C, or from D to E, is included in this calculation; and this makes it
+evident that one cannot attribute the retardation of these
+illuminations or the anticipation of the eclipses, either to any
+irregularity occurring in the movement of the little planet or to its
+eccentricity.
+
+If one considers the vast size of the diameter KL, which according to
+me is some 24 thousand diameters of the Earth, one will acknowledge
+the extreme velocity of Light. For, supposing that KL is no more than
+22 thousand of these diameters, it appears that being traversed in 22
+minutes this makes the speed a thousand diameters in one minute, that
+is 16-2/3 diameters in one second or in one beat of the pulse, which
+makes more than 11 hundred times a hundred thousand toises; since the
+diameter of the Earth contains 2,865 leagues, reckoned at 25 to the
+degree, and each each league is 2,282 Toises, according to the exact
+measurement which Mr. Picard made by order of the King in 1669. But
+Sound, as I have said above, only travels 180 toises in the same time
+of one second: hence the velocity of Light is more than six hundred
+thousand times greater than that of Sound. This, however, is quite
+another thing from being instantaneous, since there is all the
+difference between a finite thing and an infinite. Now the successive
+movement of Light being confirmed in this way, it follows, as I have
+said, that it spreads by spherical waves, like the movement of Sound.
+
+But if the one resembles the other in this respect, they differ in
+many other things; to wit, in the first production of the movement
+which causes them; in the matter in which the movement spreads; and in
+the manner in which it is propagated. As to that which occurs in the
+production of Sound, one knows that it is occasioned by the agitation
+undergone by an entire body, or by a considerable part of one, which
+shakes all the contiguous air. But the movement of the Light must
+originate as from each point of the luminous object, else we should
+not be able to perceive all the different parts of that object, as
+will be more evident in that which follows. And I do not believe that
+this movement can be better explained than by supposing that all those
+of the luminous bodies which are liquid, such as flames, and
+apparently the sun and the stars, are composed of particles which
+float in a much more subtle medium which agitates them with great
+rapidity, and makes them strike against the particles of the ether
+which surrounds them, and which are much smaller than they. But I hold
+also that in luminous solids such as charcoal or metal made red hot in
+the fire, this same movement is caused by the violent agitation of
+the particles of the metal or of the wood; those of them which are on
+the surface striking similarly against the ethereal matter. The
+agitation, moreover, of the particles which engender the light ought
+to be much more prompt and more rapid than is that of the bodies which
+cause sound, since we do not see that the tremors of a body which is
+giving out a sound are capable of giving rise to Light, even as the
+movement of the hand in the air is not capable of producing Sound.
+
+Now if one examines what this matter may be in which the movement
+coming from the luminous body is propagated, which I call Ethereal
+matter, one will see that it is not the same that serves for the
+propagation of Sound. For one finds that the latter is really that
+which we feel and which we breathe, and which being removed from any
+place still leaves there the other kind of matter that serves to
+convey Light. This may be proved by shutting up a sounding body in a
+glass vessel from which the air is withdrawn by the machine which Mr.
+Boyle has given us, and with which he has performed so many beautiful
+experiments. But in doing this of which I speak, care must be taken to
+place the sounding body on cotton or on feathers, in such a way that
+it cannot communicate its tremors either to the glass vessel which
+encloses it, or to the machine; a precaution which has hitherto been
+neglected. For then after having exhausted all the air one hears no
+Sound from the metal, though it is struck.
+
+One sees here not only that our air, which does not penetrate through
+glass, is the matter by which Sound spreads; but also that it is not
+the same air but another kind of matter in which Light spreads; since
+if the air is removed from the vessel the Light does not cease to
+traverse it as before.
+
+And this last point is demonstrated even more clearly by the
+celebrated experiment of Torricelli, in which the tube of glass from
+which the quicksilver has withdrawn itself, remaining void of air,
+transmits Light just the same as when air is in it. For this proves
+that a matter different from air exists in this tube, and that this
+matter must have penetrated the glass or the quicksilver, either one
+or the other, though they are both impenetrable to the air. And when,
+in the same experiment, one makes the vacuum after putting a little
+water above the quicksilver, one concludes equally that the said
+matter passes through glass or water, or through both.
+
+As regards the different modes in which I have said the movements of
+Sound and of Light are communicated, one may sufficiently comprehend
+how this occurs in the case of Sound if one considers that the air is
+of such a nature that it can be compressed and reduced to a much
+smaller space than that which it ordinarily occupies. And in
+proportion as it is compressed the more does it exert an effort to
+regain its volume; for this property along with its penetrability,
+which remains notwithstanding its compression, seems to prove that it
+is made up of small bodies which float about and which are agitated
+very rapidly in the ethereal matter composed of much smaller parts. So
+that the cause of the spreading of Sound is the effort which these
+little bodies make in collisions with one another, to regain freedom
+when they are a little more squeezed together in the circuit of these
+waves than elsewhere.
+
+But the extreme velocity of Light, and other properties which it has,
+cannot admit of such a propagation of motion, and I am about to show
+here the way in which I conceive it must occur. For this, it is
+needful to explain the property which hard bodies must possess to
+transmit movement from one to another.
+
+When one takes a number of spheres of equal size, made of some very
+hard substance, and arranges them in a straight line, so that they
+touch one another, one finds, on striking with a similar sphere
+against the first of these spheres, that the motion passes as in an
+instant to the last of them, which separates itself from the row,
+without one's being able to perceive that the others have been
+stirred. And even that one which was used to strike remains motionless
+with them. Whence one sees that the movement passes with an extreme
+velocity which is the greater, the greater the hardness of the
+substance of the spheres.
+
+But it is still certain that this progression of motion is not
+instantaneous, but successive, and therefore must take time. For if
+the movement, or the disposition to movement, if you will have it so,
+did not pass successively through all these spheres, they would all
+acquire the movement at the same time, and hence would all advance
+together; which does not happen. For the last one leaves the whole row
+and acquires the speed of the one which was pushed. Moreover there are
+experiments which demonstrate that all the bodies which we reckon of
+the hardest kind, such as quenched steel, glass, and agate, act as
+springs and bend somehow, not only when extended as rods but also when
+they are in the form of spheres or of other shapes. That is to say
+they yield a little in themselves at the place where they are struck,
+and immediately regain their former figure. For I have found that on
+striking with a ball of glass or of agate against a large and quite
+thick thick piece of the same substance which had a flat surface,
+slightly soiled with breath or in some other way, there remained round
+marks, of smaller or larger size according as the blow had been weak
+or strong. This makes it evident that these substances yield where
+they meet, and spring back: and for this time must be required.
+
+Now in applying this kind of movement to that which produces Light
+there is nothing to hinder us from estimating the particles of the
+ether to be of a substance as nearly approaching to perfect hardness
+and possessing a springiness as prompt as we choose. It is not
+necessary to examine here the causes of this hardness, or of that
+springiness, the consideration of which would lead us too far from our
+subject. I will say, however, in passing that we may conceive that the
+particles of the ether, notwithstanding their smallness, are in turn
+composed of other parts and that their springiness consists in the
+very rapid movement of a subtle matter which penetrates them from
+every side and constrains their structure to assume such a disposition
+as to give to this fluid matter the most overt and easy passage
+possible. This accords with the explanation which Mr. Des Cartes gives
+for the spring, though I do not, like him, suppose the pores to be in
+the form of round hollow canals. And it must not be thought that in
+this there is anything absurd or impossible, it being on the contrary
+quite credible that it is this infinite series of different sizes of
+corpuscles, having different degrees of velocity, of which Nature
+makes use to produce so many marvellous effects.
+
+But though we shall ignore the true cause of springiness we still see
+that there are many bodies which possess this property; and thus there
+is nothing strange in supposing that it exists also in little
+invisible bodies like the particles of the Ether. Also if one wishes
+to seek for any other way in which the movement of Light is
+successively communicated, one will find none which agrees better,
+with uniform progression, as seems to be necessary, than the property
+of springiness; because if this movement should grow slower in
+proportion as it is shared over a greater quantity of matter, in
+moving away from the source of the light, it could not conserve this
+great velocity over great distances. But by supposing springiness in
+the ethereal matter, its particles will have the property of equally
+rapid restitution whether they are pushed strongly or feebly; and thus
+the propagation of Light will always go on with an equal velocity.
+
+[Illustration]
+
+And it must be known that although the particles of the ether are not
+ranged thus in straight lines, as in our row of spheres, but
+confusedly, so that one of them touches several others, this does not
+hinder them from transmitting their movement and from spreading it
+always forward. As to this it is to be remarked that there is a law of
+motion serving for this propagation, and verifiable by experiment. It
+is that when a sphere, such as A here, touches several other similar
+spheres CCC, if it is struck by another sphere B in such a way as to
+exert an impulse against all the spheres CCC which touch it, it
+transmits to them the whole of its movement, and remains after that
+motionless like the sphere B. And without supposing that the ethereal
+particles are of spherical form (for I see indeed no need to suppose
+them so) one may well understand that this property of communicating
+an impulse does not fail to contribute to the aforesaid propagation
+of movement.
+
+Equality of size seems to be more necessary, because otherwise there
+ought to be some reflexion of movement backwards when it passes from a
+smaller particle to a larger one, according to the Laws of Percussion
+which I published some years ago.
+
+However, one will see hereafter that we have to suppose such an
+equality not so much as a necessity for the propagation of light as
+for rendering that propagation easier and more powerful; for it is not
+beyond the limits of probability that the particles of the ether have
+been made equal for a purpose so important as that of light, at least
+in that vast space which is beyond the region of atmosphere and which
+seems to serve only to transmit the light of the Sun and the Stars.
+
+I have then shown in what manner one may conceive Light to spread
+successively, by spherical waves, and how it is possible that this
+spreading is accomplished with as great a velocity as that which
+experiments and celestial observations demand. Whence it may be
+further remarked that although the particles are supposed to be in
+continual movement (for there are many reasons for this) the
+successive propagation of the waves cannot be hindered by this;
+because the propagation consists nowise in the transport of those
+particles but merely in a small agitation which they cannot help
+communicating to those surrounding, notwithstanding any movement which
+may act on them causing them to be changing positions amongst
+themselves.
+
+But we must consider still more particularly the origin of these
+waves, and the manner in which they spread. And, first, it follows
+from what has been said on the production of Light, that each little
+region of a luminous body, such as the Sun, a candle, or a burning
+coal, generates its own waves of which that region is the centre. Thus
+in the flame of a candle, having distinguished the points A, B, C,
+concentric circles described about each of these points represent the
+waves which come from them. And one must imagine the same about every
+point of the surface and of the part within the flame.
+
+[Illustration]
+
+But as the percussions at the centres of these waves possess no
+regular succession, it must not be supposed that the waves themselves
+follow one another at equal distances: and if the distances marked in
+the figure appear to be such, it is rather to mark the progression of
+one and the same wave at equal intervals of time than to represent
+several of them issuing from one and the same centre.
+
+After all, this prodigious quantity of waves which traverse one
+another without confusion and without effacing one another must not be
+deemed inconceivable; it being certain that one and the same particle
+of matter can serve for many waves coming from different sides or even
+from contrary directions, not only if it is struck by blows which
+follow one another closely but even for those which act on it at the
+same instant. It can do so because the spreading of the movement is
+successive. This may be proved by the row of equal spheres of hard
+matter, spoken of above. If against this row there are pushed from two
+opposite sides at the same time two similar spheres A and D, one will
+see each of them rebound with the same velocity which it had in
+striking, yet the whole row will remain in its place, although the
+movement has passed along its whole length twice over. And if these
+contrary movements happen to meet one another at the middle sphere, B,
+or at some other such as C, that sphere will yield and act as a spring
+at both sides, and so will serve at the same instant to transmit these
+two movements.
+
+[Illustration]
+
+But what may at first appear full strange and even incredible is that
+the undulations produced by such small movements and corpuscles,
+should spread to such immense distances; as for example from the Sun
+or from the Stars to us. For the force of these waves must grow feeble
+in proportion as they move away from their origin, so that the action
+of each one in particular will without doubt become incapable of
+making itself felt to our sight. But one will cease to be astonished
+by considering how at a great distance from the luminous body an
+infinitude of waves, though they have issued from different points of
+this body, unite together in such a way that they sensibly compose one
+single wave only, which, consequently, ought to have enough force to
+make itself felt. Thus this infinite number of waves which originate
+at the same instant from all points of a fixed star, big it may be as
+the Sun, make practically only one single wave which may well have
+force enough to produce an impression on our eyes. Moreover from each
+luminous point there may come many thousands of waves in the smallest
+imaginable time, by the frequent percussion of the corpuscles which
+strike the Ether at these points: which further contributes to
+rendering their action more sensible.
+
+[Illustration]
+
+There is the further consideration in the emanation of these waves,
+that each particle of matter in which a wave spreads, ought not to
+communicate its motion only to the next particle which is in the
+straight line drawn from the luminous point, but that it also imparts
+some of it necessarily to all the others which touch it and which
+oppose themselves to its movement. So it arises that around each
+particle there is made a wave of which that particle is the centre.
+Thus if DCF is a wave emanating from the luminous point A, which is
+its centre, the particle B, one of those comprised within the sphere
+DCF, will have made its particular or partial wave KCL, which will
+touch the wave DCF at C at the same moment that the principal wave
+emanating from the point A has arrived at DCF; and it is clear that it
+will be only the region C of the wave KCL which will touch the wave
+DCF, to wit, that which is in the straight line drawn through AB.
+Similarly the other particles of the sphere DCF, such as _bb_, _dd_,
+etc., will each make its own wave. But each of these waves can be
+infinitely feeble only as compared with the wave DCF, to the
+composition of which all the others contribute by the part of their
+surface which is most distant from the centre A.
+
+One sees, in addition, that the wave DCF is determined by the
+distance attained in a certain space of time by the movement which
+started from the point A; there being no movement beyond this wave,
+though there will be in the space which it encloses, namely in parts
+of the particular waves, those parts which do not touch the sphere
+DCF. And all this ought not to seem fraught with too much minuteness
+or subtlety, since we shall see in the sequel that all the properties
+of Light, and everything pertaining to its reflexion and its
+refraction, can be explained in principle by this means. This is a
+matter which has been quite unknown to those who hitherto have begun
+to consider the waves of light, amongst whom are Mr. Hooke in his
+_Micrographia_, and Father Pardies, who, in a treatise of which he let
+me see a portion, and which he was unable to complete as he died
+shortly afterward, had undertaken to prove by these waves the effects
+of reflexion and refraction. But the chief foundation, which consists
+in the remark I have just made, was lacking in his demonstrations; and
+for the rest he had opinions very different from mine, as may be will
+appear some day if his writing has been preserved.
+
+To come to the properties of Light. We remark first that each portion
+of a wave ought to spread in such a way that its extremities lie
+always between the same straight lines drawn from the luminous point.
+Thus the portion BG of the wave, having the luminous point A as its
+centre, will spread into the arc CE bounded by the straight lines ABC,
+AGE. For although the particular waves produced by the particles
+comprised within the space CAE spread also outside this space, they
+yet do not concur at the same instant to compose a wave which
+terminates the movement, as they do precisely at the circumference
+CE, which is their common tangent.
+
+And hence one sees the reason why light, at least if its rays are not
+reflected or broken, spreads only by straight lines, so that it
+illuminates no object except when the path from its source to that
+object is open along such lines.
+
+For if, for example, there were an opening BG, limited by opaque
+bodies BH, GI, the wave of light which issues from the point A will
+always be terminated by the straight lines AC, AE, as has just been
+shown; the parts of the partial waves which spread outside the space
+ACE being too feeble to produce light there.
+
+Now, however small we make the opening BG, there is always the same
+reason causing the light there to pass between straight lines; since
+this opening is always large enough to contain a great number of
+particles of the ethereal matter, which are of an inconceivable
+smallness; so that it appears that each little portion of the wave
+necessarily advances following the straight line which comes from the
+luminous point. Thus then we may take the rays of light as if they
+were straight lines.
+
+It appears, moreover, by what has been remarked touching the
+feebleness of the particular waves, that it is not needful that all
+the particles of the Ether should be equal amongst themselves, though
+equality is more apt for the propagation of the movement. For it is
+true that inequality will cause a particle by pushing against another
+larger one to strive to recoil with a part of its movement; but it
+will thereby merely generate backwards towards the luminous point some
+partial waves incapable of causing light, and not a wave compounded of
+many as CE was.
+
+Another property of waves of light, and one of the most marvellous,
+is that when some of them come from different or even from opposing
+sides, they produce their effect across one another without any
+hindrance. Whence also it comes about that a number of spectators may
+view different objects at the same time through the same opening, and
+that two persons can at the same time see one another's eyes. Now
+according to the explanation which has been given of the action of
+light, how the waves do not destroy nor interrupt one another when
+they cross one another, these effects which I have just mentioned are
+easily conceived. But in my judgement they are not at all easy to
+explain according to the views of Mr. Des Cartes, who makes Light to
+consist in a continuous pressure merely tending to movement. For this
+pressure not being able to act from two opposite sides at the same
+time, against bodies which have no inclination to approach one
+another, it is impossible so to understand what I have been saying
+about two persons mutually seeing one another's eyes, or how two
+torches can illuminate one another.
+
+
+
+
+CHAPTER II
+
+ON REFLEXION
+
+
+Having explained the effects of waves of light which spread in a
+homogeneous matter, we will examine next that which happens to them on
+encountering other bodies. We will first make evident how the
+Reflexion of light is explained by these same waves, and why it
+preserves equality of angles.
+
+Let there be a surface AB; plane and polished, of some metal, glass,
+or other body, which at first I will consider as perfectly uniform
+(reserving to myself to deal at the end of this demonstration with the
+inequalities from which it cannot be exempt), and let a line AC,
+inclined to AD, represent a portion of a wave of light, the centre of
+which is so distant that this portion AC may be considered as a
+straight line; for I consider all this as in one plane, imagining to
+myself that the plane in which this figure is, cuts the sphere of the
+wave through its centre and intersects the plane AB at right angles.
+This explanation will suffice once for all.
+
+[Illustration]
+
+The piece C of the wave AC, will in a certain space of time advance as
+far as the plane AB at B, following the straight line CB, which may be
+supposed to come from the luminous centre, and which in consequence is
+perpendicular to AC. Now in this same space of time the portion A of
+the same wave, which has been hindered from communicating its movement
+beyond the plane AB, or at least partly so, ought to have continued
+its movement in the matter which is above this plane, and this along a
+distance equal to CB, making its own partial spherical wave,
+according to what has been said above. Which wave is here represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter AN equal to CB.
+
+If one considers further the other pieces H of the wave AC, it appears
+that they will not only have reached the surface AB by straight lines
+HK parallel to CB, but that in addition they will have generated in
+the transparent air, from the centres K, K, K, particular spherical
+waves, represented here by circumferences the semi-diameters of which
+are equal to KM, that is to say to the continuations of HK as far as
+the line BG parallel to AC. But all these circumferences have as a
+common tangent the straight line BN, namely the same which is drawn
+from B as a tangent to the first of the circles, of which A is the
+centre, and AN the semi-diameter equal to BC, as is easy to see.
+
+It is then the line BN (comprised between B and the point N where the
+perpendicular from the point A falls) which is as it were formed by
+all these circumferences, and which terminates the movement which is
+made by the reflexion of the wave AC; and it is also the place where
+the movement occurs in much greater quantity than anywhere else.
+Wherefore, according to that which has been explained, BN is the
+propagation of the wave AC at the moment when the piece C of it has
+arrived at B. For there is no other line which like BN is a common
+tangent to all the aforesaid circles, except BG below the plane AB;
+which line BG would be the propagation of the wave if the movement
+could have spread in a medium homogeneous with that which is above the
+plane. And if one wishes to see how the wave AC has come successively
+to BN, one has only to draw in the same figure the straight lines KO
+parallel to BN, and the straight lines KL parallel to AC. Thus one
+will see that the straight wave AC has become broken up into all the
+OKL parts successively, and that it has become straight again at NB.
+
+Now it is apparent here that the angle of reflexion is made equal to
+the angle of incidence. For the triangles ACB, BNA being rectangular
+and having the side AB common, and the side CB equal to NA, it follows
+that the angles opposite to these sides will be equal, and therefore
+also the angles CBA, NAB. But as CB, perpendicular to CA, marks the
+direction of the incident ray, so AN, perpendicular to the wave BN,
+marks the direction of the reflected ray; hence these rays are equally
+inclined to the plane AB.
+
+But in considering the preceding demonstration, one might aver that it
+is indeed true that BN is the common tangent of the circular waves in
+the plane of this figure, but that these waves, being in truth
+spherical, have still an infinitude of similar tangents, namely all
+the straight lines which are drawn from the point B in the surface
+generated by the straight line BN about the axis BA. It remains,
+therefore, to demonstrate that there is no difficulty herein: and by
+the same argument one will see why the incident ray and the reflected
+ray are always in one and the same plane perpendicular to the
+reflecting plane. I say then that the wave AC, being regarded only as
+a line, produces no light. For a visible ray of light, however narrow
+it may be, has always some width, and consequently it is necessary, in
+representing the wave whose progression constitutes the ray, to put
+instead of a line AC some plane figure such as the circle HC in the
+following figure, by supposing, as we have done, the luminous point to
+be infinitely distant. Now it is easy to see, following the preceding
+demonstration, that each small piece of this wave HC having arrived at
+the plane AB, and there generating each one its particular wave, these
+will all have, when C arrives at B, a common plane which will touch
+them, namely a circle BN similar to CH; and this will be intersected
+at its middle and at right angles by the same plane which likewise
+intersects the circle CH and the ellipse AB.
+
+[Illustration]
+
+One sees also that the said spheres of the partial waves cannot have
+any common tangent plane other than the circle BN; so that it will be
+this plane where there will be more reflected movement than anywhere
+else, and which will therefore carry on the light in continuance from
+the wave CH.
+
+I have also stated in the preceding demonstration that the movement of
+the piece A of the incident wave is not able to communicate itself
+beyond the plane AB, or at least not wholly. Whence it is to be
+remarked that though the movement of the ethereal matter might
+communicate itself partly to that of the reflecting body, this could
+in nothing alter the velocity of progression of the waves, on which
+the angle of reflexion depends. For a slight percussion ought to
+generate waves as rapid as strong percussion in the same matter. This
+comes about from the property of bodies which act as springs, of which
+we have spoken above; namely that whether compressed little or much
+they recoil in equal times. Equally so in every reflexion of the
+light, against whatever body it may be, the angles of reflexion and
+incidence ought to be equal notwithstanding that the body might be of
+such a nature that it takes away a portion of the movement made by the
+incident light. And experiment shows that in fact there is no polished
+body the reflexion of which does not follow this rule.
+
+
+But the thing to be above all remarked in our demonstration is that it
+does not require that the reflecting surface should be considered as a
+uniform plane, as has been supposed by all those who have tried to
+explain the effects of reflexion; but only an evenness such as may be
+attained by the particles of the matter of the reflecting body being
+set near to one another; which particles are larger than those of the
+ethereal matter, as will appear by what we shall say in treating of
+the transparency and opacity of bodies. For the surface consisting
+thus of particles put together, and the ethereal particles being
+above, and smaller, it is evident that one could not demonstrate the
+equality of the angles of incidence and reflexion by similitude to
+that which happens to a ball thrown against a wall, of which writers
+have always made use. In our way, on the other hand, the thing is
+explained without difficulty. For the smallness of the particles of
+quicksilver, for example, being such that one must conceive millions
+of them, in the smallest visible surface proposed, arranged like a
+heap of grains of sand which has been flattened as much as it is
+capable of being, this surface then becomes for our purpose as even
+as a polished glass is: and, although it always remains rough with
+respect to the particles of the Ether it is evident that the centres
+of all the particular spheres of reflexion, of which we have spoken,
+are almost in one uniform plane, and that thus the common tangent can
+fit to them as perfectly as is requisite for the production of light.
+And this alone is requisite, in our method of demonstration, to cause
+equality of the said angles without the remainder of the movement
+reflected from all parts being able to produce any contrary effect.
+
+
+
+
+CHAPTER III
+
+ON REFRACTION
+
+
+In the same way as the effects of Reflexion have been explained by
+waves of light reflected at the surface of polished bodies, we will
+explain transparency and the phenomena of refraction by waves which
+spread within and across diaphanous bodies, both solids, such as
+glass, and liquids, such as water, oils, etc. But in order that it may
+not seem strange to suppose this passage of waves in the interior of
+these bodies, I will first show that one may conceive it possible in
+more than one mode.
+
+First, then, if the ethereal matter cannot penetrate transparent
+bodies at all, their own particles would be able to communicate
+successively the movement of the waves, the same as do those of the
+Ether, supposing that, like those, they are of a nature to act as a
+spring. And this is easy to conceive as regards water and other
+transparent liquids, they being composed of detached particles. But it
+may seem more difficult as regards glass and other transparent and
+hard bodies, because their solidity does not seem to permit them to
+receive movement except in their whole mass at the same time. This,
+however, is not necessary because this solidity is not such as it
+appears to us, it being probable rather that these bodies are composed
+of particles merely placed close to one another and held together by
+some pressure from without of some other matter, and by the
+irregularity of their shapes. For primarily their rarity is shown by
+the facility with which there passes through them the matter of the
+vortices of the magnet, and that which causes gravity. Further, one
+cannot say that these bodies are of a texture similar to that of a
+sponge or of light bread, because the heat of the fire makes them flow
+and thereby changes the situation of the particles amongst themselves.
+It remains then that they are, as has been said, assemblages of
+particles which touch one another without constituting a continuous
+solid. This being so, the movement which these particles receive to
+carry on the waves of light, being merely communicated from some of
+them to others, without their going for that purpose out of their
+places or without derangement, it may very well produce its effect
+without prejudicing in any way the apparent solidity of the compound.
+
+By pressure from without, of which I have spoken, must not be
+understood that of the air, which would not be sufficient, but that of
+some other more subtle matter, a pressure which I chanced upon by
+experiment long ago, namely in the case of water freed from air, which
+remains suspended in a tube open at its lower end, notwithstanding
+that the air has been removed from the vessel in which this tube is
+enclosed.
+
+One can then in this way conceive of transparency in a solid without
+any necessity that the ethereal matter which serves for light should
+pass through it, or that it should find pores in which to insinuate
+itself. But the truth is that this matter not only passes through
+solids, but does so even with great facility; of which the experiment
+of Torricelli, above cited, is already a proof. Because on the
+quicksilver and the water quitting the upper part of the glass tube,
+it appears that it is immediately filled with ethereal matter, since
+light passes across it. But here is another argument which proves this
+ready penetrability, not only in transparent bodies but also in all
+others.
+
+When light passes across a hollow sphere of glass, closed on all
+sides, it is certain that it is full of ethereal matter, as much as
+the spaces outside the sphere. And this ethereal matter, as has been
+shown above, consists of particles which just touch one another. If
+then it were enclosed in the sphere in such a way that it could not
+get out through the pores of the glass, it would be obliged to follow
+the movement of the sphere when one changes its place: and it would
+require consequently almost the same force to impress a certain
+velocity on this sphere, when placed on a horizontal plane, as if it
+were full of water or perhaps of quicksilver: because every body
+resists the velocity of the motion which one would give to it, in
+proportion to the quantity of matter which it contains, and which is
+obliged to follow this motion. But on the contrary one finds that the
+sphere resists the impress of movement only in proportion to the
+quantity of matter of the glass of which it is made. Then it must be
+that the ethereal matter which is inside is not shut up, but flows
+through it with very great freedom. We shall demonstrate hereafter
+that by this process the same penetrability may be inferred also as
+relating to opaque bodies.
+
+The second mode then of explaining transparency, and one which appears
+more probably true, is by saying that the waves of light are carried
+on in the ethereal matter, which continuously occupies the interstices
+or pores of transparent bodies. For since it passes through them
+continuously and freely, it follows that they are always full of it.
+And one may even show that these interstices occupy much more space
+than the coherent particles which constitute the bodies. For if what
+we have just said is true: that force is required to impress a certain
+horizontal velocity on bodies in proportion as they contain coherent
+matter; and if the proportion of this force follows the law of
+weights, as is confirmed by experiment, then the quantity of the
+constituent matter of bodies also follows the proportion of their
+weights. Now we see that water weighs only one fourteenth part as much
+as an equal portion of quicksilver: therefore the matter of the water
+does not occupy the fourteenth part of the space which its mass
+obtains. It must even occupy much less of it, since quicksilver is
+less heavy than gold, and the matter of gold is by no means dense, as
+follows from the fact that the matter of the vortices of the magnet
+and of that which is the cause of gravity pass very freely through it.
+
+But it may be objected here that if water is a body of so great
+rarity, and if its particles occupy so small a portion of the space of
+its apparent bulk, it is very strange how it yet resists Compression
+so strongly without permitting itself to be condensed by any force
+which one has hitherto essayed to employ, preserving even its entire
+liquidity while subjected to this pressure.
+
+This is no small difficulty. It may, however, be resolved by saying
+that the very violent and rapid motion of the subtle matter which
+renders water liquid, by agitating the particles of which it is
+composed, maintains this liquidity in spite of the pressure which
+hitherto any one has been minded to apply to it.
+
+The rarity of transparent bodies being then such as we have said, one
+easily conceives that the waves might be carried on in the ethereal
+matter which fills the interstices of the particles. And, moreover,
+one may believe that the progression of these waves ought to be a
+little slower in the interior of bodies, by reason of the small
+detours which the same particles cause. In which different velocity of
+light I shall show the cause of refraction to consist.
+
+Before doing so, I will indicate the third and last mode in which
+transparency may be conceived; which is by supposing that the movement
+of the waves of light is transmitted indifferently both in the
+particles of the ethereal matter which occupy the interstices of
+bodies, and in the particles which compose them, so that the movement
+passes from one to the other. And it will be seen hereafter that this
+hypothesis serves excellently to explain the double refraction of
+certain transparent bodies.
+
+Should it be objected that if the particles of the ether are smaller
+than those of transparent bodies (since they pass through their
+intervals), it would follow that they can communicate to them but
+little of their movement, it may be replied that the particles of
+these bodies are in turn composed of still smaller particles, and so
+it will be these secondary particles which will receive the movement
+from those of the ether.
+
+Furthermore, if the particles of transparent bodies have a recoil a
+little less prompt than that of the ethereal particles, which nothing
+hinders us from supposing, it will again follow that the progression
+of the waves of light will be slower in the interior of such bodies
+than it is outside in the ethereal matter.
+
+All this I have found as most probable for the mode in which the waves
+of light pass across transparent bodies. To which it must further be
+added in what respect these bodies differ from those which are opaque;
+and the more so since it might seem because of the easy penetration of
+bodies by the ethereal matter, of which mention has been made, that
+there would not be any body that was not transparent. For by the same
+reasoning about the hollow sphere which I have employed to prove the
+smallness of the density of glass and its easy penetrability by the
+ethereal matter, one might also prove that the same penetrability
+obtains for metals and for every other sort of body. For this sphere
+being for example of silver, it is certain that it contains some of
+the ethereal matter which serves for light, since this was there as
+well as in the air when the opening of the sphere was closed. Yet,
+being closed and placed upon a horizontal plane, it resists the
+movement which one wishes to give to it, merely according to the
+quantity of silver of which it is made; so that one must conclude, as
+above, that the ethereal matter which is enclosed does not follow the
+movement of the sphere; and that therefore silver, as well as glass,
+is very easily penetrated by this matter. Some of it is therefore
+present continuously and in quantities between the particles of silver
+and of all other opaque bodies: and since it serves for the
+propagation of light it would seem that these bodies ought also to be
+transparent, which however is not the case.
+
+Whence then, one will say, does their opacity come? Is it because the
+particles which compose them are soft; that is to say, these particles
+being composed of others that are smaller, are they capable of
+changing their figure on receiving the pressure of the ethereal
+particles, the motion of which they thereby damp, and so hinder the
+continuance of the waves of light? That cannot be: for if the
+particles of the metals are soft, how is it that polished silver and
+mercury reflect light so strongly? What I find to be most probable
+herein, is to say that metallic bodies, which are almost the only
+really opaque ones, have mixed amongst their hard particles some soft
+ones; so that some serve to cause reflexion and the others to hinder
+transparency; while, on the other hand, transparent bodies contain
+only hard particles which have the faculty of recoil, and serve
+together with those of the ethereal matter for the propagation of the
+waves of light, as has been said.
+
+[Illustration]
+
+Let us pass now to the explanation of the effects of Refraction,
+assuming, as we have done, the passage of waves of light through
+transparent bodies, and the diminution of velocity which these same
+waves suffer in them.
+
+The chief property of Refraction is that a ray of light, such as AB,
+being in the air, and falling obliquely upon the polished surface of a
+transparent body, such as FG, is broken at the point of incidence B,
+in such a way that with the straight line DBE which cuts the surface
+perpendicularly it makes an angle CBE less than ABD which it made with
+the same perpendicular when in the air. And the measure of these
+angles is found by describing, about the point B, a circle which cuts
+the radii AB, BC. For the perpendiculars AD, CE, let fall from the
+points of intersection upon the straight line DE, which are called the
+Sines of the angles ABD, CBE, have a certain ratio between themselves;
+which ratio is always the same for all inclinations of the incident
+ray, at least for a given transparent body. This ratio is, in glass,
+very nearly as 3 to 2; and in water very nearly as 4 to 3; and is
+likewise different in other diaphanous bodies.
+
+Another property, similar to this, is that the refractions are
+reciprocal between the rays entering into a transparent body and those
+which are leaving it. That is to say that if the ray AB in entering
+the transparent body is refracted into BC, then likewise CB being
+taken as a ray in the interior of this body will be refracted, on
+passing out, into BA.
+
+[Illustration]
+
+To explain then the reasons of these phenomena according to our
+principles, let AB be the straight line which represents a plane
+surface bounding the transparent substances which lie towards C and
+towards N. When I say plane, that does not signify a perfect evenness,
+but such as has been understood in treating of reflexion, and for the
+same reason. Let the line AC represent a portion of a wave of light,
+the centre of which is supposed so distant that this portion may be
+considered as a straight line. The piece C, then, of the wave AC, will
+in a certain space of time have advanced as far as the plane AB
+following the straight line CB, which may be imagined as coming from
+the luminous centre, and which consequently will cut AC at right
+angles. Now in the same time the piece A would have come to G along
+the straight line AG, equal and parallel to CB; and all the portion of
+wave AC would be at GB if the matter of the transparent body
+transmitted the movement of the wave as quickly as the matter of the
+Ether. But let us suppose that it transmits this movement less
+quickly, by one-third, for instance. Movement will then be spread from
+the point A, in the matter of the transparent body through a distance
+equal to two-thirds of CB, making its own particular spherical wave
+according to what has been said before. This wave is then represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter equal to two-thirds of CB. Then if one considers in
+order the other pieces H of the wave AC, it appears that in the same
+time that the piece C reaches B they will not only have arrived at the
+surface AB along the straight lines HK parallel to CB, but that, in
+addition, they will have generated in the diaphanous substance from
+the centres K, partial waves, represented here by circumferences the
+semi-diameters of which are equal to two-thirds of the lines KM, that
+is to say, to two-thirds of the prolongations of HK down to the
+straight line BG; for these semi-diameters would have been equal to
+entire lengths of KM if the two transparent substances had been of the
+same penetrability.
+
+Now all these circumferences have for a common tangent the straight
+line BN; namely the same line which is drawn as a tangent from the
+point B to the circumference SNR which we considered first. For it is
+easy to see that all the other circumferences will touch the same BN,
+from B up to the point of contact N, which is the same point where AN
+falls perpendicularly on BN.
+
+It is then BN, which is formed by small arcs of these circumferences,
+which terminates the movement that the wave AC has communicated within
+the transparent body, and where this movement occurs in much greater
+amount than anywhere else. And for that reason this line, in
+accordance with what has been said more than once, is the propagation
+of the wave AC at the moment when its piece C has reached B. For there
+is no other line below the plane AB which is, like BN, a common
+tangent to all these partial waves. And if one would know how the wave
+AC has come progressively to BN, it is necessary only to draw in the
+same figure the straight lines KO parallel to BN, and all the lines KL
+parallel to AC. Thus one will see that the wave CA, from being a
+straight line, has become broken in all the positions LKO
+successively, and that it has again become a straight line at BN. This
+being evident by what has already been demonstrated, there is no need
+to explain it further.
+
+Now, in the same figure, if one draws EAF, which cuts the plane AB at
+right angles at the point A, since AD is perpendicular to the wave AC,
+it will be DA which will mark the ray of incident light, and AN which
+was perpendicular to BN, the refracted ray: since the rays are nothing
+else than the straight lines along which the portions of the waves
+advance.
+
+Whence it is easy to recognize this chief property of refraction,
+namely that the Sine of the angle DAE has always the same ratio to the
+Sine of the angle NAF, whatever be the inclination of the ray DA: and
+that this ratio is the same as that of the velocity of the waves in
+the transparent substance which is towards AE to their velocity in the
+transparent substance towards AF. For, considering AB as the radius of
+a circle, the Sine of the angle BAC is BC, and the Sine of the angle
+ABN is AN. But the angle BAC is equal to DAE, since each of them added
+to CAE makes a right angle. And the angle ABN is equal to NAF, since
+each of them with BAN makes a right angle. Then also the Sine of the
+angle DAE is to the Sine of NAF as BC is to AN. But the ratio of BC to
+AN was the same as that of the velocities of light in the substance
+which is towards AE and in that which is towards AF; therefore also
+the Sine of the angle DAE will be to the Sine of the angle NAF the
+same as the said velocities of light.
+
+To see, consequently, what the refraction will be when the waves of
+light pass into a substance in which the movement travels more quickly
+than in that from which they emerge (let us again assume the ratio of
+3 to 2), it is only necessary to repeat all the same construction and
+demonstration which we have just used, merely substituting everywhere
+3/2 instead of 2/3. And it will be found by the same reasoning, in
+this other figure, that when the piece C of the wave AC shall have
+reached the surface AB at B, all the portions of the wave AC will
+have advanced as far as BN, so that BC the perpendicular on AC is to
+AN the perpendicular on BN as 2 to 3. And there will finally be this
+same ratio of 2 to 3 between the Sine of the angle BAD and the Sine of
+the angle FAN.
+
+Hence one sees the reciprocal relation of the refractions of the ray
+on entering and on leaving one and the same transparent body: namely
+that if NA falling on the external surface AB is refracted into the
+direction AD, so the ray AD will be refracted on leaving the
+transparent body into the direction AN.
+
+[Illustration]
+
+One sees also the reason for a noteworthy accident which happens in
+this refraction: which is this, that after a certain obliquity of the
+incident ray DA, it begins to be quite unable to penetrate into the
+other transparent substance. For if the angle DAQ or CBA is such that
+in the triangle ACB, CB is equal to 2/3 of AB, or is greater, then AN
+cannot form one side of the triangle ANB, since it becomes equal to or
+greater than AB: so that the portion of wave BN cannot be found
+anywhere, neither consequently can AN, which ought to be perpendicular
+to it. And thus the incident ray DA does not then pierce the surface
+AB.
+
+When the ratio of the velocities of the waves is as two to three, as
+in our example, which is that which obtains for glass and air, the
+angle DAQ must be more than 48 degrees 11 minutes in order that the
+ray DA may be able to pass by refraction. And when the ratio of the
+velocities is as 3 to 4, as it is very nearly in water and air, this
+angle DAQ must exceed 41 degrees 24 minutes. And this accords
+perfectly with experiment.
+
+But it might here be asked: since the meeting of the wave AC against
+the surface AB ought to produce movement in the matter which is on the
+other side, why does no light pass there? To which the reply is easy
+if one remembers what has been said before. For although it generates
+an infinitude of partial waves in the matter which is at the other
+side of AB, these waves never have a common tangent line (either
+straight or curved) at the same moment; and so there is no line
+terminating the propagation of the wave AC beyond the plane AB, nor
+any place where the movement is gathered together in sufficiently
+great quantity to produce light. And one will easily see the truth of
+this, namely that CB being larger than 2/3 of AB, the waves excited
+beyond the plane AB will have no common tangent if about the centres K
+one then draws circles having radii equal to 3/2 of the lengths LB to
+which they correspond. For all these circles will be enclosed in one
+another and will all pass beyond the point B.
+
+Now it is to be remarked that from the moment when the angle DAQ is
+smaller than is requisite to permit the refracted ray DA to pass into
+the other transparent substance, one finds that the interior reflexion
+which occurs at the surface AB is much augmented in brightness, as is
+easy to realize by experiment with a triangular prism; and for this
+our theory can afford this reason. When the angle DAQ is still large
+enough to enable the ray DA to pass, it is evident that the light from
+the portion AC of the wave is collected in a minimum space when it
+reaches BN. It appears also that the wave BN becomes so much the
+smaller as the angle CBA or DAQ is made less; until when the latter is
+diminished to the limit indicated a little previously, this wave BN is
+collected together always at one point. That is to say, that when the
+piece C of the wave AC has then arrived at B, the wave BN which is the
+propagation of AC is entirely reduced to the same point B. Similarly
+when the piece H has reached K, the part AH is entirely reduced to the
+same point K. This makes it evident that in proportion as the wave CA
+comes to meet the surface AB, there occurs a great quantity of
+movement along that surface; which movement ought also to spread
+within the transparent body and ought to have much re-enforced the
+partial waves which produce the interior reflexion against the surface
+AB, according to the laws of reflexion previously explained.
+
+And because a slight diminution of the angle of incidence DAQ causes
+the wave BN, however great it was, to be reduced to zero, (for this
+angle being 49 degrees 11 minutes in the glass, the angle BAN is still
+11 degrees 21 minutes, and the same angle being reduced by one degree
+only the angle BAN is reduced to zero, and so the wave BN reduced to a
+point) thence it comes about that the interior reflexion from being
+obscure becomes suddenly bright, so soon as the angle of incidence is
+such that it no longer gives passage to the refraction.
+
+Now as concerns ordinary external reflexion, that is to say which
+occurs when the angle of incidence DAQ is still large enough to enable
+the refracted ray to penetrate beyond the surface AB, this reflexion
+should occur against the particles of the substance which touches the
+transparent body on its outside. And it apparently occurs against the
+particles of the air or others mingled with the ethereal particles and
+larger than they. So on the other hand the external reflexion of these
+bodies occurs against the particles which compose them, and which are
+also larger than those of the ethereal matter, since the latter flows
+in their interstices. It is true that there remains here some
+difficulty in those experiments in which this interior reflexion
+occurs without the particles of air being able to contribute to it, as
+in vessels or tubes from which the air has been extracted.
+
+Experience, moreover, teaches us that these two reflexions are of
+nearly equal force, and that in different transparent bodies they are
+so much the stronger as the refraction of these bodies is the greater.
+Thus one sees manifestly that the reflexion of glass is stronger than
+that of water, and that of diamond stronger than that of glass.
+
+I will finish this theory of refraction by demonstrating a remarkable
+proposition which depends on it; namely, that a ray of light in order
+to go from one point to another, when these points are in different
+media, is refracted in such wise at the plane surface which joins
+these two media that it employs the least possible time: and exactly
+the same happens in the case of reflexion against a plane surface. Mr.
+Fermat was the first to propound this property of refraction, holding
+with us, and directly counter to the opinion of Mr. Des Cartes, that
+light passes more slowly through glass and water than through air.
+But he assumed besides this a constant ratio of Sines, which we have
+just proved by these different degrees of velocity alone: or rather,
+what is equivalent, he assumed not only that the velocities were
+different but that the light took the least time possible for its
+passage, and thence deduced the constant ratio of the Sines. His
+demonstration, which may be seen in his printed works, and in the
+volume of letters of Mr. Des Cartes, is very long; wherefore I give
+here another which is simpler and easier.
+
+[Illustration]
+
+Let KF be the plane surface; A the point in the medium which the light
+traverses more easily, as the air; C the point in the other which is
+more difficult to penetrate, as water. And suppose that a ray has come
+from A, by B, to C, having been refracted at B according to the law
+demonstrated a little before; that is to say that, having drawn PBQ,
+which cuts the plane at right angles, let the sine of the angle ABP
+have to the sine of the angle CBQ the same ratio as the velocity of
+light in the medium where A is to the velocity of light in the medium
+where C is. It is to be shown that the time of passage of light along
+AB and BC taken together, is the shortest that can be. Let us assume
+that it may have come by other lines, and, in the first place, along
+AF, FC, so that the point of refraction F may be further from B than
+the point A; and let AO be a line perpendicular to AB, and FO parallel
+to AB; BH perpendicular to FO, and FG to BC.
+
+Since then the angle HBF is equal to PBA, and the angle BFG equal to
+QBC, it follows that the sine of the angle HBF will also have the same
+ratio to the sine of BFG, as the velocity of light in the medium A is
+to its velocity in the medium C. But these sines are the straight
+lines HF, BG, if we take BF as the semi-diameter of a circle. Then
+these lines HF, BG, will bear to one another the said ratio of the
+velocities. And, therefore, the time of the light along HF, supposing
+that the ray had been OF, would be equal to the time along BG in the
+interior of the medium C. But the time along AB is equal to the time
+along OH; therefore the time along OF is equal to the time along AB,
+BG. Again the time along FC is greater than that along GC; then the
+time along OFC will be longer than that along ABC. But AF is longer
+than OF, then the time along AFC will by just so much more exceed the
+time along ABC.
+
+Now let us assume that the ray has come from A to C along AK, KC; the
+point of refraction K being nearer to A than the point B is; and let
+CN be the perpendicular upon BC, KN parallel to BC: BM perpendicular
+upon KN, and KL upon BA.
+
+Here BL and KM are the sines of angles BKL, KBM; that is to say, of
+the angles PBA, QBC; and therefore they are to one another as the
+velocity of light in the medium A is to the velocity in the medium C.
+Then the time along LB is equal to the time along KM; and since the
+time along BC is equal to the time along MN, the time along LBC will
+be equal to the time along KMN. But the time along AK is longer than
+that along AL: hence the time along AKN is longer than that along ABC.
+And KC being longer than KN, the time along AKC will exceed, by as
+much more, the time along ABC. Hence it appears that the time along
+ABC is the shortest possible; which was to be proven.
+
+
+
+
+CHAPTER IV
+
+ON THE REFRACTION OF THE AIR
+
+
+We have shown how the movement which constitutes light spreads by
+spherical waves in any homogeneous matter. And it is evident that when
+the matter is not homogeneous, but of such a constitution that the
+movement is communicated in it more rapidly toward one side than
+toward another, these waves cannot be spherical: but that they must
+acquire their figure according to the different distances over which
+the successive movement passes in equal times.
+
+It is thus that we shall in the first place explain the refractions
+which occur in the air, which extends from here to the clouds and
+beyond. The effects of which refractions are very remarkable; for by
+them we often see objects which the rotundity of the Earth ought
+otherwise to hide; such as Islands, and the tops of mountains when one
+is at sea. Because also of them the Sun and the Moon appear as risen
+before in fact they have, and appear to set later: so that at times
+the Moon has been seen eclipsed while the Sun appeared still above the
+horizon. And so also the heights of the Sun and of the Moon, and those
+of all the Stars always appear a little greater than they are in
+reality, because of these same refractions, as Astronomers know. But
+there is one experiment which renders this refraction very evident;
+which is that of fixing a telescope on some spot so that it views an
+object, such as a steeple or a house, at a distance of half a league
+or more. If then you look through it at different hours of the day,
+leaving it always fixed in the same way, you will see that the same
+spots of the object will not always appear at the middle of the
+aperture of the telescope, but that generally in the morning and in
+the evening, when there are more vapours near the Earth, these objects
+seem to rise higher, so that the half or more of them will no longer
+be visible; and so that they seem lower toward mid-day when these
+vapours are dissipated.
+
+Those who consider refraction to occur only in the surfaces which
+separate transparent bodies of different nature, would find it
+difficult to give a reason for all that I have just related; but
+according to our Theory the thing is quite easy. It is known that the
+air which surrounds us, besides the particles which are proper to it
+and which float in the ethereal matter as has been explained, is full
+also of particles of water which are raised by the action of heat; and
+it has been ascertained further by some very definite experiments that
+as one mounts up higher the density of air diminishes in proportion.
+Now whether the particles of water and those of air take part, by
+means of the particles of ethereal matter, in the movement which
+constitutes light, but have a less prompt recoil than these, or
+whether the encounter and hindrance which these particles of air and
+water offer to the propagation of movement of the ethereal progress,
+retard the progression, it follows that both kinds of particles flying
+amidst the ethereal particles, must render the air, from a great
+height down to the Earth, gradually less easy for the spreading of the
+waves of light.
+
+[Illustration]
+
+Whence the configuration of the waves ought to become nearly such as
+this figure represents: namely, if A is a light, or the visible point
+of a steeple, the waves which start from it ought to spread more
+widely upwards and less widely downwards, but in other directions more
+or less as they approximate to these two extremes. This being so, it
+necessarily follows that every line intersecting one of these waves at
+right angles will pass above the point A, always excepting the one
+line which is perpendicular to the horizon.
+
+[Illustration]
+
+Let BC be the wave which brings the light to the spectator who is at
+B, and let BD be the straight line which intersects this wave at right
+angles. Now because the ray or straight line by which we judge the
+spot where the object appears to us is nothing else than the
+perpendicular to the wave that reaches our eye, as will be understood
+by what was said above, it is manifest that the point A will be
+perceived as being in the line BD, and therefore higher than in fact it
+is.
+
+Similarly if the Earth be AB, and the top of the Atmosphere CD, which
+probably is not a well defined spherical surface (since we know that
+the air becomes rare in proportion as one ascends, for above there is
+so much less of it to press down upon it), the waves of light from the
+sun coming, for instance, in such a way that so long as they have not
+reached the Atmosphere CD the straight line AE intersects them
+perpendicularly, they ought, when they enter the Atmosphere, to
+advance more quickly in elevated regions than in regions nearer to the
+Earth. So that if CA is the wave which brings the light to the
+spectator at A, its region C will be the furthest advanced; and the
+straight line AF, which intersects this wave at right angles, and
+which determines the apparent place of the Sun, will pass above the
+real Sun, which will be seen along the line AE. And so it may occur
+that when it ought not to be visible in the absence of vapours,
+because the line AE encounters the rotundity of the Earth, it will be
+perceived in the line AF by refraction. But this angle EAF is scarcely
+ever more than half a degree because the attenuation of the vapours
+alters the waves of light but little. Furthermore these refractions
+are not altogether constant in all weathers, particularly at small
+elevations of 2 or 3 degrees; which results from the different
+quantity of aqueous vapours rising above the Earth.
+
+And this same thing is the cause why at certain times a distant object
+will be hidden behind another less distant one, and yet may at another
+time be able to be seen, although the spot from which it is viewed is
+always the same. But the reason for this effect will be still more
+evident from what we are going to remark touching the curvature of
+rays. It appears from the things explained above that the progression
+or propagation of a small part of a wave of light is properly what one
+calls a ray. Now these rays, instead of being straight as they are in
+homogeneous media, ought to be curved in an atmosphere of unequal
+penetrability. For they necessarily follow from the object to the eye
+the line which intersects at right angles all the progressions of the
+waves, as in the first figure the line AEB does, as will be shown
+hereafter; and it is this line which determines what interposed bodies
+would or would not hinder us from seeing the object. For although the
+point of the steeple A appears raised to D, it would yet not appear to
+the eye B if the tower H was between the two, because it crosses the
+curve AEB. But the tower E, which is beneath this curve, does not
+hinder the point A from being seen. Now according as the air near the
+Earth exceeds in density that which is higher, the curvature of the
+ray AEB becomes greater: so that at certain times it passes above the
+summit E, which allows the point A to be perceived by the eye at B;
+and at other times it is intercepted by the same tower E which hides A
+from this same eye.
+
+[Illustration]
+
+But to demonstrate this curvature of the rays conformably to all our
+preceding Theory, let us imagine that AB is a small portion of a wave
+of light coming from the side C, which we may consider as a straight
+line. Let us also suppose that it is perpendicular to the Horizon, the
+portion B being nearer to the Earth than the portion A; and that
+because the vapours are less hindering at A than at B, the particular
+wave which comes from the point A spreads through a certain space AD
+while the particular wave which starts from the point B spreads
+through a shorter space BE; AD and BE being parallel to the Horizon.
+Further, supposing the straight lines FG, HI, etc., to be drawn from
+an infinitude of points in the straight line AB and to terminate on
+the line DE (which is straight or may be considered as such), let the
+different penetrabilities at the different heights in the air between
+A and B be represented by all these lines; so that the particular
+wave, originating from the point F, will spread across the space FG,
+and that from the point H across the space HI, while that from the
+point A spreads across the space AD.
+
+Now if about the centres A, B, one describes the circles DK, EL, which
+represent the spreading of the waves which originate from these two
+points, and if one draws the straight line KL which touches these two
+circles, it is easy to see that this same line will be the common
+tangent to all the other circles drawn about the centres F, H, etc.;
+and that all the points of contact will fall within that part of this
+line which is comprised between the perpendiculars AK, BL. Then it
+will be the line KL which will terminate the movement of the
+particular waves originating from the points of the wave AB; and this
+movement will be stronger between the points KL, than anywhere else at
+the same instant, since an infinitude of circumferences concur to form
+this straight line; and consequently KL will be the propagation of the
+portion of wave AB, as has been said in explaining reflexion and
+ordinary refraction. Now it appears that AK and BL dip down toward the
+side where the air is less easy to penetrate: for AK being longer than
+BL, and parallel to it, it follows that the lines AB and KL, being
+prolonged, would meet at the side L. But the angle K is a right angle:
+hence KAB is necessarily acute, and consequently less than DAB. If one
+investigates in the same way the progression of the portion of the
+wave KL, one will find that after a further time it has arrived at MN
+in such a manner that the perpendiculars KM, LN, dip down even more
+than do AK, BL. And this suffices to show that the ray will continue
+along the curved line which intersects all the waves at right angles,
+as has been said.
+
+
+
+
+CHAPTER V
+
+ON THE STRANGE REFRACTION OF ICELAND CRYSTAL
+
+
+1.
+
+There is brought from Iceland, which is an Island in the North Sea, in
+the latitude of 66 degrees, a kind of Crystal or transparent stone,
+very remarkable for its figure and other qualities, but above all for
+its strange refractions. The causes of this have seemed to me to be
+worthy of being carefully investigated, the more so because amongst
+transparent bodies this one alone does not follow the ordinary rules
+with respect to rays of light. I have even been under some necessity
+to make this research, because the refractions of this Crystal seemed
+to overturn our preceding explanation of regular refraction; which
+explanation, on the contrary, they strongly confirm, as will be seen
+after they have been brought under the same principle. In Iceland are
+found great lumps of this Crystal, some of which I have seen of 4 or 5
+pounds. But it occurs also in other countries, for I have had some of
+the same sort which had been found in France near the town of Troyes
+in Champagne, and some others which came from the Island of Corsica,
+though both were less clear and only in little bits, scarcely capable
+of letting any effect of refraction be observed.
+
+2. The first knowledge which the public has had about it is due to Mr.
+Erasmus Bartholinus, who has given a description of Iceland Crystal
+and of its chief phenomena. But here I shall not desist from giving my
+own, both for the instruction of those who may not have seen his book,
+and because as respects some of these phenomena there is a slight
+difference between his observations and those which I have made: for I
+have applied myself with great exactitude to examine these properties
+of refraction, in order to be quite sure before undertaking to explain
+the causes of them.
+
+3. As regards the hardness of this stone, and the property which it
+has of being easily split, it must be considered rather as a species
+of Talc than of Crystal. For an iron spike effects an entrance into it
+as easily as into any other Talc or Alabaster, to which it is equal in
+gravity.
+
+[Illustration]
+
+4. The pieces of it which are found have the figure of an oblique
+parallelepiped; each of the six faces being a parallelogram; and it
+admits of being split in three directions parallel to two of these
+opposed faces. Even in such wise, if you will, that all the six faces
+are equal and similar rhombuses. The figure here added represents a
+piece of this Crystal. The obtuse angles of all the parallelograms, as
+C, D, here, are angles of 101 degrees 52 minutes, and consequently
+the acute angles, such as A and B, are of 78 degrees 8 minutes.
+
+5. Of the solid angles there are two opposite to one another, such as
+C and E, which are each composed of three equal obtuse plane angles.
+The other six are composed of two acute angles and one obtuse. All
+that I have just said has been likewise remarked by Mr. Bartholinus in
+the aforesaid treatise; if we differ it is only slightly about the
+values of the angles. He recounts moreover some other properties of
+this Crystal; to wit, that when rubbed against cloth it attracts
+straws and other light things as do amber, diamond, glass, and Spanish
+wax. Let a piece be covered with water for a day or more, the surface
+loses its natural polish. When aquafortis is poured on it it produces
+ebullition, especially, as I have found, if the Crystal has been
+pulverized. I have also found by experiment that it may be heated to
+redness in the fire without being in anywise altered or rendered less
+transparent; but a very violent fire calcines it nevertheless. Its
+transparency is scarcely less than that of water or of Rock Crystal,
+and devoid of colour. But rays of light pass through it in another
+fashion and produce those marvellous refractions the causes of which I
+am now going to try to explain; reserving for the end of this Treatise
+the statement of my conjectures touching the formation and
+extraordinary configuration of this Crystal.
+
+6. In all other transparent bodies that we know there is but one sole
+and simple refraction; but in this substance there are two different
+ones. The effect is that objects seen through it, especially such as
+are placed right against it, appear double; and that a ray of
+sunlight, falling on one of its surfaces, parts itself into two rays
+and traverses the Crystal thus.
+
+7. It is again a general law in all other transparent bodies that the
+ray which falls perpendicularly on their surface passes straight on
+without suffering refraction, and that an oblique ray is always
+refracted. But in this Crystal the perpendicular ray suffers
+refraction, and there are oblique rays which pass through it quite
+straight.
+
+[Illustration]
+
+8. But in order to explain these phenomena more particularly, let
+there be, in the first place, a piece ABFE of the same Crystal, and
+let the obtuse angle ACB, one of the three which constitute the
+equilateral solid angle C, be divided into two equal parts by the
+straight line CG, and let it be conceived that the Crystal is
+intersected by a plane which passes through this line and through the
+side CF, which plane will necessarily be perpendicular to the surface
+AB; and its section in the Crystal will form a parallelogram GCFH. We
+will call this section the principal section of the Crystal.
+
+9. Now if one covers the surface AB, leaving there only a small
+aperture at the point K, situated in the straight line CG, and if one
+exposes it to the sun, so that his rays face it perpendicularly above,
+then the ray IK will divide itself at the point K into two, one of
+which will continue to go on straight by KL, and the other will
+separate itself along the straight line KM, which is in the plane
+GCFH, and which makes with KL an angle of about 6 degrees 40 minutes,
+tending from the side of the solid angle C; and on emerging from the
+other side of the Crystal it will turn again parallel to JK, along MZ.
+And as, in this extraordinary refraction, the point M is seen by the
+refracted ray MKI, which I consider as going to the eye at I, it
+necessarily follows that the point L, by virtue of the same
+refraction, will be seen by the refracted ray LRI, so that LR will be
+parallel to MK if the distance from the eye KI is supposed very great.
+The point L appears then as being in the straight line IRS; but the
+same point appears also, by ordinary refraction, to be in the straight
+line IK, hence it is necessarily judged to be double. And similarly if
+L be a small hole in a sheet of paper or other substance which is laid
+against the Crystal, it will appear when turned towards daylight as if
+there were two holes, which will seem the wider apart from one another
+the greater the thickness of the Crystal.
+
+10. Again, if one turns the Crystal in such wise that an incident ray
+NO, of sunlight, which I suppose to be in the plane continued from
+GCFH, makes with GC an angle of 73 degrees and 20 minutes, and is
+consequently nearly parallel to the edge CF, which makes with FH an
+angle of 70 degrees 57 minutes, according to the calculation which I
+shall put at the end, it will divide itself at the point O into two
+rays, one of which will continue along OP in a straight line with NO,
+and will similarly pass out of the other side of the crystal without
+any refraction; but the other will be refracted and will go along OQ.
+And it must be noted that it is special to the plane through GCF and
+to those which are parallel to it, that all incident rays which are in
+one of these planes continue to be in it after they have entered the
+Crystal and have become double; for it is quite otherwise for rays in
+all other planes which intersect the Crystal, as we shall see
+afterwards.
+
+11. I recognized at first by these experiments and by some others that
+of the two refractions which the ray suffers in this Crystal, there is
+one which follows the ordinary rules; and it is this to which the rays
+KL and OQ belong. This is why I have distinguished this ordinary
+refraction from the other; and having measured it by exact
+observation, I found that its proportion, considered as to the Sines
+of the angles which the incident and refracted rays make with the
+perpendicular, was very precisely that of 5 to 3, as was found also by
+Mr. Bartholinus, and consequently much greater than that of Rock
+Crystal, or of glass, which is nearly 3 to 2.
+
+[Illustration]
+
+12. The mode of making these observations exactly is as follows. Upon
+a leaf of paper fixed on a thoroughly flat table there is traced a
+black line AB, and two others, CED and KML, which cut it at right
+angles and are more or less distant from one another according as it
+is desired to examine a ray that is more or less oblique. Then place
+the Crystal upon the intersection E so that the line AB concurs with
+that which bisects the obtuse angle of the lower surface, or with some
+line parallel to it. Then by placing the eye directly above the line
+AB it will appear single only; and one will see that the portion
+viewed through the Crystal and the portions which appear outside it,
+meet together in a straight line: but the line CD will appear double,
+and one can distinguish the image which is due to regular refraction
+by the circumstance that when one views it with both eyes it seems
+raised up more than the other, or again by the circumstance that, when
+the Crystal is turned around on the paper, this image remains
+stationary, whereas the other image shifts and moves entirely around.
+Afterwards let the eye be placed at I (remaining always in the plane
+perpendicular through AB) so that it views the image which is formed
+by regular refraction of the line CD making a straight line with the
+remainder of that line which is outside the Crystal. And then, marking
+on the surface of the Crystal the point H where the intersection E
+appears, this point will be directly above E. Then draw back the eye
+towards O, keeping always in the plane perpendicular through AB, so
+that the image of the line CD, which is formed by ordinary refraction,
+may appear in a straight line with the line KL viewed without
+refraction; and then mark on the Crystal the point N where the point
+of intersection E appears.
+
+13. Then one will know the length and position of the lines NH, EM,
+and of HE, which is the thickness of the Crystal: which lines being
+traced separately upon a plan, and then joining NE and NM which cuts
+HE at P, the proportion of the refraction will be that of EN to NP,
+because these lines are to one another as the sines of the angles NPH,
+NEP, which are equal to those which the incident ray ON and its
+refraction NE make with the perpendicular to the surface. This
+proportion, as I have said, is sufficiently precisely as 5 to 3, and
+is always the same for all inclinations of the incident ray.
+
+14. The same mode of observation has also served me for examining the
+extraordinary or irregular refraction of this Crystal. For, the point
+H having been found and marked, as aforesaid, directly above the point
+E, I observed the appearance of the line CD, which is made by the
+extraordinary refraction; and having placed the eye at Q, so that this
+appearance made a straight line with the line KL viewed without
+refraction, I ascertained the triangles REH, RES, and consequently the
+angles RSH, RES, which the incident and the refracted ray make with
+the perpendicular.
+
+15. But I found in this refraction that the ratio of FR to RS was not
+constant, like the ordinary refraction, but that it varied with the
+varying obliquity of the incident ray.
+
+16. I found also that when QRE made a straight line, that is, when the
+incident ray entered the Crystal without being refracted (as I
+ascertained by the circumstance that then the point E viewed by the
+extraordinary refraction appeared in the line CD, as seen without
+refraction) I found, I say, then that the angle QRG was 73 degrees 20
+minutes, as has been already remarked; and so it is not the ray
+parallel to the edge of the Crystal, which crosses it in a straight
+line without being refracted, as Mr. Bartholinus believed, since that
+inclination is only 70 degrees 57 minutes, as was stated above. And
+this is to be noted, in order that no one may search in vain for the
+cause of the singular property of this ray in its parallelism to the
+edges mentioned.
+
+[Illustration]
+
+17. Finally, continuing my observations to discover the nature of
+this refraction, I learned that it obeyed the following remarkable
+rule. Let the parallelogram GCFH, made by the principal section of the
+Crystal, as previously determined, be traced separately. I found then
+that always, when the inclinations of two rays which come from
+opposite sides, as VK, SK here, are equal, their refractions KX and KT
+meet the bottom line HF in such wise that points X and T are equally
+distant from the point M, where the refraction of the perpendicular
+ray IK falls; and this occurs also for refractions in other sections
+of this Crystal. But before speaking of those, which have also other
+particular properties, we will investigate the causes of the phenomena
+which I have already reported.
+
+It was after having explained the refraction of ordinary transparent
+bodies by means of the spherical emanations of light, as above, that I
+resumed my examination of the nature of this Crystal, wherein I had
+previously been unable to discover anything.
+
+18. As there were two different refractions, I conceived that there
+were also two different emanations of waves of light, and that one
+could occur in the ethereal matter extending through the body of the
+Crystal. Which matter, being present in much larger quantity than is
+that of the particles which compose it, was alone capable of causing
+transparency, according to what has been explained heretofore. I
+attributed to this emanation of waves the regular refraction which is
+observed in this stone, by supposing these waves to be ordinarily of
+spherical form, and having a slower progression within the Crystal
+than they have outside it; whence proceeds refraction as I have
+demonstrated.
+
+19. As to the other emanation which should produce the irregular
+refraction, I wished to try what Elliptical waves, or rather
+spheroidal waves, would do; and these I supposed would spread
+indifferently both in the ethereal matter diffused throughout the
+crystal and in the particles of which it is composed, according to the
+last mode in which I have explained transparency. It seemed to me that
+the disposition or regular arrangement of these particles could
+contribute to form spheroidal waves (nothing more being required for
+this than that the successive movement of light should spread a little
+more quickly in one direction than in the other) and I scarcely
+doubted that there were in this crystal such an arrangement of equal
+and similar particles, because of its figure and of its angles with
+their determinate and invariable measure. Touching which particles,
+and their form and disposition, I shall, at the end of this Treatise,
+propound my conjectures and some experiments which confirm them.
+
+20. The double emission of waves of light, which I had imagined,
+became more probable to me after I had observed a certain phenomenon
+in the ordinary [Rock] Crystal, which occurs in hexagonal form, and
+which, because of this regularity, seems also to be composed of
+particles, of definite figure, and ranged in order. This was, that
+this crystal, as well as that from Iceland, has a double refraction,
+though less evident. For having had cut from it some well polished
+Prisms of different sections, I remarked in all, in viewing through
+them the flame of a candle or the lead of window panes, that
+everything appeared double, though with images not very distant from
+one another. Whence I understood the reason why this substance, though
+so transparent, is useless for Telescopes, when they have ever so
+little length.
+
+21. Now this double refraction, according to my Theory hereinbefore
+established, seemed to demand a double emission of waves of light,
+both of them spherical (for both the refractions are regular) and
+those of one series a little slower only than the others. For thus the
+phenomenon is quite naturally explained, by postulating substances
+which serve as vehicle for these waves, as I have done in the case of
+Iceland Crystal. I had then less trouble after that in admitting two
+emissions of waves in one and the same body. And since it might have
+been objected that in composing these two kinds of crystal of equal
+particles of a certain figure, regularly piled, the interstices which
+these particles leave and which contain the ethereal matter would
+scarcely suffice to transmit the waves of light which I have localized
+there, I removed this difficulty by regarding these particles as being
+of a very rare texture, or rather as composed of other much smaller
+particles, between which the ethereal matter passes quite freely.
+This, moreover, necessarily follows from that which has been already
+demonstrated touching the small quantity of matter of which the bodies
+are built up.
+
+22. Supposing then these spheroidal waves besides the spherical ones,
+I began to examine whether they could serve to explain the phenomena
+of the irregular refraction, and how by these same phenomena I could
+determine the figure and position of the spheroids: as to which I
+obtained at last the desired success, by proceeding as follows.
+
+[Illustration]
+
+23. I considered first the effect of waves so formed, as respects the
+ray which falls perpendicularly on the flat surface of a transparent
+body in which they should spread in this manner. I took AB for the
+exposed region of the surface. And, since a ray perpendicular to a
+plane, and coming from a very distant source of light, is nothing
+else, according to the precedent Theory, than the incidence of a
+portion of the wave parallel to that plane, I supposed the straight
+line RC, parallel and equal to AB, to be a portion of a wave of light,
+in which an infinitude of points such as RH_h_C come to meet the
+surface AB at the points AK_k_B. Then instead of the hemispherical
+partial waves which in a body of ordinary refraction would spread from
+each of these last points, as we have above explained in treating of
+refraction, these must here be hemi-spheroids. The axes (or rather the
+major diameters) of these I supposed to be oblique to the plane AB, as
+is AV the semi-axis or semi-major diameter of the spheroid SVT, which
+represents the partial wave coming from the point A, after the wave RC
+has reached AB. I say axis or major diameter, because the same ellipse
+SVT may be considered as the section of a spheroid of which the axis
+is AZ perpendicular to AV. But, for the present, without yet deciding
+one or other, we will consider these spheroids only in those sections
+of them which make ellipses in the plane of this figure. Now taking a
+certain space of time during which the wave SVT has spread from A, it
+would needs be that from all the other points K_k_B there should
+proceed, in the same time, waves similar to SVT and similarly
+situated. And the common tangent NQ of all these semi-ellipses would
+be the propagation of the wave RC which fell on AB, and would be the
+place where this movement occurs in much greater amount than anywhere
+else, being made up of arcs of an infinity of ellipses, the centres of
+which are along the line AB.
+
+24. Now it appeared that this common tangent NQ was parallel to AB,
+and of the same length, but that it was not directly opposite to it,
+since it was comprised between the lines AN, BQ, which are diameters
+of ellipses having A and B for centres, conjugate with respect to
+diameters which are not in the straight line AB. And in this way I
+comprehended, a matter which had seemed to me very difficult, how a
+ray perpendicular to a surface could suffer refraction on entering a
+transparent body; seeing that the wave RC, having come to the aperture
+AB, went on forward thence, spreading between the parallel lines AN,
+BQ, yet itself remaining always parallel to AB, so that here the light
+does not spread along lines perpendicular to its waves, as in ordinary
+refraction, but along lines cutting the waves obliquely.
+
+[Illustration]
+
+25. Inquiring subsequently what might be the position and form of
+these spheroids in the crystal, I considered that all the six faces
+produced precisely the same refractions. Taking, then, the
+parallelopiped AFB, of which the obtuse solid angle C is contained
+between the three equal plane angles, and imagining in it the three
+principal sections, one of which is perpendicular to the face DC and
+passes through the edge CF, another perpendicular to the face BF
+passing through the edge CA, and the third perpendicular to the face
+AF passing through the edge BC; I knew that the refractions of the
+incident rays belonging to these three planes were all similar. But
+there could be no position of the spheroid which would have the same
+relation to these three sections except that in which the axis was
+also the axis of the solid angle C. Consequently I saw that the axis
+of this angle, that is to say the straight line which traversed the
+crystal from the point C with equal inclination to the edges CF, CA,
+CB was the line which determined the position of the axis of all the
+spheroidal waves which one imagined to originate from some point,
+taken within or on the surface of the crystal, since all these
+spheroids ought to be alike, and have their axes parallel to one
+another.
+
+26. Considering after this the plane of one of these three sections,
+namely that through GCF, the angle of which is 109 degrees 3 minutes,
+since the angle F was shown above to be 70 degrees 57 minutes; and,
+imagining a spheroidal wave about the centre C, I knew, because I have
+just explained it, that its axis must be in the same plane, the half
+of which axis I have marked CS in the next figure: and seeking by
+calculation (which will be given with others at the end of this
+discourse) the value of the angle CGS, I found it 45 degrees 20
+minutes.
+
+[Illustration]
+
+27. To know from this the form of this spheroid, that is to say the
+proportion of the semi-diameters CS, CP, of its elliptical section,
+which are perpendicular to one another, I considered that the point M
+where the ellipse is touched by the straight line FH, parallel to CG,
+ought to be so situated that CM makes with the perpendicular CL an
+angle of 6 degrees 40 minutes; since, this being so, this ellipse
+satisfies what has been said about the refraction of the ray
+perpendicular to the surface CG, which is inclined to the
+perpendicular CL by the same angle. This, then, being thus disposed,
+and taking CM at 100,000 parts, I found by the calculation which will
+be given at the end, the semi-major diameter CP to be 105,032, and the
+semi-axis CS to be 93,410, the ratio of which numbers is very nearly 9
+to 8; so that the spheroid was of the kind which resembles a
+compressed sphere, being generated by the revolution of an ellipse
+about its smaller diameter. I found also the value of CG the
+semi-diameter parallel to the tangent ML to be 98,779.
+
+[Illustration]
+
+28. Now passing to the investigation of the refractions which
+obliquely incident rays must undergo, according to our hypothesis of
+spheroidal waves, I saw that these refractions depended on the ratio
+between the velocity of movement of the light outside the crystal in
+the ether, and that within the crystal. For supposing, for example,
+this proportion to be such that while the light in the crystal forms
+the spheroid GSP, as I have just said, it forms outside a sphere the
+semi-diameter of which is equal to the line N which will be determined
+hereafter, the following is the way of finding the refraction of the
+incident rays. Let there be such a ray RC falling upon the surface
+CK. Make CO perpendicular to RC, and across the angle KCO adjust OK,
+equal to N and perpendicular to CO; then draw KI, which touches the
+Ellipse GSP, and from the point of contact I join IC, which will be
+the required refraction of the ray RC. The demonstration of this is,
+it will be seen, entirely similar to that of which we made use in
+explaining ordinary refraction. For the refraction of the ray RC is
+nothing else than the progression of the portion C of the wave CO,
+continued in the crystal. Now the portions H of this wave, during the
+time that O came to K, will have arrived at the surface CK along the
+straight lines H_x_, and will moreover have produced in the crystal
+around the centres _x_ some hemi-spheroidal partial waves similar to
+the hemi-spheroidal GSP_g_, and similarly disposed, and of which the
+major and minor diameters will bear the same proportions to the lines
+_xv_ (the continuations of the lines H_x_ up to KB parallel to CO)
+that the diameters of the spheroid GSP_g_ bear to the line CB, or N.
+And it is quite easy to see that the common tangent of all these
+spheroids, which are here represented by Ellipses, will be the
+straight line IK, which consequently will be the propagation of the
+wave CO; and the point I will be that of the point C, conformably with
+that which has been demonstrated in ordinary refraction.
+
+Now as to finding the point of contact I, it is known that one must
+find CD a third proportional to the lines CK, CG, and draw DI parallel
+to CM, previously determined, which is the conjugate diameter to CG;
+for then, by drawing KI it touches the Ellipse at I.
+
+29. Now as we have found CI the refraction of the ray RC, similarly
+one will find C_i_ the refraction of the ray _r_C, which comes from
+the opposite side, by making C_o_ perpendicular to _r_C and following
+out the rest of the construction as before. Whence one sees that if
+the ray _r_C is inclined equally with RC, the line C_d_ will
+necessarily be equal to CD, because C_k_ is equal to CK, and C_g_ to
+CG. And in consequence I_i_ will be cut at E into equal parts by the
+line CM, to which DI and _di_ are parallel. And because CM is the
+conjugate diameter to CG, it follows that _i_I will be parallel to
+_g_G. Therefore if one prolongs the refracted rays CI, C_i_, until
+they meet the tangent ML at T and _t_, the distances MT, M_t_, will
+also be equal. And so, by our hypothesis, we explain perfectly the
+phenomenon mentioned above; to wit, that when there are two rays
+equally inclined, but coming from opposite sides, as here the rays RC,
+_rc_, their refractions diverge equally from the line followed by the
+refraction of the ray perpendicular to the surface, by considering
+these divergences in the direction parallel to the surface of the
+crystal.
+
+30. To find the length of the line N, in proportion to CP, CS, CG, it
+must be determined by observations of the irregular refraction which
+occurs in this section of the crystal; and I find thus that the ratio
+of N to GC is just a little less than 8 to 5. And having regard to
+some other observations and phenomena of which I shall speak
+afterwards, I put N at 156,962 parts, of which the semi-diameter CG is
+found to contain 98,779, making this ratio 8 to 5-1/29. Now this
+proportion, which there is between the line N and CG, may be called
+the Proportion of the Refraction; similarly as in glass that of 3 to
+2, as will be manifest when I shall have explained a short process in
+the preceding way to find the irregular refractions.
+
+31. Supposing then, in the next figure, as previously, the surface of
+the crystal _g_G, the Ellipse GP_g_, and the line N; and CM the
+refraction of the perpendicular ray FC, from which it diverges by 6
+degrees 40 minutes. Now let there be some other ray RC, the refraction
+of which must be found.
+
+About the centre C, with semi-diameter CG, let the circumference _g_RG
+be described, cutting the ray RC at R; and let RV be the perpendicular
+on CG. Then as the line N is to CG let CV be to CD, and let DI be
+drawn parallel to CM, cutting the Ellipse _g_MG at I; then joining CI,
+this will be the required refraction of the ray RC. Which is
+demonstrated thus.
+
+[Illustration]
+
+Let CO be perpendicular to CR, and across the angle OCG let OK be
+adjusted, equal to N and perpendicular to CO, and let there be drawn
+the straight line KI, which if it is demonstrated to be a tangent to
+the Ellipse at I, it will be evident by the things heretofore
+explained that CI is the refraction of the ray RC. Now since the angle
+RCO is a right angle, it is easy to see that the right-angled
+triangles RCV, KCO, are similar. As then, CK is to KO, so also is RC
+to CV. But KO is equal to N, and RC to CG: then as CK is to N so will
+CG be to CV. But as N is to CG, so, by construction, is CV to CD. Then
+as CK is to CG so is CG to CD. And because DI is parallel to CM, the
+conjugate diameter to CG, it follows that KI touches the Ellipse at I;
+which remained to be shown.
+
+32. One sees then that as there is in the refraction of ordinary
+media a certain constant proportion between the sines of the angles
+which the incident ray and the refracted ray make with the
+perpendicular, so here there is such a proportion between CV and CD or
+IE; that is to say between the Sine of the angle which the incident
+ray makes with the perpendicular, and the horizontal intercept, in the
+Ellipse, between the refraction of this ray and the diameter CM. For
+the ratio of CV to CD is, as has been said, the same as that of N to
+the semi-diameter CG.
+
+33. I will add here, before passing away, that in comparing together
+the regular and irregular refraction of this crystal, there is this
+remarkable fact, that if ABPS be the spheroid by which light spreads
+in the Crystal in a certain space of time (which spreading, as has
+been said, serves for the irregular refraction), then the inscribed
+sphere BVST is the extension in the same space of time of the light
+which serves for the regular refraction.
+
+[Illustration]
+
+For we have stated before this, that the line N being the radius of a
+spherical wave of light in air, while in the crystal it spread through
+the spheroid ABPS, the ratio of N to CS will be 156,962 to 93,410. But
+it has also been stated that the proportion of the regular refraction
+was 5 to 3; that is to say, that N being the radius of a spherical
+wave of light in air, its extension in the crystal would, in the same
+space of time, form a sphere the radius of which would be to N as 3 to
+5. Now 156,962 is to 93,410 as 5 to 3 less 1/41. So that it is
+sufficiently nearly, and may be exactly, the sphere BVST, which the
+light describes for the regular refraction in the crystal, while it
+describes the spheroid BPSA for the irregular refraction, and while it
+describes the sphere of radius N in air outside the crystal.
+
+Although then there are, according to what we have supposed, two
+different propagations of light within the crystal, it appears that it
+is only in directions perpendicular to the axis BS of the spheroid
+that one of these propagations occurs more rapidly than the other; but
+that they have an equal velocity in the other direction, namely, in
+that parallel to the same axis BS, which is also the axis of the
+obtuse angle of the crystal.
+
+[Illustration]
+
+34. The proportion of the refraction being what we have just seen, I
+will now show that there necessarily follows thence that notable
+property of the ray which falling obliquely on the surface of the
+crystal enters it without suffering refraction. For supposing the same
+things as before, and that the ray makes with the same surface _g_G
+the angle RCG of 73 degrees 20 minutes, inclining to the same side as
+the crystal (of which ray mention has been made above); if one
+investigates, by the process above explained, the refraction CI, one
+will find that it makes exactly a straight line with RC, and that thus
+this ray is not deviated at all, conformably with experiment. This is
+proved as follows by calculation.
+
+CG or CR being, as precedently, 98,779; CM being 100,000; and the
+angle RCV 73 degrees 20 minutes, CV will be 28,330. But because CI is
+the refraction of the ray RC, the proportion of CV to CD is 156,962 to
+98,779, namely, that of N to CG; then CD is 17,828.
+
+Now the rectangle _g_DC is to the square of DI as the square of CG is
+to the square of CM; hence DI or CE will be 98,353. But as CE is to
+EI, so will CM be to MT, which will then be 18,127. And being added to
+ML, which is 11,609 (namely the sine of the angle LCM, which is 6
+degrees 40 minutes, taking CM 100,000 as radius) we get LT 27,936; and
+this is to LC 99,324 as CV to VR, that is to say, as 29,938, the
+tangent of the complement of the angle RCV, which is 73 degrees 20
+minutes, is to the radius of the Tables. Whence it appears that RCIT
+is a straight line; which was to be proved.
+
+35. Further it will be seen that the ray CI in emerging through the
+opposite surface of the crystal, ought to pass out quite straight,
+according to the following demonstration, which proves that the
+reciprocal relation of refraction obtains in this crystal the same as
+in other transparent bodies; that is to say, that if a ray RC in
+meeting the surface of the crystal CG is refracted as CI, the ray CI
+emerging through the opposite parallel surface of the crystal, which
+I suppose to be IB, will have its refraction IA parallel to the ray
+RC.
+
+[Illustration]
+
+Let the same things be supposed as before; that is to say, let CO,
+perpendicular to CR, represent a portion of a wave the continuation of
+which in the crystal is IK, so that the piece C will be continued on
+along the straight line CI, while O comes to K. Now if one takes a
+second period of time equal to the first, the piece K of the wave IK
+will, in this second period, have advanced along the straight line KB,
+equal and parallel to CI, because every piece of the wave CO, on
+arriving at the surface CK, ought to go on in the crystal the same as
+the piece C; and in this same time there will be formed in the air
+from the point I a partial spherical wave having a semi-diameter IA
+equal to KO, since KO has been traversed in an equal time. Similarly,
+if one considers some other point of the wave IK, such as _h_, it will
+go along _hm_, parallel to CI, to meet the surface IB, while the point
+K traverses K_l_ equal to _hm_; and while this accomplishes the
+remainder _l_B, there will start from the point _m_ a partial wave the
+semi-diameter of which, _mn_, will have the same ratio to _l_B as IA
+to KB. Whence it is evident that this wave of semi-diameter _mn_, and
+the other of semi-diameter IA will have the same tangent BA. And
+similarly for all the partial spherical waves which will be formed
+outside the crystal by the impact of all the points of the wave IK
+against the surface of the Ether IB. It is then precisely the tangent
+BA which will be the continuation of the wave IK, outside the crystal,
+when the piece K has reached B. And in consequence IA, which is
+perpendicular to BA, will be the refraction of the ray CI on emerging
+from the crystal. Now it is clear that IA is parallel to the incident
+ray RC, since IB is equal to CK, and IA equal to KO, and the angles A
+and O are right angles.
+
+It is seen then that, according to our hypothesis, the reciprocal
+relation of refraction holds good in this crystal as well as in
+ordinary transparent bodies; as is thus in fact found by observation.
+
+36. I pass now to the consideration of other sections of the crystal,
+and of the refractions there produced, on which, as will be seen, some
+other very remarkable phenomena depend.
+
+Let ABH be a parallelepiped of crystal, and let the top surface AEHF
+be a perfect rhombus, the obtuse angles of which are equally divided
+by the straight line EF, and the acute angles by the straight line AH
+perpendicular to FE.
+
+The section which we have hitherto considered is that which passes
+through the lines EF, EB, and which at the same time cuts the plane
+AEHF at right angles. Refractions in this section have this in common
+with the refractions in ordinary media that the plane which is drawn
+through the incident ray and which also intersects the surface of the
+crystal at right angles, is that in which the refracted ray also is
+found. But the refractions which appertain to every other section of
+this crystal have this strange property that the refracted ray always
+quits the plane of the incident ray perpendicular to the surface, and
+turns away towards the side of the slope of the crystal. For which
+fact we shall show the reason, in the first place, for the section
+through AH; and we shall show at the same time how one can determine
+the refraction, according to our hypothesis. Let there be, then, in
+the plane which passes through AH, and which is perpendicular to the
+plane AFHE, the incident ray RC; it is required to find its refraction
+in the crystal.
+
+[Illustration]
+
+37. About the centre C, which I suppose to be in the intersection of
+AH and FE, let there be imagined a hemi-spheroid QG_qg_M, such as the
+light would form in spreading in the crystal, and let its section by
+the plane AEHF form the Ellipse QG_qg_, the major diameter of which
+Q_q_, which is in the line AH, will necessarily be one of the major
+diameters of the spheroid; because the axis of the spheroid being in
+the plane through FEB, to which QC is perpendicular, it follows that
+QC is also perpendicular to the axis of the spheroid, and consequently
+QC_q_ one of its major diameters. But the minor diameter of this
+Ellipse, G_g_, will bear to Q_q_ the proportion which has been defined
+previously, Article 27, between CG and the major semi-diameter of the
+spheroid, CP, namely, that of 98,779 to 105,032.
+
+Let the line N be the length of the travel of light in air during the
+time in which, within the crystal, it makes, from the centre C, the
+spheroid QC_qg_M. Then having drawn CO perpendicular to the ray CR and
+situate in the plane through CR and AH, let there be adjusted, across
+the angle ACO, the straight line OK equal to N and perpendicular to
+CO, and let it meet the straight line AH at K. Supposing consequently
+that CL is perpendicular to the surface of the crystal AEHF, and that
+CM is the refraction of the ray which falls perpendicularly on this
+same surface, let there be drawn a plane through the line CM and
+through KCH, making in the spheroid the semi-ellipse QM_q_, which will
+be given, since the angle MCL is given of value 6 degrees 40 minutes.
+And it is certain, according to what has been explained above, Article
+27, that a plane which would touch the spheroid at the point M, where
+I suppose the straight line CM to meet the surface, would be parallel
+to the plane QG_q_. If then through the point K one now draws KS
+parallel to G_g_, which will be parallel also to QX, the tangent to
+the Ellipse QG_q_ at Q; and if one conceives a plane passing through
+KS and touching the spheroid, the point of contact will necessarily be
+in the Ellipse QM_q_, because this plane through KS, as well as the
+plane which touches the spheroid at the point M, are parallel to QX,
+the tangent of the spheroid: for this consequence will be demonstrated
+at the end of this Treatise. Let this point of contact be at I, then
+making KC, QC, DC proportionals, draw DI parallel to CM; also join CI.
+I say that CI will be the required refraction of the ray RC. This will
+be manifest if, in considering CO, which is perpendicular to the ray
+RC, as a portion of the wave of light, we can demonstrate that the
+continuation of its piece C will be found in the crystal at I, when O
+has arrived at K.
+
+38. Now as in the Chapter on Reflexion, in demonstrating that the
+incident and reflected rays are always in the same plane perpendicular
+to the reflecting surface, we considered the breadth of the wave of
+light, so, similarly, we must here consider the breadth of the wave CO
+in the diameter G_g_. Taking then the breadth C_c_ on the side toward
+the angle E, let the parallelogram CO_oc_ be taken as a portion of a
+wave, and let us complete the parallelograms CK_kc_, CI_ic_, Kl_ik_,
+OK_ko_. In the time then that the line O_o_ arrives at the surface of
+the crystal at K_k_, all the points of the wave CO_oc_ will have
+arrived at the rectangle K_c_ along lines parallel to OK; and from the
+points of their incidences there will originate, beyond that, in the
+crystal partial hemi-spheroids, similar to the hemi-spheroid QM_q_,
+and similarly disposed. These hemi-spheroids will necessarily all
+touch the plane of the parallelogram KI_ik_ at the same instant that
+O_o_ has reached K_k_. Which is easy to comprehend, since, of these
+hemi-spheroids, all those which have their centres along the line CK,
+touch this plane in the line KI (for this is to be shown in the same
+way as we have demonstrated the refraction of the oblique ray in the
+principal section through EF) and all those which have their centres
+in the line C_c_ will touch the same plane KI in the line I_i_; all
+these being similar to the hemi-spheroid QM_q_. Since then the
+parallelogram K_i_ is that which touches all these spheroids, this
+same parallelogram will be precisely the continuation of the wave
+CO_oc_ in the crystal, when O_o_ has arrived at K_k_, because it forms
+the termination of the movement and because of the quantity of
+movement which occurs more there than anywhere else: and thus it
+appears that the piece C of the wave CO_oc_ has its continuation at I;
+that is to say, that the ray RC is refracted as CI.
+
+From this it is to be noted that the proportion of the refraction for
+this section of the crystal is that of the line N to the semi-diameter
+CQ; by which one will easily find the refractions of all incident
+rays, in the same way as we have shown previously for the case of the
+section through FE; and the demonstration will be the same. But it
+appears that the said proportion of the refraction is less here than
+in the section through FEB; for it was there the same as the ratio of
+N to CG, that is to say, as 156,962 to 98,779, very nearly as 8 to 5;
+and here it is the ratio of N to CQ the major semi-diameter of the
+spheroid, that is to say, as 156,962 to 105,032, very nearly as 3 to
+2, but just a little less. Which still agrees perfectly with what one
+finds by observation.
+
+39. For the rest, this diversity of proportion of refraction produces
+a very singular effect in this Crystal; which is that when it is
+placed upon a sheet of paper on which there are letters or anything
+else marked, if one views it from above with the two eyes situated in
+the plane of the section through EF, one sees the letters raised up by
+this irregular refraction more than when one puts one's eyes in the
+plane of section through AH: and the difference of these elevations
+appears by comparison with the other ordinary refraction of the
+crystal, the proportion of which is as 5 to 3, and which always raises
+the letters equally, and higher than the irregular refraction does.
+For one sees the letters and the paper on which they are written, as
+on two different stages at the same time; and in the first position of
+the eyes, namely, when they are in the plane through AH these two
+stages are four times more distant from one another than when the eyes
+are in the plane through EF.
+
+We will show that this effect follows from the refractions; and it
+will enable us at the same time to ascertain the apparent place of a
+point of an object placed immediately under the crystal, according to
+the different situation of the eyes.
+
+40. Let us see first by how much the irregular refraction of the plane
+through AH ought to lift the bottom of the crystal. Let the plane of
+this figure represent separately the section through Q_q_ and CL, in
+which section there is also the ray RC, and let the semi-elliptic
+plane through Q_q_ and CM be inclined to the former, as previously, by
+an angle of 6 degrees 40 minutes; and in this plane CI is then the
+refraction of the ray RC.
+
+[Illustration]
+
+If now one considers the point I as at the bottom of the crystal, and
+that it is viewed by the rays ICR, _Icr_, refracted equally at the
+points C_c_, which should be equally distant from D, and that these
+rays meet the two eyes at R_r_; it is certain that the point I will
+appear raised to S where the straight lines RC, _rc_, meet; which
+point S is in DP, perpendicular to Q_q_. And if upon DP there is drawn
+the perpendicular IP, which will lie at the bottom of the crystal, the
+length SP will be the apparent elevation of the point I above the
+bottom.
+
+Let there be described on Q_q_ a semicircle cutting the ray CR at B,
+from which BV is drawn perpendicular to Q_q_; and let the proportion
+of the refraction for this section be, as before, that of the line N
+to the semi-diameter CQ.
+
+Then as N is to CQ so is VC to CD, as appears by the method of finding
+the refraction which we have shown above, Article 31; but as VC is to
+CD, so is VB to DS. Then as N is to CQ, so is VB to DS. Let ML be
+perpendicular to CL. And because I suppose the eyes R_r_ to be distant
+about a foot or so from the crystal, and consequently the angle RS_r_
+very small, VB may be considered as equal to the semi-diameter CQ, and
+DP as equal to CL; then as N is to CQ so is CQ to DS. But N is valued
+at 156,962 parts, of which CM contains 100,000 and CQ 105,032. Then DS
+will have 70,283. But CL is 99,324, being the sine of the complement
+of the angle MCL which is 6 degrees 40 minutes; CM being supposed as
+radius. Then DP, considered as equal to CL, will be to DS as 99,324 to
+70,283. And so the elevation of the point I by the refraction of this
+section is known.
+
+[Illustration]
+
+41. Now let there be represented the other section through EF in the
+figure before the preceding one; and let CM_g_ be the semi-ellipse,
+considered in Articles 27 and 28, which is made by cutting a
+spheroidal wave having centre C. Let the point I, taken in this
+ellipse, be imagined again at the bottom of the Crystal; and let it be
+viewed by the refracted rays ICR, I_cr_, which go to the two eyes; CR
+and _cr_ being equally inclined to the surface of the crystal G_g_.
+This being so, if one draws ID parallel to CM, which I suppose to be
+the refraction of the perpendicular ray incident at the point C, the
+distances DC, D_c_, will be equal, as is easy to see by that which has
+been demonstrated in Article 28. Now it is certain that the point I
+should appear at S where the straight lines RC, _rc_, meet when
+prolonged; and that this point will fall in the line DP perpendicular
+to G_g_. If one draws IP perpendicular to this DP, it will be the
+distance PS which will mark the apparent elevation of the point I. Let
+there be described on G_g_ a semicircle cutting CR at B, from which
+let BV be drawn perpendicular to G_g_; and let N to GC be the
+proportion of the refraction in this section, as in Article 28. Since
+then CI is the refraction of the radius BC, and DI is parallel to CM,
+VC must be to CD as N to GC, according to what has been demonstrated
+in Article 31. But as VC is to CD so is BV to DS. Let ML be drawn
+perpendicular to CL. And because I consider, again, the eyes to be
+distant above the crystal, BV is deemed equal to the semi-diameter CG;
+and hence DS will be a third proportional to the lines N and CG: also
+DP will be deemed equal to CL. Now CG consisting of 98,778 parts, of
+which CM contains 100,000, N is taken as 156,962. Then DS will be
+62,163. But CL is also determined, and contains 99,324 parts, as has
+been said in Articles 34 and 40. Then the ratio of PD to DS will be as
+99,324 to 62,163. And thus one knows the elevation of the point at the
+bottom I by the refraction of this section; and it appears that this
+elevation is greater than that by the refraction of the preceding
+section, since the ratio of PD to DS was there as 99,324 to 70,283.
+
+[Illustration]
+
+But by the regular refraction of the crystal, of which we have above
+said that the proportion is 5 to 3, the elevation of the point I, or
+P, from the bottom, will be 2/5 of the height DP; as appears by this
+figure, where the point P being viewed by the rays PCR, P_cr_,
+refracted equally at the surface C_c_, this point must needs appear
+to be at S, in the perpendicular PD where the lines RC, _rc_, meet
+when prolonged: and one knows that the line PC is to CS as 5 to 3,
+since they are to one another as the sine of the angle CSP or DSC is
+to the sine of the angle SPC. And because the ratio of PD to DS is
+deemed the same as that of PC to CS, the two eyes Rr being supposed
+very far above the crystal, the elevation PS will thus be 2/5 of PD.
+
+[Illustration]
+
+42. If one takes a straight line AB for the thickness of the crystal,
+its point B being at the bottom, and if one divides it at the points
+C, D, E, according to the proportions of the elevations found, making
+AE 3/5 of AB, AB to AC as 99,324 to 70,283, and AB to AD as 99,324 to
+62,163, these points will divide AB as in this figure. And it will be
+found that this agrees perfectly with experiment; that is to say by
+placing the eyes above in the plane which cuts the crystal according
+to the shorter diameter of the rhombus, the regular refraction will
+lift up the letters to E; and one will see the bottom, and the letters
+over which it is placed, lifted up to D by the irregular refraction.
+But by placing the eyes above in the plane which cuts the crystal
+according to the longer diameter of the rhombus, the regular
+refraction will lift the letters to E as before; but the irregular
+refraction will make them, at the same time, appear lifted up only to
+C; and in such a way that the interval CE will be quadruple the
+interval ED, which one previously saw.
+
+
+43. I have only to make the remark here that in both the positions of
+the eyes the images caused by the irregular refraction do not appear
+directly below those which proceed from the regular refraction, but
+they are separated from them by being more distant from the
+equilateral solid angle of the Crystal. That follows, indeed, from all
+that has been hitherto demonstrated about the irregular refraction;
+and it is particularly shown by these last demonstrations, from which
+one sees that the point I appears by irregular refraction at S in the
+perpendicular line DP, in which line also the image of the point P
+ought to appear by regular refraction, but not the image of the point
+I, which will be almost directly above the same point, and higher than
+S.
+
+But as to the apparent elevation of the point I in other positions of
+the eyes above the crystal, besides the two positions which we have
+just examined, the image of that point by the irregular refraction
+will always appear between the two heights of D and C, passing from
+one to the other as one turns one's self around about the immovable
+crystal, while looking down from above. And all this is still found
+conformable to our hypothesis, as any one can assure himself after I
+shall have shown here the way of finding the irregular refractions
+which appear in all other sections of the crystal, besides the two
+which we have considered. Let us suppose one of the faces of the
+crystal, in which let there be the Ellipse HDE, the centre C of which
+is also the centre of the spheroid HME in which the light spreads, and
+of which the said Ellipse is the section. And let the incident ray be
+RC, the refraction of which it is required to find.
+
+Let there be taken a plane passing through the ray RC and which is
+perpendicular to the plane of the ellipse HDE, cutting it along the
+straight line BCK; and having in the same plane through RC made CO
+perpendicular to CR, let OK be adjusted across the angle OCK, so as
+to be perpendicular to OC and equal to the line N, which I suppose to
+measure the travel of the light in air during the time that it spreads
+in the crystal through the spheroid HDEM. Then in the plane of the
+Ellipse HDE let KT be drawn, through the point K, perpendicular to
+BCK. Now if one conceives a plane drawn through the straight line KT
+and touching the spheroid HME at I, the straight line CI will be the
+refraction of the ray RC, as is easy to deduce from that which has
+been demonstrated in Article 36.
+
+[Illustration]
+
+But it must be shown how one can determine the point of contact I. Let
+there be drawn parallel to the line KT a line HF which touches the
+Ellipse HDE, and let this point of contact be at H. And having drawn a
+straight line along CH to meet KT at T, let there be imagined a plane
+passing through the same CH and through CM (which I suppose to be the
+refraction of the perpendicular ray), which makes in the spheroid the
+elliptical section HME. It is certain that the plane which will pass
+through the straight line KT, and which will touch the spheroid, will
+touch it at a point in the Ellipse HME, according to the Lemma which
+will be demonstrated at the end of the Chapter. Now this point is
+necessarily the point I which is sought, since the plane drawn through
+TK can touch the spheroid at one point only. And this point I is easy
+to determine, since it is needful only to draw from the point T, which
+is in the plane of this Ellipse, the tangent TI, in the way shown
+previously. For the Ellipse HME is given, and its conjugate
+semi-diameters are CH and CM; because a straight line drawn through M,
+parallel to HE, touches the Ellipse HME, as follows from the fact that
+a plane taken through M, and parallel to the plane HDE, touches the
+spheroid at that point M, as is seen from Articles 27 and 23. For the
+rest, the position of this ellipse, with respect to the plane through
+the ray RC and through CK, is also given; from which it will be easy
+to find the position of CI, the refraction corresponding to the ray
+RC.
+
+Now it must be noted that the same ellipse HME serves to find the
+refractions of any other ray which may be in the plane through RC and
+CK. Because every plane, parallel to the straight line HF, or TK,
+which will touch the spheroid, will touch it in this ellipse,
+according to the Lemma quoted a little before.
+
+I have investigated thus, in minute detail, the properties of the
+irregular refraction of this Crystal, in order to see whether each
+phenomenon that is deduced from our hypothesis accords with that which
+is observed in fact. And this being so it affords no slight proof of
+the truth of our suppositions and principles. But what I am going to
+add here confirms them again marvellously. It is this: that there are
+different sections of this Crystal, the surfaces of which, thereby
+produced, give rise to refractions precisely such as they ought to be,
+and as I had foreseen them, according to the preceding Theory.
+
+In order to explain what these sections are, let ABKF _be_ the
+principal section through the axis of the crystal ACK, in which there
+will also be the axis SS of a spheroidal wave of light spreading in
+the crystal from the centre C; and the straight line which cuts SS
+through the middle and at right angles, namely PP, will be one of the
+major diameters.
+
+[Illustration: {Section ABKF}]
+
+Now as in the natural section of the crystal, made by a plane parallel
+to two opposite faces, which plane is here represented by the line GG,
+the refraction of the surfaces which are produced by it will be
+governed by the hemi-spheroids GNG, according to what has been
+explained in the preceding Theory. Similarly, cutting the Crystal
+through NN, by a plane perpendicular to the parallelogram ABKF, the
+refraction of the surfaces will be governed by the hemi-spheroids NGN.
+And if one cuts it through PP, perpendicularly to the said
+parallelogram, the refraction of the surfaces ought to be governed by
+the hemi-spheroids PSP, and so for others. But I saw that if the plane
+NN was almost perpendicular to the plane GG, making the angle NCG,
+which is on the side A, an angle of 90 degrees 40 minutes, the
+hemi-spheroids NGN would become similar to the hemi-spheroids GNG,
+since the planes NN and GG were equally inclined by an angle of 45
+degrees 20 minutes to the axis SS. In consequence it must needs be, if
+our theory is true, that the surfaces which the section through NN
+produces should effect the same refractions as the surfaces of the
+section through GG. And not only the surfaces of the section NN but
+all other sections produced by planes which might be inclined to the
+axis at an angle equal to 45 degrees 20 minutes. So that there are an
+infinitude of planes which ought to produce precisely the same
+refractions as the natural surfaces of the crystal, or as the section
+parallel to any one of those surfaces which are made by cleavage.
+
+I saw also that by cutting it by a plane taken through PP, and
+perpendicular to the axis SS, the refraction of the surfaces ought to
+be such that the perpendicular ray should suffer thereby no deviation;
+and that for oblique rays there would always be an irregular
+refraction, differing from the regular, and by which objects placed
+beneath the crystal would be less elevated than by that other
+refraction.
+
+That, similarly, by cutting the crystal by any plane through the axis
+SS, such as the plane of the figure is, the perpendicular ray ought to
+suffer no refraction; and that for oblique rays there were different
+measures for the irregular refraction according to the situation of
+the plane in which the incident ray was.
+
+Now these things were found in fact so; and, after that, I could not
+doubt that a similar success could be met with everywhere. Whence I
+concluded that one might form from this crystal solids similar to
+those which are its natural forms, which should produce, at all their
+surfaces, the same regular and irregular refractions as the natural
+surfaces, and which nevertheless would cleave in quite other ways, and
+not in directions parallel to any of their faces. That out of it one
+would be able to fashion pyramids, having their base square,
+pentagonal, hexagonal, or with as many sides as one desired, all the
+surfaces of which should have the same refractions as the natural
+surfaces of the crystal, except the base, which will not refract the
+perpendicular ray. These surfaces will each make an angle of 45
+degrees 20 minutes with the axis of the crystal, and the base will be
+the section perpendicular to the axis.
+
+That, finally, one could also fashion out of it triangular prisms, or
+prisms with as many sides as one would, of which neither the sides nor
+the bases would refract the perpendicular ray, although they would yet
+all cause double refraction for oblique rays. The cube is included
+amongst these prisms, the bases of which are sections perpendicular to
+the axis of the crystal, and the sides are sections parallel to the
+same axis.
+
+From all this it further appears that it is not at all in the
+disposition of the layers of which this crystal seems to be composed,
+and according to which it splits in three different senses, that the
+cause resides of its irregular refraction; and that it would be in
+vain to wish to seek it there.
+
+But in order that any one who has some of this stone may be able to
+find, by his own experience, the truth of what I have just advanced, I
+will state here the process of which I have made use to cut it, and to
+polish it. Cutting is easy by the slicing wheels of lapidaries, or in
+the way in which marble is sawn: but polishing is very difficult, and
+by employing the ordinary means one more often depolishes the surfaces
+than makes them lucent.
+
+After many trials, I have at last found that for this service no plate
+of metal must be used, but a piece of mirror glass made matt and
+depolished. Upon this, with fine sand and water, one smoothes the
+crystal little by little, in the same way as spectacle glasses, and
+polishes it simply by continuing the work, but ever reducing the
+material. I have not, however, been able to give it perfect clarity
+and transparency; but the evenness which the surfaces acquire enables
+one to observe in them the effects of refraction better than in those
+made by cleaving the stone, which always have some inequality.
+
+Even when the surface is only moderately smoothed, if one rubs it over
+with a little oil or white of egg, it becomes quite transparent, so
+that the refraction is discerned in it quite distinctly. And this aid
+is specially necessary when it is wished to polish the natural
+surfaces to remove the inequalities; because one cannot render them
+lucent equally with the surfaces of other sections, which take a
+polish so much the better the less nearly they approximate to these
+natural planes.
+
+Before finishing the treatise on this Crystal, I will add one more
+marvellous phenomenon which I discovered after having written all the
+foregoing. For though I have not been able till now to find its cause,
+I do not for that reason wish to desist from describing it, in order
+to give opportunity to others to investigate it. It seems that it will
+be necessary to make still further suppositions besides those which I
+have made; but these will not for all that cease to keep their
+probability after having been confirmed by so many tests.
+
+[Illustration]
+
+The phenomenon is, that by taking two pieces of this crystal and
+applying them one over the other, or rather holding them with a space
+between the two, if all the sides of one are parallel to those of the
+other, then a ray of light, such as AB, is divided into two in the
+first piece, namely into BD and BC, following the two refractions,
+regular and irregular. On penetrating thence into the other piece
+each ray will pass there without further dividing itself in two; but
+that one which underwent the regular refraction, as here DG, will
+undergo again only a regular refraction at GH; and the other, CE, an
+irregular refraction at EF. And the same thing occurs not only in this
+disposition, but also in all those cases in which the principal
+section of each of the pieces is situated in one and the same plane,
+without it being needful for the two neighbouring surfaces to be
+parallel. Now it is marvellous why the rays CE and DG, incident from
+the air on the lower crystal, do not divide themselves the same as the
+first ray AB. One would say that it must be that the ray DG in passing
+through the upper piece has lost something which is necessary to move
+the matter which serves for the irregular refraction; and that
+likewise CE has lost that which was necessary to move the matter
+which serves for regular refraction: but there is yet another thing
+which upsets this reasoning. It is that when one disposes the two
+crystals in such a way that the planes which constitute the principal
+sections intersect one another at right angles, whether the
+neighbouring surfaces are parallel or not, then the ray which has come
+by the regular refraction, as DG, undergoes only an irregular
+refraction in the lower piece; and on the contrary the ray which has
+come by the irregular refraction, as CE, undergoes only a regular
+refraction.
+
+But in all the infinite other positions, besides those which I have
+just stated, the rays DG, CE, divide themselves anew each one into
+two, by refraction in the lower crystal so that from the single ray AB
+there are four, sometimes of equal brightness, sometimes some much
+less bright than others, according to the varying agreement in the
+positions of the crystals: but they do not appear to have all together
+more light than the single ray AB.
+
+When one considers here how, while the rays CE, DG, remain the same,
+it depends on the position that one gives to the lower piece, whether
+it divides them both in two, or whether it does not divide them, and
+yet how the ray AB above is always divided, it seems that one is
+obliged to conclude that the waves of light, after having passed
+through the first crystal, acquire a certain form or disposition in
+virtue of which, when meeting the texture of the second crystal, in
+certain positions, they can move the two different kinds of matter
+which serve for the two species of refraction; and when meeting the
+second crystal in another position are able to move only one of these
+kinds of matter. But to tell how this occurs, I have hitherto found
+nothing which satisfies me.
+
+Leaving then to others this research, I pass to what I have to say
+touching the cause of the extraordinary figure of this crystal, and
+why it cleaves easily in three different senses, parallel to any one
+of its surfaces.
+
+There are many bodies, vegetable, mineral, and congealed salts, which
+are formed with certain regular angles and figures. Thus among flowers
+there are many which have their leaves disposed in ordered polygons,
+to the number of 3, 4, 5, or 6 sides, but not more. This well deserves
+to be investigated, both as to the polygonal figure, and as to why it
+does not exceed the number 6.
+
+Rock Crystal grows ordinarily in hexagonal bars, and diamonds are
+found which occur with a square point and polished surfaces. There is
+a species of small flat stones, piled up directly upon one another,
+which are all of pentagonal figure with rounded angles, and the sides
+a little folded inwards. The grains of gray salt which are formed from
+sea water affect the figure, or at least the angle, of the cube; and
+in the congelations of other salts, and in that of sugar, there are
+found other solid angles with perfectly flat faces. Small snowflakes
+almost always fall in little stars with 6 points, and sometimes in
+hexagons with straight sides. And I have often observed, in water
+which is beginning to freeze, a kind of flat and thin foliage of ice,
+the middle ray of which throws out branches inclined at an angle of 60
+degrees. All these things are worthy of being carefully investigated
+to ascertain how and by what artifice nature there operates. But it is
+not now my intention to treat fully of this matter. It seems that in
+general the regularity which occurs in these productions comes from
+the arrangement of the small invisible equal particles of which they
+are composed. And, coming to our Iceland Crystal, I say that if there
+were a pyramid such as ABCD, composed of small rounded corpuscles, not
+spherical but flattened spheroids, such as would be made by the
+rotation of the ellipse GH around its lesser diameter EF (of which the
+ratio to the greater diameter is very nearly that of 1 to the square
+root of 8)--I say that then the solid angle of the point D would be
+equal to the obtuse and equilateral angle of this Crystal. I say,
+further, that if these corpuscles were lightly stuck together, on
+breaking this pyramid it would break along faces parallel to those
+that make its point: and by this means, as it is easy to see, it would
+produce prisms similar to those of the same crystal as this other
+figure represents. The reason is that when broken in this fashion a
+whole layer separates easily from its neighbouring layer since each
+spheroid has to be detached only from the three spheroids of the next
+layer; of which three there is but one which touches it on its
+flattened surface, and the other two at the edges. And the reason why
+the surfaces separate sharp and polished is that if any spheroid of
+the neighbouring surface would come out by attaching itself to the
+surface which is being separated, it would be needful for it to detach
+itself from six other spheroids which hold it locked, and four of
+which press it by these flattened surfaces. Since then not only the
+angles of our crystal but also the manner in which it splits agree
+precisely with what is observed in the assemblage composed of such
+spheroids, there is great reason to believe that the particles are
+shaped and ranged in the same way.
+
+[Illustration: {Pyramid and section of spheroids}]
+
+There is even probability enough that the prisms of this crystal are
+produced by the breaking up of pyramids, since Mr. Bartholinus relates
+that he occasionally found some pieces of triangularly pyramidal
+figure. But when a mass is composed interiorly only of these little
+spheroids thus piled up, whatever form it may have exteriorly, it is
+certain, by the same reasoning which I have just explained, that if
+broken it would produce similar prisms. It remains to be seen whether
+there are other reasons which confirm our conjecture, and whether
+there are none which are repugnant to it.
+
+[Illustration: {paralleloid arrangement of spheroids with planes of
+potential cleavage}]
+
+It may be objected that this crystal, being so composed, might be
+capable of cleavage in yet two more fashions; one of which would be
+along planes parallel to the base of the pyramid, that is to say to
+the triangle ABC; the other would be parallel to a plane the trace of
+which is marked by the lines GH, HK, KL. To which I say that both the
+one and the other, though practicable, are more difficult than those
+which were parallel to any one of the three planes of the pyramid; and
+that therefore, when striking on the crystal in order to break it, it
+ought always to split rather along these three planes than along the
+two others. When one has a number of spheroids of the form above
+described, and ranges them in a pyramid, one sees why the two methods
+of division are more difficult. For in the case of that division which
+would be parallel to the base, each spheroid would be obliged to
+detach itself from three others which it touches upon their flattened
+surfaces, which hold more strongly than the contacts at the edges. And
+besides that, this division will not occur along entire layers,
+because each of the spheroids of a layer is scarcely held at all by
+the 6 of the same layer that surround it, since they only touch it at
+the edges; so that it adheres readily to the neighbouring layer, and
+the others to it, for the same reason; and this causes uneven
+surfaces. Also one sees by experiment that when grinding down the
+crystal on a rather rough stone, directly on the equilateral solid
+angle, one verily finds much facility in reducing it in this
+direction, but much difficulty afterwards in polishing the surface
+which has been flattened in this manner.
+
+As for the other method of division along the plane GHKL, it will be
+seen that each spheroid would have to detach itself from four of the
+neighbouring layer, two of which touch it on the flattened surfaces,
+and two at the edges. So that this division is likewise more difficult
+than that which is made parallel to one of the surfaces of the
+crystal; where, as we have said, each spheroid is detached from only
+three of the neighbouring layer: of which three there is one only
+which touches it on the flattened surface, and the other two at the
+edges only.
+
+However, that which has made me know that in the crystal there are
+layers in this last fashion, is that in a piece weighing half a pound
+which I possess, one sees that it is split along its length, as is the
+above-mentioned prism by the plane GHKL; as appears by colours of the
+Iris extending throughout this whole plane although the two pieces
+still hold together. All this proves then that the composition of the
+crystal is such as we have stated. To which I again add this
+experiment; that if one passes a knife scraping along any one of the
+natural surfaces, and downwards as it were from the equilateral obtuse
+angle, that is to say from the apex of the pyramid, one finds it quite
+hard; but by scraping in the opposite sense an incision is easily
+made. This follows manifestly from the situation of the small
+spheroids; over which, in the first manner, the knife glides; but in
+the other manner it seizes them from beneath almost as if they were
+the scales of a fish.
+
+I will not undertake to say anything touching the way in which so many
+corpuscles all equal and similar are generated, nor how they are set
+in such beautiful order; whether they are formed first and then
+assembled, or whether they arrange themselves thus in coming into
+being and as fast as they are produced, which seems to me more
+probable. To develop truths so recondite there would be needed a
+knowledge of nature much greater than that which we have. I will add
+only that these little spheroids could well contribute to form the
+spheroids of the waves of light, here above supposed, these as well as
+those being similarly situated, and with their axes parallel.
+
+
+_Calculations which have been supposed in this Chapter_.
+
+Mr. Bartholinus, in his treatise of this Crystal, puts at 101 degrees
+the obtuse angles of the faces, which I have stated to be 101 degrees
+52 minutes. He states that he measured these angles directly on the
+crystal, which is difficult to do with ultimate exactitude, because
+the edges such as CA, CB, in this figure, are generally worn, and not
+quite straight. For more certainty, therefore, I preferred to measure
+actually the obtuse angle by which the faces CBDA, CBVF, are inclined
+to one another, namely the angle OCN formed by drawing CN
+perpendicular to FV, and CO perpendicular to DA. This angle OCN I
+found to be 105 degrees; and its supplement CNP, to be 75 degrees, as
+it should be.
+
+[Illustration]
+
+To find from this the obtuse angle BCA, I imagined a sphere having its
+centre at C, and on its surface a spherical triangle, formed by the
+intersection of three planes which enclose the solid angle C. In this
+equilateral triangle, which is ABF in this other figure, I see that
+each of the angles should be 105 degrees, namely equal to the angle
+OCN; and that each of the sides should be of as many degrees as the
+angle ACB, or ACF, or BCF. Having then drawn the arc FQ perpendicular
+to the side AB, which it divides equally at Q, the triangle FQA has a
+right angle at Q, the angle A 105 degrees, and F half as much, namely
+52 degrees 30 minutes; whence the hypotenuse AF is found to be 101
+degrees 52 minutes. And this arc AF is the measure of the angle ACF in
+the figure of the crystal.
+
+[Illustration]
+
+In the same figure, if the plane CGHF cuts the crystal so that it
+divides the obtuse angles ACB, MHV, in the middle, it is stated, in
+Article 10, that the angle CFH is 70 degrees 57 minutes. This again is
+easily shown in the same spherical triangle ABF, in which it appears
+that the arc FQ is as many degrees as the angle GCF in the crystal,
+the supplement of which is the angle CFH. Now the arc FQ is found to
+be 109 degrees 3 minutes. Then its supplement, 70 degrees 57 minutes,
+is the angle CFH.
+
+It was stated, in Article 26, that the straight line CS, which in the
+preceding figure is CH, being the axis of the crystal, that is to say
+being equally inclined to the three sides CA, CB, CF, the angle GCH is
+45 degrees 20 minutes. This is also easily calculated by the same
+spherical triangle. For by drawing the other arc AD which cuts BF
+equally, and intersects FQ at S, this point will be the centre of the
+triangle. And it is easy to see that the arc SQ is the measure of the
+angle GCH in the figure which represents the crystal. Now in the
+triangle QAS, which is right-angled, one knows also the angle A, which
+is 52 degrees 30 minutes, and the side AQ 50 degrees 56 minutes;
+whence the side SQ is found to be 45 degrees 20 minutes.
+
+In Article 27 it was required to show that PMS being an ellipse the
+centre of which is C, and which touches the straight line MD at M so
+that the angle MCL which CM makes with CL, perpendicular on DM, is 6
+degrees 40 minutes, and its semi-minor axis CS making with CG (which
+is parallel to MD) an angle GCS of 45 degrees 20 minutes, it was
+required to show, I say, that, CM being 100,000 parts, PC the
+semi-major diameter of this ellipse is 105,032 parts, and CS, the
+semi-minor diameter, 93,410.
+
+Let CP and CS be prolonged and meet the tangent DM at D and Z; and
+from the point of contact M let MN and MO be drawn as perpendiculars
+to CP and CS. Now because the angles SCP, GCL, are right angles, the
+angle PCL will be equal to GCS which was 45 degrees 20 minutes. And
+deducting the angle LCM, which is 6 degrees 40 minutes, from LCP,
+which is 45 degrees 20 minutes, there remains MCP, 38 degrees 40
+minutes. Considering then CM as a radius of 100,000 parts, MN, the
+sine of 38 degrees 40 minutes, will be 62,479. And in the right-angled
+triangle MND, MN will be to ND as the radius of the Tables is to the
+tangent of 45 degrees 20 minutes (because the angle NMD is equal to
+DCL, or GCS); that is to say as 100,000 to 101,170: whence results ND
+63,210. But NC is 78,079 of the same parts, CM being 100,000, because
+NC is the sine of the complement of the angle MCP, which was 38
+degrees 40 minutes. Then the whole line DC is 141,289; and CP, which
+is a mean proportional between DC and CN, since MD touches the
+Ellipse, will be 105,032.
+
+[Illustration]
+
+Similarly, because the angle OMZ is equal to CDZ, or LCZ, which is 44
+degrees 40 minutes, being the complement of GCS, it follows that, as
+the radius of the Tables is to the tangent of 44 degrees 40 minutes,
+so will OM 78,079 be to OZ 77,176. But OC is 62,479 of these same
+parts of which CM is 100,000, because it is equal to MN, the sine of
+the angle MCP, which is 38 degrees 40 minutes. Then the whole line CZ
+is 139,655; and CS, which is a mean proportional between CZ and CO
+will be 93,410.
+
+At the same place it was stated that GC was found to be 98,779 parts.
+To prove this, let PE be drawn in the same figure parallel to DM, and
+meeting CM at E. In the right-angled triangle CLD the side CL is
+99,324 (CM being 100,000), because CL is the sine of the complement of
+the angle LCM, which is 6 degrees 40 minutes. And since the angle LCD
+is 45 degrees 20 minutes, being equal to GCS, the side LD is found to
+be 100,486: whence deducting ML 11,609 there will remain MD 88,877.
+Now as CD (which was 141,289) is to DM 88,877, so will CP 105,032 be
+to PE 66,070. But as the rectangle MEH (or rather the difference of
+the squares on CM and CE) is to the square on MC, so is the square on
+PE to the square on C_g_; then also as the difference of the squares
+on DC and CP to the square on CD, so also is the square on PE to the
+square on _g_C. But DP, CP, and PE are known; hence also one knows GC,
+which is 98,779.
+
+
+_Lemma which has been supposed_.
+
+If a spheroid is touched by a straight line, and also by two or more
+planes which are parallel to this line, though not parallel to one
+another, all the points of contact of the line, as well as of the
+planes, will be in one and the same ellipse made by a plane which
+passes through the centre of the spheroid.
+
+Let LED be the spheroid touched by the line BM at the point B, and
+also by the planes parallel to this line at the points O and A. It is
+required to demonstrate that the points B, O, and A are in one and the
+same Ellipse made in the spheroid by a plane which passes through its
+centre.
+
+[Illustration]
+
+Through the line BM, and through the points O and A, let there be
+drawn planes parallel to one another, which, in cutting the spheroid
+make the ellipses LBD, POP, QAQ; which will all be similar and
+similarly disposed, and will have their centres K, N, R, in one and
+the same diameter of the spheroid, which will also be the diameter of
+the ellipse made by the section of the plane that passes through the
+centre of the spheroid, and which cuts the planes of the three said
+Ellipses at right angles: for all this is manifest by proposition 15
+of the book of Conoids and Spheroids of Archimedes. Further, the two
+latter planes, which are drawn through the points O and A, will also,
+by cutting the planes which touch the spheroid in these same points,
+generate straight lines, as OH and AS, which will, as is easy to see,
+be parallel to BM; and all three, BM, OH, AS, will touch the Ellipses
+LBD, POP, QAQ in these points, B, O, A; since they are in the planes
+of these ellipses, and at the same time in the planes which touch the
+spheroid. If now from these points B, O, A, there are drawn the
+straight lines BK, ON, AR, through the centres of the same ellipses,
+and if through these centres there are drawn also the diameters LD,
+PP, QQ, parallel to the tangents BM, OH, AS; these will be conjugate
+to the aforesaid BK, ON, AR. And because the three ellipses are
+similar and similarly disposed, and have their diameters LD, PP, QQ
+parallel, it is certain that their conjugate diameters BK, ON, AR,
+will also be parallel. And the centres K, N, R being, as has been
+stated, in one and the same diameter of the spheroid, these parallels
+BK, ON, AR will necessarily be in one and the same plane, which passes
+through this diameter of the spheroid, and, in consequence, the points
+R, O, A are in one and the same ellipse made by the intersection of
+this plane. Which was to be proved. And it is manifest that the
+demonstration would be the same if, besides the points O, A, there had
+been others in which the spheroid had been touched by planes parallel
+to the straight line BM.
+
+
+
+
+CHAPTER VI
+
+ON THE FIGURES OF THE TRANSPARENT BODIES
+
+Which serve for Refraction and for Reflexion
+
+
+After having explained how the properties of reflexion and refraction
+follow from what we have supposed concerning the nature of light, and
+of opaque bodies, and of transparent media, I will here set forth a
+very easy and natural way of deducing, from the same principles, the
+true figures which serve, either by reflexion or by refraction, to
+collect or disperse the rays of light, as may be desired. For though I
+do not see yet that there are means of making use of these figures, so
+far as relates to Refraction, not only because of the difficulty of
+shaping the glasses of Telescopes with the requisite exactitude
+according to these figures, but also because there exists in
+refraction itself a property which hinders the perfect concurrence of
+the rays, as Mr. Newton has very well proved by experiment, I will yet
+not desist from relating the invention, since it offers itself, so to
+speak, of itself, and because it further confirms our Theory of
+refraction, by the agreement which here is found between the refracted
+ray and the reflected ray. Besides, it may occur that some one in the
+future will discover in it utilities which at present are not seen.
+
+[Illustration]
+
+To proceed then to these figures, let us suppose first that it is
+desired to find a surface CDE which shall reassemble at a point B rays
+coming from another point A; and that the summit of the surface shall
+be the given point D in the straight line AB. I say that, whether by
+reflexion or by refraction, it is only necessary to make this surface
+such that the path of the light from the point A to all points of the
+curved line CDE, and from these to the point of concurrence (as here
+the path along the straight lines AC, CB, along AL, LB, and along AD,
+DB), shall be everywhere traversed in equal times: by which principle
+the finding of these curves becomes very easy.
+
+[Illustration]
+
+So far as relates to the reflecting surface, since the sum of the
+lines AC, CB ought to be equal to that of AD, DB, it appears that DCE
+ought to be an ellipse; and for refraction, the ratio of the
+velocities of waves of light in the media A and B being supposed to be
+known, for example that of 3 to 2 (which is the same, as we have
+shown, as the ratio of the Sines in the refraction), it is only
+necessary to make DH equal to 3/2 of DB; and having after that
+described from the centre A some arc FC, cutting DB at F, then
+describe another from centre B with its semi-diameter BX equal to 2/3
+of FH; and the point of intersection of the two arcs will be one of
+the points required, through which the curve should pass. For this
+point, having been found in this fashion, it is easy forthwith to
+demonstrate that the time along AC, CB, will be equal to the time
+along AD, DB.
+
+For assuming that the line AD represents the time which the light
+takes to traverse this same distance AD in air, it is evident that DH,
+equal to 3/2 of DB, will represent the time of the light along DB in
+the medium, because it needs here more time in proportion as its speed
+is slower. Therefore the whole line AH will represent the time along
+AD, DB. Similarly the line AC or AF will represent the time along AC;
+and FH being by construction equal to 3/2 of CB, it will represent the
+time along CB in the medium; and in consequence the whole line AH will
+represent also the time along AC, CB. Whence it appears that the time
+along AC, CB, is equal to the time along AD, DB. And similarly it can
+be shown if L and K are other points in the curve CDE, that the times
+along AL, LB, and along AK, KB, are always represented by the line AH,
+and therefore equal to the said time along AD, DB.
+
+In order to show further that the surfaces, which these curves will
+generate by revolution, will direct all the rays which reach them from
+the point A in such wise that they tend towards B, let there be
+supposed a point K in the curve, farther from D than C is, but such
+that the straight line AK falls from outside upon the curve which
+serves for the refraction; and from the centre B let the arc KS be
+described, cutting BD at S, and the straight line CB at R; and from
+the centre A describe the arc DN meeting AK at N.
+
+Since the sums of the times along AK, KB, and along AC, CB are equal,
+if from the former sum one deducts the time along KB, and if from the
+other one deducts the time along RB, there will remain the time along
+AK as equal to the time along the two parts AC, CR. Consequently in
+the time that the light has come along AK it will also have come along
+AC and will in addition have made, in the medium from the centre C, a
+partial spherical wave, having a semi-diameter equal to CR. And this
+wave will necessarily touch the circumference KS at R, since CB cuts
+this circumference at right angles. Similarly, having taken any other
+point L in the curve, one can show that in the same time as the light
+passes along AL it will also have come along AL and in addition will
+have made a partial wave, from the centre L, which will touch the same
+circumference KS. And so with all other points of the curve CDE. Then
+at the moment that the light reaches K the arc KRS will be the
+termination of the movement, which has spread from A through DCK. And
+thus this same arc will constitute in the medium the propagation of
+the wave emanating from A; which wave may be represented by the arc
+DN, or by any other nearer the centre A. But all the pieces of the arc
+KRS are propagated successively along straight lines which are
+perpendicular to them, that is to say, which tend to the centre B (for
+that can be demonstrated in the same way as we have proved above that
+the pieces of spherical waves are propagated along the straight lines
+coming from their centre), and these progressions of the pieces of the
+waves constitute the rays themselves of light. It appears then that
+all these rays tend here towards the point B.
+
+One might also determine the point C, and all the others, in this
+curve which serves for the refraction, by dividing DA at G in such a
+way that DG is 2/3 of DA, and describing from the centre B any arc CX
+which cuts BD at N, and another from the centre A with its
+semi-diameter AF equal to 3/2 of GX; or rather, having described, as
+before, the arc CX, it is only necessary to make DF equal to 3/2 of
+DX, and from-the centre A to strike the arc FC; for these two
+constructions, as may be easily known, come back to the first one
+which was shown before. And it is manifest by the last method that
+this curve is the same that Mr. Des Cartes has given in his Geometry,
+and which he calls the first of his Ovals.
+
+It is only a part of this oval which serves for the refraction,
+namely, the part DK, ending at K, if AK is the tangent. As to the,
+other part, Des Cartes has remarked that it could serve for
+reflexions, if there were some material of a mirror of such a nature
+that by its means the force of the rays (or, as we should say, the
+velocity of the light, which he could not say, since he held that the
+movement of light was instantaneous) could be augmented in the
+proportion of 3 to 2. But we have shown that in our way of explaining
+reflexion, such a thing could not arise from the matter of the mirror,
+and it is entirely impossible.
+
+[Illustration]
+
+[Illustration]
+
+From what has been demonstrated about this oval, it will be easy to
+find the figure which serves to collect to a point incident parallel
+rays. For by supposing just the same construction, but the point A
+infinitely distant, giving parallel rays, our oval becomes a true
+Ellipse, the construction of which differs in no way from that of the
+oval, except that FC, which previously was an arc of a circle, is here
+a straight line, perpendicular to DB. For the wave of light DN, being
+likewise represented by a straight line, it will be seen that all the
+points of this wave, travelling as far as the surface KD along lines
+parallel to DB, will advance subsequently towards the point B, and
+will arrive there at the same time. As for the Ellipse which served
+for reflexion, it is evident that it will here become a parabola,
+since its focus A may be regarded as infinitely distant from the
+other, B, which is here the focus of the parabola, towards which all
+the reflexions of rays parallel to AB tend. And the demonstration of
+these effects is just the same as the preceding.
+
+But that this curved line CDE which serves for refraction is an
+Ellipse, and is such that its major diameter is to the distance
+between its foci as 3 to 2, which is the proportion of the refraction,
+can be easily found by the calculus of Algebra. For DB, which is
+given, being called _a_; its undetermined perpendicular DT being
+called _x_; and TC _y_; FB will be _a - y_; CB will be sqrt(_xx + aa
+-2ay + yy_). But the nature of the curve is such that 2/3 of TC
+together with CB is equal to DB, as was stated in the last
+construction: then the equation will be between _(2/3)y + sqrt(xx + aa
+- 2ay + yy)_ and _a_; which being reduced, gives _(6/5)ay - yy_ equal
+to _(9/5)xx_; that is to say that having made DO equal to 6/5 of DB,
+the rectangle DFO is equal to 9/5 of the square on FC. Whence it is
+seen that DC is an ellipse, of which the axis DO is to the parameter
+as 9 to 5; and therefore the square on DO is to the square of the
+distance between the foci as 9 to 9 - 5, that is to say 4; and finally
+the line DO will be to this distance as 3 to 2.
+
+[Illustration]
+
+Again, if one supposes the point B to be infinitely distant, in lieu
+of our first oval we shall find that CDE is a true Hyperbola; which
+will make those rays become parallel which come from the point A. And
+in consequence also those which are parallel within the transparent
+body will be collected outside at the point A. Now it must be remarked
+that CX and KS become straight lines perpendicular to BA, because they
+represent arcs of circles the centre of which is infinitely distant.
+And the intersection of the perpendicular CX with the arc FC will give
+the point C, one of those through which the curve ought to pass. And
+this operates so that all the parts of the wave of light DN, coming to
+meet the surface KDE, will advance thence along parallels to KS and
+will arrive at this straight line at the same time; of which the proof
+is again the same as that which served for the first oval. Besides one
+finds by a calculation as easy as the preceding one, that CDE is here
+a hyperbola of which the axis DO is 4/5 of AD, and the parameter
+equal to AD. Whence it is easily proved that DO is to the distance
+between the foci as 3 to 2.
+
+[Illustration]
+
+These are the two cases in which Conic sections serve for refraction,
+and are the same which are explained, in his _Dioptrique_, by Des
+Cartes, who first found out the use of these lines in relation to
+refraction, as also that of the Ovals the first of which we have
+already set forth. The second oval is that which serves for rays that
+tend to a given point; in which oval, if the apex of the surface which
+receives the rays is D, it will happen that the other apex will be
+situated between B and A, or beyond A, according as the ratio of AD to
+DB is given of greater or lesser value. And in this latter case it is
+the same as that which Des Cartes calls his 3rd oval.
+
+Now the finding and construction of this second oval is the same as
+that of the first, and the demonstration of its effect likewise. But
+it is worthy of remark that in one case this oval becomes a perfect
+circle, namely when the ratio of AD to DB is the same as the ratio of
+the refractions, here as 3 to 2, as I observed a long time ago. The
+4th oval, serving only for impossible reflexions, there is no need to
+set it forth.
+
+[Illustration]
+
+As for the manner in which Mr. Des Cartes discovered these lines,
+since he has given no explanation of it, nor any one else since that I
+know of, I will say here, in passing, what it seems to me it must have
+been. Let it be proposed to find the surface generated by the
+revolution of the curve KDE, which, receiving the incident rays coming
+to it from the point A, shall deviate them toward the point B. Then
+considering this other curve as already known, and that its apex D is
+in the straight line AB, let us divide it up into an infinitude of
+small pieces by the points G, C, F; and having drawn from each of
+these points, straight lines towards A to represent the incident rays,
+and other straight lines towards B, let there also be described with
+centre A the arcs GL, CM, FN, DO, cutting the rays that come from A at
+L, M, N, O; and from the points K, G, C, F, let there be described
+the arcs KQ, GR, CS, FT cutting the rays towards B at Q, R, S, T; and
+let us suppose that the straight line HKZ cuts the curve at K at
+right-angles.
+
+[Illustration]
+
+Then AK being an incident ray, and KB its refraction within the
+medium, it needs must be, according to the law of refraction which was
+known to Mr. Des Cartes, that the sine of the angle ZKA should be to
+the sine of the angle HKB as 3 to 2, supposing that this is the
+proportion of the refraction of glass; or rather, that the sine of the
+angle KGL should have this same ratio to the sine of the angle GKQ,
+considering KG, GL, KQ as straight lines because of their smallness.
+But these sines are the lines KL and GQ, if GK is taken as the radius
+of the circle. Then LK ought to be to GQ as 3 to 2; and in the same
+ratio MG to CR, NC to FS, OF to DT. Then also the sum of all the
+antecedents to all the consequents would be as 3 to 2. Now by
+prolonging the arc DO until it meets AK at X, KX is the sum of the
+antecedents. And by prolonging the arc KQ till it meets AD at Y, the
+sum of the consequents is DY. Then KX ought to be to DY as 3 to 2.
+Whence it would appear that the curve KDE was of such a nature that
+having drawn from some point which had been assumed, such as K, the
+straight lines KA, KB, the excess by which AK surpasses AD should be
+to the excess of DB over KB, as 3 to 2. For it can similarly be
+demonstrated, by taking any other point in the curve, such as G, that
+the excess of AG over AD, namely VG, is to the excess of BD over DG,
+namely DP, in this same ratio of 3 to 2. And following this principle
+Mr. Des Cartes constructed these curves in his _Geometric_; and he
+easily recognized that in the case of parallel rays, these curves
+became Hyperbolas and Ellipses.
+
+Let us now return to our method and let us see how it leads without
+difficulty to the finding of the curves which one side of the glass
+requires when the other side is of a given figure; a figure not only
+plane or spherical, or made by one of the conic sections (which is the
+restriction with which Des Cartes proposed this problem, leaving the
+solution to those who should come after him) but generally any figure
+whatever: that is to say, one made by the revolution of any given
+curved line to which one must merely know how to draw straight lines
+as tangents.
+
+Let the given figure be that made by the revolution of some curve such
+as AK about the axis AV, and that this side of the glass receives rays
+coming from the point L. Furthermore, let the thickness AB of the
+middle of the glass be given, and the point F at which one desires the
+rays to be all perfectly reunited, whatever be the first refraction
+occurring at the surface AK.
+
+I say that for this the sole requirement is that the outline BDK which
+constitutes the other surface shall be such that the path of the
+light from the point L to the surface AK, and from thence to the
+surface BDK, and from thence to the point F, shall be traversed
+everywhere in equal times, and in each case in a time equal to that
+which the light employs, to pass along the straight line LF of which
+the part AB is within the glass.
+
+[Illustration]
+
+Let LG be a ray falling on the arc AK. Its refraction GV will be given
+by means of the tangent which will be drawn at the point G. Now in GV
+the point D must be found such that FD together with 3/2 of DG and the
+straight line GL, may be equal to FB together with 3/2 of BA and the
+straight line AL; which, as is clear, make up a given length. Or
+rather, by deducting from each the length of LG, which is also given,
+it will merely be needful to adjust FD up to the straight line VG in
+such a way that FD together with 3/2 of DG is equal to a given
+straight line, which is a quite easy plane problem: and the point D
+will be one of those through which the curve BDK ought to pass. And
+similarly, having drawn another ray LM, and found its refraction MO,
+the point N will be found in this line, and so on as many times as one
+desires.
+
+To demonstrate the effect of the curve, let there be described about
+the centre L the circular arc AH, cutting LG at H; and about the
+centre F the arc BP; and in AB let AS be taken equal to 2/3 of HG; and
+SE equal to GD. Then considering AH as a wave of light emanating from
+the point L, it is certain that during the time in which its piece H
+arrives at G the piece A will have advanced within the transparent
+body only along AS; for I suppose, as above, the proportion of the
+refraction to be as 3 to 2. Now we know that the piece of wave which
+is incident on G, advances thence along the line GD, since GV is the
+refraction of the ray LG. Then during the time that this piece of wave
+has taken from G to D, the other piece which was at S has reached E,
+since GD, SE are equal. But while the latter will advance from E to B,
+the piece of wave which was at D will have spread into the air its
+partial wave, the semi-diameter of which, DC (supposing this wave to
+cut the line DF at C), will be 3/2 of EB, since the velocity of light
+outside the medium is to that inside as 3 to 2. Now it is easy to show
+that this wave will touch the arc BP at this point C. For since, by
+construction, FD + 3/2 DG + GL are equal to FB + 3/2 BA + AL; on
+deducting the equals LH, LA, there will remain FD + 3/2 DG + GH equal
+to FB + 3/2 BA. And, again, deducting from one side GH, and from the
+other side 3/2 of AS, which are equal, there will remain FD with 3/2
+DG equal to FB with 3/2 of BS. But 3/2 of DG are equal to 3/2 of ES;
+then FD is equal to FB with 3/2 of BE. But DC was equal to 3/2 of EB;
+then deducting these equal lengths from one side and from the other,
+there will remain CF equal to FB. And thus it appears that the wave,
+the semi-diameter of which is DC, touches the arc BP at the moment
+when the light coming from the point L has arrived at B along the line
+LB. It can be demonstrated similarly that at this same moment the
+light that has come along any other ray, such as LM, MN, will have
+propagated the movement which is terminated at the arc BP. Whence it
+follows, as has been often said, that the propagation of the wave AH,
+after it has passed through the thickness of the glass, will be the
+spherical wave BP, all the pieces of which ought to advance along
+straight lines, which are the rays of light, to the centre F. Which
+was to be proved. Similarly these curved lines can be found in all the
+cases which can be proposed, as will be sufficiently shown by one or
+two examples which I will add.
+
+Let there be given the surface of the glass AK, made by the revolution
+about the axis BA of the line AK, which may be straight or curved. Let
+there be also given in the axis the point L and the thickness BA of
+the glass; and let it be required to find the other surface KDB, which
+receiving rays that are parallel to AB will direct them in such wise
+that after being again refracted at the given surface AK they will all
+be reassembled at the point L.
+
+[Illustration]
+
+From the point L let there be drawn to some point of the given line
+AK the straight line LG, which, being considered as a ray of light,
+its refraction GD will then be found. And this line being then
+prolonged at one side or the other will meet the straight line BL, as
+here at V. Let there then be erected on AB the perpendicular BC, which
+will represent a wave of light coming from the infinitely distant
+point F, since we have supposed the rays to be parallel. Then all the
+parts of this wave BC must arrive at the same time at the point L; or
+rather all the parts of a wave emanating from the point L must arrive
+at the same time at the straight line BC. And for that, it is
+necessary to find in the line VGD the point D such that having drawn
+DC parallel to AB, the sum of CD, plus 3/2 of DG, plus GL may be equal
+to 3/2 of AB, plus AL: or rather, on deducting from both sides GL,
+which is given, CD plus 3/2 of DG must be equal to a given length;
+which is a still easier problem than the preceding construction. The
+point D thus found will be one of those through which the curve ought
+to pass; and the proof will be the same as before. And by this it will
+be proved that the waves which come from the point L, after having
+passed through the glass KAKB, will take the form of straight lines,
+as BC; which is the same thing as saying that the rays will become
+parallel. Whence it follows reciprocally that parallel rays falling on
+the surface KDB will be reassembled at the point L.
+
+[Illustration]
+
+Again, let there be given the surface AK, of any desired form,
+generated by revolution about the axis AB, and let the thickness of
+the glass at the middle be AB. Also let the point L be given in the
+axis behind the glass; and let it be supposed that the rays which fall
+on the surface AK tend to this point, and that it is required to find
+the surface BD, which on their emergence from the glass turns them as
+if they came from the point F in front of the glass.
+
+Having taken any point G in the line AK, and drawing the straight line
+IGL, its part GI will represent one of the incident rays, the
+refraction of which, GV, will then be found: and it is in this line
+that we must find the point D, one of those through which the curve DG
+ought to pass. Let us suppose that it has been found: and about L as
+centre let there be described GT, the arc of a circle cutting the
+straight line AB at T, in case the distance LG is greater than LA; for
+otherwise the arc AH must be described about the same centre, cutting
+the straight line LG at H. This arc GT (or AH, in the other case) will
+represent an incident wave of light, the rays of which tend towards
+L. Similarly, about the centre F let there be described the circular
+arc DQ, which will represent a wave emanating from the point F.
+
+Then the wave TG, after having passed through the glass, must form the
+wave QD; and for this I observe that the time taken by the light along
+GD in the glass must be equal to that taken along the three, TA, AB,
+and BQ, of which AB alone is within the glass. Or rather, having taken
+AS equal to 2/3 of AT, I observe that 3/2 of GD ought to be equal to
+3/2 of SB, plus BQ; and, deducting both of them from FD or FQ, that FD
+less 3/2 of GD ought to be equal to FB less 3/2 of SB. And this last
+difference is a given length: and all that is required is to draw the
+straight line FD from the given point F to meet VG so that it may be
+thus. Which is a problem quite similar to that which served for the
+first of these constructions, where FD plus 3/2 of GD had to be equal
+to a given length.
+
+In the demonstration it is to be observed that, since the arc BC falls
+within the glass, there must be conceived an arc RX, concentric with
+it and on the other side of QD. Then after it shall have been shown
+that the piece G of the wave GT arrives at D at the same time that the
+piece T arrives at Q, which is easily deduced from the construction,
+it will be evident as a consequence that the partial wave generated at
+the point D will touch the arc RX at the moment when the piece Q shall
+have come to R, and that thus this arc will at the same moment be the
+termination of the movement that comes from the wave TG; whence all
+the rest may be concluded.
+
+Having shown the method of finding these curved lines which serve for
+the perfect concurrence of the rays, there remains to be explained a
+notable thing touching the uncoordinated refraction of spherical,
+plane, and other surfaces: an effect which if ignored might cause some
+doubt concerning what we have several times said, that rays of light
+are straight lines which intersect at right angles the waves which
+travel along them.
+
+[Illustration]
+
+For in the case of rays which, for example, fall parallel upon a
+spherical surface AFE, intersecting one another, after refraction, at
+different points, as this figure represents; what can the waves of
+light be, in this transparent body, which are cut at right angles by
+the converging rays? For they can not be spherical. And what will
+these waves become after the said rays begin to intersect one another?
+It will be seen in the solution of this difficulty that something very
+remarkable comes to pass herein, and that the waves do not cease to
+persist though they do not continue entire, as when they cross the
+glasses designed according to the construction we have seen.
+
+According to what has been shown above, the straight line AD, which
+has been drawn at the summit of the sphere, at right angles to the
+axis parallel to which the rays come, represents the wave of light;
+and in the time taken by its piece D to reach the spherical surface
+AGE at E, its other parts will have met the same surface at F, G, H,
+etc., and will have also formed spherical partial waves of which these
+points are the centres. And the surface EK which all those waves will
+touch, will be the continuation of the wave AD in the sphere at the
+moment when the piece D has reached E. Now the line EK is not an arc
+of a circle, but is a curved line formed as the evolute of another
+curve ENC, which touches all the rays HL, GM, FO, etc., that are the
+refractions of the parallel rays, if we imagine laid over the
+convexity ENC a thread which in unwinding describes at its end E the
+said curve EK. For, supposing that this curve has been thus described,
+we will show that the said waves formed from the centres F, G, H,
+etc., will all touch it.
+
+It is certain that the curve EK and all the others described by the
+evolution of the curve ENC, with different lengths of thread, will cut
+all the rays HL, GM, FO, etc., at right angles, and in such wise that
+the parts of them intercepted between two such curves will all be
+equal; for this follows from what has been demonstrated in our
+treatise _de Motu Pendulorum_. Now imagining the incident rays as
+being infinitely near to one another, if we consider two of them, as
+RG, TF, and draw GQ perpendicular to RG, and if we suppose the curve
+FS which intersects GM at P to have been described by evolution from
+the curve NC, beginning at F, as far as which the thread is supposed
+to extend, we may assume the small piece FP as a straight line
+perpendicular to the ray GM, and similarly the arc GF as a straight
+line. But GM being the refraction of the ray RG, and FP being
+perpendicular to it, QF must be to GP as 3 to 2, that is to say in the
+proportion of the refraction; as was shown above in explaining the
+discovery of Des Cartes. And the same thing occurs in all the small
+arcs GH, HA, etc., namely that in the quadrilaterals which enclose
+them the side parallel to the axis is to the opposite side as 3 to 2.
+Then also as 3 to 2 will the sum of the one set be to the sum of the
+other; that is to say, TF to AS, and DE to AK, and BE to SK or DV,
+supposing V to be the intersection of the curve EK and the ray FO.
+But, making FB perpendicular to DE, the ratio of 3 to 2 is also that
+of BE to the semi-diameter of the spherical wave which emanated from
+the point F while the light outside the transparent body traversed the
+space BE. Then it appears that this wave will intersect the ray FM at
+the same point V where it is intersected at right angles by the curve
+EK, and consequently that the wave will touch this curve. In the same
+way it can be proved that the same will apply to all the other waves
+above mentioned, originating at the points G, H, etc.; to wit, that
+they will touch the curve EK at the moment when the piece D of the
+wave ED shall have reached E.
+
+Now to say what these waves become after the rays have begun to cross
+one another: it is that from thence they fold back and are composed of
+two contiguous parts, one being a curve formed as evolute of the curve
+ENC in one sense, and the other as evolute of the same curve in the
+opposite sense. Thus the wave KE, while advancing toward the meeting
+place becomes _abc_, whereof the part _ab_ is made by the evolute
+_b_C, a portion of the curve ENC, while the end C remains attached;
+and the part _bc_ by the evolute of the portion _b_E while the end E
+remains attached. Consequently the same wave becomes _def_, then
+_ghk_, and finally CY, from whence it subsequently spreads without any
+fold, but always along curved lines which are evolutes of the curve
+ENC, increased by some straight line at the end C.
+
+There is even, in this curve, a part EN which is straight, N being the
+point where the perpendicular from the centre X of the sphere falls
+upon the refraction of the ray DE, which I now suppose to touch the
+sphere. The folding of the waves of light begins from the point N up
+to the end of the curve C, which point is formed by taking AC to CX in
+the proportion of the refraction, as here 3 to 2.
+
+As many other points as may be desired in the curve NC are found by a
+Theorem which Mr. Barrow has demonstrated in section 12 of his
+_Lectiones Opticae_, though for another purpose. And it is to be noted
+that a straight line equal in length to this curve can be given. For
+since it together with the line NE is equal to the line CK, which is
+known, since DE is to AK in the proportion of the refraction, it
+appears that by deducting EN from CK the remainder will be equal to
+the curve NC.
+
+Similarly the waves that are folded back in reflexion by a concave
+spherical mirror can be found. Let ABC be the section, through the
+axis, of a hollow hemisphere, the centre of which is D, its axis being
+DB, parallel to which I suppose the rays of light to come. All the
+reflexions of those rays which fall upon the quarter-circle AB will
+touch a curved line AFE, of which line the end E is at the focus of
+the hemisphere, that is to say, at the point which divides the
+semi-diameter BD into two equal parts. The points through which this
+curve ought to pass are found by taking, beyond A, some arc AO, and
+making the arc OP double the length of it; then dividing the chord OP
+at F in such wise that the part FP is three times the part FO; for
+then F is one of the required points.
+
+[Illustration]
+
+And as the parallel rays are merely perpendiculars to the waves which
+fall on the concave surface, which waves are parallel to AD, it will
+be found that as they come successively to encounter the surface AB,
+they form on reflexion folded waves composed of two curves which
+originate from two opposite evolutions of the parts of the curve AFE.
+So, taking AD as an incident wave, when the part AG shall have met the
+surface AI, that is to say when the piece G shall have reached I, it
+will be the curves HF, FI, generated as evolutes of the curves FA, FE,
+both beginning at F, which together constitute the propagation of the
+part AG. And a little afterwards, when the part AK has met the surface
+AM, the piece K having come to M, then the curves LN, NM, will
+together constitute the propagation of that part. And thus this folded
+wave will continue to advance until the point N has reached the focus
+E. The curve AFE can be seen in smoke, or in flying dust, when a
+concave mirror is held opposite the sun. And it should be known that
+it is none other than that curve which is described by the point E on
+the circumference of the circle EB, when that circle is made to roll
+within another whose semi-diameter is ED and whose centre is D. So
+that it is a kind of Cycloid, of which, however, the points can be
+found geometrically.
+
+Its length is exactly equal to 3/4 of the diameter of the sphere, as
+can be found and demonstrated by means of these waves, nearly in the
+same way as the mensuration of the preceding curve; though it may also
+be demonstrated in other ways, which I omit as outside the subject.
+The area AOBEFA, comprised between the arc of the quarter-circle, the
+straight line BE, and the curve EFA, is equal to the fourth part of
+the quadrant DAB.
+
+
+
+
+
+INDEX
+
+Archimedes, 104.
+
+Atmospheric refraction, 45.
+
+Barrow, Isaac, 126.
+
+Bartholinus, Erasmus, 53, 54, 57, 60, 97, 99.
+
+Boyle, Hon. Robert, 11.
+
+Cassini, Jacques, iii.
+
+Caustic Curves, 123.
+
+Crystals, see Iceland Crystal, Rock Crystal.
+
+Crystals, configuration of, 95.
+
+Descartes, Rénê, 3, 5, 7, 14, 22, 42, 43, 109, 113.
+
+Double Refraction, discovery of, 54, 81, 93.
+
+Elasticity, 12, 14.
+
+Ether, the, or Ethereal matter, 11, 14, 16, 28.
+
+Extraordinary refraction, 55, 56.
+
+Fermat, principle of, 42.
+
+Figures of transparent bodies, 105.
+
+Hooke, Robert, 20.
+
+Iceland Crystal, 2, 52 sqq.
+
+Iceland Crystal, Cutting and Polishing of, 91, 92, 98.
+
+Leibnitz, G.W., vi.
+
+Light, nature of, 3.
+
+Light, velocity of, 4, 15.
+
+Molecular texture of bodies, 27, 95.
+
+Newton, Sir Isaac, vi, 106.
+
+Opacity, 34.
+
+Ovals, Cartesian, 107, 113.
+
+Pardies, Rev. Father, 20.
+
+Rays, definition of, 38, 49.
+
+Reflexion, 22.
+
+Refraction, 28, 34.
+
+Rock Crystal, 54, 57, 62, 95.
+
+Römer, Olaf, v, 7.
+
+Roughness of surfaces, 27.
+
+Sines, law of, 1, 35, 38, 43.
+
+Spheres, elasticity of, 15.
+
+Spheroidal waves in crystals, 63.
+
+Spheroids, lemma about, 103.
+
+Sound, speed of, 7, 10, 12.
+
+Telescopes, lenses for, 62, 105.
+
+Torricelli's experiment, 12, 30.
+
+Transparency, explanation of, 28, 31, 32.
+
+Waves, no regular succession of, 17.
+
+Waves, principle of wave envelopes, 19, 24.
+
+Waves, principle of elementary wave fronts, 19.
+
+Waves, propagation of light as, 16, 63.
+
+
+
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+<title>The Project Gutenberg eBook of Treatise on Light, by Christiaan Huygens</title>
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+<h1>The Project Gutenberg eBook, Treatise on Light, by Christiaan Huygens,
+Translated by Silvanus P. Thompson</h1>
+<pre>
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at <a href = "https://www.gutenberg.org">www.gutenberg.org</a></pre>
+<p>Title: Treatise on Light</p>
+<p>Author: Christiaan Huygens</p>
+<p>Release Date: January 18, 2005  [eBook #14725]</p>
+<p>Language: English</p>
+<p>Character set encoding: ISO-8859-1</p>
+<p>***START OF THE PROJECT GUTENBERG EBOOK TREATISE ON LIGHT***</p>
+<p>&nbsp;</p>
+<h3>E-text prepared by Clare Boothby, Stephen Schulze,<br />
+    and the Project Gutenberg Online Distributed Proofreading Team</h3>
+<p>&nbsp;</p>
+<hr class="full" />
+<p>&nbsp;</p>
+
+<h1><a name="Page_iii" id="Page_iii" /><b>TREATISE ON LIGHT</b></h1>
+
+
+<p class="center">In which are explained<br />
+The causes of that which occurs<br />
+<b>In REFLEXION, &amp; in REFRACTION</b></p>
+
+<p class="center">And particularly<br />
+<b>In the strange REFRACTION</b><br />
+<b>OF ICELAND CRYSTAL</b></p>
+
+
+<h3>By</h3>
+
+<h2><b>CHRISTIAAN HUYGENS</b></h2>
+
+
+<p class="center">Rendered into English</p>
+
+<p class="center">By</p>
+
+<p class="center"><b>SILVANUS P. THOMPSON</b></p>
+
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+<h6>University of Chicago Press</h6>
+<p><a name="Page_iv" id="Page_iv" /></p>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+
+
+<div class="pagenum">[Pg v]<a name="Page_v" id="Page_v" /></div>
+<div class="figcenter" style="width: 600px;">
+<img src="images/prefhead.png" width="600" height="150" alt="" title="" />
+</div>
+<h2>PREFACE</h2>
+
+
+<div style="width: 147px; float: left; margin-right: .2em;">
+<img src="images/pref.png" width="147" height="150" alt="I" title="I" />
+</div><p> wrote this Treatise during my sojourn in France twelve years ago,
+and I communicated it in the year 1678 to the learned persons who then
+composed the Royal Academy of Science, to the membership of which the
+King had done me the honour of calling, me. Several of that body who
+are still alive will remember having been present when I read it, and
+above the rest those amongst them who applied themselves particularly
+to the study of Mathematics; of whom I cannot cite more than the
+celebrated gentlemen Cassini, R&ouml;mer, and De la Hire. And, although I
+have since corrected and changed some parts, the copies which I had
+made of it at that time may serve for proof that I have yet added
+nothing to it save some conjectures touching the formation of Iceland
+Crystal, and a novel observation on the refraction of Rock Crystal. I
+have desired to relate these particulars to make known how long I have
+meditated the things which now I publish, and not for the purpose of
+detracting from the merit of those who, without having seen anything
+that I have written, may be found to have treated <span class="pagenum">[Pg vi]</span><a name="Page_vi" id="Page_vi" />of like matters: as
+has in fact occurred to two eminent Geometricians, Messieurs Newton
+and Leibnitz, with respect to the Problem of the figure of glasses for
+collecting rays when one of the surfaces is given.</p>
+
+<p>One may ask why I have so long delayed to bring this work to the
+light. The reason is that I wrote it rather carelessly in the Language
+in which it appears, with the intention of translating it into Latin,
+so doing in order to obtain greater attention to the thing. After
+which I proposed to myself to give it out along with another Treatise
+on Dioptrics, in which I explain the effects of Telescopes and those
+things which belong more to that Science. But the pleasure of novelty
+being past, I have put off from time to time the execution of this
+design, and I know not when I shall ever come to an end if it, being
+often turned aside either by business or by some new study.
+Considering which I have finally judged that it was better worth while
+to publish this writing, such as it is, than to let it run the risk,
+by waiting longer, of remaining lost.</p>
+
+<p>There will be seen in it demonstrations of those kinds which do not
+produce as great a certitude as those of Geometry, and which even
+differ much therefrom, since whereas the Geometers prove their
+Propositions by fixed and incontestable Principles, here the
+Principles are verified by the conclusions to be drawn from them; the
+nature of these things not allowing of this being done otherwise.</p>
+
+<p>It is always possible to attain thereby to a degree of probability
+which very often is scarcely less than complete proof. To wit, when
+things which have been demonstrated by the Principles that have been
+assumed correspond perfectly to the phenomena which experiment has
+brought under observation; especially when there are a great number of
+<span class="pagenum">[Pg vii]</span><a name="Page_vii" id="Page_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 fact corresponds to our prevision.
+But if all these proofs of probability are met with in that which I
+propose to discuss, as it seems to me they are, this ought to be a
+very strong confirmation of the success of my inquiry; and it must be
+ill if the facts are not pretty much as I represent them. I would
+believe then that those who love to know the Causes of things and who
+are able to admire the marvels of Light, will find some satisfaction
+in these various speculations regarding it, and in the new explanation
+of its famous property which is the main foundation of the
+construction of our eyes and of those great inventions which extend so
+vastly the use of them.</p>
+
+<p>I hope also that there will be some who by following these beginnings
+will penetrate much further into this question than I have been able
+to do, since the subject must be far from being exhausted. This
+appears from the passages which I have indicated where I leave certain
+difficulties without having resolved them, and still more from matters
+which I have not touched at all, such as Luminous Bodies of several
+sorts, and all that concerns Colours; in which no one until now can
+boast of having succeeded. Finally, there remains much more to be
+investigated touching the nature of Light which I do not pretend to
+have disclosed, and I shall owe much in return to him who shall be
+able to supplement that which is here lacking to me in knowledge. The
+Hague. The 8 January 1690.<span class="pagenum">[Pg viii]</span><a name="Page_viii" id="Page_viii" /></p>
+
+
+<div class="pagenum">[Pg ix]<a name="Page_ix" id="Page_ix" /></div>
+<div class="figcenter" style="width: 600px;">
+<img src="images/tranhead.png" width="600" height="151" alt="." title="" />
+</div>
+<h2><a name="NOTE_BY_THE_TRANSLATOR" id="NOTE_BY_THE_TRANSLATOR" />NOTE BY THE TRANSLATOR</h2>
+
+
+<div style="width: 150px; float: left; margin-right: .2em;">
+<img src="images/trans.png" width="150" height="150" alt="C" title="C" />
+</div><p>onsidering the great influence which this Treatise has exercised in
+the development of the Science of Optics, it seems strange that two
+centuries should have passed before an English edition of the work
+appeared. Perhaps the circumstance is due to the mistaken zeal with
+which formerly everything that conflicted with the cherished ideas of
+Newton was denounced by his followers. The Treatise on Light of
+Huygens has, however, withstood the test of time: and even now the
+exquisite skill with which he applied his conception of the
+propagation of waves of light to unravel the intricacies of the
+phenomena of the double refraction of crystals, and of the refraction
+of the atmosphere, will excite the admiration of the student of
+Optics. It is true that his wave theory was far from the complete
+doctrine as subsequently developed by Thomas Young and Augustin
+Fresnel, and belonged rather to geometrical than to physical Optics.
+If Huygens had no conception of transverse vibrations, of the
+principle of interference, or of the existence of the ordered sequence
+of waves in trains, he nevertheless attained to a remarkably clear
+understanding of the prin<span class="pagenum">[Pg x]</span><a name="Page_x" id="Page_x" />ciples of wave-propagation; and his
+exposition of the subject marks an epoch in the treatment of Optical
+problems. It has been needful in preparing this translation to
+exercise care lest one should import into the author's text ideas of
+subsequent date, by using words that have come to imply modern
+conceptions. Hence the adoption of as literal a rendering as possible.
+A few of the author's terms need explanation. He uses the word
+&quot;refraction,&quot; for example, both for the phenomenon or process usually
+so denoted, and for the result of that process: thus the refracted ray
+he habitually terms &quot;the refraction&quot; of the incident ray. When a
+wave-front, or, as he terms it, a &quot;wave,&quot; has passed from some initial
+position to a subsequent one, he terms the wave-front in its
+subsequent position &quot;the continuation&quot; of the wave. He also speaks of
+the envelope of a set of elementary waves, formed by coalescence of
+those elementary wave-fronts, as &quot;the termination&quot; of the wave; and
+the elementary wave-fronts he terms &quot;particular&quot; waves. Owing to the
+circumstance that the French word <i>rayon</i> possesses the double
+signification of ray of light and radius of a circle, he avoids its
+use in the latter sense and speaks always of the semi-diameter, not of
+the radius. His speculations as to the ether, his suggestive views of
+the structure of crystalline bodies, and his explanation of opacity,
+slight as they are, will possibly surprise the reader by their seeming
+modernness. And none can read his investigation of the phenomena found
+in Iceland spar without marvelling at his insight and sagacity.</p>
+
+<div style="margin-left: 80%;"><p>S.P.T.</p>
+
+<p><i>June</i>, 1912.</p></div>
+
+
+
+<hr style="width: 65%;" />
+<div class="pagenum">[Pg xi]<a name="Page_xi" id="Page_xi" /></div>
+
+<h2><a name="TABLE_OF_MATTERS" id="TABLE_OF_MATTERS" />TABLE OF MATTERS</h2>
+
+<h3><i>Contained in this Treatise</i></h3>
+
+
+<table border="0" cellpadding="4" cellspacing="0" summary="" width="500">
+<tr><td><a href="#CHAPTER_I"><b>CHAP. I. On Rays Propagated in Straight Lines.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>That Light is produced by a certain movement.</i></td><td align='left'><a href="#Page_3">p.&nbsp;3</a></td></tr>
+<tr><td align='left'><i>That no substance passes from the luminous object to the eyes.</i></td><td align='left'><a href="#Page_3">p.&nbsp;3</a></td></tr>
+<tr><td align='left'><i>That Light spreads spherically, almost as Sound does.</i></td><td align='left'><a href="#Page_4">p.&nbsp;4</a></td></tr>
+<tr><td align='left'><i>Whether Light takes time to spread.</i></td><td align='left'><a href="#Page_4">p.&nbsp;4</a></td></tr>
+<tr><td align='left'><i>Experience seeming to prove that it passes instantaneously.</i></td><td align='left'><a href="#Page_5">p.&nbsp;5</a></td></tr>
+<tr><td align='left'><i>Experience proving that it takes time.</i></td><td align='left'><a href="#Page_8">p.&nbsp;8</a></td></tr>
+<tr><td align='left'><i>How much its speed is greater than that of Sound.</i></td><td align='left'><a href="#Page_10">p.&nbsp;10</a></td></tr>
+<tr><td align='left'><i>In what the emission of Light differs from that of Sound.</i></td><td align='left'><a href="#Page_10">p.&nbsp;10</a></td></tr>
+<tr><td align='left'><i>That it is not the same medium which serves for Light and Sound.</i></td><td align='left'><a href="#Page_11">p.&nbsp;11</a></td></tr>
+<tr><td align='left'><i>How Sound is propagated.</i></td><td align='left'><a href="#Page_12">p.&nbsp;12</a></td></tr>
+<tr><td align='left'><i>How Light is propagated.</i></td><td align='left'><a href="#Page_14">p.&nbsp;14</a></td></tr>
+<tr><td align='left'><i>Detailed Remarks on the propagation of Light.</i></td><td align='left'><a href="#Page_15">p.&nbsp;15</a></td></tr>
+<tr><td align='left'><i>Why Rays are propagated only in straight lines.</i></td><td align='left'><a href="#Page_20">p.&nbsp;20</a></td></tr>
+<tr><td align='left'><i>How Light coming in different directions can cross itself.</i></td><td align='left'><a href="#Page_22">p.&nbsp;22</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_II"><b>CHAP. II. On Reflexion.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>Demonstration of equality of angles of incidence and reflexion.</i></td><td align='left'><a href="#Page_23">p.&nbsp;23</a> </td></tr>
+<tr><td align='left'><i>Why the incident and reflected rays are in the same plane perpendicular to the reflecting surface.</i></td><td align='left'><a href="#Page_25">p.&nbsp;25</a></td></tr>
+<tr><td align='left'><i>That it is not needful for the reflecting surface to be perfectly flat to attain equality of the angles of incidence and reflexion.</i></td><td align='left'><a href="#Page_27">p.&nbsp;27</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_III"><b>CHAP. III. On Refraction.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>That bodies may be transparent without any substance passing through them.</i></td><td align='left'><a href="#Page_29">p.&nbsp;29</a></td></tr>
+<tr><td align='left'><i>Proof that the ethereal matter passes through transparent bodies.</i></td><td align='left'><a href="#Page_30">p.&nbsp;30</a></td></tr>
+<tr><td align='left'><i>How this matter passing through can render them transparent.</i></td><td align='left'><a href="#Page_31">p.&nbsp;31</a></td></tr>
+<tr><td align='left'><i>That the most solid bodies in appearance are of a very loose texture.</i></td><td align='left'><a href="#Page_31">p.&nbsp;31</a></td></tr>
+<tr><td align='left'><i>That Light spreads more slowly in water and in glass than in air.</i></td><td align='left'><a href="#Page_32">p.&nbsp;32</a></td></tr>
+<tr><td align='left'><i>Third hypothesis to explain transparency, and the retardation which Light suffers.</i></td><td align='left'><a href="#Page_32">p.&nbsp;32</a></td></tr>
+<tr><td align='left'><i>On that which makes bodies opaque.</i></td><td align='left'><a href="#Page_34">p.&nbsp;34</a></td></tr>
+<tr><td align='left'><i>Demonstration why Refraction obeys the known proportion of Sines.</i></td><td align='left'><a href="#Page_35">p.&nbsp;35</a></td></tr>
+<tr><td align='left'><i>Why the incident and refracted Rays produce one another reciprocally.</i></td><td align='left'><a href="#Page_39">p.&nbsp;39</a></td></tr>
+<tr><td align='left'><i>Why Reflexion within a triangular glass prism is suddenly augmented when the Light can no longer penetrate.</i></td><td align='left'><a href="#Page_40">p.&nbsp;40</a></td></tr>
+<tr><td align='left'><i>That bodies which cause greater Refraction also cause stronger Reflexion.</i></td><td align='left'><a href="#Page_42">p.&nbsp;42</a></td></tr>
+<tr><td align='left'><i>Demonstration of the Theorem of Mr. Fermat.</i></td><td align='left'><a href="#Page_43">p.&nbsp;43</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_IV"><b>CHAP. IV. On the Refraction of the Air.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>That the emanations of Light in the air are not spherical.</i></td><td align='left'><a href="#Page_45">p.&nbsp;45</a></td></tr>
+<tr><td align='left'><i>How consequently some objects appear higher than they are.</i></td><td align='left'><a href="#Page_47">p.&nbsp;47</a></td></tr>
+<tr><td align='left'><i>How the Sun may appear on the Horizon before he has risen.</i></td><td align='left'><a href="#Page_49">p.&nbsp;49</a></td></tr>
+<tr><td align='left'><i>That the rays of light become curved in the Air of the Atmosphere, and what effects this produces.</i></td><td align='left'><a href="#Page_50">p.&nbsp;50</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_V"><b>CHAP. V. On the Strange Refraction of Iceland Crystal.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>That this Crystal grows also in other countries.</i></td><td align='left'><a href="#Page_52">p.&nbsp;52</a></td></tr>
+<tr><td align='left'><i>Who first-wrote about it.</i></td><td align='left'><a href="#Page_53">p.&nbsp;53</a></td></tr>
+<tr><td align='left'><i>Description of Iceland Crystal; its substance, shape, and properties.</i></td><td align='left'><a href="#Page_53">p.&nbsp;53</a></td></tr>
+<tr><td align='left'><i>That it has two different Refractions.</i></td><td align='left'><a href="#Page_54">p.&nbsp;54</a></td></tr>
+<tr><td align='left'><i>That the ray perpendicular to the surface suffers refraction, and that some rays inclined to the surface pass without suffering refraction.</i></td><td align='left'><a href="#Page_55">p.&nbsp;55</a></td></tr>
+<tr><td align='left'><i>Observation of the refractions in this Crystal.</i></td><td align='left'><a href="#Page_56">p.&nbsp;56</a></td></tr>
+<tr><td align='left'><i>That there is a Regular and an Irregular Refraction.</i></td><td align='left'><a href="#Page_57">p.&nbsp;57</a></td></tr>
+<tr><td align='left'><i>The way of measuring the two Refractions of Iceland Crystal.</i></td><td align='left'><a href="#Page_57">p.&nbsp;57</a></td></tr>
+<tr><td align='left'><i>Remarkable properties of the Irregular Refraction.</i></td><td align='left'><a href="#Page_60">p.&nbsp;60</a></td></tr>
+<tr><td align='left'><i>Hypothesis to explain the double Refraction.</i></td><td align='left'><a href="#Page_61">p.&nbsp;61</a></td></tr>
+<tr><td align='left'><i>That Rock Crystal has also a double Refraction.</i></td><td align='left'><a href="#Page_62">p.&nbsp;62</a></td></tr>
+<tr><td align='left'><i>Hypothesis of emanations of Light, within Iceland Crystal, of spheroidal form, for the Irregular Refraction.</i></td><td align='left'><a href="#Page_63">p.&nbsp;63</a></td></tr>
+<tr><td align='left'><i>How a perpendicular ray can suffer Refraction.</i></td><td align='left'><a href="#Page_64">p.&nbsp;64</a></td></tr>
+<tr><td align='left'><i>How the position and form of the spheroidal emanations in this Crystal can be defined.</i></td><td align='left'><a href="#Page_65">p.&nbsp;65</a></td></tr>
+<tr><td align='left'><i>Explanation of the Irregular Refraction by these spheroidal emanations.</i></td><td align='left'><a href="#Page_67">p.&nbsp;67</a></td></tr>
+<tr><td align='left'><i>Easy way to find the Irregular Refraction of each incident ray.</i></td><td align='left'><a href="#Page_70">p.&nbsp;70</a></td></tr>
+<tr><td align='left'><i>Demonstration of the oblique ray which traverses the Crystal without being refracted.</i></td><td align='left'><a href="#Page_73">p.&nbsp;73</a></td></tr>
+<tr><td align='left'><i>Other irregularities of Refraction explained.</i></td><td align='left'><a href="#Page_76">p.&nbsp;76</a></td></tr>
+<tr><td align='left'><i>That an object placed beneath the Crystal appears double, in two images of different heights.</i></td><td align='left'><a href="#Page_81">p.&nbsp;81</a></td></tr>
+<tr><td align='left'><i>Why the apparent heights of one of the images change on changing the position of the eyes above the Crystal.</i></td><td align='left'><a href="#Page_85">p.&nbsp;85</a></td></tr>
+<tr><td align='left'><i>Of the different sections of this Crystal which produce yet other refractions, and confirm all this Theory.</i></td><td align='left'><a href="#Page_88">p.&nbsp;88</a></td></tr>
+<tr><td align='left'><i>Particular way of polishing the surfaces after it has been cut.</i></td><td align='left'><a href="#Page_91">p.&nbsp;91</a></td></tr>
+<tr><td align='left'><i>Surprising phenomenon touching the rays which pass through two separated pieces; the cause of which is not explained.</i></td><td align='left'><a href="#Page_92">p.&nbsp;92</a></td></tr>
+<tr><td align='left'><i>Probable conjecture on the internal composition of Iceland Crystal, and of what figure its particles are.</i></td><td align='left'><a href="#Page_95">p.&nbsp;95</a></td></tr>
+<tr><td align='left'><i>Tests to confirm this conjecture.</i></td><td align='left'><a href="#Page_97">p.&nbsp;97</a></td></tr>
+<tr><td align='left'><i>Calculations which have been supposed in this Chapter.</i></td><td align='left'><a href="#Page_99">p.&nbsp;99</a></td></tr>
+</table>
+</td></tr>
+<tr><td>
+<a href="#CHAPTER_VI"><b>CHAP. VI. On the Figures of transparent bodies which serve for Refraction and for Reflexion.</b></a>
+</td></tr>
+<tr><td>
+<table border="0" cellpadding="2" cellspacing="0" summary="" width="450">
+<tr><td align='left'><i>General and easy rule to find these Figures.</i></td><td align='left'><a href="#Page_106">p.&nbsp;106</a></td></tr>
+<tr><td align='left'><i>Invention of the Ovals of Mr. Des Cartes for Dioptrics.</i></td><td align='left'><a href="#Page_109">p.&nbsp;109</a></td></tr>
+<tr><td align='left'><i>How he was able to find these Lines.</i></td><td align='left'><a href="#Page_114">p.&nbsp;114</a></td></tr>
+<tr><td align='left'><i>Way of finding the surface of a glass for perfect refraction, when the other surface is given.</i></td><td align='left'><a href="#Page_116">p.&nbsp;116</a></td></tr>
+<tr><td align='left'><i>Remark on what happens to rays refracted at a spherical surface.</i></td><td align='left'><a href="#Page_123">p.&nbsp;123</a></td></tr>
+<tr><td align='left'><i>Remark on the curved line which is formed by reflexion in a spherical concave mirror.</i></td><td align='left'><a href="#Page_126">p.&nbsp;126</a></td></tr>
+</table>
+</td></tr>
+</table>
+
+<hr style="width: 65%;" />
+<div><span class="pagenum">[Pg 1]</span><a name="Page_1" id="Page_1" /></div>
+<div class="figcenter" style="width: 600px;">
+<img src="images/ch01head.png" width="600" height="137" alt="" title="" />
+</div>
+<h1>TREATISE ON LIGHT</h1>
+
+
+<h2><a name="CHAPTER_I" id="CHAPTER_I" />CHAPTER I</h2>
+
+<h3>ON RAYS PROPAGATED IN STRAIGHT LINES</h3>
+
+
+<div style="width: 154px; float: left; margin-right: .2em;">
+<img src="images/ch01.png" width="154" height="150" alt="A" title="A" />
+</div><p>s happens in all the sciences in which Geometry is applied to matter,
+the demonstrations concerning Optics are founded on truths drawn from
+experience. Such are that the rays of light are propagated in straight
+lines; that the angles of reflexion and of incidence are equal; and
+that in refraction the ray is bent according to the law of sines, now
+so well known, and which is no less certain than the preceding laws.</p>
+
+<p>The majority of those who have written touching the various parts of
+Optics have contented themselves with presuming these truths. But
+some, more inquiring, have desired to investigate the origin and the
+causes, considering these to be in themselves wonderful effects of
+Nature. In which they advanced some ingenious things, but not however
+such that the most intelligent folk do not wish for better and more
+satisfactory explanations. Wherefore I here desire to propound what I
+have meditated on the sub<span class="pagenum">[Pg 2]</span><a name="Page_2" id="Page_2" />ject, 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.</p>
+
+<p>I shall therefore essay in this book, to give, in accordance with the
+principles accepted in the Philosophy of the present day, some clearer
+and more probable reasons, firstly of these properties of light
+propagated rectilinearly; secondly of light which is reflected on
+meeting other bodies. Then I shall explain the phenomena of those rays
+which are said to suffer refraction on passing through transparent
+bodies of different sorts; and in this part I shall also explain the
+effects of the refraction of the air by the different densities of the
+Atmosphere.</p>
+
+<p>Thereafter I shall examine the causes of the strange refraction of a
+certain kind of Crystal which is brought from Iceland. And finally I
+shall treat of the various shapes of transparent and reflecting bodies
+by which rays are collected at a point or are turned aside in various
+ways. From this it will be seen with what facility, following our new
+Theory, we find not only the Ellipses, Hyperbolas, and <span class="pagenum">[Pg 3]</span><a name="Page_3" id="Page_3" />other curves
+which Mr. Des Cartes has ingeniously invented for this purpose; but
+also those which the surface of a glass lens ought to possess when its
+other surface is given as spherical or plane, or of any other figure
+that may be.</p>
+
+<p>It is inconceivable to doubt that light consists in the motion of some
+sort of matter. For whether one considers its production, one sees
+that here upon the Earth it is chiefly engendered by fire and flame
+which contain without doubt bodies that are in rapid motion, since
+they dissolve and melt many other bodies, even the most solid; or
+whether one considers its effects, one sees that when light is
+collected, as by concave mirrors, it has the property of burning as a
+fire does, that is to say it disunites the particles of bodies. This
+is assuredly the mark of motion, at least in the true Philosophy, in
+which one conceives the causes of all natural effects in terms of
+mechanical motions. This, in my opinion, we must necessarily do, or
+else renounce all hopes of ever comprehending anything in Physics.</p>
+
+<p>And as, according to this Philosophy, one holds as certain that the
+sensation of sight is excited only by the impression of some movement
+of a kind of matter which acts on the nerves at the back of our eyes,
+there is here yet one reason more for believing that light consists in
+a movement of the matter which exists between us and the luminous
+body.</p>
+
+<p>Further, when one considers the extreme speed with which light spreads
+on every side, and how, when it comes from different regions, even
+from those directly opposite, the rays traverse one another without
+hindrance, one may well understand that when we see a luminous object,
+it cannot be by any transport of matter coming to us from this object,
+<span class="pagenum">[Pg 4]</span><a name="Page_4" id="Page_4" />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.</p>
+
+<p>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&mdash;which we are now going to examine&mdash;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.</p>
+
+<p>To see then whether the spreading of light takes time, let us consider
+first whether there are any facts of experience which can convince us
+to the contrary. As to those which can be made here on the Earth, by
+striking lights at great distances, although they prove that light
+takes no sensible time to pass over these distances, one may say with
+good <span class="pagenum">[Pg 5]</span><a name="Page_5" id="Page_5" />reason that they are too small, and that the only conclusion to
+be drawn from them is that the passage of light is extremely rapid.
+Mr. Des Cartes, who was of opinion that it is instantaneous, founded
+his views, not without reason, upon a better basis of experience,
+drawn from the Eclipses of the Moon; which, nevertheless, as I shall
+show, is not at all convincing. I will set it forth, in a way a little
+different from his, in order to make the conclusion more
+comprehensible.</p>
+
+<div class="figcenter" style="width: 400px;">
+<img src="images/pg005.png" width="400" height="196" alt="" title="" />
+</div>
+
+<p>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.</p>
+
+<p>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, <span class="pagenum">[Pg 6]</span><a name="Page_6" id="Page_6" />that the
+Moon, arriving at C, will be obscured: but this obscuration or
+interruption of the light will not reach the Earth till after another
+hour. Let us suppose that the Earth in these two hours will have
+arrived at E. The Earth then, being at E, will see the Eclipsed Moon
+at C, which it left an hour before, and at the same time will see the
+sun at A. For it being immovable, as I suppose with Copernicus, and
+the light moving always in straight lines, it must always appear where
+it is. But one has always observed, we are told, that the eclipsed
+Moon appears at the point of the Ecliptic opposite to the Sun; and yet
+here it would appear in arrear of that point by an amount equal to the
+angle GEC, the supplement of AEC. This, however, is contrary to
+experience, since the angle GEC would be very sensible, and about 33
+degrees. Now according to our computation, which is given in the
+Treatise on the causes of the phenomena of Saturn, the distance BA
+between the Earth and the Sun is about twelve thousand diameters of
+the Earth, and hence four hundred times greater than BC the distance
+of the Moon, which is 30 diameters. Then the angle ECB will be nearly
+four hundred times greater than BAE, which is five minutes; namely,
+the path which the earth travels in two hours along its orbit; and
+thus the angle BCE will be nearly 33 degrees; and likewise the angle
+CEG, which is greater by five minutes.</p>
+
+<p>But it must be noted that the speed of light in this argument has been
+assumed such that it takes a time of one hour to make the passage from
+here to the Moon. If one supposes that for this it requires only one
+minute of time, then it is manifest that the angle CEG will only be 33
+minutes; and if it requires only ten seconds of time, <span class="pagenum">[Pg 7]</span><a name="Page_7" id="Page_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.</p>
+
+<p>It is true that we are here supposing a strange velocity that would be
+a hundred thousand times greater than that of Sound. For Sound,
+according to what I have observed, travels about 180 Toises in the
+time of one Second, or in about one beat of the pulse. But this
+supposition ought not to seem to be an impossibility; since it is not
+a question of the transport of a body with so great a speed, but of a
+successive movement which is passed on from some bodies to others. I
+have then made no difficulty, in meditating on these things, in
+supposing that the emanation of light is accomplished with time,
+seeing that in this way all its phenomena can be explained, and that
+in following the contrary opinion everything is incomprehensible. For
+it has always seemed tome that even Mr. Des Cartes, whose aim has been
+to treat all the subjects of Physics intelligibly, and who assuredly
+has succeeded in this better than any one before him, has said nothing
+that is not full of difficulties, or even inconceivable, in dealing
+with Light and its properties.</p>
+
+<p>But that which I employed only as a hypothesis, has recently received
+great seemingness as an established truth by the ingenious proof of
+Mr. R&ouml;mer 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.</p>
+
+<p><span class="pagenum">[Pg 8]</span><a name="Page_8" id="Page_8" />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.</p>
+
+<div class="figleft" style="width: 166px;">
+<img src="images/pg008.png" width="166" height="400" alt="" title="" />
+</div>
+
+<p>Let it be then supposed, the Earth being at B some time before the
+last quadrature, that one has seen the said Satellite emerge from the
+shadow; it must needs be, if the Earth remains at the same place,
+that, after 42-1/2 hours, one would again see a similar emergence,
+because that is the time in which it makes the round of its orbit, and
+when it would come again into opposition to the Sun. And if the Earth,
+for instance, were to remain always at B during 30 revolutions of this
+Satellite, one would see it again emerge from the shadow after 30
+times 42-1/2 hours. But the Earth having been carried along during
+this time to C, increasing thus its distance from Jupiter, it follows
+that if Light requires time for its passage the illumination of the
+little planet will be perceived later at <span class="pagenum">[Pg 9]</span><a name="Page_9" id="Page_9" />C than it would have been at
+B, and that there must be added to this time of 30 times 42-1/2 hours
+that which the Light has required to traverse the space MC, the
+difference of the spaces CH, BH. Similarly at the other quadrature
+when the earth has come to E from D while approaching toward Jupiter,
+the immersions of the Satellite ought to be observed at E earlier than
+they would have been seen if the Earth had remained at D.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>If one considers the vast size of the diameter KL, which according to
+me is some 24 thousand diameters of the Earth, one will acknowledge
+the extreme velocity of Light. For, supposing that KL is no more than
+22 thousand of these diameters, it appears that being traversed in 22
+minutes this makes the speed a thousand diameters in one minute, that
+is 16-2/3 diameters in one second or in one beat of the pulse, which
+makes more than 11 hundred times a hundred thousand toises; since the
+diameter of the Earth contains 2,865 leagues, reckoned at 25 to the
+degree, and each <span class="pagenum">[Pg 10]</span><a name="Page_10" id="Page_10" />each league is 2,282 Toises, according to the exact
+measurement which Mr. Picard made by order of the King in 1669. But
+Sound, as I have said above, only travels 180 toises in the same time
+of one second: hence the velocity of Light is more than six hundred
+thousand times greater than that of Sound. This, however, is quite
+another thing from being instantaneous, since there is all the
+difference between a finite thing and an infinite. Now the successive
+movement of Light being confirmed in this way, it follows, as I have
+said, that it spreads by spherical waves, like the movement of Sound.</p>
+
+<p>But if the one resembles the other in this respect, they differ in
+many other things; to wit, in the first production of the movement
+which causes them; in the matter in which the movement spreads; and in
+the manner in which it is propagated. As to that which occurs in the
+production of Sound, one knows that it is occasioned by the agitation
+undergone by an entire body, or by a considerable part of one, which
+shakes all the contiguous air. But the movement of the Light must
+originate as from each point of the luminous object, else we should
+not be able to perceive all the different parts of that object, as
+will be more evident in that which follows. And I do not believe that
+this movement can be better explained than by supposing that all those
+of the luminous bodies which are liquid, such as flames, and
+apparently the sun and the stars, are composed of particles which
+float in a much more subtle medium which agitates them with great
+rapidity, and makes them strike against the particles of the ether
+which surrounds them, and which are much smaller than they. But I hold
+also that in luminous solids such as charcoal or metal made red hot in
+the fire, this same movement is caused by the violent <span class="pagenum">[Pg 11]</span><a name="Page_11" id="Page_11" />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.</p>
+
+<p>Now if one examines what this matter may be in which the movement
+coming from the luminous body is propagated, which I call Ethereal
+matter, one will see that it is not the same that serves for the
+propagation of Sound. For one finds that the latter is really that
+which we feel and which we breathe, and which being removed from any
+place still leaves there the other kind of matter that serves to
+convey Light. This may be proved by shutting up a sounding body in a
+glass vessel from which the air is withdrawn by the machine which Mr.
+Boyle has given us, and with which he has performed so many beautiful
+experiments. But in doing this of which I speak, care must be taken to
+place the sounding body on cotton or on feathers, in such a way that
+it cannot communicate its tremors either to the glass vessel which
+encloses it, or to the machine; a precaution which has hitherto been
+neglected. For then after having exhausted all the air one hears no
+Sound from the metal, though it is struck.</p>
+
+<p>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 <span class="pagenum">[Pg 12]</span><a name="Page_12" id="Page_12" />removed from the vessel the Light does not cease to
+traverse it as before.</p>
+
+<p>And this last point is demonstrated even more clearly by the
+celebrated experiment of Torricelli, in which the tube of glass from
+which the quicksilver has withdrawn itself, remaining void of air,
+transmits Light just the same as when air is in it. For this proves
+that a matter different from air exists in this tube, and that this
+matter must have penetrated the glass or the quicksilver, either one
+or the other, though they are both impenetrable to the air. And when,
+in the same experiment, one makes the vacuum after putting a little
+water above the quicksilver, one concludes equally that the said
+matter passes through glass or water, or through both.</p>
+
+<p>As regards the different modes in which I have said the movements of
+Sound and of Light are communicated, one may sufficiently comprehend
+how this occurs in the case of Sound if one considers that the air is
+of such a nature that it can be compressed and reduced to a much
+smaller space than that which it ordinarily occupies. And in
+proportion as it is compressed the more does it exert an effort to
+regain its volume; for this property along with its penetrability,
+which remains notwithstanding its compression, seems to prove that it
+is made up of small bodies which float about and which are agitated
+very rapidly in the ethereal matter composed of much smaller parts. So
+that the cause of the spreading of Sound is the effort which these
+little bodies make in collisions with one another, to regain freedom
+when they are a little more squeezed together in the circuit of these
+waves than elsewhere.</p>
+
+<p>But the extreme velocity of Light, and other properties which it has,
+cannot admit of such a propagation of motion, <span class="pagenum">[Pg 13]</span><a name="Page_13" id="Page_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 property which hard bodies must possess to
+transmit movement from one to another.</p>
+
+<p>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.</p>
+
+<p>But it is still certain that this progression of motion is not
+instantaneous, but successive, and therefore must take time. For if
+the movement, or the disposition to movement, if you will have it so,
+did not pass successively through all these spheres, they would all
+acquire the movement at the same time, and hence would all advance
+together; which does not happen. For the last one leaves the whole row
+and acquires the speed of the one which was pushed. Moreover there are
+experiments which demonstrate that all the bodies which we reckon of
+the hardest kind, such as quenched steel, glass, and agate, act as
+springs and bend somehow, not only when extended as rods but also when
+they are in the form of spheres or of other shapes. That is to say
+they yield a little in themselves at the place where they are struck,
+and immediately regain their former figure. For I have found that on
+striking with a ball of glass or of agate against a large and quite
+thick <span class="pagenum">[Pg 14]</span><a name="Page_14" id="Page_14" />thick piece of the same substance which had a flat surface,
+slightly soiled with breath or in some other way, there remained round
+marks, of smaller or larger size according as the blow had been weak
+or strong. This makes it evident that these substances yield where
+they meet, and spring back: and for this time must be required.</p>
+
+<p>Now in applying this kind of movement to that which produces Light
+there is nothing to hinder us from estimating the particles of the
+ether to be of a substance as nearly approaching to perfect hardness
+and possessing a springiness as prompt as we choose. It is not
+necessary to examine here the causes of this hardness, or of that
+springiness, the consideration of which would lead us too far from our
+subject. I will say, however, in passing that we may conceive that the
+particles of the ether, notwithstanding their smallness, are in turn
+composed of other parts and that their springiness consists in the
+very rapid movement of a subtle matter which penetrates them from
+every side and constrains their structure to assume such a disposition
+as to give to this fluid matter the most overt and easy passage
+possible. This accords with the explanation which Mr. Des Cartes gives
+for the spring, though I do not, like him, suppose the pores to be in
+the form of round hollow canals. And it must not be thought that in
+this there is anything absurd or impossible, it being on the contrary
+quite credible that it is this infinite series of different sizes of
+corpuscles, having different degrees of velocity, of which Nature
+makes use to produce so many marvellous effects.</p>
+
+<p>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 <span class="pagenum">[Pg 15]</span><a name="Page_15" id="Page_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 communicated, one will find none which agrees better,
+with uniform progression, as seems to be necessary, than the property
+of springiness; because if this movement should grow slower in
+proportion as it is shared over a greater quantity of matter, in
+moving away from the source of the light, it could not conserve this
+great velocity over great distances. But by supposing springiness in
+the ethereal matter, its particles will have the property of equally
+rapid restitution whether they are pushed strongly or feebly; and thus
+the propagation of Light will always go on with an equal velocity.</p>
+
+<div class="figleft" style="width: 131px;">
+<img src="images/pg015.png" width="131" height="200" alt="" title="" />
+</div>
+
+<p>And it must be known that although the particles of the ether are not
+ranged thus in straight lines, as in our row of spheres, but
+confusedly, so that one of them touches several others, this does not
+hinder them from transmitting their movement and from spreading it
+always forward. As to this it is to be remarked that there is a law of
+motion serving for this propagation, and verifiable by experiment. It
+is that when a sphere, such as A here, touches several other similar
+spheres CCC, if it is struck by another sphere B in such a way as to
+exert an impulse against all the spheres CCC which touch it, it
+transmits to them the whole of its movement, and remains after that
+motionless like the sphere B. And without supposing that the ethereal
+particles are of spherical form (for I see indeed no need to suppose
+them so) one may well understand that this property of communicating
+an impulse <span class="pagenum">[Pg 16]</span><a name="Page_16" id="Page_16" />does not fail to contribute to the aforesaid propagation
+of movement.</p>
+
+<p>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.</p>
+
+<p>However, one will see hereafter that we have to suppose such an
+equality not so much as a necessity for the propagation of light as
+for rendering that propagation easier and more powerful; for it is not
+beyond the limits of probability that the particles of the ether have
+been made equal for a purpose so important as that of light, at least
+in that vast space which is beyond the region of atmosphere and which
+seems to serve only to transmit the light of the Sun and the Stars.</p>
+
+<div class="figright" style="width: 182px;">
+<img src="images/pg017.png" width="182" height="300" alt="" title="" />
+</div>
+
+<p>I have then shown in what manner one may conceive Light to spread
+successively, by spherical waves, and how it is possible that this
+spreading is accomplished with as great a velocity as that which
+experiments and celestial observations demand. Whence it may be
+further remarked that although the particles are supposed to be in
+continual movement (for there are many reasons for this) the
+successive propagation of the waves cannot be hindered by this;
+because the propagation consists nowise in the transport of those
+particles but merely in a small agitation which they cannot help
+communicating to those surrounding, notwithstanding any movement which
+may act on them causing them to be changing positions amongst
+themselves.</p>
+
+<p>But we must consider still more particularly the origin of these
+waves, and the manner in which they spread. And, first, it follows
+from what has been said on the production <span class="pagenum">[Pg 17]</span><a name="Page_17" id="Page_17" />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.</p>
+
+<p>But as the percussions at the centres of these waves possess no
+regular succession, it must not be supposed that the waves themselves
+follow one another at equal distances: and if the distances marked in
+the figure appear to be such, it is rather to mark the progression of
+one and the same wave at equal intervals of time than to represent
+several of them issuing from one and the same centre.</p>
+
+<p>After all, this prodigious quantity of waves which traverse one
+another without confusion and without effacing one another must not be
+deemed inconceivable; it being certain that one and the same particle
+of matter can serve for many waves coming from different sides or even
+from contrary directions, not only if it is struck by blows which
+follow one another closely but even for those which act on it at the
+same instant. It can do so because the spreading of the movement is
+successive. This may be proved by the row of equal spheres of hard
+matter, spoken of above. If against this row there are pushed from two
+opposite sides at the same time two similar spheres A and <span class="pagenum">[Pg 18]</span><a name="Page_18" id="Page_18" />D, one will
+see each of them rebound with the same velocity which it had in
+striking, yet the whole row will remain in its place, although the
+movement has passed along its whole length twice over. And if these
+contrary movements happen to meet one another at the middle sphere, B,
+or at some other such as C, that sphere will yield and act as a spring
+at both sides, and so will serve at the same instant to transmit these
+two movements.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg018.png" width="500" height="59" alt="" title="" />
+</div>
+
+<p>But what may at first appear full strange and even incredible is that
+the undulations produced by such small movements and corpuscles,
+should spread to such immense distances; as for example from the Sun
+or from the Stars to us. For the force of these waves must grow feeble
+in proportion as they move away from their origin, so that the action
+of each one in particular will without doubt become incapable of
+making itself felt to our sight. But one will cease to be astonished
+by considering how at a great distance from the luminous body an
+infinitude of waves, though they have issued from different points of
+this body, unite together in such a way that they sensibly compose one
+single wave only, which, consequently, ought to have enough force to
+make itself felt. Thus this infinite number of waves which originate
+at the same instant from all points of a fixed star, big it may be as
+the Sun, make practically only one single wave which may well have
+force enough to produce an impression on our eyes. Moreover from each
+luminous point there may come many thousands of waves in the smallest
+imaginable time, by the frequent percussion of the corpuscles which
+strike the <span class="pagenum">[Pg 19]</span><a name="Page_19" id="Page_19" />Ether at these points: which further contributes to
+rendering their action more sensible.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg019.png" width="300" height="284" alt="" title="" />
+</div>
+
+<p>There is the further consideration in the emanation of these waves,
+that each particle of matter in which a wave spreads, ought not to
+communicate its motion only to the next particle which is in the
+straight line drawn from the luminous point, but that it also imparts
+some of it necessarily to all the others which touch it and which
+oppose themselves to its movement. So it arises that around each
+particle there is made a wave of which that particle is the centre.
+Thus if DCF is a wave emanating from the luminous point A, which is
+its centre, the particle B, one of those comprised within the sphere
+DCF, will have made its particular or partial wave KCL, which will
+touch the wave DCF at C at the same moment that the principal wave
+emanating from the point A has arrived at DCF; and it is clear that it
+will be only the region C of the wave KCL which will touch the wave
+DCF, to wit, that which is in the straight line drawn through AB.
+Similarly the other particles of the sphere DCF, such as <i>bb</i>, <i>dd</i>,
+etc., will each make its own wave. But each of these waves can be
+infinitely feeble only as compared with the wave DCF, to the
+composition of which all the others contribute by the part of their
+surface which is most distant from the centre A.</p>
+
+<p><span class="pagenum">[Pg 20]</span><a name="Page_20" id="Page_20" />One sees, in addition, that the wave DCF is determined by the
+distance attained in a certain space of time by the movement which
+started from the point A; there being no movement beyond this wave,
+though there will be in the space which it encloses, namely in parts
+of the particular waves, those parts which do not touch the sphere
+DCF. And all this ought not to seem fraught with too much minuteness
+or subtlety, since we shall see in the sequel that all the properties
+of Light, and everything pertaining to its reflexion and its
+refraction, can be explained in principle by this means. This is a
+matter which has been quite unknown to those who hitherto have begun
+to consider the waves of light, amongst whom are Mr. Hooke in his
+<i>Micrographia</i>, and Father Pardies, who, in a treatise of which he let
+me see a portion, and which he was unable to complete as he died
+shortly afterward, had undertaken to prove by these waves the effects
+of reflexion and refraction. But the chief foundation, which consists
+in the remark I have just made, was lacking in his demonstrations; and
+for the rest he had opinions very different from mine, as may be will
+appear some day if his writing has been preserved.</p>
+
+<p>To come to the properties of Light. We remark first that each portion
+of a wave ought to spread in such a way that its extremities lie
+always between the same straight lines drawn from the luminous point.
+Thus the portion BG of the wave, having the luminous point A as its
+centre, will spread into the arc CE bounded by the straight lines ABC,
+AGE. For although the particular waves produced by the particles
+comprised within the space CAE spread also outside this space, they
+yet do not concur at the same instant to compose a wave which
+terminates the <span class="pagenum">[Pg 21]</span><a name="Page_21" id="Page_21" />movement, as they do precisely at the circumference
+CE, which is their common tangent.</p>
+
+<p>And hence one sees the reason why light, at least if its rays are not
+reflected or broken, spreads only by straight lines, so that it
+illuminates no object except when the path from its source to that
+object is open along such lines.</p>
+
+<p>For if, for example, there were an opening BG, limited by opaque
+bodies BH, GI, the wave of light which issues from the point A will
+always be terminated by the straight lines AC, AE, as has just been
+shown; the parts of the partial waves which spread outside the space
+ACE being too feeble to produce light there.</p>
+
+<p>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.</p>
+
+<p>It appears, moreover, by what has been remarked touching the
+feebleness of the particular waves, that it is not needful that all
+the particles of the Ether should be equal amongst themselves, though
+equality is more apt for the propagation of the movement. For it is
+true that inequality will cause a particle by pushing against another
+larger one to strive to recoil with a part of its movement; but it
+will thereby merely generate backwards towards the luminous point some
+partial waves incapable of causing light, and not a wave compounded of
+many as CE was.</p>
+
+<p>Another property of waves of light, and one of the most <span class="pagenum">[Pg 22]</span><a name="Page_22" id="Page_22" />marvellous,
+is that when some of them come from different or even from opposing
+sides, they produce their effect across one another without any
+hindrance. Whence also it comes about that a number of spectators may
+view different objects at the same time through the same opening, and
+that two persons can at the same time see one another's eyes. Now
+according to the explanation which has been given of the action of
+light, how the waves do not destroy nor interrupt one another when
+they cross one another, these effects which I have just mentioned are
+easily conceived. But in my judgement they are not at all easy to
+explain according to the views of Mr. Des Cartes, who makes Light to
+consist in a continuous pressure merely tending to movement. For this
+pressure not being able to act from two opposite sides at the same
+time, against bodies which have no inclination to approach one
+another, it is impossible so to understand what I have been saying
+about two persons mutually seeing one another's eyes, or how two
+torches can illuminate one another.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_II" id="CHAPTER_II" />CHAPTER II</h2>
+
+<h3>ON REFLEXION</h3>
+
+
+<div style="width: 158px; float: left; margin-right: .2em;">
+<img src="images/ch02.png" width="158" height="150" alt="H" title="H" />
+</div><p>aving explained the effects of waves of light which spread in a
+homogeneous matter, we will examine next that which happens to them on
+encountering other bodies. We will first make evident how the
+Reflexion of light is explained by these same waves, and why it
+preserves equality of angles.</p>
+
+<p><span class="pagenum">[Pg 23]</span><a name="Page_23" id="Page_23" />Let there be a surface AB; plane and polished, of some metal, glass,
+or other body, which at first I will consider as perfectly uniform
+(reserving to myself to deal at the end of this demonstration with the
+inequalities from which it cannot be exempt), and let a line AC,
+inclined to AD, represent a portion of a wave of light, the centre of
+which is so distant that this portion AC may be considered as a
+straight line; for I consider all this as in one plane, imagining to
+myself that the plane in which this figure is, cuts the sphere of the
+wave through its centre and intersects the plane AB at right angles.
+This explanation will suffice once for all.</p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg023.png" width="350" height="352" alt="" title="" />
+</div>
+
+<p>The piece C of the wave AC, will in a certain space of time advance as
+far as the plane AB at B, following the straight line CB, which may be
+supposed to come from the luminous centre, and which in consequence is
+perpendicular to AC. Now in this same space of time the portion A of
+the same wave, which has been hindered from communicating its movement
+beyond the plane AB, or at least partly so, ought to have continued
+its movement in the matter which is above this plane, and this along a
+distance equal to CB, making its <span class="pagenum">[Pg 24]</span><a name="Page_24" id="Page_24" />own partial spherical wave,
+according to what has been said above. Which wave is here represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter AN equal to CB.</p>
+
+<p>If one considers further the other pieces H of the wave AC, it appears
+that they will not only have reached the surface AB by straight lines
+HK parallel to CB, but that in addition they will have generated in
+the transparent air, from the centres K, K, K, particular spherical
+waves, represented here by circumferences the semi-diameters of which
+are equal to KM, that is to say to the continuations of HK as far as
+the line BG parallel to AC. But all these circumferences have as a
+common tangent the straight line BN, namely the same which is drawn
+from B as a tangent to the first of the circles, of which A is the
+centre, and AN the semi-diameter equal to BC, as is easy to see.</p>
+
+<p>It is then the line BN (comprised between B and the point N where the
+perpendicular from the point A falls) which is as it were formed by
+all these circumferences, and which terminates the movement which is
+made by the reflexion of the wave AC; and it is also the place where
+the movement occurs in much greater quantity than anywhere else.
+Wherefore, according to that which has been explained, BN is the
+propagation of the wave AC at the moment when the piece C of it has
+arrived at B. For there is no other line which like BN is a common
+tangent to all the aforesaid circles, except BG below the plane AB;
+which line BG would be the propagation of the wave if the movement
+could have spread in a medium homogeneous with that which is above the
+plane. And if one wishes to see how the wave AC has come successively
+to BN, one has only to draw in the same figure the straight lines KO
+<span class="pagenum">[Pg 25]</span><a name="Page_25" id="Page_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.</p>
+
+<p>Now it is apparent here that the angle of reflexion is made equal to
+the angle of incidence. For the triangles ACB, BNA being rectangular
+and having the side AB common, and the side CB equal to NA, it follows
+that the angles opposite to these sides will be equal, and therefore
+also the angles CBA, NAB. But as CB, perpendicular to CA, marks the
+direction of the incident ray, so AN, perpendicular to the wave BN,
+marks the direction of the reflected ray; hence these rays are equally
+inclined to the plane AB.</p>
+
+<p>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. <span class="pagenum">[Pg 26]</span><a name="Page_26" id="Page_26" />Now it is easy to see, following the preceding
+demonstration, that each small piece of this wave HC having arrived at
+the plane AB, and there generating each one its particular wave, these
+will all have, when C arrives at B, a common plane which will touch
+them, namely a circle BN similar to CH; and this will be intersected
+at its middle and at right angles by the same plane which likewise
+intersects the circle CH and the ellipse AB.</p>
+
+<div class="figcenter" style="width: 400px;">
+<img src="images/pg026.png" width="400" height="214" alt="" title="" />
+</div>
+
+<p>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.</p>
+
+<p>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
+<span class="pagenum">[Pg 27]</span><a name="Page_27" id="Page_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 act as springs, of which
+we have spoken above; namely that whether compressed little or much
+they recoil in equal times. Equally so in every reflexion of the
+light, against whatever body it may be, the angles of reflexion and
+incidence ought to be equal notwithstanding that the body might be of
+such a nature that it takes away a portion of the movement made by the
+incident light. And experiment shows that in fact there is no polished
+body the reflexion of which does not follow this rule.</p>
+
+
+<p>But the thing to be above all remarked in our demonstration is that it
+does not require that the reflecting surface should be considered as a
+uniform plane, as has been supposed by all those who have tried to
+explain the effects of reflexion; but only an evenness such as may be
+attained by the particles of the matter of the reflecting body being
+set near to one another; which particles are larger than those of the
+ethereal matter, as will appear by what we shall say in treating of
+the transparency and opacity of bodies. For the surface consisting
+thus of particles put together, and the ethereal particles being
+above, and smaller, it is evident that one could not demonstrate the
+equality of the angles of incidence and reflexion by similitude to
+that which happens to a ball thrown against a wall, of which writers
+have always made use. In our way, on the other hand, the thing is
+explained without difficulty. For the smallness of the particles of
+quicksilver, for example, being such that one must conceive millions
+of them, in the smallest visible surface proposed, arranged like a
+heap of grains of sand which has been flattened as much as it is
+capable of being, <span class="pagenum">[Pg 28]</span><a name="Page_28" id="Page_28" />this surface then becomes for our purpose as even
+as a polished glass is: and, although it always remains rough with
+respect to the particles of the Ether it is evident that the centres
+of all the particular spheres of reflexion, of which we have spoken,
+are almost in one uniform plane, and that thus the common tangent can
+fit to them as perfectly as is requisite for the production of light.
+And this alone is requisite, in our method of demonstration, to cause
+equality of the said angles without the remainder of the movement
+reflected from all parts being able to produce any contrary effect.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_III" id="CHAPTER_III" />CHAPTER III</h2>
+
+<h3>ON REFRACTION</h3>
+
+
+<div style="width: 145px; float: left; margin-right: .2em;">
+<img src="images/ch03.png" width="145" height="150" alt="I" title="I" />
+</div><p>n the same way as the effects of Reflexion have been explained by
+waves of light reflected at the surface of polished bodies, we will
+explain transparency and the phenomena of refraction by waves which
+spread within and across diaphanous bodies, both solids, such as
+glass, and liquids, such as water, oils, etc. But in order that it may
+not seem strange to suppose this passage of waves in the interior of
+these bodies, I will first show that one may conceive it possible in
+more than one mode.</p>
+
+<p>First, then, if the ethereal matter cannot penetrate transparent
+bodies at all, their own particles would be able to communicate
+successively the movement of the waves, the same as do those of the
+Ether, supposing that, like those, they are of a nature to act as a
+spring. And this is <span class="pagenum">[Pg 29]</span><a name="Page_29" id="Page_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 transparent and
+hard bodies, because their solidity does not seem to permit them to
+receive movement except in their whole mass at the same time. This,
+however, is not necessary because this solidity is not such as it
+appears to us, it being probable rather that these bodies are composed
+of particles merely placed close to one another and held together by
+some pressure from without of some other matter, and by the
+irregularity of their shapes. For primarily their rarity is shown by
+the facility with which there passes through them the matter of the
+vortices of the magnet, and that which causes gravity. Further, one
+cannot say that these bodies are of a texture similar to that of a
+sponge or of light bread, because the heat of the fire makes them flow
+and thereby changes the situation of the particles amongst themselves.
+It remains then that they are, as has been said, assemblages of
+particles which touch one another without constituting a continuous
+solid. This being so, the movement which these particles receive to
+carry on the waves of light, being merely communicated from some of
+them to others, without their going for that purpose out of their
+places or without derangement, it may very well produce its effect
+without prejudicing in any way the apparent solidity of the compound.</p>
+
+<p>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
+<span class="pagenum">[Pg 30]</span><a name="Page_30" id="Page_30" />that the air has been removed from the vessel in which this tube is
+enclosed.</p>
+
+<p>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.</p>
+
+<p>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 <span class="pagenum">[Pg 31]</span><a name="Page_31" id="Page_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.</p>
+
+<p>The second mode then of explaining transparency, and one which appears
+more probably true, is by saying that the waves of light are carried
+on in the ethereal matter, which continuously occupies the interstices
+or pores of transparent bodies. For since it passes through them
+continuously and freely, it follows that they are always full of it.
+And one may even show that these interstices occupy much more space
+than the coherent particles which constitute the bodies. For if what
+we have just said is true: that force is required to impress a certain
+horizontal velocity on bodies in proportion as they contain coherent
+matter; and if the proportion of this force follows the law of
+weights, as is confirmed by experiment, then the quantity of the
+constituent matter of bodies also follows the proportion of their
+weights. Now we see that water weighs only one fourteenth part as much
+as an equal portion of quicksilver: therefore the matter of the water
+does not occupy the fourteenth part of the space which its mass
+obtains. It must even occupy much less of it, since quicksilver is
+less heavy than gold, and the matter of gold is by no means dense, as
+follows from the fact that the matter of the vortices of the magnet
+and of that which is the cause of gravity pass very freely through it.</p>
+
+<p>But it may be objected here that if water is a body of so great
+rarity, and if its particles occupy so small a portion of the space of
+its apparent bulk, it is very strange how it yet resists Compression
+so strongly without permitting itself to be condensed by any force
+which one has <span class="pagenum">[Pg 32]</span><a name="Page_32" id="Page_32" />hitherto essayed to employ, preserving even its entire
+liquidity while subjected to this pressure.</p>
+
+<p>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.</p>
+
+<p>The rarity of transparent bodies being then such as we have said, one
+easily conceives that the waves might be carried on in the ethereal
+matter which fills the interstices of the particles. And, moreover,
+one may believe that the progression of these waves ought to be a
+little slower in the interior of bodies, by reason of the small
+detours which the same particles cause. In which different velocity of
+light I shall show the cause of refraction to consist.</p>
+
+<p>Before doing so, I will indicate the third and last mode in which
+transparency may be conceived; which is by supposing that the movement
+of the waves of light is transmitted indifferently both in the
+particles of the ethereal matter which occupy the interstices of
+bodies, and in the particles which compose them, so that the movement
+passes from one to the other. And it will be seen hereafter that this
+hypothesis serves excellently to explain the double refraction of
+certain transparent bodies.</p>
+
+<p>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 <span class="pagenum">[Pg 33]</span><a name="Page_33" id="Page_33" />these secondary particles which will receive the movement
+from those of the ether.</p>
+
+<p>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.</p>
+
+<p>All this I have found as most probable for the mode in which the waves
+of light pass across transparent bodies. To which it must further be
+added in what respect these bodies differ from those which are opaque;
+and the more so since it might seem because of the easy penetration of
+bodies by the ethereal matter, of which mention has been made, that
+there would not be any body that was not transparent. For by the same
+reasoning about the hollow sphere which I have employed to prove the
+smallness of the density of glass and its easy penetrability by the
+ethereal matter, one might also prove that the same penetrability
+obtains for metals and for every other sort of body. For this sphere
+being for example of silver, it is certain that it contains some of
+the ethereal matter which serves for light, since this was there as
+well as in the air when the opening of the sphere was closed. Yet,
+being closed and placed upon a horizontal plane, it resists the
+movement which one wishes to give to it, merely according to the
+quantity of silver of which it is made; so that one must conclude, as
+above, that the ethereal matter which is enclosed does not follow the
+movement of the sphere; and that therefore silver, as well as glass,
+is very easily penetrated by this matter. Some of it is therefore
+present continuously and in quantities between the particles of silver
+and of all other opaque <span class="pagenum">[Pg 34]</span><a name="Page_34" id="Page_34" />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.</p>
+
+<p>Whence then, one will say, does their opacity come? Is it because the
+particles which compose them are soft; that is to say, these particles
+being composed of others that are smaller, are they capable of
+changing their figure on receiving the pressure of the ethereal
+particles, the motion of which they thereby damp, and so hinder the
+continuance of the waves of light? That cannot be: for if the
+particles of the metals are soft, how is it that polished silver and
+mercury reflect light so strongly? What I find to be most probable
+herein, is to say that metallic bodies, which are almost the only
+really opaque ones, have mixed amongst their hard particles some soft
+ones; so that some serve to cause reflexion and the others to hinder
+transparency; while, on the other hand, transparent bodies contain
+only hard particles which have the faculty of recoil, and serve
+together with those of the ethereal matter for the propagation of the
+waves of light, as has been said.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg034.png" width="300" height="283" alt="" title="" />
+</div>
+
+<p>Let us pass now to the explanation of the effects of Refraction,
+assuming, as we have done, the passage of waves of light through
+transparent bodies, and the diminution of velocity which these same
+waves suffer in them.</p>
+
+<p>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 <span class="pagenum">[Pg 35]</span><a name="Page_35" id="Page_35" />broken at the point of incidence B,
+in such a way that with the straight line DBE which cuts the surface
+perpendicularly it makes an angle CBE less than ABD which it made with
+the same perpendicular when in the air. And the measure of these
+angles is found by describing, about the point B, a circle which cuts
+the radii AB, BC. For the perpendiculars AD, CE, let fall from the
+points of intersection upon the straight line DE, which are called the
+Sines of the angles ABD, CBE, have a certain ratio between themselves;
+which ratio is always the same for all inclinations of the incident
+ray, at least for a given transparent body. This ratio is, in glass,
+very nearly as 3 to 2; and in water very nearly as 4 to 3; and is
+likewise different in other diaphanous bodies.</p>
+
+<p>Another property, similar to this, is that the refractions are
+reciprocal between the rays entering into a transparent body and those
+which are leaving it. That is to say that if the ray AB in entering
+the transparent body is refracted into BC, then likewise CB being
+taken as a ray in the interior of this body will be refracted, on
+passing out, into BA.</p>
+
+<div class="figleft" style="width: 400px;">
+<img src="images/pg035.png" width="400" height="298" alt="" title="" />
+</div>
+
+<p>To explain then the reasons of these phenomena according to our
+principles, let AB be the straight line which <span class="pagenum">[Pg 36]</span><a name="Page_36" id="Page_36" />represents a plane
+surface bounding the transparent substances which lie towards C and
+towards N. When I say plane, that does not signify a perfect evenness,
+but such as has been understood in treating of reflexion, and for the
+same reason. Let the line AC represent a portion of a wave of light,
+the centre of which is supposed so distant that this portion may be
+considered as a straight line. The piece C, then, of the wave AC, will
+in a certain space of time have advanced as far as the plane AB
+following the straight line CB, which may be imagined as coming from
+the luminous centre, and which consequently will cut AC at right
+angles. Now in the same time the piece A would have come to G along
+the straight line AG, equal and parallel to CB; and all the portion of
+wave AC would be at GB if the matter of the transparent body
+transmitted the movement of the wave as quickly as the matter of the
+Ether. But let us suppose that it transmits this movement less
+quickly, by one-third, for instance. Movement will then be spread from
+the point A, in the matter of the transparent body through a distance
+equal to two-thirds of CB, making its own particular spherical wave
+according to what has been said before. This wave is then represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter equal to two-thirds of CB. Then if one considers in
+order the other pieces H of the wave AC, it appears that in the same
+time that the piece C reaches B they will not only have arrived at the
+surface AB along the straight lines HK parallel to CB, but that, in
+addition, they will have generated in the diaphanous substance from
+the centres K, partial waves, represented here by circumferences the
+semi-diameters of which are equal to two-thirds of the lines KM, that
+is to say, to <span class="pagenum">[Pg 37]</span><a name="Page_37" id="Page_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.</p>
+
+<p>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.</p>
+
+<p>It is then BN, which is formed by small arcs of these circumferences,
+which terminates the movement that the wave AC has communicated within
+the transparent body, and where this movement occurs in much greater
+amount than anywhere else. And for that reason this line, in
+accordance with what has been said more than once, is the propagation
+of the wave AC at the moment when its piece C has reached B. For there
+is no other line below the plane AB which is, like BN, a common
+tangent to all these partial waves. And if one would know how the wave
+AC has come progressively to BN, it is necessary only to draw in the
+same figure the straight lines KO parallel to BN, and all the lines KL
+parallel to AC. Thus one will see that the wave CA, from being a
+straight line, has become broken in all the positions LKO
+successively, and that it has again become a straight line at BN. This
+being evident by what has already been demonstrated, there is no need
+to explain it further.</p>
+
+<p>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 <span class="pagenum">[Pg 38]</span><a name="Page_38" id="Page_38" />mark the ray of incident light, and AN which
+was perpendicular to BN, the refracted ray: since the rays are nothing
+else than the straight lines along which the portions of the waves
+advance.</p>
+
+<p>Whence it is easy to recognize this chief property of refraction,
+namely that the Sine of the angle DAE has always the same ratio to the
+Sine of the angle NAF, whatever be the inclination of the ray DA: and
+that this ratio is the same as that of the velocity of the waves in
+the transparent substance which is towards AE to their velocity in the
+transparent substance towards AF. For, considering AB as the radius of
+a circle, the Sine of the angle BAC is BC, and the Sine of the angle
+ABN is AN. But the angle BAC is equal to DAE, since each of them added
+to CAE makes a right angle. And the angle ABN is equal to NAF, since
+each of them with BAN makes a right angle. Then also the Sine of the
+angle DAE is to the Sine of NAF as BC is to AN. But the ratio of BC to
+AN was the same as that of the velocities of light in the substance
+which is towards AE and in that which is towards AF; therefore also
+the Sine of the angle DAE will be to the Sine of the angle NAF the
+same as the said velocities of light.</p>
+
+<p>To see, consequently, what the refraction will be when the waves of
+light pass into a substance in which the movement travels more quickly
+than in that from which they emerge (let us again assume the ratio of
+3 to 2), it is only necessary to repeat all the same construction and
+demonstration which we have just used, merely substituting everywhere
+3/2 instead of 2/3. And it will be found by the same reasoning, in
+this other figure, that when the piece C of the wave AC shall have
+reached the surface AB at B, <span class="pagenum">[Pg 39]</span><a name="Page_39" id="Page_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.</p>
+
+<p>Hence one sees the reciprocal relation of the refractions of the ray
+on entering and on leaving one and the same transparent body: namely
+that if NA falling on the external surface AB is refracted into the
+direction AD, so the ray AD will be refracted on leaving the
+transparent body into the direction AN.</p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg039.png" width="350" height="311" alt="" title="" />
+</div>
+
+<p>One sees also the reason for a noteworthy accident which happens in
+this refraction: which is this, that after a certain obliquity of the
+incident ray DA, it begins to be quite unable to penetrate into the
+other transparent substance. For if the angle DAQ or CBA is such that
+in the triangle ACB, CB is equal to 2/3 of AB, or is greater, then AN
+cannot form one side of the triangle ANB, since it becomes equal to or
+greater than AB: so that the portion of wave BN cannot be found
+anywhere, neither consequently can AN, which ought to be perpendicular
+to it. And thus the incident ray DA does not then pierce the surface
+AB.</p>
+
+<p><span class="pagenum">[Pg 40]</span><a name="Page_40" id="Page_40" />When the ratio of the velocities of the waves is as two to three, as
+in our example, which is that which obtains for glass and air, the
+angle DAQ must be more than 48 degrees 11 minutes in order that the
+ray DA may be able to pass by refraction. And when the ratio of the
+velocities is as 3 to 4, as it is very nearly in water and air, this
+angle DAQ must exceed 41 degrees 24 minutes. And this accords
+perfectly with experiment.</p>
+
+<p>But it might here be asked: since the meeting of the wave AC against
+the surface AB ought to produce movement in the matter which is on the
+other side, why does no light pass there? To which the reply is easy
+if one remembers what has been said before. For although it generates
+an infinitude of partial waves in the matter which is at the other
+side of AB, these waves never have a common tangent line (either
+straight or curved) at the same moment; and so there is no line
+terminating the propagation of the wave AC beyond the plane AB, nor
+any place where the movement is gathered together in sufficiently
+great quantity to produce light. And one will easily see the truth of
+this, namely that CB being larger than 2/3 of AB, the waves excited
+beyond the plane AB will have no common tangent if about the centres K
+one then draws circles having radii equal to 3/2 of the lengths LB to
+which they correspond. For all these circles will be enclosed in one
+another and will all pass beyond the point B.</p>
+
+<p>Now it is to be remarked that from the moment when the angle DAQ is
+smaller than is requisite to permit the refracted ray DA to pass into
+the other transparent substance, one finds that the interior reflexion
+which occurs at the surface AB is much augmented in brightness, as <span class="pagenum">[Pg 41]</span><a name="Page_41" id="Page_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 collected in a minimum space when it
+reaches BN. It appears also that the wave BN becomes so much the
+smaller as the angle CBA or DAQ is made less; until when the latter is
+diminished to the limit indicated a little previously, this wave BN is
+collected together always at one point. That is to say, that when the
+piece C of the wave AC has then arrived at B, the wave BN which is the
+propagation of AC is entirely reduced to the same point B. Similarly
+when the piece H has reached K, the part AH is entirely reduced to the
+same point K. This makes it evident that in proportion as the wave CA
+comes to meet the surface AB, there occurs a great quantity of
+movement along that surface; which movement ought also to spread
+within the transparent body and ought to have much re-enforced the
+partial waves which produce the interior reflexion against the surface
+AB, according to the laws of reflexion previously explained.</p>
+
+<p>And because a slight diminution of the angle of incidence DAQ causes
+the wave BN, however great it was, to be reduced to zero, (for this
+angle being 49 degrees 11 minutes in the glass, the angle BAN is still
+11 degrees 21 minutes, and the same angle being reduced by one degree
+only the angle BAN is reduced to zero, and so the wave BN reduced to a
+point) thence it comes about that the interior reflexion from being
+obscure becomes suddenly bright, so soon as the angle of incidence is
+such that it no longer gives passage to the refraction.</p>
+
+<p><span class="pagenum">[Pg 42]</span><a name="Page_42" id="Page_42" />Now as concerns ordinary external reflexion, that is to say which
+occurs when the angle of incidence DAQ is still large enough to enable
+the refracted ray to penetrate beyond the surface AB, this reflexion
+should occur against the particles of the substance which touches the
+transparent body on its outside. And it apparently occurs against the
+particles of the air or others mingled with the ethereal particles and
+larger than they. So on the other hand the external reflexion of these
+bodies occurs against the particles which compose them, and which are
+also larger than those of the ethereal matter, since the latter flows
+in their interstices. It is true that there remains here some
+difficulty in those experiments in which this interior reflexion
+occurs without the particles of air being able to contribute to it, as
+in vessels or tubes from which the air has been extracted.</p>
+
+<p>Experience, moreover, teaches us that these two reflexions are of
+nearly equal force, and that in different transparent bodies they are
+so much the stronger as the refraction of these bodies is the greater.
+Thus one sees manifestly that the reflexion of glass is stronger than
+that of water, and that of diamond stronger than that of glass.</p>
+
+<p>I will finish this theory of refraction by demonstrating a remarkable
+proposition which depends on it; namely, that a ray of light in order
+to go from one point to another, when these points are in different
+media, is refracted in such wise at the plane surface which joins
+these two media that it employs the least possible time: and exactly
+the same happens in the case of reflexion against a plane surface. Mr.
+Fermat was the first to propound this property of refraction, holding
+with us, and directly counter to the opinion of Mr. Des Cartes, that
+light passes <span class="pagenum">[Pg 43]</span><a name="Page_43" id="Page_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
+demonstration, which may be seen in his printed works, and in the
+volume of letters of Mr. Des Cartes, is very long; wherefore I give
+here another which is simpler and easier.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg043.png" width="350" height="320" alt="" title="" />
+</div>
+
+<p>Let KF be the plane surface; A the point in the medium which the light
+traverses more easily, as the air; C the point in the other which is
+more difficult to penetrate, as water. And suppose that a ray has come
+from A, by B, to C, having been refracted at B according to the law
+demonstrated a little before; that is to say that, having drawn PBQ,
+which cuts the plane at right angles, let the sine of the angle ABP
+have to the sine of the angle CBQ the same ratio as the velocity of
+light in the medium where A is to the velocity of light in the medium
+where C is. It is to be shown that the time of passage of light along
+AB and BC taken together, is the shortest that can be. Let us assume
+that it may have come by other lines, and, in the first place, along
+AF, FC, so <span class="pagenum">[Pg 44]</span><a name="Page_44" id="Page_44" />that the point of refraction F may be further from B than
+the point A; and let AO be a line perpendicular to AB, and FO parallel
+to AB; BH perpendicular to FO, and FG to BC.</p>
+
+<p>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.</p>
+
+<p>Now let us assume that the ray has come from A to C along AK, KC; the
+point of refraction K being nearer to A than the point B is; and let
+CN be the perpendicular upon BC, KN parallel to BC: BM perpendicular
+upon KN, and KL upon BA.</p>
+
+<p>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 <span class="pagenum">[Pg 45]</span><a name="Page_45" id="Page_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.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_IV" id="CHAPTER_IV" />CHAPTER IV</h2>
+
+<h3>ON THE REFRACTION OF THE AIR</h3>
+
+
+<div style="width: 208px; float: left; margin-right: .2em;">
+<img src="images/ch04.png" width="208" height="150" alt="W" title="W" />
+</div><p>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
+movement is communicated in it more rapidly toward one side than
+toward another, these waves cannot be spherical: but that they must
+acquire their figure according to the different distances over which
+the successive movement passes in equal times.</p>
+
+<p>It is thus that we shall in the first place explain the refractions
+which occur in the air, which extends from here to the clouds and
+beyond. The effects of which refractions are very remarkable; for by
+them we often see objects which the rotundity of the Earth ought
+otherwise to hide; such as Islands, and the tops of mountains when one
+is at sea. Because also of them the Sun and the Moon appear as risen
+before in fact they have, and appear to set <span class="pagenum">[Pg 46]</span><a name="Page_46" id="Page_46" />later: so that at times
+the Moon has been seen eclipsed while the Sun appeared still above the
+horizon. And so also the heights of the Sun and of the Moon, and those
+of all the Stars always appear a little greater than they are in
+reality, because of these same refractions, as Astronomers know. But
+there is one experiment which renders this refraction very evident;
+which is that of fixing a telescope on some spot so that it views an
+object, such as a steeple or a house, at a distance of half a league
+or more. If then you look through it at different hours of the day,
+leaving it always fixed in the same way, you will see that the same
+spots of the object will not always appear at the middle of the
+aperture of the telescope, but that generally in the morning and in
+the evening, when there are more vapours near the Earth, these objects
+seem to rise higher, so that the half or more of them will no longer
+be visible; and so that they seem lower toward mid-day when these
+vapours are dissipated.</p>
+
+<p>Those who consider refraction to occur only in the surfaces which
+separate transparent bodies of different nature, would find it
+difficult to give a reason for all that I have just related; but
+according to our Theory the thing is quite easy. It is known that the
+air which surrounds us, besides the particles which are proper to it
+and which float in the ethereal matter as has been explained, is full
+also of particles of water which are raised by the action of heat; and
+it has been ascertained further by some very definite experiments that
+as one mounts up higher the density of air diminishes in proportion.
+Now whether the particles of water and those of air take part, by
+means of the particles of ethereal matter, in the movement which
+constitutes light, but have a less prompt recoil than these, <span class="pagenum">[Pg 47]</span><a name="Page_47" id="Page_47" />or
+whether the encounter and hindrance which these particles of air and
+water offer to the propagation of movement of the ethereal progress,
+retard the progression, it follows that both kinds of particles flying
+amidst the ethereal particles, must render the air, from a great
+height down to the Earth, gradually less easy for the spreading of the
+waves of light.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg047.png" width="500" height="302" alt="" title="" />
+</div>
+
+<p>Whence the configuration of the waves ought to become nearly such as
+this figure represents: namely, if A is a light, or the visible point
+of a steeple, the waves which start from it ought to spread more
+widely upwards and less widely downwards, but in other directions more
+or less as they approximate to these two extremes. This being so, it
+necessarily follows that every line intersecting one of these waves at
+right angles will pass above the point A, always excepting the one
+line which is perpendicular to the horizon.<span class="pagenum">[Pg 48]</span><a name="Page_48" id="Page_48" /></p>
+
+<div class="figcenter" style="width: 400px;">
+<img src="images/pg048.png" width="400" height="371" alt="" title="" />
+</div>
+
+<p>Let BC be the wave which brings the light to the spectator who is at
+B, and let BD be the straight line which intersects this wave at right
+angles. Now because the ray or straight line by which we judge the
+spot where the object appears to us is nothing else than the
+perpendicular to the wave that reaches our eye, as will be understood
+by what was said above, it is manifest that the point A will be
+perceived as being in the line BD, and therefore higher than in fact it
+is.</p>
+
+<p>Similarly if the Earth be AB, and the top of the Atmosphere CD, which
+probably is not a well defined spherical surface (since we know that
+the air becomes rare in proportion as one ascends, for above there is
+so much less of it to press down upon it), the waves of light from the
+sun coming, for instance, in such a way that so long as they have not
+reached the Atmosphere CD the straight line AE intersects them
+perpendicularly, they ought, when they enter the Atmosphere, to
+advance more quickly in elevated regions than in regions nearer to the
+Earth. So that if <span class="pagenum">[Pg 49]</span><a name="Page_49" id="Page_49" />CA is the wave which brings the light to the
+spectator at A, its region C will be the furthest advanced; and the
+straight line AF, which intersects this wave at right angles, and
+which determines the apparent place of the Sun, will pass above the
+real Sun, which will be seen along the line AE. And so it may occur
+that when it ought not to be visible in the absence of vapours,
+because the line AE encounters the rotundity of the Earth, it will be
+perceived in the line AF by refraction. But this angle EAF is scarcely
+ever more than half a degree because the attenuation of the vapours
+alters the waves of light but little. Furthermore these refractions
+are not altogether constant in all weathers, particularly at small
+elevations of 2 or 3 degrees; which results from the different
+quantity of aqueous vapours rising above the Earth.</p>
+
+<p>And this same thing is the cause why at certain times a distant object
+will be hidden behind another less distant one, and yet may at another
+time be able to be seen, although the spot from which it is viewed is
+always the same. But the reason for this effect will be still more
+evident from what we are going to remark touching the curvature of
+rays. It appears from the things explained above that the progression
+or propagation of a small part of a wave of light is properly what one
+calls a ray. Now these rays, instead of being straight as they are in
+homogeneous media, ought to be curved in an atmosphere of unequal
+penetrability. For they necessarily follow from the object to the eye
+the line which intersects at right angles all the progressions of the
+waves, as in the first figure the line AEB does, as will be shown
+hereafter; and it is this line which determines what interposed bodies
+would or would not hinder us from seeing the object. For <span class="pagenum">[Pg 50]</span><a name="Page_50" id="Page_50" />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.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg050.png" width="350" height="367" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 51]</span><a name="Page_51" id="Page_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 penetrabilities at the different heights in the air between
+A and B be represented by all these lines; so that the particular
+wave, originating from the point F, will spread across the space FG,
+and that from the point H across the space HI, while that from the
+point A spreads across the space AD.</p>
+
+<p>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 <span class="pagenum">[Pg 52]</span><a name="Page_52" id="Page_52" />arrived at MN
+in such a manner that the perpendiculars KM, LN, dip down even more
+than do AK, BL. And this suffices to show that the ray will continue
+along the curved line which intersects all the waves at right angles,
+as has been said.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_V" id="CHAPTER_V" />CHAPTER V</h2>
+
+<h3>ON THE STRANGE REFRACTION OF ICELAND CRYSTAL</h3>
+
+
+<p>1.</p>
+
+<div style="width: 156px; float: left; margin-right: .2em;">
+<img src="images/ch05.png" width="156" height="150" alt="T" title="T" />
+</div><p>here is brought from Iceland, which is an Island in the North Sea, in
+the latitude of 66 degrees, a kind of Crystal or transparent stone,
+very remarkable for its figure and other qualities, but above all for
+its strange refractions. The causes of this have seemed to me to be
+worthy of being carefully investigated, the more so because amongst
+transparent bodies this one alone does not follow the ordinary rules
+with respect to rays of light. I have even been under some necessity
+to make this research, because the refractions of this Crystal seemed
+to overturn our preceding explanation of regular refraction; which
+explanation, on the contrary, they strongly confirm, as will be seen
+after they have been brought under the same principle. In Iceland are
+found great lumps of this Crystal, some of which I have seen of 4 or 5
+pounds. But it occurs also in other countries, for I have had some of
+the same sort which had been found in France near the town of Troyes
+in Champagne, and some others which came from the Island of Corsica,
+though both were <span class="pagenum">[Pg 53]</span><a name="Page_53" id="Page_53" />less clear and only in little bits, scarcely capable
+of letting any effect of refraction be observed.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg053.png" width="300" height="255" alt="" title="" />
+</div>
+
+<p>4. The pieces of it which are found have the figure of an oblique
+parallelepiped; each of the six faces being a parallelogram; and it
+admits of being split in three directions parallel to two of these
+opposed faces. Even in such wise, if you will, that all the six faces
+are equal and similar rhombuses. The figure here added represents a
+piece of this Crystal. The obtuse angles of all the parallelograms, as
+C, D, here, are angles of 101 degrees 52 minutes, <span class="pagenum">[Pg 54]</span><a name="Page_54" id="Page_54" />and consequently
+the acute angles, such as A and B, are of 78 degrees 8 minutes.</p>
+
+<p>5. Of the solid angles there are two opposite to one another, such as
+C and E, which are each composed of three equal obtuse plane angles.
+The other six are composed of two acute angles and one obtuse. All
+that I have just said has been likewise remarked by Mr. Bartholinus in
+the aforesaid treatise; if we differ it is only slightly about the
+values of the angles. He recounts moreover some other properties of
+this Crystal; to wit, that when rubbed against cloth it attracts
+straws and other light things as do amber, diamond, glass, and Spanish
+wax. Let a piece be covered with water for a day or more, the surface
+loses its natural polish. When aquafortis is poured on it it produces
+ebullition, especially, as I have found, if the Crystal has been
+pulverized. I have also found by experiment that it may be heated to
+redness in the fire without being in anywise altered or rendered less
+transparent; but a very violent fire calcines it nevertheless. Its
+transparency is scarcely less than that of water or of Rock Crystal,
+and devoid of colour. But rays of light pass through it in another
+fashion and produce those marvellous refractions the causes of which I
+am now going to try to explain; reserving for the end of this Treatise
+the statement of my conjectures touching the formation and
+extraordinary configuration of this Crystal.</p>
+
+<p>6. In all other transparent bodies that we know there is but one sole
+and simple refraction; but in this substance there are two different
+ones. The effect is that objects seen through it, especially such as
+are placed right against it, appear double; and that a ray of
+sunlight, falling on one of its surfaces, parts itself into two rays
+and traverses the Crystal thus.</p>
+
+<p><span class="pagenum">[Pg 55]</span><a name="Page_55" id="Page_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
+refracted. But in this Crystal the perpendicular ray suffers
+refraction, and there are oblique rays which pass through it quite
+straight.</p>
+
+<div class="figcenter" style="width: 450px;">
+<img src="images/pg055.png" width="450" height="444" alt="" title="" />
+</div>
+
+<p>8. But in order to explain these phenomena more particularly, let
+there be, in the first place, a piece ABFE of the same Crystal, and
+let the obtuse angle ACB, one of the three which constitute the
+equilateral solid angle C, be divided into two equal parts by the
+straight line CG, and let it be conceived that the Crystal is
+intersected by a plane which passes through this line and through the
+side CF, which plane will necessarily be perpendicular to <span class="pagenum">[Pg 56]</span><a name="Page_56" id="Page_56" />the surface
+AB; and its section in the Crystal will form a parallelogram GCFH. We
+will call this section the principal section of the Crystal.</p>
+
+<p>9. Now if one covers the surface AB, leaving there only a small
+aperture at the point K, situated in the straight line CG, and if one
+exposes it to the sun, so that his rays face it perpendicularly above,
+then the ray IK will divide itself at the point K into two, one of
+which will continue to go on straight by KL, and the other will
+separate itself along the straight line KM, which is in the plane
+GCFH, and which makes with KL an angle of about 6 degrees 40 minutes,
+tending from the side of the solid angle C; and on emerging from the
+other side of the Crystal it will turn again parallel to JK, along MZ.
+And as, in this extraordinary refraction, the point M is seen by the
+refracted ray MKI, which I consider as going to the eye at I, it
+necessarily follows that the point L, by virtue of the same
+refraction, will be seen by the refracted ray LRI, so that LR will be
+parallel to MK if the distance from the eye KI is supposed very great.
+The point L appears then as being in the straight line IRS; but the
+same point appears also, by ordinary refraction, to be in the straight
+line IK, hence it is necessarily judged to be double. And similarly if
+L be a small hole in a sheet of paper or other substance which is laid
+against the Crystal, it will appear when turned towards daylight as if
+there were two holes, which will seem the wider apart from one another
+the greater the thickness of the Crystal.</p>
+
+<p>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 <span class="pagenum">[Pg 57]</span><a name="Page_57" id="Page_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 calculation which I
+shall put at the end, it will divide itself at the point O into two
+rays, one of which will continue along OP in a straight line with NO,
+and will similarly pass out of the other side of the crystal without
+any refraction; but the other will be refracted and will go along OQ.
+And it must be noted that it is special to the plane through GCF and
+to those which are parallel to it, that all incident rays which are in
+one of these planes continue to be in it after they have entered the
+Crystal and have become double; for it is quite otherwise for rays in
+all other planes which intersect the Crystal, as we shall see
+afterwards.</p>
+
+<p>11. I recognized at first by these experiments and by some others that
+of the two refractions which the ray suffers in this Crystal, there is
+one which follows the ordinary rules; and it is this to which the rays
+KL and OQ belong. This is why I have distinguished this ordinary
+refraction from the other; and having measured it by exact
+observation, I found that its proportion, considered as to the Sines
+of the angles which the incident and refracted rays make with the
+perpendicular, was very precisely that of 5 to 3, as was found also by
+Mr. Bartholinus, and consequently much greater than that of Rock
+Crystal, or of glass, which is nearly 3 to 2.</p>
+
+<div class="figright" style="width: 400px;">
+<img src="images/pg058.png" width="400" height="307" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 58]</span><a name="Page_58" id="Page_58" />as it
+is desired to examine a ray that is more or less oblique. Then place
+the Crystal upon the intersection E so that the line AB concurs with
+that which bisects the obtuse angle of the lower surface, or with some
+line parallel to it. Then by placing the eye directly above the line
+AB it will appear single only; and one will see that the portion
+viewed through the Crystal and the portions which appear outside it,
+meet together in a straight line: but the line CD will appear double,
+and one can distinguish the image which is due to regular refraction
+by the circumstance that when one views it with both eyes it seems
+raised up more than the other, or again by the circumstance that, when
+the Crystal is turned around on the paper, this image remains
+stationary, whereas the other image shifts and moves entirely around.
+Afterwards let the eye be placed at I (remaining <span class="pagenum">[Pg 59]</span><a name="Page_59" id="Page_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.</p>
+
+<p>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.</p>
+
+<p>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, <span class="pagenum">[Pg 60]</span><a name="Page_60" id="Page_60" />RES, which the incident and the refracted ray make with
+the perpendicular.</p>
+
+<p>15. But I found in this refraction that the ratio of FR to RS was not
+constant, like the ordinary refraction, but that it varied with the
+varying obliquity of the incident ray.</p>
+
+<p>16. I found also that when QRE made a straight line, that is, when the
+incident ray entered the Crystal without being refracted (as I
+ascertained by the circumstance that then the point E viewed by the
+extraordinary refraction appeared in the line CD, as seen without
+refraction) I found, I say, then that the angle QRG was 73 degrees 20
+minutes, as has been already remarked; and so it is not the ray
+parallel to the edge of the Crystal, which crosses it in a straight
+line without being refracted, as Mr. Bartholinus believed, since that
+inclination is only 70 degrees 57 minutes, as was stated above. And
+this is to be noted, in order that no one may search in vain for the
+cause of the singular property of this ray in its parallelism to the
+edges mentioned.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg060.png" width="350" height="347" alt="" title="" />
+</div>
+
+<p>17. Finally, continuing my observations to discover the <span class="pagenum">[Pg 61]</span><a name="Page_61" id="Page_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 refractions KX and KT
+meet the bottom line HF in such wise that points X and T are equally
+distant from the point M, where the refraction of the perpendicular
+ray IK falls; and this occurs also for refractions in other sections
+of this Crystal. But before speaking of those, which have also other
+particular properties, we will investigate the causes of the phenomena
+which I have already reported.</p>
+
+<p>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.</p>
+
+<p>18. As there were two different refractions, I conceived that there
+were also two different emanations of waves of light, and that one
+could occur in the ethereal matter extending through the body of the
+Crystal. Which matter, being present in much larger quantity than is
+that of the particles which compose it, was alone capable of causing
+transparency, according to what has been explained heretofore. I
+attributed to this emanation of waves the regular refraction which is
+observed in this stone, by supposing these waves to be ordinarily of
+spherical form, and having a slower progression within the Crystal
+than they have outside it; whence proceeds refraction as I have
+demonstrated.</p>
+
+<p>19. As to the other emanation which should produce <span class="pagenum">[Pg 62]</span><a name="Page_62" id="Page_62" />the irregular
+refraction, I wished to try what Elliptical waves, or rather
+spheroidal waves, would do; and these I supposed would spread
+indifferently both in the ethereal matter diffused throughout the
+crystal and in the particles of which it is composed, according to the
+last mode in which I have explained transparency. It seemed to me that
+the disposition or regular arrangement of these particles could
+contribute to form spheroidal waves (nothing more being required for
+this than that the successive movement of light should spread a little
+more quickly in one direction than in the other) and I scarcely
+doubted that there were in this crystal such an arrangement of equal
+and similar particles, because of its figure and of its angles with
+their determinate and invariable measure. Touching which particles,
+and their form and disposition, I shall, at the end of this Treatise,
+propound my conjectures and some experiments which confirm them.</p>
+
+<p>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.</p>
+
+<p><span class="pagenum">[Pg 63]</span><a name="Page_63" id="Page_63" />21. Now this double refraction, according to my Theory hereinbefore
+established, seemed to demand a double emission of waves of light,
+both of them spherical (for both the refractions are regular) and
+those of one series a little slower only than the others. For thus the
+phenomenon is quite naturally explained, by postulating substances
+which serve as vehicle for these waves, as I have done in the case of
+Iceland Crystal. I had then less trouble after that in admitting two
+emissions of waves in one and the same body. And since it might have
+been objected that in composing these two kinds of crystal of equal
+particles of a certain figure, regularly piled, the interstices which
+these particles leave and which contain the ethereal matter would
+scarcely suffice to transmit the waves of light which I have localized
+there, I removed this difficulty by regarding these particles as being
+of a very rare texture, or rather as composed of other much smaller
+particles, between which the ethereal matter passes quite freely.
+This, moreover, necessarily follows from that which has been already
+demonstrated touching the small quantity of matter of which the bodies
+are built up.</p>
+
+<p>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.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg064.png" width="300" height="253" alt="" title="" />
+</div>
+
+<p>23. I considered first the effect of waves so formed, as respects the
+ray which falls perpendicularly on the flat surface of a transparent
+body in which they should spread in this manner. I took AB for the
+exposed region of the surface. And, since a ray perpendicular to a
+plane, and <span class="pagenum">[Pg 64]</span><a name="Page_64" id="Page_64" />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<i>h</i>C come to meet the
+surface AB at the points AK<i>k</i>B. Then instead of the hemispherical
+partial waves which in a body of ordinary refraction would spread from
+each of these last points, as we have above explained in treating of
+refraction, these must here be hemi-spheroids. The axes (or rather the
+major diameters) of these I supposed to be oblique to the plane AB, as
+is AV the semi-axis or semi-major diameter of the spheroid SVT, which
+represents the partial wave coming from the point A, after the wave RC
+has reached AB. I say axis or major diameter, because the same ellipse
+SVT may be considered as the section of a spheroid of which the axis
+is AZ perpendicular to AV. But, for the present, without yet deciding
+one or other, we will consider these spheroids only in those sections
+of them which make ellipses in the plane of this figure. Now taking a
+certain space of time during which the wave SVT has spread from A, it
+would needs be that from all the other points K<i>k</i>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 <span class="pagenum">[Pg 65]</span><a name="Page_65" id="Page_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.</p>
+
+<p>24. Now it appeared that this common tangent NQ was parallel to AB,
+and of the same length, but that it was not directly opposite to it,
+since it was comprised between the lines AN, BQ, which are diameters
+of ellipses having A and B for centres, conjugate with respect to
+diameters which are not in the straight line AB. And in this way I
+comprehended, a matter which had seemed to me very difficult, how a
+ray perpendicular to a surface could suffer refraction on entering a
+transparent body; seeing that the wave RC, having come to the aperture
+AB, went on forward thence, spreading between the parallel lines AN,
+BQ, yet itself remaining always parallel to AB, so that here the light
+does not spread along lines perpendicular to its waves, as in ordinary
+refraction, but along lines cutting the waves obliquely.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg065.png" width="300" height="253" alt="" title="" />
+</div>
+
+<p>25. Inquiring subsequently what might be the position and form of
+these spheroids in the crystal, I considered that all the six faces
+produced precisely the same refractions. Taking, then, the
+parallelopiped AFB, of which the obtuse solid angle C is contained
+between the three equal plane angles, and imagining in it the three
+principal sections, one of which is perpendicular to the face DC and
+passes through the edge CF, another perpendicular to the face BF
+passing through the edge <span class="pagenum">[Pg 66]</span><a name="Page_66" id="Page_66" />CA, and the third perpendicular to the face
+AF passing through the edge BC; I knew that the refractions of the
+incident rays belonging to these three planes were all similar. But
+there could be no position of the spheroid which would have the same
+relation to these three sections except that in which the axis was
+also the axis of the solid angle C. Consequently I saw that the axis
+of this angle, that is to say the straight line which traversed the
+crystal from the point C with equal inclination to the edges CF, CA,
+CB was the line which determined the position of the axis of all the
+spheroidal waves which one imagined to originate from some point,
+taken within or on the surface of the crystal, since all these
+spheroids ought to be alike, and have their axes parallel to one
+another.</p>
+
+<p>26. Considering after this the plane of one of these three sections,
+namely that through GCF, the angle of which is 109 degrees 3 minutes,
+since the angle F was shown above to be 70 degrees 57 minutes; and,
+imagining a spheroidal wave about the centre C, I knew, because I have
+just explained it, that its axis must be in the same plane, the half
+of which axis I have marked CS in the next figure: and seeking by
+calculation (which will be given with others at the end of this
+discourse) the value of the angle CGS, I found it 45 degrees 20
+minutes.</p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg067.png" width="350" height="234" alt="" title="" />
+</div>
+
+<p>27. To know from this the form of this spheroid, that is to say the
+proportion of the semi-diameters CS, CP, of its elliptical section,
+which are perpendicular to one another, I considered that the point M
+where the ellipse is touched by the straight line FH, parallel to CG,
+ought to be so situated that CM makes with the perpendicular CL an
+angle of 6 degrees 40 minutes; since, this being so, this ellipse
+satisfies what has been said about the refraction of <span class="pagenum">[Pg 67]</span><a name="Page_67" id="Page_67" />the ray
+perpendicular to the surface CG, which is inclined to the
+perpendicular CL by the same angle. This, then, being thus disposed,
+and taking CM at 100,000 parts, I found by the calculation which will
+be given at the end, the semi-major diameter CP to be 105,032, and the
+semi-axis CS to be 93,410, the ratio of which numbers is very nearly 9
+to 8; so that the spheroid was of the kind which resembles a
+compressed sphere, being generated by the revolution of an ellipse
+about its smaller diameter. I found also the value of CG the
+semi-diameter parallel to the tangent ML to be 98,779.</p>
+
+<div class="figleft" style="width: 500px;">
+<img src="images/pg068.png" width="500" height="314" alt="" title="" />
+</div>
+
+<p>28. Now passing to the investigation of the refractions which
+obliquely incident rays must undergo, according to our hypothesis of
+spheroidal waves, I saw that these refractions depended on the ratio
+between the velocity of movement of the light outside the crystal in
+the ether, and that within the crystal. For supposing, for example,
+this proportion to be such that while the light in the crystal forms
+the spheroid GSP, as I have just said, it forms outside a sphere the
+semi-diameter of which is equal to the line N which will be determined
+hereafter, the following is the way of finding the refraction of the
+incident rays. Let there be such a ray RC falling upon the <span class="pagenum">[Pg 68]</span><a name="Page_68" id="Page_68" />surface
+CK. Make CO perpendicular to RC, and across the angle KCO adjust OK,
+equal to N and perpendicular to CO; then draw KI, which touches the
+Ellipse GSP, and from the point of contact I join IC, which will be
+the required refraction of the ray RC. The demonstration of this is,
+it will be seen, entirely similar to that of which we made use in
+explaining ordinary refraction. For the refraction of the ray RC is
+nothing else than the progression of the portion C of the wave CO,
+continued in the crystal. Now the portions H of this wave, during the
+time that O came to K, will have arrived at the surface CK along the
+straight lines H<i>x</i>, and will moreover have produced in the crystal
+around the centres <i>x</i> some hemi-spheroidal partial waves similar to
+the hemi-spheroidal GSP<i>g</i>, and similarly disposed, and of which the
+major <span class="pagenum">[Pg 69]</span><a name="Page_69" id="Page_69" />and minor diameters will bear the same proportions to the lines
+<i>xv</i> (the continuations of the lines H<i>x</i> up to KB parallel to CO)
+that the diameters of the spheroid GSP<i>g</i> 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.</p>
+
+<p>Now as to finding the point of contact I, it is known that one must
+find CD a third proportional to the lines CK, CG, and draw DI parallel
+to CM, previously determined, which is the conjugate diameter to CG;
+for then, by drawing KI it touches the Ellipse at I.</p>
+
+<p>29. Now as we have found CI the refraction of the ray RC, similarly
+one will find C<i>i</i> the refraction of the ray <i>r</i>C, which comes from
+the opposite side, by making C<i>o</i> perpendicular to <i>r</i>C and following
+out the rest of the construction as before. Whence one sees that if
+the ray <i>r</i>C is inclined equally with RC, the line C<i>d</i> will
+necessarily be equal to CD, because C<i>k</i> is equal to CK, and C<i>g</i> to
+CG. And in consequence I<i>i</i> will be cut at E into equal parts by the
+line CM, to which DI and <i>di</i> are parallel. And because CM is the
+conjugate diameter to CG, it follows that <i>i</i>I will be parallel to
+<i>g</i>G. Therefore if one prolongs the refracted rays CI, C<i>i</i>, until
+they meet the tangent ML at T and <i>t</i>, the distances MT, M<i>t</i>, will
+also be equal. And so, by our hypothesis, we explain perfectly the
+phenomenon mentioned above; to wit, that when there are two rays
+equally inclined, but coming from opposite sides, as here the rays RC,
+<i>rc</i>, their refractions diverge equally from the line <span class="pagenum">[Pg 70]</span><a name="Page_70" id="Page_70" />followed by the
+refraction of the ray perpendicular to the surface, by considering
+these divergences in the direction parallel to the surface of the
+crystal.</p>
+
+<p>30. To find the length of the line N, in proportion to CP, CS, CG, it
+must be determined by observations of the irregular refraction which
+occurs in this section of the crystal; and I find thus that the ratio
+of N to GC is just a little less than 8 to 5. And having regard to
+some other observations and phenomena of which I shall speak
+afterwards, I put N at 156,962 parts, of which the semi-diameter CG is
+found to contain 98,779, making this ratio 8 to 5-1/29. Now this
+proportion, which there is between the line N and CG, may be called
+the Proportion of the Refraction; similarly as in glass that of 3 to
+2, as will be manifest when I shall have explained a short process in
+the preceding way to find the irregular refractions.</p>
+
+<p>31. Supposing then, in the next figure, as previously, the surface of
+the crystal <i>g</i>G, the Ellipse GP<i>g</i>, and the line N; and CM the
+refraction of the perpendicular ray FC, from which it diverges by 6
+degrees 40 minutes. Now let there be some other ray RC, the refraction
+of which must be found.</p>
+
+<p>About the centre C, with semi-diameter CG, let the circumference <i>g</i>RG
+be described, cutting the ray RC at R; and let RV be the perpendicular
+on CG. Then as the line N is to CG let CV be to CD, and let DI be
+drawn parallel to CM, cutting the Ellipse <i>g</i>MG at I; then joining CI,
+this will be the required refraction of the ray RC. Which is
+demonstrated thus.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg071.png" width="500" height="364" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 71]</span><a name="Page_71" id="Page_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 refraction of the ray RC. Now since the angle
+RCO is a right angle, it is easy to see that the right-angled
+triangles RCV, KCO, are similar. As then, CK is to KO, so also is RC
+to CV. But KO is equal to N, and RC to CG: then as CK is to N so will
+CG be to CV. But as N is to CG, so, by construction, is CV to CD. Then
+as CK is to CG so is CG to CD. And because DI is parallel to CM, the
+conjugate diameter to CG, it follows that KI touches the Ellipse at I;
+which remained to be shown.</p>
+
+<p>32. One sees then that as there is in the refraction of <span class="pagenum">[Pg 72]</span><a name="Page_72" id="Page_72" />ordinary
+media a certain constant proportion between the sines of the angles
+which the incident ray and the refracted ray make with the
+perpendicular, so here there is such a proportion between CV and CD or
+IE; that is to say between the Sine of the angle which the incident
+ray makes with the perpendicular, and the horizontal intercept, in the
+Ellipse, between the refraction of this ray and the diameter CM. For
+the ratio of CV to CD is, as has been said, the same as that of N to
+the semi-diameter CG.</p>
+
+<p>33. I will add here, before passing away, that in comparing together
+the regular and irregular refraction of this crystal, there is this
+remarkable fact, that if ABPS be the spheroid by which light spreads
+in the Crystal in a certain space of time (which spreading, as has
+been said, serves for the irregular refraction), then the inscribed
+sphere BVST is the extension in the same space of time of the light
+which serves for the regular refraction.</p>
+
+<div class="figright" style="width: 250px;">
+<img src="images/pg072.png" width="250" height="314" alt="" title="" />
+</div>
+
+<p>For we have stated before this, that the line N being the radius of a
+spherical wave of light in air, while in the crystal it spread through
+the spheroid ABPS, the ratio of N to CS will be 156,962 to 93,410. But
+it has also been stated that the proportion of the regular refraction
+was 5 to 3; that is to say, that N being the radius of a spherical
+wave of light in air, its extension in the crystal would, in the same
+space of time, form a sphere the radius of which would be to N as 3 to
+5. Now 156,962 is to 93,410 as 5 to 3 less 1/41. So that it is
+sufficiently nearly, and may <span class="pagenum">[Pg 73]</span><a name="Page_73" id="Page_73" />be exactly, the sphere BVST, which the
+light describes for the regular refraction in the crystal, while it
+describes the spheroid BPSA for the irregular refraction, and while it
+describes the sphere of radius N in air outside the crystal.</p>
+
+<p>Although then there are, according to what we have supposed, two
+different propagations of light within the crystal, it appears that it
+is only in directions perpendicular to the axis BS of the spheroid
+that one of these propagations occurs more rapidly than the other; but
+that they have an equal velocity in the other direction, namely, in
+that parallel to the same axis BS, which is also the axis of the
+obtuse angle of the crystal.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg073.png" width="300" height="336" alt="" title="" />
+</div>
+
+<p>34. The proportion of the refraction being what we have just seen, I
+will now show that there necessarily follows thence that notable
+property of the ray which falling obliquely on the surface of the
+crystal enters it without suffering refraction. For supposing the same
+things as before, and that the ray makes with the same surface <i>g</i>G
+the angle RCG of <span class="pagenum">[Pg 74]</span><a name="Page_74" id="Page_74" />73 degrees 20 minutes, inclining to the same side as
+the crystal (of which ray mention has been made above); if one
+investigates, by the process above explained, the refraction CI, one
+will find that it makes exactly a straight line with RC, and that thus
+this ray is not deviated at all, conformably with experiment. This is
+proved as follows by calculation.</p>
+
+<p>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.</p>
+
+<p>Now the rectangle <i>g</i>DC is to the square of DI as the square of CG is
+to the square of CM; hence DI or CE will be 98,353. But as CE is to
+EI, so will CM be to MT, which will then be 18,127. And being added to
+ML, which is 11,609 (namely the sine of the angle LCM, which is 6
+degrees 40 minutes, taking CM 100,000 as radius) we get LT 27,936; and
+this is to LC 99,324 as CV to VR, that is to say, as 29,938, the
+tangent of the complement of the angle RCV, which is 73 degrees 20
+minutes, is to the radius of the Tables. Whence it appears that RCIT
+is a straight line; which was to be proved.</p>
+
+<p>35. Further it will be seen that the ray CI in emerging through the
+opposite surface of the crystal, ought to pass out quite straight,
+according to the following demonstration, which proves that the
+reciprocal relation of refraction obtains in this crystal the same as
+in other transparent bodies; that is to say, that if a ray RC in
+meeting the surface of the crystal CG is refracted as CI, the ray CI
+emerging through the opposite parallel surface of the <span class="pagenum">[Pg 75]</span><a name="Page_75" id="Page_75" />crystal, which
+I suppose to be IB, will have its refraction IA parallel to the ray
+RC.</p>
+
+<div class="figright" style="width: 400px;">
+<img src="images/pg075.png" width="400" height="337" alt="" title="" />
+</div>
+
+<p>Let the same things be supposed as before; that is to say, let CO,
+perpendicular to CR, represent a portion of a wave the continuation of
+which in the crystal is IK, so that the piece C will be continued on
+along the straight line CI, while O comes to K. Now if one takes a
+second period of time equal to the first, the piece K of the wave IK
+will, in this second period, have advanced along the straight line KB,
+equal and parallel to CI, because every piece of the wave CO, on
+arriving at the surface CK, ought to go on in the crystal the same as
+the piece C; and in this same time there will be formed in the air
+from the point I a partial spherical wave having a semi-diameter IA
+equal to KO, since KO has been traversed in an equal time. Similarly,
+if one considers some other point of the wave IK, such as <i>h</i>, it will
+go along <i>hm</i>, parallel to CI, to meet the surface IB, while the point
+K traverses K<i>l</i> equal to <i>hm</i>; and while this accomplishes the
+remainder <i>l</i>B, there will start from the point <i>m</i> a partial wave the
+semi-diameter of which, <i>mn</i>, will have the same ratio to <i>l</i>B as IA
+to <span class="pagenum">[Pg 76]</span><a name="Page_76" id="Page_76" />KB. Whence it is evident that this wave of semi-diameter <i>mn</i>, and
+the other of semi-diameter IA will have the same tangent BA. And
+similarly for all the partial spherical waves which will be formed
+outside the crystal by the impact of all the points of the wave IK
+against the surface of the Ether IB. It is then precisely the tangent
+BA which will be the continuation of the wave IK, outside the crystal,
+when the piece K has reached B. And in consequence IA, which is
+perpendicular to BA, will be the refraction of the ray CI on emerging
+from the crystal. Now it is clear that IA is parallel to the incident
+ray RC, since IB is equal to CK, and IA equal to KO, and the angles A
+and O are right angles.</p>
+
+<p>It is seen then that, according to our hypothesis, the reciprocal
+relation of refraction holds good in this crystal as well as in
+ordinary transparent bodies; as is thus in fact found by observation.</p>
+
+<p>36. I pass now to the consideration of other sections of the crystal,
+and of the refractions there produced, on which, as will be seen, some
+other very remarkable phenomena depend.</p>
+
+<p>Let ABH be a parallelepiped of crystal, and let the top surface AEHF
+be a perfect rhombus, the obtuse angles of which are equally divided
+by the straight line EF, and the acute angles by the straight line AH
+perpendicular to FE.</p>
+
+<p>The section which we have hitherto considered is that which passes
+through the lines EF, EB, and which at the same time cuts the plane
+AEHF at right angles. Refractions in this section have this in common
+with the refractions in ordinary media that the plane which is drawn
+through the incident ray and which also intersects the <span class="pagenum">[Pg 77]</span><a name="Page_77" id="Page_77" />surface of the
+crystal at right angles, is that in which the refracted ray also is
+found. But the refractions which appertain to every other section of
+this crystal have this strange property that the refracted ray always
+quits the plane of the incident ray perpendicular to the surface, and
+turns away towards the side of the slope of the crystal. For which
+fact we shall show the reason, in the first place, for the section
+through AH; and we shall show at the same time how one can determine
+the refraction, according to our hypothesis. Let there be, then, in
+the plane which passes through AH, and which is perpendicular to the
+plane AFHE, the incident ray RC; it is required to find its refraction
+in the crystal.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg077.png" width="500" height="432" alt="" title="" />
+</div>
+
+<p><span class="pagenum">[Pg 78]</span><a name="Page_78" id="Page_78" />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<i>qg</i>M, such as the
+light would form in spreading in the crystal, and let its section by
+the plane AEHF form the Ellipse QG<i>qg</i>, the major diameter of which
+Q<i>q</i>, which is in the line AH, will necessarily be one of the major
+diameters of the spheroid; because the axis of the spheroid being in
+the plane through FEB, to which QC is perpendicular, it follows that
+QC is also perpendicular to the axis of the spheroid, and consequently
+QC<i>q</i> one of its major diameters. But the minor diameter of this
+Ellipse, G<i>g</i>, will bear to Q<i>q</i> 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.</p>
+
+<p>Let the line N be the length of the travel of light in air during the
+time in which, within the crystal, it makes, from the centre C, the
+spheroid QC<i>qg</i>M. Then having drawn CO perpendicular to the ray CR and
+situate in the plane through CR and AH, let there be adjusted, across
+the angle ACO, the straight line OK equal to N and perpendicular to
+CO, and let it meet the straight line AH at K. Supposing consequently
+that CL is perpendicular to the surface of the crystal AEHF, and that
+CM is the refraction of the ray which falls perpendicularly on this
+same surface, let there be drawn a plane through the line CM and
+through KCH, making in the spheroid the semi-ellipse QM<i>q</i>, 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 <span class="pagenum">[Pg 79]</span><a name="Page_79" id="Page_79" />straight line CM to meet the surface, would be parallel
+to the plane QG<i>q</i>. If then through the point K one now draws KS
+parallel to G<i>g</i>, which will be parallel also to QX, the tangent to
+the Ellipse QG<i>q</i> 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<i>q</i>, because this plane through KS, as well as the
+plane which touches the spheroid at the point M, are parallel to QX,
+the tangent of the spheroid: for this consequence will be demonstrated
+at the end of this Treatise. Let this point of contact be at I, then
+making KC, QC, DC proportionals, draw DI parallel to CM; also join CI.
+I say that CI will be the required refraction of the ray RC. This will
+be manifest if, in considering CO, which is perpendicular to the ray
+RC, as a portion of the wave of light, we can demonstrate that the
+continuation of its piece C will be found in the crystal at I, when O
+has arrived at K.</p>
+
+<p>38. Now as in the Chapter on Reflexion, in demonstrating that the
+incident and reflected rays are always in the same plane perpendicular
+to the reflecting surface, we considered the breadth of the wave of
+light, so, similarly, we must here consider the breadth of the wave CO
+in the diameter G<i>g</i>. Taking then the breadth C<i>c</i> on the side toward
+the angle E, let the parallelogram CO<i>oc</i> be taken as a portion of a
+wave, and let us complete the parallelograms CK<i>kc</i>, CI<i>ic</i>, Kl<i>ik</i>,
+OK<i>ko</i>. In the time then that the line O<i>o</i> arrives at the surface of
+the crystal at K<i>k</i>, all the points of the wave CO<i>oc</i> will have
+arrived at the rectangle K<i>c</i> 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 <span class="pagenum">[Pg 80]</span><a name="Page_80" id="Page_80" />hemi-spheroid QM<i>q</i>,
+and similarly disposed. These hemi-spheroids will necessarily all
+touch the plane of the parallelogram KI<i>ik</i> at the same instant that
+O<i>o</i> has reached K<i>k</i>. Which is easy to comprehend, since, of these
+hemi-spheroids, all those which have their centres along the line CK,
+touch this plane in the line KI (for this is to be shown in the same
+way as we have demonstrated the refraction of the oblique ray in the
+principal section through EF) and all those which have their centres
+in the line C<i>c</i> will touch the same plane KI in the line I<i>i</i>; all
+these being similar to the hemi-spheroid QM<i>q</i>. Since then the
+parallelogram K<i>i</i> is that which touches all these spheroids, this
+same parallelogram will be precisely the continuation of the wave
+CO<i>oc</i> in the crystal, when O<i>o</i> has arrived at K<i>k</i>, because it forms
+the termination of the movement and because of the quantity of
+movement which occurs more there than anywhere else: and thus it
+appears that the piece C of the wave CO<i>oc</i> has its continuation at I;
+that is to say, that the ray RC is refracted as CI.</p>
+
+<p>From this it is to be noted that the proportion of the refraction for
+this section of the crystal is that of the line N to the semi-diameter
+CQ; by which one will easily find the refractions of all incident
+rays, in the same way as we have shown previously for the case of the
+section through FE; and the demonstration will be the same. But it
+appears that the said proportion of the refraction is less here than
+in the section through FEB; for it was there the same as the ratio of
+N to CG, that is to say, as 156,962 to 98,779, very nearly as 8 to 5;
+and here it is the ratio of N to CQ the major semi-diameter of the
+spheroid, that is to say, as 156,962 to 105,032, very nearly <span class="pagenum">[Pg 81]</span><a name="Page_81" id="Page_81" />as 3 to
+2, but just a little less. Which still agrees perfectly with what one
+finds by observation.</p>
+
+<p>39. For the rest, this diversity of proportion of refraction produces
+a very singular effect in this Crystal; which is that when it is
+placed upon a sheet of paper on which there are letters or anything
+else marked, if one views it from above with the two eyes situated in
+the plane of the section through EF, one sees the letters raised up by
+this irregular refraction more than when one puts one's eyes in the
+plane of section through AH: and the difference of these elevations
+appears by comparison with the other ordinary refraction of the
+crystal, the proportion of which is as 5 to 3, and which always raises
+the letters equally, and higher than the irregular refraction does.
+For one sees the letters and the paper on which they are written, as
+on two different stages at the same time; and in the first position of
+the eyes, namely, when they are in the plane through AH these two
+stages are four times more distant from one another than when the eyes
+are in the plane through EF.</p>
+
+<p>We will show that this effect follows from the refractions; and it
+will enable us at the same time to ascertain the apparent place of a
+point of an object placed immediately under the crystal, according to
+the different situation of the eyes.</p>
+
+<p>40. Let us see first by how much the irregular refraction of the plane
+through AH ought to lift the bottom of the crystal. Let the plane of
+this figure represent separately the section through Q<i>q</i> and CL, in
+which section there is also the ray RC, and let the semi-elliptic
+plane through Q<i>q</i> and CM be inclined to the former, as previously, by
+an angle of 6 degrees 40 minutes; and in this plane CI is then the
+refraction of the ray RC.<span class="pagenum">[Pg 82]</span><a name="Page_82" id="Page_82" /></p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg082.png" width="350" height="336" alt="" title="" />
+</div>
+
+<p>If now one considers the point I as at the bottom of the crystal, and
+that it is viewed by the rays ICR, <i>Icr</i>, refracted equally at the
+points C<i>c</i>, which should be equally distant from D, and that these
+rays meet the two eyes at R<i>r</i>; it is certain that the point I will
+appear raised to S where the straight lines RC, <i>rc</i>, meet; which
+point S is in DP, perpendicular to Q<i>q</i>. 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.</p>
+
+<p>Let there be described on Q<i>q</i> a semicircle cutting the ray CR at B,
+from which BV is drawn perpendicular to Q<i>q</i>; and let the proportion
+of the refraction for this section be, as before, that of the line N
+to the semi-diameter CQ.</p>
+
+<p>Then as N is to CQ so is VC to CD, as appears by the method of finding
+the refraction which we have shown above, Article 31; but as VC is to
+CD, so is VB to DS. Then as N is to CQ, so is VB to DS. Let ML be
+perpendicular to CL. And because I suppose the eyes R<i>r</i> to be distant
+about a foot or so from the crystal, and consequently the angle RS<i>r</i>
+very small, VB may be considered as equal to the semi-diameter CQ, and
+DP as equal to CL; then as N is to <span class="pagenum">[Pg 83]</span><a name="Page_83" id="Page_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, considered as equal to CL, will be to DS as 99,324 to
+70,283. And so the elevation of the point I by the refraction of this
+section is known.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg083.png" width="350" height="370" alt="" title="" />
+</div>
+
+<p>41. Now let there be represented the other section through EF in the
+figure before the preceding one; and let CM<i>g</i> be the semi-ellipse,
+considered in Articles 27 and 28, which is made by cutting a
+spheroidal wave having centre C. Let the point I, taken in this
+ellipse, be imagined again at the bottom of the Crystal; and let it be
+viewed by the refracted rays ICR, I<i>cr</i>, which go to the two eyes; CR
+and <i>cr</i> being equally inclined to the surface of the crystal G<i>g</i>.
+This being so, if one draws ID parallel to CM, which I suppose to be
+the refraction of the perpendicular ray incident at the point C, the
+distances DC, D<i>c</i>, will be equal, as is easy to see by that which has
+been demonstrated in Article 28. Now it is certain that the point I
+should appear at S where the straight lines RC, <i>rc</i>, meet when
+prolonged; and that this point will fall in the <span class="pagenum">[Pg 84]</span><a name="Page_84" id="Page_84" />line DP perpendicular
+to G<i>g</i>. If one draws IP perpendicular to this DP, it will be the
+distance PS which will mark the apparent elevation of the point I. Let
+there be described on G<i>g</i> a semicircle cutting CR at B, from which
+let BV be drawn perpendicular to G<i>g</i>; and let N to GC be the
+proportion of the refraction in this section, as in Article 28. Since
+then CI is the refraction of the radius BC, and DI is parallel to CM,
+VC must be to CD as N to GC, according to what has been demonstrated
+in Article 31. But as VC is to CD so is BV to DS. Let ML be drawn
+perpendicular to CL. And because I consider, again, the eyes to be
+distant above the crystal, BV is deemed equal to the semi-diameter CG;
+and hence DS will be a third proportional to the lines N and CG: also
+DP will be deemed equal to CL. Now CG consisting of 98,778 parts, of
+which CM contains 100,000, N is taken as 156,962. Then DS will be
+62,163. But CL is also determined, and contains 99,324 parts, as has
+been said in Articles 34 and 40. Then the ratio of PD to DS will be as
+99,324 to 62,163. And thus one knows the elevation of the point at the
+bottom I by the refraction of this section; and it appears that this
+elevation is greater than that by the refraction of the preceding
+section, since the ratio of PD to DS was there as 99,324 to 70,283.</p>
+
+<div class="figleft" style="width: 150px;">
+<img src="images/pg084.png" width="150" height="287" alt="" title="" />
+</div>
+
+<p>But by the regular refraction of the crystal, of which we have above
+said that the proportion is 5 to 3, the elevation of the point I, or
+P, from the bottom, will be 2/5 of the height DP; as appears by this
+figure, where the point P being viewed by the rays PCR, P<i>cr</i>,
+refracted equally <span class="pagenum">[Pg 85]</span><a name="Page_85" id="Page_85" />at the surface C<i>c</i>, this point must needs appear
+to be at S, in the perpendicular PD where the lines RC, <i>rc</i>, meet
+when prolonged: and one knows that the line PC is to CS as 5 to 3,
+since they are to one another as the sine of the angle CSP or DSC is
+to the sine of the angle SPC. And because the ratio of PD to DS is
+deemed the same as that of PC to CS, the two eyes Rr being supposed
+very far above the crystal, the elevation PS will thus be 2/5 of PD.</p>
+
+<div class="figright" style="width: 75px;">
+<img src="images/pg085.png" width="75" height="325" alt="" title="" />
+</div>
+
+<p>42. If one takes a straight line AB for the thickness of the crystal,
+its point B being at the bottom, and if one divides it at the points
+C, D, E, according to the proportions of the elevations found, making
+AE 3/5 of AB, AB to AC as 99,324 to 70,283, and AB to AD as 99,324 to
+62,163, these points will divide AB as in this figure. And it will be
+found that this agrees perfectly with experiment; that is to say by
+placing the eyes above in the plane which cuts the crystal according
+to the shorter diameter of the rhombus, the regular refraction will
+lift up the letters to E; and one will see the bottom, and the letters
+over which it is placed, lifted up to D by the irregular refraction.
+But by placing the eyes above in the plane which cuts the crystal
+according to the longer diameter of the rhombus, the regular
+refraction will lift the letters to E as before; but the irregular
+refraction will make them, at the same time, appear lifted up only to
+C; and in such a way that the interval CE will be quadruple the
+interval ED, which one previously saw.</p>
+
+
+<p>43. I have only to make the remark here that in both the positions of
+the eyes the images caused by the irregular refraction do not appear
+directly below those which proceed <span class="pagenum">[Pg 86]</span><a name="Page_86" id="Page_86" />from the regular refraction, but
+they are separated from them by being more distant from the
+equilateral solid angle of the Crystal. That follows, indeed, from all
+that has been hitherto demonstrated about the irregular refraction;
+and it is particularly shown by these last demonstrations, from which
+one sees that the point I appears by irregular refraction at S in the
+perpendicular line DP, in which line also the image of the point P
+ought to appear by regular refraction, but not the image of the point
+I, which will be almost directly above the same point, and higher than
+S.</p>
+
+<p>But as to the apparent elevation of the point I in other positions of
+the eyes above the crystal, besides the two positions which we have
+just examined, the image of that point by the irregular refraction
+will always appear between the two heights of D and C, passing from
+one to the other as one turns one's self around about the immovable
+crystal, while looking down from above. And all this is still found
+conformable to our hypothesis, as any one can assure himself after I
+shall have shown here the way of finding the irregular refractions
+which appear in all other sections of the crystal, besides the two
+which we have considered. Let us suppose one of the faces of the
+crystal, in which let there be the Ellipse HDE, the centre C of which
+is also the centre of the spheroid HME in which the light spreads, and
+of which the said Ellipse is the section. And let the incident ray be
+RC, the refraction of which it is required to find.</p>
+
+<p>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, <span class="pagenum">[Pg 87]</span><a name="Page_87" id="Page_87" />let OK be adjusted across the angle OCK, so as
+to be perpendicular to OC and equal to the line N, which I suppose to
+measure the travel of the light in air during the time that it spreads
+in the crystal through the spheroid HDEM. Then in the plane of the
+Ellipse HDE let KT be drawn, through the point K, perpendicular to
+BCK. Now if one conceives a plane drawn through the straight line KT
+and touching the spheroid HME at I, the straight line CI will be the
+refraction of the ray RC, as is easy to deduce from that which has
+been demonstrated in Article 36.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg087.png" width="350" height="268" alt="" title="" />
+</div>
+
+<p>But it must be shown how one can determine the point of contact I. Let
+there be drawn parallel to the line KT a line HF which touches the
+Ellipse HDE, and let this point of contact be at H. And having drawn a
+straight line along CH to meet KT at T, let there be imagined a plane
+passing through the same CH and through CM (which I suppose to be the
+refraction of the perpendicular ray), which makes in the spheroid the
+elliptical section HME. It is certain that the plane which will pass
+through the straight line KT, and which will touch the spheroid, will
+touch it at a point in the Ellipse HME, according to the Lemma which
+will be demonstrated at the end of the <span class="pagenum">[Pg 88]</span><a name="Page_88" id="Page_88" />Chapter. Now this point is
+necessarily the point I which is sought, since the plane drawn through
+TK can touch the spheroid at one point only. And this point I is easy
+to determine, since it is needful only to draw from the point T, which
+is in the plane of this Ellipse, the tangent TI, in the way shown
+previously. For the Ellipse HME is given, and its conjugate
+semi-diameters are CH and CM; because a straight line drawn through M,
+parallel to HE, touches the Ellipse HME, as follows from the fact that
+a plane taken through M, and parallel to the plane HDE, touches the
+spheroid at that point M, as is seen from Articles 27 and 23. For the
+rest, the position of this ellipse, with respect to the plane through
+the ray RC and through CK, is also given; from which it will be easy
+to find the position of CI, the refraction corresponding to the ray
+RC.</p>
+
+<p>Now it must be noted that the same ellipse HME serves to find the
+refractions of any other ray which may be in the plane through RC and
+CK. Because every plane, parallel to the straight line HF, or TK,
+which will touch the spheroid, will touch it in this ellipse,
+according to the Lemma quoted a little before.</p>
+
+<p>I have investigated thus, in minute detail, the properties of the
+irregular refraction of this Crystal, in order to see whether each
+phenomenon that is deduced from our hypothesis accords with that which
+is observed in fact. And this being so it affords no slight proof of
+the truth of our suppositions and principles. But what I am going to
+add here confirms them again marvellously. It is this: that there are
+different sections of this Crystal, the surfaces of which, thereby
+produced, give rise to refractions precisely such as they ought to be,
+and as I had foreseen them, according to the preceding Theory.</p>
+
+<p><span class="pagenum">[Pg 89]</span><a name="Page_89" id="Page_89" />In order to explain what these sections are, let ABKF <i>be</i> the
+principal section through the axis of the crystal ACK, in which there
+will also be the axis SS of a spheroidal wave of light spreading in
+the crystal from the centre C; and the straight line which cuts SS
+through the middle and at right angles, namely PP, will be one of the
+major diameters.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg089.png" width="300" height="213" alt="{Section ABKF}" title="" />
+</div>
+
+<p>Now as in the natural section of the crystal, made by a plane parallel
+to two opposite faces, which plane is here represented by the line GG,
+the refraction of the surfaces which are produced by it will be
+governed by the hemi-spheroids GNG, according to what has been
+explained in the preceding Theory. Similarly, cutting the Crystal
+through NN, by a plane perpendicular to the parallelogram ABKF, the
+refraction of the surfaces will be governed by the hemi-spheroids NGN.
+And if one cuts it through PP, perpendicularly to the said
+parallelogram, the refraction of the surfaces ought to be governed by
+the hemi-spheroids PSP, and so for others. But I saw that if the plane
+NN was almost perpendicular to the plane GG, making the angle NCG,
+which is on the side A, an angle of 90 degrees 40 minutes, the
+hemi-spheroids NGN would become similar to the hemi-spheroids GNG,
+since the planes NN and GG were equally inclined by an angle of 45
+degrees 20 minutes to the axis SS. In consequence it must needs be, if
+our theory is true, that the surfaces which the section through <span class="pagenum">[Pg 90]</span><a name="Page_90" id="Page_90" />NN
+produces should effect the same refractions as the surfaces of the
+section through GG. And not only the surfaces of the section NN but
+all other sections produced by planes which might be inclined to the
+axis at an angle equal to 45 degrees 20 minutes. So that there are an
+infinitude of planes which ought to produce precisely the same
+refractions as the natural surfaces of the crystal, or as the section
+parallel to any one of those surfaces which are made by cleavage.</p>
+
+<p>I saw also that by cutting it by a plane taken through PP, and
+perpendicular to the axis SS, the refraction of the surfaces ought to
+be such that the perpendicular ray should suffer thereby no deviation;
+and that for oblique rays there would always be an irregular
+refraction, differing from the regular, and by which objects placed
+beneath the crystal would be less elevated than by that other
+refraction.</p>
+
+<p>That, similarly, by cutting the crystal by any plane through the axis
+SS, such as the plane of the figure is, the perpendicular ray ought to
+suffer no refraction; and that for oblique rays there were different
+measures for the irregular refraction according to the situation of
+the plane in which the incident ray was.</p>
+
+<p>Now these things were found in fact so; and, after that, I could not
+doubt that a similar success could be met with everywhere. Whence I
+concluded that one might form from this crystal solids similar to
+those which are its natural forms, which should produce, at all their
+surfaces, the same regular and irregular refractions as the natural
+surfaces, and which nevertheless would cleave in quite other ways, and
+not in directions parallel to any of their faces. That out of it one
+would be able to fashion pyramids, having their base square,
+pentagonal, hexagonal, or with as many sides <span class="pagenum">[Pg 91]</span><a name="Page_91" id="Page_91" />as one desired, all the
+surfaces of which should have the same refractions as the natural
+surfaces of the crystal, except the base, which will not refract the
+perpendicular ray. These surfaces will each make an angle of 45
+degrees 20 minutes with the axis of the crystal, and the base will be
+the section perpendicular to the axis.</p>
+
+<p>That, finally, one could also fashion out of it triangular prisms, or
+prisms with as many sides as one would, of which neither the sides nor
+the bases would refract the perpendicular ray, although they would yet
+all cause double refraction for oblique rays. The cube is included
+amongst these prisms, the bases of which are sections perpendicular to
+the axis of the crystal, and the sides are sections parallel to the
+same axis.</p>
+
+<p>From all this it further appears that it is not at all in the
+disposition of the layers of which this crystal seems to be composed,
+and according to which it splits in three different senses, that the
+cause resides of its irregular refraction; and that it would be in
+vain to wish to seek it there.</p>
+
+<p>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.</p>
+
+<p>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 <span class="pagenum">[Pg 92]</span><a name="Page_92" id="Page_92" />way as spectacle glasses, and
+polishes it simply by continuing the work, but ever reducing the
+material. I have not, however, been able to give it perfect clarity
+and transparency; but the evenness which the surfaces acquire enables
+one to observe in them the effects of refraction better than in those
+made by cleaving the stone, which always have some inequality.</p>
+
+<p>Even when the surface is only moderately smoothed, if one rubs it over
+with a little oil or white of egg, it becomes quite transparent, so
+that the refraction is discerned in it quite distinctly. And this aid
+is specially necessary when it is wished to polish the natural
+surfaces to remove the inequalities; because one cannot render them
+lucent equally with the surfaces of other sections, which take a
+polish so much the better the less nearly they approximate to these
+natural planes.</p>
+
+<p>Before finishing the treatise on this Crystal, I will add one more
+marvellous phenomenon which I discovered after having written all the
+foregoing. For though I have not been able till now to find its cause,
+I do not for that reason wish to desist from describing it, in order
+to give opportunity to others to investigate it. It seems that it will
+be necessary to make still further suppositions besides those which I
+have made; but these will not for all that cease to keep their
+probability after having been confirmed by so many tests.</p>
+
+<div class="figcenter" style="width: 450px;">
+<img src="images/pg093.png" width="450" height="360" alt="" title="" />
+
+</div>
+
+<p>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,
+<span class="pagenum">[Pg 93]</span><a name="Page_93" id="Page_93" />regular and irregular. On penetrating thence into the other piece
+each ray will pass there without further dividing itself in two; but
+that one which underwent the regular refraction, as here DG, will
+undergo again only a regular refraction at GH; and the other, CE, an
+irregular refraction at EF. And the same thing occurs not only in this
+disposition, but also in all those cases in which the principal
+section of each of the pieces is situated in one and the same plane,
+without it being needful for the two neighbouring surfaces to be
+parallel. Now it is marvellous why the rays CE and DG, incident from
+the air on the lower crystal, do not divide themselves the same as the
+first ray AB. One would say that it must be that the ray DG in passing
+through the upper piece has lost something which is necessary to move
+the matter which serves for the irregular refraction; and that
+likewise CE has lost that which <span class="pagenum">[Pg 94]</span><a name="Page_94" id="Page_94" />was necessary to move the matter
+which serves for regular refraction: but there is yet another thing
+which upsets this reasoning. It is that when one disposes the two
+crystals in such a way that the planes which constitute the principal
+sections intersect one another at right angles, whether the
+neighbouring surfaces are parallel or not, then the ray which has come
+by the regular refraction, as DG, undergoes only an irregular
+refraction in the lower piece; and on the contrary the ray which has
+come by the irregular refraction, as CE, undergoes only a regular
+refraction.</p>
+
+<p>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.</p>
+
+<p>When one considers here how, while the rays CE, DG, remain the same,
+it depends on the position that one gives to the lower piece, whether
+it divides them both in two, or whether it does not divide them, and
+yet how the ray AB above is always divided, it seems that one is
+obliged to conclude that the waves of light, after having passed
+through the first crystal, acquire a certain form or disposition in
+virtue of which, when meeting the texture of the second crystal, in
+certain positions, they can move the two different kinds of matter
+which serve for the two species of refraction; and when meeting the
+second crystal in another position are able to move only one of these
+kinds of matter. But to tell how this occurs, I have hitherto found
+nothing which satisfies me.</p>
+
+<p><span class="pagenum">[Pg 95]</span><a name="Page_95" id="Page_95" />Leaving then to others this research, I pass to what I have to say
+touching the cause of the extraordinary figure of this crystal, and
+why it cleaves easily in three different senses, parallel to any one
+of its surfaces.</p>
+
+<p>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.</p>
+
+<p>Rock Crystal grows ordinarily in hexagonal bars, and diamonds are
+found which occur with a square point and polished surfaces. There is
+a species of small flat stones, piled up directly upon one another,
+which are all of pentagonal figure with rounded angles, and the sides
+a little folded inwards. The grains of gray salt which are formed from
+sea water affect the figure, or at least the angle, of the cube; and
+in the congelations of other salts, and in that of sugar, there are
+found other solid angles with perfectly flat faces. Small snowflakes
+almost always fall in little stars with 6 points, and sometimes in
+hexagons with straight sides. And I have often observed, in water
+which is beginning to freeze, a kind of flat and thin foliage of ice,
+the middle ray of which throws out branches inclined at an angle of 60
+degrees. All these things are worthy of being carefully investigated
+to ascertain how and by what artifice nature there operates. But it is
+not now my intention to treat fully of this matter. It seems that in
+general the regularity which occurs in these productions comes from
+the arrangement of the small invisible equal particles of which they
+are composed. And, coming to our Iceland Crystal, I say <span class="pagenum">[Pg 96]</span><a name="Page_96" id="Page_96" />that if there
+were a pyramid such as ABCD, composed of small rounded corpuscles, not
+spherical but flattened spheroids, such as would be made by the
+rotation of the ellipse GH around its lesser diameter EF (of which the
+ratio to the greater diameter is very nearly that of 1 to the square
+root of 8)&mdash;I say that then the solid angle of the point D would be
+equal to the obtuse and equilateral angle of this Crystal. I say,
+further, that if these corpuscles were lightly stuck together, on
+breaking this pyramid it would break along faces parallel to those
+that make its point: and by this means, as it is easy to see, it would
+produce prisms similar to those of the same crystal as this other
+figure represents. The reason is that when broken in this fashion a
+whole layer separates easily from its neighbouring layer since each
+spheroid has to be detached only from the three spheroids of the next
+layer; of which three there is but one which touches it on its
+flattened surface, and the other two at the edges. And the reason why
+the surfaces separate sharp and polished is that if any spheroid of
+the neighbouring surface would come out by attaching itself to the
+surface which is being separated, it would be needful for it to detach
+itself from six other spheroids which hold it locked, and four of
+which press it by these flattened surfaces. Since then not only the
+angles of our crystal but also the manner in which it splits agree
+precisely with what is observed in the assemblage composed of such
+spheroids, there is great reason to believe that the particles are
+shaped and ranged in the same way.</p>
+
+<div class="figright" style="width: 200px;">
+<img src="images/pg096.png" width="200" height="310" alt="{Pyramid and section of spheroids}" title="" />
+</div>
+
+<p><span class="pagenum">[Pg 97]</span><a name="Page_97" id="Page_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.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg097.png" width="300" height="228" alt="{paralleloid arrangement of spheroids with planes of
+potential cleavage}" title="" />
+</div>
+
+<p>It may be objected that this crystal, being so composed, might be
+capable of cleavage in yet two more fashions; one of which would be
+along planes parallel to the base of the pyramid, that is to say to
+the triangle ABC; the other would be parallel to a plane the trace of
+which is marked by the lines GH, HK, KL. To which I say that both the
+one and the other, though practicable, are more difficult than those
+which were parallel to any one of the three planes of the pyramid; and
+that therefore, when striking on the crystal in order to break it, it
+ought always to split rather along these three planes than along the
+two others. When one has a number of spheroids of the form above
+described, and ranges them in a pyramid, one sees why the two methods
+of division are more difficult. For in the case of that division which
+would be parallel to the base, <span class="pagenum">[Pg 98]</span><a name="Page_98" id="Page_98" />each spheroid would be obliged to
+detach itself from three others which it touches upon their flattened
+surfaces, which hold more strongly than the contacts at the edges. And
+besides that, this division will not occur along entire layers,
+because each of the spheroids of a layer is scarcely held at all by
+the 6 of the same layer that surround it, since they only touch it at
+the edges; so that it adheres readily to the neighbouring layer, and
+the others to it, for the same reason; and this causes uneven
+surfaces. Also one sees by experiment that when grinding down the
+crystal on a rather rough stone, directly on the equilateral solid
+angle, one verily finds much facility in reducing it in this
+direction, but much difficulty afterwards in polishing the surface
+which has been flattened in this manner.</p>
+
+<p>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.</p>
+
+<p>However, that which has made me know that in the crystal there are
+layers in this last fashion, is that in a piece weighing half a pound
+which I possess, one sees that it is split along its length, as is the
+above-mentioned prism by the plane GHKL; as appears by colours of the
+Iris extending throughout this whole plane although the two pieces
+still hold together. All this proves then that the composition of the
+crystal is such as we have stated. To <span class="pagenum">[Pg 99]</span><a name="Page_99" id="Page_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 incision is easily
+made. This follows manifestly from the situation of the small
+spheroids; over which, in the first manner, the knife glides; but in
+the other manner it seizes them from beneath almost as if they were
+the scales of a fish.</p>
+
+<p>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.</p>
+
+
+<p><i>Calculations which have been supposed in this Chapter</i>.</p>
+
+<p>Mr. Bartholinus, in his treatise of this Crystal, puts at 101 degrees
+the obtuse angles of the faces, which I have stated to be 101 degrees
+52 minutes. He states that he measured these angles directly on the
+crystal, which is difficult to do with ultimate exactitude, because
+the edges such as CA, CB, in this figure, are generally worn, and not
+quite straight. For more certainty, therefore, I preferred to measure
+actually the obtuse angle by which the faces <span class="pagenum">[Pg 100]</span><a name="Page_100" id="Page_100" />CBDA, CBVF, are inclined
+to one another, namely the angle OCN formed by drawing CN
+perpendicular to FV, and CO perpendicular to DA. This angle OCN I
+found to be 105 degrees; and its supplement CNP, to be 75 degrees, as
+it should be.</p>
+
+<div class="figleft" style="width: 350px;">
+<img src="images/pg100.png" width="350" height="235" alt="" title="" />
+</div>
+
+<p>To find from this the obtuse angle BCA, I imagined a sphere having its
+centre at C, and on its surface a spherical triangle, formed by the
+intersection of three planes which enclose the solid angle C. In this
+equilateral triangle, which is ABF in this other figure, I see that
+each of the angles should be 105 degrees, namely equal to the angle
+OCN; and that each of the sides should be of as many degrees as the
+angle ACB, or ACF, or BCF. Having then drawn the arc FQ perpendicular
+to the side AB, which it divides equally at Q, the triangle FQA has a
+right angle at Q, the angle A 105 degrees, and F half as much, namely
+52 degrees 30 minutes; whence the hypotenuse AF is found to be 101
+degrees 52 minutes. And this arc AF is the measure of the angle ACF in
+the figure of the crystal.</p>
+
+<div class="figright" style="width: 150px;">
+<img src="images/pg100a.png" width="150" height="165" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 101]</span><a name="Page_101" id="Page_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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Let CP and CS be prolonged and meet the tangent DM at D and Z; and
+from the point of contact M let MN and MO be drawn as perpendiculars
+to CP and CS. Now because the angles SCP, GCL, are right angles, the
+<span class="pagenum">[Pg 102]</span><a name="Page_102" id="Page_102" />angle PCL will be equal to GCS which was 45 degrees 20 minutes. And
+deducting the angle LCM, which is 6 degrees 40 minutes, from LCP,
+which is 45 degrees 20 minutes, there remains MCP, 38 degrees 40
+minutes. Considering then CM as a radius of 100,000 parts, MN, the
+sine of 38 degrees 40 minutes, will be 62,479. And in the right-angled
+triangle MND, MN will be to ND as the radius of the Tables is to the
+tangent of 45 degrees 20 minutes (because the angle NMD is equal to
+DCL, or GCS); that is to say as 100,000 to 101,170: whence results ND
+63,210. But NC is 78,079 of the same parts, CM being 100,000, because
+NC is the sine of the complement of the angle MCP, which was 38
+degrees 40 minutes. Then the whole line DC is 141,289; and CP, which
+is a mean proportional between DC and CN, since MD touches the
+Ellipse, will be 105,032.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg102.png" width="300" height="300" alt="" title="" />
+</div>
+
+<p>Similarly, because the angle OMZ is equal to CDZ, or LCZ, which is 44
+degrees 40 minutes, being the complement of GCS, it follows that, as
+the radius of the Tables is to the tangent of 44 degrees 40 minutes,
+so will OM 78,079 be to OZ 77,176. But OC is 62,479 of these same
+parts of which CM is 100,000, because it is equal to MN, the sine of
+the angle MCP, which is 38 degrees 40 minutes. Then the whole line CZ
+is 139,655; and CS, which is a mean proportional between CZ and CO
+will be 93,410.</p>
+
+<p><span class="pagenum">[Pg 103]</span><a name="Page_103" id="Page_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 complement of
+the angle LCM, which is 6 degrees 40 minutes. And since the angle LCD
+is 45 degrees 20 minutes, being equal to GCS, the side LD is found to
+be 100,486: whence deducting ML 11,609 there will remain MD 88,877.
+Now as CD (which was 141,289) is to DM 88,877, so will CP 105,032 be
+to PE 66,070. But as the rectangle MEH (or rather the difference of
+the squares on CM and CE) is to the square on MC, so is the square on
+PE to the square on C<i>g</i>; 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 <i>g</i>C. But DP, CP, and PE are known; hence also one knows GC,
+which is 98,779.</p>
+
+
+<p><i>Lemma which has been supposed</i>.</p>
+
+<p>If a spheroid is touched by a straight line, and also by two or more
+planes which are parallel to this line, though not parallel to one
+another, all the points of contact of the line, as well as of the
+planes, will be in one and the same ellipse made by a plane which
+passes through the centre of the spheroid.</p>
+
+<p>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.<span class="pagenum">[Pg 104]</span><a name="Page_104" id="Page_104" /></p>
+
+<div class="figright" style="width: 350px;">
+<img src="images/pg104.png" width="350" height="400" alt="" title="" />
+</div>
+
+<p>Through the line BM, and through the points O and A, let there be
+drawn planes parallel to one another, which, in cutting the spheroid
+make the ellipses LBD, POP, QAQ; which will all be similar and
+similarly disposed, and will have their centres K, N, R, in one and
+the same diameter of the spheroid, which will also be the diameter of
+the ellipse made by the section of the plane that passes through the
+centre of the spheroid, and which cuts the planes of the three said
+Ellipses at right angles: for all this is manifest by proposition 15
+of the book of Conoids and Spheroids of Archimedes. Further, the two
+latter planes, which are drawn through the points O and A, will also,
+by cutting the planes which touch the spheroid in these same points,
+generate straight lines, as OH and AS, which will, as is easy to see,
+be parallel to BM; and all three, BM, OH, AS, will touch the Ellipses
+LBD, POP, QAQ in these points, B, O, A; since they are in the planes
+of these ellipses, and at the same time in the planes which touch the
+spheroid. If now from these points B, O, A, there are drawn the
+straight lines BK, ON, AR, through the centres of the same ellipses,
+and if through these centres there are drawn also the diameters LD,
+PP, QQ, parallel to the tangents BM, OH, AS; these will be conjugate
+to the aforesaid BK, ON, AR. And because the three ellipses are
+similar and similarly <span class="pagenum">[Pg 105]</span><a name="Page_105" id="Page_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 intersection of
+this plane. Which was to be proved. And it is manifest that the
+demonstration would be the same if, besides the points O, A, there had
+been others in which the spheroid had been touched by planes parallel
+to the straight line BM.</p>
+
+
+
+<hr style="width: 65%;" />
+<h2><a name="CHAPTER_VI" id="CHAPTER_VI" />CHAPTER VI</h2>
+
+<h3>ON THE FIGURES OF THE TRANSPARENT BODIES</h3>
+
+<h4><i>Which serve for Refraction and for Reflexion</i>.</h4>
+
+<div style="width: 154px; float: left; margin-right: .2em;">
+<img src="images/ch06.png" width="154" height="150" alt="A" title="A" />
+</div><p>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 refraction, to
+collect or disperse the rays of light, as may be desired. For though I
+do not see yet that there are means of making use of these figures, so
+far as relates to Refraction, not only because of the difficulty of
+shaping the glasses of Telescopes with the requisite<span class="pagenum">[Pg 106]</span><a name="Page_106" id="Page_106" /> exactitude
+according to these figures, but also because there exists in
+refraction itself a property which hinders the perfect concurrence of
+the rays, as Mr. Newton has very well proved by experiment, I will yet
+not desist from relating the invention, since it offers itself, so to
+speak, of itself, and because it further confirms our Theory of
+refraction, by the agreement which here is found between the refracted
+ray and the reflected ray. Besides, it may occur that some one in the
+future will discover in it utilities which at present are not seen.</p>
+
+<div class="figcenter" style="width: 500px;">
+<img src="images/pg106.png" width="500" height="191" alt="" title="" />
+</div>
+
+<p>To proceed then to these figures, let us suppose first that it is
+desired to find a surface CDE which shall reassemble at a point B rays
+coming from another point A; and that the summit of the surface shall
+be the given point D in the straight line AB. I say that, whether by
+reflexion or by refraction, it is only necessary to make this surface
+such that the path of the light from the point A to all points of the
+curved line CDE, and from these to the point of concurrence (as here
+the path along the straight lines AC, CB, along AL, LB, and along AD,
+DB), shall be everywhere traversed in equal times: by which principle
+the finding of these curves becomes very easy.<span class="pagenum">[Pg 107]</span><a name="Page_107" id="Page_107" /></p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg107.png" width="300" height="236" alt="" title="" />
+</div>
+
+<p>So far as relates to the reflecting surface, since the sum of the
+lines AC, CB ought to be equal to that of AD, DB, it appears that DCE
+ought to be an ellipse; and for refraction, the ratio of the
+velocities of waves of light in the media A and B being supposed to be
+known, for example that of 3 to 2 (which is the same, as we have
+shown, as the ratio of the Sines in the refraction), it is only
+necessary to make DH equal to 3/2 of DB; and having after that
+described from the centre A some arc FC, cutting DB at F, then
+describe another from centre B with its semi-diameter BX equal to 2/3
+of FH; and the point of intersection of the two arcs will be one of
+the points required, through which the curve should pass. For this
+point, having been found in this fashion, it is easy forthwith to
+demonstrate that the time along AC, CB, will be equal to the time
+along AD, DB.</p>
+
+<p>For assuming that the line AD represents the time which the light
+takes to traverse this same distance AD in air, it is evident that DH,
+equal to 3/2 of DB, will represent the time of the light along DB in
+the medium, because it needs here more time in proportion as its speed
+is slower. Therefore the whole line AH will represent the time along
+AD, DB. Similarly the line AC or AF will represent the time along AC;
+and FH being by construction equal to 3/2 of CB, it will represent the
+time along CB in the medium; and in consequence the whole line AH will
+represent also the time along AC, CB. Whence it appears that the <span class="pagenum">[Pg 108]</span><a name="Page_108" id="Page_108" />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.</p>
+
+<p>In order to show further that the surfaces, which these curves will
+generate by revolution, will direct all the rays which reach them from
+the point A in such wise that they tend towards B, let there be
+supposed a point K in the curve, farther from D than C is, but such
+that the straight line AK falls from outside upon the curve which
+serves for the refraction; and from the centre B let the arc KS be
+described, cutting BD at S, and the straight line CB at R; and from
+the centre A describe the arc DN meeting AK at N.</p>
+
+<p>Since the sums of the times along AK, KB, and along AC, CB are equal,
+if from the former sum one deducts the time along KB, and if from the
+other one deducts the time along RB, there will remain the time along
+AK as equal to the time along the two parts AC, CR. Consequently in
+the time that the light has come along AK it will also have come along
+AC and will in addition have made, in the medium from the centre C, a
+partial spherical wave, having a semi-diameter equal to CR. And this
+wave will necessarily touch the circumference KS at R, since CB cuts
+this circumference at right angles. Similarly, having taken any other
+point L in the curve, one can show that in the same time as the light
+passes along AL it will also have come along AL and in addition will
+have made a partial wave, from the centre L, which will touch the same
+circumference KS. And so with all other points of the curve CDE. Then
+at the moment that the light reaches K the arc KRS will be the
+termination <span class="pagenum">[Pg 109]</span><a name="Page_109" id="Page_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.</p>
+
+<p>One might also determine the point C, and all the others, in this
+curve which serves for the refraction, by dividing DA at G in such a
+way that DG is 2/3 of DA, and describing from the centre B any arc CX
+which cuts BD at N, and another from the centre A with its
+semi-diameter AF equal to 3/2 of GX; or rather, having described, as
+before, the arc CX, it is only necessary to make DF equal to 3/2 of
+DX, and from-the centre A to strike the arc FC; for these two
+constructions, as may be easily known, come back to the first one
+which was shown before. And it is manifest by the last method that
+this curve is the same that Mr. Des Cartes has given in his Geometry,
+and which he calls the first of his Ovals.</p>
+
+<p>It is only a part of this oval which serves for the refraction,
+namely, the part DK, ending at K, if AK is the tangent. As to the,
+other part, Des Cartes has remarked that it could serve for
+reflexions, if there were some material of a mirror of such a nature
+that by its <span class="pagenum">[Pg 110]</span><a name="Page_110" id="Page_110" />means the force of the rays (or, as we should say, the
+velocity of the light, which he could not say, since he held that the
+movement of light was instantaneous) could be augmented in the
+proportion of 3 to 2. But we have shown that in our way of explaining
+reflexion, such a thing could not arise from the matter of the mirror,
+and it is entirely impossible.</p>
+
+<div class="figcenter" style="width: 400px;">
+<img src="images/pg110.png" width="400" height="439" alt="" title="" />
+</div>
+
+<p>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 <span class="pagenum">[Pg 111]</span><a name="Page_111" id="Page_111" />construction of which differs in no way from that of the
+oval, except that FC, which previously was an arc of a circle, is here
+a straight line, perpendicular to DB. For the wave of light DN, being
+likewise represented by a straight line, it will be seen that all the
+points of this wave, travelling as far as the surface KD along lines
+parallel to DB, will advance subsequently towards the point B, and
+will arrive there at the same time. As for the Ellipse which served
+for reflexion, it is evident that it will here become a parabola,
+since its focus A may be regarded as infinitely distant from the
+other, B, which is here the focus of the parabola, towards which all
+the reflexions of rays parallel to AB tend. And the demonstration of
+these effects is just the same as the preceding.</p>
+
+<p>But that this curved line CDE which serves for refraction is an
+Ellipse, and is such that its major diameter is to the distance
+between its foci as 3 to 2, which is the proportion of the refraction,
+can be easily found by the calculus of Algebra. For DB, which is
+given, being called <i>a</i>; its undetermined perpendicular DT being
+called <i>x</i>; and TC <i>y</i>; FB will be <i>a - y</i>; CB will be
+sqrt(<i>xx + aa - 2ay + yy</i>). But the nature of the curve is such that
+2/3 of TC together with CB is equal to DB, as was stated in the last
+construction: then the equation will be between <i>(2/3)y +
+sqrt(xx + aa - 2ay + yy)</i> and <i>a</i>; which being reduced, gives
+<i>(6/5)ay - yy</i> equal to <i>(9/5)xx</i>; that is to say that
+having made DO equal to 6/5 of DB, the rectangle DFO is equal to 9/5
+of the square on FC. Whence it is seen that DC is an ellipse, of which
+the axis DO is to the parameter as 9 to 5; and therefore the square on
+DO is to the square of the distance between the foci as 9 to 9 - 5,
+that is to say 4; and finally the line DO will be to this distance as
+3 to 2.<span class="pagenum">[Pg 112]</span><a name="Page_112" id="Page_112" /></p>
+
+<div class="figleft" style="width: 400px;">
+<img src="images/pg112.png" width="400" height="307" alt="" title="" />
+</div>
+
+<p>Again, if one supposes the point B to be infinitely distant, in lieu
+of our first oval we shall find that CDE is a true Hyperbola; which
+will make those rays become parallel which come from the point A. And
+in consequence also those which are parallel within the transparent
+body will be collected outside at the point A. Now it must be remarked
+that CX and KS become straight lines perpendicular to BA, because they
+represent arcs of circles the centre of which is infinitely distant.
+And the intersection of the perpendicular CX with the arc FC will give
+the point C, one of those through which the curve ought to pass. And
+this operates so that all the parts of the wave of light DN, coming to
+meet the surface KDE, will advance thence along parallels to KS and
+will arrive at this straight line at the same time; of which the proof
+is again the same as that which served for the first oval. Besides one
+finds by a calculation as easy as the preceding one, that CDE is here
+a hyperbola of which the axis DO <span class="pagenum">[Pg 113]</span><a name="Page_113" id="Page_113" />is 4/5 of AD, and the parameter
+equal to AD. Whence it is easily proved that DO is to the distance
+between the foci as 3 to 2.</p>
+
+<div class="figright" style="width: 400px;">
+<img src="images/pg113.png" width="400" height="316" alt="" title="" />
+</div>
+
+<p>These are the two cases in which Conic sections serve for refraction,
+and are the same which are explained, in his <i>Dioptrique</i>, by Des
+Cartes, who first found out the use of these lines in relation to
+refraction, as also that of the Ovals the first of which we have
+already set forth. The second oval is that which serves for rays that
+tend to a given point; in which oval, if the apex of the surface which
+receives the rays is D, it will happen that the other apex will be
+situated between B and A, or beyond A, according as the ratio of AD to
+DB is given of greater or lesser value. And in this latter case it is
+the same as that which Des Cartes calls his 3rd oval.</p>
+
+<p>Now the finding and construction of this second oval is <span class="pagenum">[Pg 114]</span><a name="Page_114" id="Page_114" />the same as
+that of the first, and the demonstration of its effect likewise. But
+it is worthy of remark that in one case this oval becomes a perfect
+circle, namely when the ratio of AD to DB is the same as the ratio of
+the refractions, here as 3 to 2, as I observed a long time ago. The
+4th oval, serving only for impossible reflexions, there is no need to
+set it forth.</p>
+
+<div class="figleft" style="width: 400px;">
+<img src="images/pg114.png" width="400" height="316" alt="" title="" />
+</div>
+
+<p>As for the manner in which Mr. Des Cartes discovered these lines,
+since he has given no explanation of it, nor any one else since that I
+know of, I will say here, in passing, what it seems to me it must have
+been. Let it be proposed to find the surface generated by the
+revolution of the curve KDE, which, receiving the incident rays coming
+to it from the point A, shall deviate them toward the point B. Then
+considering this other curve as already known, and that its apex D is
+in the straight line AB, let us divide it up into an infinitude of
+small pieces by the points G, C, F; and having drawn from each of
+these points, straight lines towards A to represent the incident rays,
+and other straight lines towards B, let there also be described with
+centre A the arcs GL, CM, FN, DO, cutting the rays that come from A at
+L, M, N, O; and from the points K, G, C, F, <span class="pagenum">[Pg 115]</span><a name="Page_115" id="Page_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.</p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/pg115.png" width="600" height="274" alt="" title="" />
+</div>
+
+<p>Then AK being an incident ray, and KB its refraction within the
+medium, it needs must be, according to the law of refraction which was
+known to Mr. Des Cartes, that the sine of the angle ZKA should be to
+the sine of the angle HKB as 3 to 2, supposing that this is the
+proportion of the refraction of glass; or rather, that the sine of the
+angle KGL should have this same ratio to the sine of the angle GKQ,
+considering KG, GL, KQ as straight lines because of their smallness.
+But these sines are the lines KL and GQ, if GK is taken as the radius
+of the circle. Then LK ought to be to GQ as 3 to 2; and in the same
+ratio MG to CR, NC to FS, OF to DT. Then also the sum of all the
+antecedents to all the consequents would be as 3 to 2. Now by
+prolonging the arc DO until it meets AK at X, KX is the sum of the
+antecedents. And by prolonging the arc KQ till it meets AD at Y, the
+sum of <span class="pagenum">[Pg 116]</span><a name="Page_116" id="Page_116" />the consequents is DY. Then KX ought to be to DY as 3 to 2.
+Whence it would appear that the curve KDE was of such a nature that
+having drawn from some point which had been assumed, such as K, the
+straight lines KA, KB, the excess by which AK surpasses AD should be
+to the excess of DB over KB, as 3 to 2. For it can similarly be
+demonstrated, by taking any other point in the curve, such as G, that
+the excess of AG over AD, namely VG, is to the excess of BD over DG,
+namely DP, in this same ratio of 3 to 2. And following this principle
+Mr. Des Cartes constructed these curves in his <i>Geometric</i>; and he
+easily recognized that in the case of parallel rays, these curves
+became Hyperbolas and Ellipses.</p>
+
+<p>Let us now return to our method and let us see how it leads without
+difficulty to the finding of the curves which one side of the glass
+requires when the other side is of a given figure; a figure not only
+plane or spherical, or made by one of the conic sections (which is the
+restriction with which Des Cartes proposed this problem, leaving the
+solution to those who should come after him) but generally any figure
+whatever: that is to say, one made by the revolution of any given
+curved line to which one must merely know how to draw straight lines
+as tangents.</p>
+
+<p>Let the given figure be that made by the revolution of some curve such
+as AK about the axis AV, and that this side of the glass receives rays
+coming from the point L. Furthermore, let the thickness AB of the
+middle of the glass be given, and the point F at which one desires the
+rays to be all perfectly reunited, whatever be the first refraction
+occurring at the surface AK.</p>
+
+<p>I say that for this the sole requirement is that the outline BDK which
+constitutes the other surface shall be <span class="pagenum">[Pg 117]</span><a name="Page_117" id="Page_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
+everywhere in equal times, and in each case in a time equal to that
+which the light employs, to pass along the straight line LF of which
+the part AB is within the glass.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg117.png" width="300" height="677" alt="" title="" />
+</div>
+
+<p>Let LG be a ray falling on the arc AK. Its refraction GV will be given
+by means of the tangent which will be drawn at the point G. Now in GV
+the point D must be found such that FD together with 3/2 of DG and the
+straight line <span class="pagenum">[Pg 118]</span><a name="Page_118" id="Page_118" />GL, may be equal to FB together with 3/2 of BA and the
+straight line AL; which, as is clear, make up a given length. Or
+rather, by deducting from each the length of LG, which is also given,
+it will merely be needful to adjust FD up to the straight line VG in
+such a way that FD together with 3/2 of DG is equal to a given
+straight line, which is a quite easy plane problem: and the point D
+will be one of those through which the curve BDK ought to pass. And
+similarly, having drawn another ray LM, and found its refraction MO,
+the point N will be found in this line, and so on as many times as one
+desires.</p>
+
+<p>To demonstrate the effect of the curve, let there be described about
+the centre L the circular arc AH, cutting LG at H; and about the
+centre F the arc BP; and in AB let AS be taken equal to 2/3 of HG; and
+SE equal to GD. Then considering AH as a wave of light emanating from
+the point L, it is certain that during the time in which its piece H
+arrives at G the piece A will have advanced within the transparent
+body only along AS; for I suppose, as above, the proportion of the
+refraction to be as 3 to 2. Now we know that the piece of wave which
+is incident on G, advances thence along the line GD, since GV is the
+refraction of the ray LG. Then during the time that this piece of wave
+has taken from G to D, the other piece which was at S has reached E,
+since GD, SE are equal. But while the latter will advance from E to B,
+the piece of wave which was at D will have spread into the air its
+partial wave, the semi-diameter of which, DC (supposing this wave to
+cut the line DF at C), will be 3/2 of EB, since the velocity of light
+outside the medium is to that inside as 3 to 2. Now it is easy to show
+that this wave will touch the arc BP at this point C. For since, by
+construction, FD + <span class="pagenum">[Pg 119]</span><a name="Page_119" id="Page_119" />3/2 DG + GL are equal to FB + 3/2 BA + AL; on
+deducting the equals LH, LA, there will remain FD + 3/2 DG + GH equal
+to FB + 3/2 BA. And, again, deducting from one side GH, and from the
+other side 3/2 of AS, which are equal, there will remain FD with 3/2
+DG equal to FB with 3/2 of BS. But 3/2 of DG are equal to 3/2 of ES;
+then FD is equal to FB with 3/2 of BE. But DC was equal to 3/2 of EB;
+then deducting these equal lengths from one side and from the other,
+there will remain CF equal to FB. And thus it appears that the wave,
+the semi-diameter of which is DC, touches the arc BP at the moment
+when the light coming from the point L has arrived at B along the line
+LB. It can be demonstrated similarly that at this same moment the
+light that has come along any other ray, such as LM, MN, will have
+propagated the movement which is terminated at the arc BP. Whence it
+follows, as has been often said, that the propagation of the wave AH,
+after it has passed through the thickness of the glass, will be the
+spherical wave BP, all the pieces of which ought to advance along
+straight lines, which are the rays of light, to the centre F. Which
+was to be proved. Similarly these curved lines can be found in all the
+cases which can be proposed, as will be sufficiently shown by one or
+two examples which I will add.</p>
+
+<p>Let there be given the surface of the glass AK, made by the revolution
+about the axis BA of the line AK, which may be straight or curved. Let
+there be also given in the axis the point L and the thickness BA of
+the glass; and let it be required to find the other surface KDB, which
+receiving rays that are parallel to AB will direct them in such wise
+that after being again refracted at the given surface AK they will all
+be reassembled at the point L.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg120.png" width="300" height="416" alt="" title="" />
+</div>
+
+<p>From the point L let there be drawn to some point of <span class="pagenum">[Pg 120]</span><a name="Page_120" id="Page_120" />the given line
+AK the straight line LG, which, being considered as a ray of light,
+its refraction GD will then be found. And this line being then
+prolonged at one side or the other will meet the straight line BL, as
+here at V. Let there then be erected on AB the perpendicular BC, which
+will represent a wave of light coming from the infinitely distant
+point F, since we have supposed the rays to be parallel. Then all the
+parts of this wave BC must arrive at the same time at the point L; or
+rather all the parts of a wave emanating from the point L must arrive
+at the same time at the straight line BC. And for that, it is
+necessary to find in the line VGD the point D such that having drawn
+DC parallel to AB, the sum of CD, plus 3/2 of DG, plus GL may be equal
+to 3/2 of AB, plus AL: or rather, on deducting from both sides GL,
+which is given, CD plus 3/2 of DG must be equal to a given length;
+which is a still easier problem than the preceding construction. The
+point D thus found will be one of those through which the curve ought
+to pass; and the proof will be the same as before. And by this it will
+be proved that the waves which come from the point L, after having
+passed through the glass KAKB, will take <span class="pagenum">[Pg 121]</span><a name="Page_121" id="Page_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 parallel rays falling on
+the surface KDB will be reassembled at the point L.</p>
+
+<div class="figright" style="width: 250px;">
+<img src="images/pg121.png" width="250" height="383" alt="" title="" />
+</div>
+
+<p>Again, let there be given the surface AK, of any desired form,
+generated by revolution about the axis AB, and let the thickness of
+the glass at the middle be AB. Also let the point L be given in the
+axis behind the glass; and let it be supposed that the rays which fall
+on the surface AK tend to this point, and that it is required to find
+the surface BD, which on their emergence from the glass turns them as
+if they came from the point F in front of the glass.</p>
+
+<p>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 <span class="pagenum">[Pg 122]</span><a name="Page_122" id="Page_122" />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.</p>
+
+<p>Then the wave TG, after having passed through the glass, must form the
+wave QD; and for this I observe that the time taken by the light along
+GD in the glass must be equal to that taken along the three, TA, AB,
+and BQ, of which AB alone is within the glass. Or rather, having taken
+AS equal to 2/3 of AT, I observe that 3/2 of GD ought to be equal to
+3/2 of SB, plus BQ; and, deducting both of them from FD or FQ, that FD
+less 3/2 of GD ought to be equal to FB less 3/2 of SB. And this last
+difference is a given length: and all that is required is to draw the
+straight line FD from the given point F to meet VG so that it may be
+thus. Which is a problem quite similar to that which served for the
+first of these constructions, where FD plus 3/2 of GD had to be equal
+to a given length.</p>
+
+<p>In the demonstration it is to be observed that, since the arc BC falls
+within the glass, there must be conceived an arc RX, concentric with
+it and on the other side of QD. Then after it shall have been shown
+that the piece G of the wave GT arrives at D at the same time that the
+piece T arrives at Q, which is easily deduced from the construction,
+it will be evident as a consequence that the partial wave generated at
+the point D will touch the arc RX at the moment when the piece Q shall
+have come to R, and that thus this arc will at the same moment be the
+termination of the movement that comes from the wave TG; whence all
+the rest may be concluded.</p>
+
+<p>Having shown the method of finding these curved lines which serve for
+the perfect concurrence of the rays, <span class="pagenum">[Pg 123]</span><a name="Page_123" id="Page_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 angles the waves which
+travel along them.</p>
+
+<div class="figleft" style="width: 300px;">
+<img src="images/pg123.png" width="300" height="464" alt="" title="" />
+</div>
+
+<p>For in the case of rays which, for example, fall parallel upon a
+spherical surface AFE, intersecting one another, after refraction, at
+different points, as this figure represents; what can the waves of
+light be, in this transparent body, which are cut at right angles by
+the converging rays? For they can not be spherical. And what will
+these waves become after the said rays begin to intersect one another?
+It will be seen in the solution of this difficulty that something very
+remarkable comes to pass herein, and that the waves do not cease to
+persist though they do not continue entire, as when they cross the
+glasses designed according to the construction we have seen.</p>
+
+<p><span class="pagenum">[Pg 124]</span><a name="Page_124" id="Page_124" />According to what has been shown above, the straight line AD, which
+has been drawn at the summit of the sphere, at right angles to the
+axis parallel to which the rays come, represents the wave of light;
+and in the time taken by its piece D to reach the spherical surface
+AGE at E, its other parts will have met the same surface at F, G, H,
+etc., and will have also formed spherical partial waves of which these
+points are the centres. And the surface EK which all those waves will
+touch, will be the continuation of the wave AD in the sphere at the
+moment when the piece D has reached E. Now the line EK is not an arc
+of a circle, but is a curved line formed as the evolute of another
+curve ENC, which touches all the rays HL, GM, FO, etc., that are the
+refractions of the parallel rays, if we imagine laid over the
+convexity ENC a thread which in unwinding describes at its end E the
+said curve EK. For, supposing that this curve has been thus described,
+we will show that the said waves formed from the centres F, G, H,
+etc., will all touch it.</p>
+
+<p>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 <i>de Motu Pendulorum</i>. 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 <span class="pagenum">[Pg 125]</span><a name="Page_125" id="Page_125" />to the ray GM, and similarly the arc GF as a straight
+line. But GM being the refraction of the ray RG, and FP being
+perpendicular to it, QF must be to GP as 3 to 2, that is to say in the
+proportion of the refraction; as was shown above in explaining the
+discovery of Des Cartes. And the same thing occurs in all the small
+arcs GH, HA, etc., namely that in the quadrilaterals which enclose
+them the side parallel to the axis is to the opposite side as 3 to 2.
+Then also as 3 to 2 will the sum of the one set be to the sum of the
+other; that is to say, TF to AS, and DE to AK, and BE to SK or DV,
+supposing V to be the intersection of the curve EK and the ray FO.
+But, making FB perpendicular to DE, the ratio of 3 to 2 is also that
+of BE to the semi-diameter of the spherical wave which emanated from
+the point F while the light outside the transparent body traversed the
+space BE. Then it appears that this wave will intersect the ray FM at
+the same point V where it is intersected at right angles by the curve
+EK, and consequently that the wave will touch this curve. In the same
+way it can be proved that the same will apply to all the other waves
+above mentioned, originating at the points G, H, etc.; to wit, that
+they will touch the curve EK at the moment when the piece D of the
+wave ED shall have reached E.</p>
+
+<p>Now to say what these waves become after the rays have begun to cross
+one another: it is that from thence they fold back and are composed of
+two contiguous parts, one being a curve formed as evolute of the curve
+ENC in one sense, and the other as evolute of the same curve in the
+opposite sense. Thus the wave KE, while advancing toward the meeting
+place becomes <i>abc</i>, whereof the part <i>ab</i> is made by the evolute
+<i>b</i>C, a portion of the curve <span class="pagenum">[Pg 126]</span><a name="Page_126" id="Page_126" />ENC, while the end C remains attached;
+and the part <i>bc</i> by the evolute of the portion <i>b</i>E while the end E
+remains attached. Consequently the same wave becomes <i>def</i>, then
+<i>ghk</i>, 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.</p>
+
+<p>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.</p>
+
+<p>As many other points as may be desired in the curve NC are found by a
+Theorem which Mr. Barrow has demonstrated in section 12 of his
+<i>Lectiones Opticae</i>, 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.</p>
+
+<p>Similarly the waves that are folded back in reflexion by a concave
+spherical mirror can be found. Let ABC be the section, through the
+axis, of a hollow hemisphere, the centre of which is D, its axis being
+DB, parallel to which I suppose the rays of light to come. All the
+reflexions of those rays which fall upon the quarter-circle AB will
+touch a curved line AFE, of which line the end E is at the focus of
+the hemisphere, that is to say, at the point which divides the
+semi-diameter BD into two equal parts. <span class="pagenum">[Pg 127]</span><a name="Page_127" id="Page_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 points.</p>
+
+<div class="figright" style="width: 300px;">
+<img src="images/pg127.png" width="300" height="232" alt="" title="" />
+</div>
+
+<p>And as the parallel rays are merely perpendiculars to the waves which
+fall on the concave surface, which waves are parallel to AD, it will
+be found that as they come successively to encounter the surface AB,
+they form on reflexion folded waves composed of two curves which
+originate from two opposite evolutions of the parts of the curve AFE.
+So, taking AD as an incident wave, when the part AG shall have met the
+surface AI, that is to say when the piece G shall have reached I, it
+will be the curves HF, FI, generated as evolutes of the curves FA, FE,
+both beginning at F, which together constitute the propagation of the
+part AG. And a little afterwards, when the part AK has met the surface
+AM, the piece K having come to M, then the curves LN, NM, will
+together constitute the propagation of that part. And thus this folded
+wave will continue to advance until the point N has reached the focus
+E. The curve AFE can be seen in smoke, or in flying dust, when a
+concave mirror is held opposite the sun. And it should be known that
+it is none other than that curve which is described <span class="pagenum">[Pg 128]</span><a name="Page_128" id="Page_128" />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.</p>
+
+<p>Its length is exactly equal to 3/4 of the diameter of the sphere, as
+can be found and demonstrated by means of these waves, nearly in the
+same way as the mensuration of the preceding curve; though it may also
+be demonstrated in other ways, which I omit as outside the subject.
+The area AOBEFA, comprised between the arc of the quarter-circle, the
+straight line BE, and the curve EFA, is equal to the fourth part of
+the quadrant DAB.</p>
+
+<h2>END.</h2>
+
+
+
+<hr style="width: 65%;" />
+<div class="pagenum">[Pg 129]<a name="Page_129" id="Page_129" /><a name="INDEX" id="INDEX" /></div>
+<h2>INDEX</h2>
+
+<p>
+<i>Archimedes</i>, <a href="#Page_104">104</a>.<br />
+<i>Atmospheric refraction</i>, <a href="#Page_45">45</a>.<br />
+<br />
+<i>Barrow, Isaac</i>, <a href="#Page_126">126</a>.<br />
+<i>Bartholinus, Erasmus</i>, <a href="#Page_53">53</a>, <a href="#Page_54">54</a>, <a href="#Page_57">57</a>, <a href="#Page_60">60</a>, <a href="#Page_97">97</a>, <a href="#Page_99">99</a>.<br />
+<i>Boyle, Hon. Robert,</i> <a href="#Page_11">11</a>.<br />
+<br />
+<i>Cassini, Jacques</i>, <a href="#Page_iii">iii</a>.<br />
+<i>Caustic Curves</i>, <a href="#Page_123">123</a>.<br />
+<i>Crystals</i>, see <i>Iceland Crystal, Rock Crystal</i>.<br />
+<i>Crystals, configuration of</i>, <a href="#Page_95">95</a>.<br />
+<br />
+<i>Descartes, R&eacute;n&ecirc;</i>, <a href="#Page_3">3</a>, <a href="#Page_5">5</a>, <a href="#Page_7">7</a>, <a href="#Page_14">14</a>, <a href="#Page_22">22</a>, <a href="#Page_42">42</a>, <a href="#Page_43">43</a>, <a href="#Page_109">109</a>, <a href="#Page_113">113</a>.<br />
+<i>Double Refraction, discovery of</i>, <a href="#Page_54">54</a>, <a href="#Page_81">81</a>, <a href="#Page_93">93</a>.<br />
+<br />
+<i>Elasticity</i>, <a href="#Page_12">12</a>, <a href="#Page_14">14</a>.<br />
+<i>Ether, the, or Ethereal matter</i>, <a href="#Page_11">11</a>, <a href="#Page_14">14</a>, <a href="#Page_16">16</a>, <a href="#Page_28">28</a>.<br />
+<i>Extraordinary refraction</i>, <a href="#Page_55">55</a>, <a href="#Page_56">56</a>.<br />
+<br />
+<i>Fermat, principle of</i>, <a href="#Page_42">42</a>.<br />
+<i>Figures of transparent bodies</i>, <a href="#Page_105">105</a>.<br />
+<br />
+<i>Hooke, Robert</i>, <a href="#Page_20">20</a>.<br />
+<br />
+<i>Iceland Crystal</i>, <a href="#Page_2">2</a>, <a href="#Page_52">52</a> sqq.<br />
+<i>Iceland Crystal, Cutting and Polishing of</i>, <a href="#Page_91">91</a>, <a href="#Page_92">92</a>, <a href="#Page_98">98</a>.<br />
+<br />
+<i>Leibnitz, G.W.</i>, <a href="#Page_vi">vi</a>.<br />
+<i>Light, nature of</i>, <a href="#Page_3">3</a>.<br />
+<i>Light, velocity of</i>, <a href="#Page_4">4</a>, <a href="#Page_15">15</a>.<br />
+<br />
+<i>Molecular texture of bodies</i>, <a href="#Page_27">27</a>, <a href="#Page_95">95</a>.<br />
+<br />
+<i>Newton, Sir Isaac</i>, <a href="#Page_vi">vi</a>, <a href="#Page_106">106</a>.<br />
+<br />
+<i>Opacity</i>, <a href="#Page_34">34</a>.<br />
+<i>Ovals, Cartesian</i>, <a href="#Page_107">107</a>, <a href="#Page_113">113</a>.<br />
+<br />
+<i>Pardies, Rev. Father</i>, <a href="#Page_20">20</a>.<br />
+<i>Rays, definition of</i>, <a href="#Page_38">38</a>, <a href="#Page_49">49</a>.<br />
+<i>Reflexion</i>, <a href="#Page_22">22</a>.<br />
+<i>Refraction,</i> <a href="#Page_28">28</a>, <a href="#Page_34">34</a>.<br />
+<i>Rock Crystal</i>, <a href="#Page_54">54</a>, <a href="#Page_57">57</a>, <a href="#Page_62">62</a>, <a href="#Page_95">95</a>.<br />
+<i>R&ouml;mer, Olaf</i>, <a href="#Page_v">v</a>, <a href="#Page_7">7</a>.<br />
+<i>Roughness of surfaces</i>, <a href="#Page_27">27</a>.<br />
+<br />
+<i>Sines, law of</i>, <a href="#Page_1">1</a>, <a href="#Page_35">35</a>, <a href="#Page_38">38</a>, <a href="#Page_43">43</a>.<br />
+<i>Spheres, elasticity of</i>, <a href="#Page_15">15</a>.<br />
+<i>Spheroidal waves in crystals</i>, <a href="#Page_63">63</a>.<br />
+<i>Spheroids, lemma about</i>, <a href="#Page_103">103</a>.<br />
+<i>Sound, speed of</i>, <a href="#Page_7">7</a>, <a href="#Page_10">10</a>, <a href="#Page_12">12</a>.<br />
+<br />
+<i>Telescopes, lenses for</i>, <a href="#Page_62">62</a>, <a href="#Page_105">105</a>.<br />
+<i>Torricelli's experiment</i>, <a href="#Page_12">12</a>, <a href="#Page_30">30</a>.<br />
+<i>Transparency, explanation of</i>, <a href="#Page_28">28</a>, <a href="#Page_31">31</a>, <a href="#Page_32">32</a>.<br />
+<br />
+<i>Waves, no regular succession of</i>, <a href="#Page_17">17</a>.<br />
+<i>Waves, principle of wave envelopes</i>, <a href="#Page_19">19</a>, <a href="#Page_24">24</a>.<br />
+<i>Waves, principle of elementary wave fronts</i>, <a href="#Page_19">19</a>.<br />
+<i>Waves, propagation of light as</i>, <a href="#Page_16">16</a>, <a href="#Page_63">63</a>.<br />
+</p>
+
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@@ -0,0 +1,4174 @@
+The Project Gutenberg eBook, Treatise on Light, by Christiaan Huygens,
+Translated by Silvanus P. Thompson
+
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+
+
+
+Title: Treatise on Light
+
+Author: Christiaan Huygens
+
+Release Date: January 18, 2005  [eBook #14725]
+
+Language: English
+
+Character set encoding: ISO-646-US (US-ASCII)
+
+
+***START OF THE PROJECT GUTENBERG EBOOK TREATISE ON LIGHT***
+
+
+E-text prepared by Clare Boothby, Stephen Schulze, and the Project
+Gutenberg Online Distributed Proofreading Team
+
+
+
+Note: Project Gutenberg also has an HTML version of this
+      file which includes the original illustrations.
+      See 14725-h.htm or 14725-h.zip:
+      (https://www.gutenberg.org/dirs/1/4/7/2/14725/14725-h/14725-h.htm)
+      or
+      (https://www.gutenberg.org/dirs/1/4/7/2/14725/14725-h.zip)
+
+
+
+
+
+TREATISE ON LIGHT
+
+In which are explained
+The causes of that which occurs
+In REFLEXION, & in REFRACTION
+
+And particularly
+In the strange REFRACTION
+OF ICELAND CRYSTAL
+
+by
+
+CHRISTIAAN HUYGENS
+
+Rendered into English by
+
+SILVANUS P. THOMPSON
+
+University of Chicago Press
+
+
+
+
+
+
+
+PREFACE
+
+
+I wrote this Treatise during my sojourn in France twelve years ago,
+and I communicated it in the year 1678 to the learned persons who then
+composed the Royal Academy of Science, to the membership of which the
+King had done me the honour of calling, me. Several of that body who
+are still alive will remember having been present when I read it, and
+above the rest those amongst them who applied themselves particularly
+to the study of Mathematics; of whom I cannot cite more than the
+celebrated gentlemen Cassini, Roemer, and De la Hire. And, although I
+have since corrected and changed some parts, the copies which I had
+made of it at that time may serve for proof that I have yet added
+nothing to it save some conjectures touching the formation of Iceland
+Crystal, and a novel observation on the refraction of Rock Crystal. I
+have desired to relate these particulars to make known how long I have
+meditated the things which now I publish, and not for the purpose of
+detracting from the merit of those who, without having seen anything
+that I have written, may be found to have treated of like matters: as
+has in fact occurred to two eminent Geometricians, Messieurs Newton
+and Leibnitz, with respect to the Problem of the figure of glasses for
+collecting rays when one of the surfaces is given.
+
+One may ask why I have so long delayed to bring this work to the
+light. The reason is that I wrote it rather carelessly in the Language
+in which it appears, with the intention of translating it into Latin,
+so doing in order to obtain greater attention to the thing. After
+which I proposed to myself to give it out along with another Treatise
+on Dioptrics, in which I explain the effects of Telescopes and those
+things which belong more to that Science. But the pleasure of novelty
+being past, I have put off from time to time the execution of this
+design, and I know not when I shall ever come to an end if it, being
+often turned aside either by business or by some new study.
+Considering which I have finally judged that it was better worth while
+to publish this writing, such as it is, than to let it run the risk,
+by waiting longer, of remaining lost.
+
+There will be seen in it demonstrations of those kinds which do not
+produce as great a certitude as those of Geometry, and which even
+differ much therefrom, since whereas the Geometers prove their
+Propositions by fixed and incontestable Principles, here the
+Principles are verified by the conclusions to be drawn from them; the
+nature of these things not allowing of this being done otherwise.
+
+It is always possible to attain thereby to a degree of probability
+which very often is scarcely less than complete proof. To wit, when
+things which have been demonstrated by the Principles that have been
+assumed correspond perfectly to the phenomena which experiment has
+brought under observation; especially when there are a great number of
+them, and further, principally, when one can imagine and foresee new
+phenomena which ought to follow from the hypotheses which one employs,
+and when one finds that therein the fact corresponds to our prevision.
+But if all these proofs of probability are met with in that which I
+propose to discuss, as it seems to me they are, this ought to be a
+very strong confirmation of the success of my inquiry; and it must be
+ill if the facts are not pretty much as I represent them. I would
+believe then that those who love to know the Causes of things and who
+are able to admire the marvels of Light, will find some satisfaction
+in these various speculations regarding it, and in the new explanation
+of its famous property which is the main foundation of the
+construction of our eyes and of those great inventions which extend so
+vastly the use of them.
+
+I hope also that there will be some who by following these beginnings
+will penetrate much further into this question than I have been able
+to do, since the subject must be far from being exhausted. This
+appears from the passages which I have indicated where I leave certain
+difficulties without having resolved them, and still more from matters
+which I have not touched at all, such as Luminous Bodies of several
+sorts, and all that concerns Colours; in which no one until now can
+boast of having succeeded. Finally, there remains much more to be
+investigated touching the nature of Light which I do not pretend to
+have disclosed, and I shall owe much in return to him who shall be
+able to supplement that which is here lacking to me in knowledge. The
+Hague. The 8 January 1690.
+
+
+
+
+NOTE BY THE TRANSLATOR
+
+
+Considering the great influence which this Treatise has exercised in
+the development of the Science of Optics, it seems strange that two
+centuries should have passed before an English edition of the work
+appeared. Perhaps the circumstance is due to the mistaken zeal with
+which formerly everything that conflicted with the cherished ideas of
+Newton was denounced by his followers. The Treatise on Light of
+Huygens has, however, withstood the test of time: and even now the
+exquisite skill with which he applied his conception of the
+propagation of waves of light to unravel the intricacies of the
+phenomena of the double refraction of crystals, and of the refraction
+of the atmosphere, will excite the admiration of the student of
+Optics. It is true that his wave theory was far from the complete
+doctrine as subsequently developed by Thomas Young and Augustin
+Fresnel, and belonged rather to geometrical than to physical Optics.
+If Huygens had no conception of transverse vibrations, of the
+principle of interference, or of the existence of the ordered sequence
+of waves in trains, he nevertheless attained to a remarkably clear
+understanding of the principles of wave-propagation; and his
+exposition of the subject marks an epoch in the treatment of Optical
+problems. It has been needful in preparing this translation to
+exercise care lest one should import into the author's text ideas of
+subsequent date, by using words that have come to imply modern
+conceptions. Hence the adoption of as literal a rendering as possible.
+A few of the author's terms need explanation. He uses the word
+"refraction," for example, both for the phenomenon or process usually
+so denoted, and for the result of that process: thus the refracted ray
+he habitually terms "the refraction" of the incident ray. When a
+wave-front, or, as he terms it, a "wave," has passed from some initial
+position to a subsequent one, he terms the wave-front in its
+subsequent position "the continuation" of the wave. He also speaks of
+the envelope of a set of elementary waves, formed by coalescence of
+those elementary wave-fronts, as "the termination" of the wave; and
+the elementary wave-fronts he terms "particular" waves. Owing to the
+circumstance that the French word _rayon_ possesses the double
+signification of ray of light and radius of a circle, he avoids its
+use in the latter sense and speaks always of the semi-diameter, not of
+the radius. His speculations as to the ether, his suggestive views of
+the structure of crystalline bodies, and his explanation of opacity,
+slight as they are, will possibly surprise the reader by their seeming
+modernness. And none can read his investigation of the phenomena found
+in Iceland spar without marvelling at his insight and sagacity.
+
+S.P.T.
+
+June, 1912.
+
+
+
+
+TABLE OF MATTERS
+
+Contained in this Treatise
+
+
+CHAPTER I.
+On Rays Propagated in Straight Lines.
+
+  That Light is produced by a certain movement.
+
+  That no substance passes from the luminous object to the eyes.
+
+  That Light spreads spherically, almost as Sound does.
+
+  Whether Light takes time to spread.
+
+  Experience seeming to prove that it passes instantaneously.
+
+  Experience proving that it takes time.
+
+  How much its speed is greater than that of Sound.
+
+  In what the emission of Light differs from that of Sound.
+
+  That it is not the same medium which serves for Light and Sound.
+
+  How Sound is propagated.
+
+  How Light is propagated.
+
+  Detailed Remarks on the propagation of Light.
+
+  Why Rays are propagated only in straight lines.
+
+  How Light coming in different directions can cross itself.
+
+CHAPTER II.
+On Reflexion.
+
+  Demonstration of equality of angles of incidence and reflexion.
+
+  Why the incident and reflected rays are in the same plane
+  perpendicular to the reflecting surface.
+
+  That it is not needful for the reflecting surface to be perfectly
+  flat to attain equality of the angles of incidence and reflexion.
+
+CHAPTER III.
+On Refraction.
+
+  That bodies may be transparent without any substance passing through
+  them.
+
+  Proof that the ethereal matter passes through transparent bodies.
+
+  How this matter passing through can render them transparent.
+
+  That the most solid bodies in appearance are of a very loose texture.
+
+  That Light spreads more slowly in water and in glass than in air.
+
+  Third hypothesis to explain transparency, and the retardation which
+  Light suffers.
+
+  On that which makes bodies opaque.
+
+  Demonstration why Refraction obeys the known proportion of Sines.
+
+  Why the incident and refracted Rays produce one another reciprocally.
+
+  Why Reflexion within a triangular glass prism is suddenly augmented
+  when the Light can no longer penetrate.
+
+  That bodies which cause greater Refraction also cause stronger
+  Reflexion.
+
+  Demonstration of the Theorem of Mr. Fermat.
+
+CHAPTER IV.
+On the Refraction of the Air.
+
+  That the emanations of Light in the air are not spherical.
+
+  How consequently some objects appear higher than they are.
+
+  How the Sun may appear on the Horizon before he has risen.
+
+  That the rays of light become curved in the Air of the Atmosphere,
+  and what effects this produces.
+
+CHAPTER V.
+On the Strange Refraction of Iceland Crystal.
+
+  That this Crystal grows also in other countries.
+
+  Who first-wrote about it.
+
+  Description of Iceland Crystal; its substance, shape, and properties.
+
+  That it has two different Refractions.
+
+  That the ray perpendicular to the surface suffers refraction, and
+  that some rays inclined to the surface pass without suffering
+  refraction.
+
+  Observation of the refractions in this Crystal.
+
+  That there is a Regular and an Irregular Refraction.
+
+  The way of measuring the two Refractions of Iceland Crystal.
+
+  Remarkable properties of the Irregular Refraction.
+
+  Hypothesis to explain the double Refraction.
+
+  That Rock Crystal has also a double Refraction.
+
+  Hypothesis of emanations of Light, within Iceland Crystal, of
+  spheroidal form, for the Irregular Refraction.
+
+  How a perpendicular ray can suffer Refraction.
+
+  How the position and form of the spheroidal emanations in this
+  Crystal can be defined.
+
+  Explanation of the Irregular Refraction by these spheroidal
+  emanations.
+
+  Easy way to find the Irregular Refraction of each incident ray.
+
+  Demonstration of the oblique ray which traverses the Crystal without
+  being refracted.
+
+  Other irregularities of Refraction explained.
+
+  That an object placed beneath the Crystal appears double, in two
+  images of different heights.
+
+  Why the apparent heights of one of the images change on changing the
+  position of the eyes above the Crystal.
+
+  Of the different sections of this Crystal which produce yet other
+  refractions, and confirm all this Theory.
+
+  Particular way of polishing the surfaces after it has been cut.
+
+  Surprising phenomenon touching the rays which pass through two
+  separated pieces; the cause of which is not explained.
+
+  Probable conjecture on the internal composition of Iceland Crystal,
+  and of what figure its particles are.
+
+  Tests to confirm this conjecture.
+
+  Calculations which have been supposed in this Chapter.
+
+CHAPTER VI.
+On the Figures of transparent bodies which serve for Refraction and
+for Reflexion.
+
+  General and easy rule to find these Figures.
+
+  Invention of the Ovals of Mr. Des Cartes for Dioptrics.
+
+  How he was able to find these Lines.
+
+  Way of finding the surface of a glass for perfect refraction, when
+  the other surface is given.
+
+  Remark on what happens to rays refracted at a spherical surface.
+
+  Remark on the curved line which is formed by reflexion in a spherical
+  concave mirror.
+
+
+
+
+
+CHAPTER I
+
+ON RAYS PROPAGATED IN STRAIGHT LINES
+
+
+As happens in all the sciences in which Geometry is applied to matter,
+the demonstrations concerning Optics are founded on truths drawn from
+experience. Such are that the rays of light are propagated in straight
+lines; that the angles of reflexion and of incidence are equal; and
+that in refraction the ray is bent according to the law of sines, now
+so well known, and which is no less certain than the preceding laws.
+
+The majority of those who have written touching the various parts of
+Optics have contented themselves with presuming these truths. But
+some, more inquiring, have desired to investigate the origin and the
+causes, considering these to be in themselves wonderful effects of
+Nature. In which they advanced some ingenious things, but not however
+such that the most intelligent folk do not wish for better and more
+satisfactory explanations. Wherefore I here desire to propound what I
+have meditated on the subject, so as to contribute as much as I can
+to the explanation of this department of Natural Science, which, not
+without reason, is reputed to be one of its most difficult parts. I
+recognize myself to be much indebted to those who were the first to
+begin to dissipate the strange obscurity in which these things were
+enveloped, and to give us hope that they might be explained by
+intelligible reasoning. But, on the other hand I am astonished also
+that even here these have often been willing to offer, as assured and
+demonstrative, reasonings which were far from conclusive. For I do not
+find that any one has yet given a probable explanation of the first
+and most notable phenomena of light, namely why it is not propagated
+except in straight lines, and how visible rays, coming from an
+infinitude of diverse places, cross one another without hindering one
+another in any way.
+
+I shall therefore essay in this book, to give, in accordance with the
+principles accepted in the Philosophy of the present day, some clearer
+and more probable reasons, firstly of these properties of light
+propagated rectilinearly; secondly of light which is reflected on
+meeting other bodies. Then I shall explain the phenomena of those rays
+which are said to suffer refraction on passing through transparent
+bodies of different sorts; and in this part I shall also explain the
+effects of the refraction of the air by the different densities of the
+Atmosphere.
+
+Thereafter I shall examine the causes of the strange refraction of a
+certain kind of Crystal which is brought from Iceland. And finally I
+shall treat of the various shapes of transparent and reflecting bodies
+by which rays are collected at a point or are turned aside in various
+ways. From this it will be seen with what facility, following our new
+Theory, we find not only the Ellipses, Hyperbolas, and other curves
+which Mr. Des Cartes has ingeniously invented for this purpose; but
+also those which the surface of a glass lens ought to possess when its
+other surface is given as spherical or plane, or of any other figure
+that may be.
+
+It is inconceivable to doubt that light consists in the motion of some
+sort of matter. For whether one considers its production, one sees
+that here upon the Earth it is chiefly engendered by fire and flame
+which contain without doubt bodies that are in rapid motion, since
+they dissolve and melt many other bodies, even the most solid; or
+whether one considers its effects, one sees that when light is
+collected, as by concave mirrors, it has the property of burning as a
+fire does, that is to say it disunites the particles of bodies. This
+is assuredly the mark of motion, at least in the true Philosophy, in
+which one conceives the causes of all natural effects in terms of
+mechanical motions. This, in my opinion, we must necessarily do, or
+else renounce all hopes of ever comprehending anything in Physics.
+
+And as, according to this Philosophy, one holds as certain that the
+sensation of sight is excited only by the impression of some movement
+of a kind of matter which acts on the nerves at the back of our eyes,
+there is here yet one reason more for believing that light consists in
+a movement of the matter which exists between us and the luminous
+body.
+
+Further, when one considers the extreme speed with which light spreads
+on every side, and how, when it comes from different regions, even
+from those directly opposite, the rays traverse one another without
+hindrance, one may well understand that when we see a luminous object,
+it cannot be by any transport of matter coming to us from this object,
+in the way in which a shot or an arrow traverses the air; for
+assuredly that would too greatly impugn these two properties of light,
+especially the second of them. It is then in some other way that light
+spreads; and that which can lead us to comprehend it is the knowledge
+which we have of the spreading of Sound in the air.
+
+We know that by means of the air, which is an invisible and impalpable
+body, Sound spreads around the spot where it has been produced, by a
+movement which is passed on successively from one part of the air to
+another; and that the spreading of this movement, taking place equally
+rapidly on all sides, ought to form spherical surfaces ever enlarging
+and which strike our ears. Now there is no doubt at all that light
+also comes from the luminous body to our eyes by some movement
+impressed on the matter which is between the two; since, as we have
+already seen, it cannot be by the transport of a body which passes
+from one to the other. If, in addition, light takes time for its
+passage--which we are now going to examine--it will follow that this
+movement, impressed on the intervening matter, is successive; and
+consequently it spreads, as Sound does, by spherical surfaces and
+waves: for I call them waves from their resemblance to those which are
+seen to be formed in water when a stone is thrown into it, and which
+present a successive spreading as circles, though these arise from
+another cause, and are only in a flat surface.
+
+To see then whether the spreading of light takes time, let us consider
+first whether there are any facts of experience which can convince us
+to the contrary. As to those which can be made here on the Earth, by
+striking lights at great distances, although they prove that light
+takes no sensible time to pass over these distances, one may say with
+good reason that they are too small, and that the only conclusion to
+be drawn from them is that the passage of light is extremely rapid.
+Mr. Des Cartes, who was of opinion that it is instantaneous, founded
+his views, not without reason, upon a better basis of experience,
+drawn from the Eclipses of the Moon; which, nevertheless, as I shall
+show, is not at all convincing. I will set it forth, in a way a little
+different from his, in order to make the conclusion more
+comprehensible.
+
+[Illustration]
+
+Let A be the place of the sun, BD a part of the orbit or annual path
+of the Earth: ABC a straight line which I suppose to meet the orbit of
+the Moon, which is represented by the circle CD, at C.
+
+Now if light requires time, for example one hour, to traverse the
+space which is between the Earth and the Moon, it will follow that the
+Earth having arrived at B, the shadow which it casts, or the
+interruption of the light, will not yet have arrived at the point C,
+but will only arrive there an hour after. It will then be one hour
+after, reckoning from the moment when the Earth was at B, that the
+Moon, arriving at C, will be obscured: but this obscuration or
+interruption of the light will not reach the Earth till after another
+hour. Let us suppose that the Earth in these two hours will have
+arrived at E. The Earth then, being at E, will see the Eclipsed Moon
+at C, which it left an hour before, and at the same time will see the
+sun at A. For it being immovable, as I suppose with Copernicus, and
+the light moving always in straight lines, it must always appear where
+it is. But one has always observed, we are told, that the eclipsed
+Moon appears at the point of the Ecliptic opposite to the Sun; and yet
+here it would appear in arrear of that point by an amount equal to the
+angle GEC, the supplement of AEC. This, however, is contrary to
+experience, since the angle GEC would be very sensible, and about 33
+degrees. Now according to our computation, which is given in the
+Treatise on the causes of the phenomena of Saturn, the distance BA
+between the Earth and the Sun is about twelve thousand diameters of
+the Earth, and hence four hundred times greater than BC the distance
+of the Moon, which is 30 diameters. Then the angle ECB will be nearly
+four hundred times greater than BAE, which is five minutes; namely,
+the path which the earth travels in two hours along its orbit; and
+thus the angle BCE will be nearly 33 degrees; and likewise the angle
+CEG, which is greater by five minutes.
+
+But it must be noted that the speed of light in this argument has been
+assumed such that it takes a time of one hour to make the passage from
+here to the Moon. If one supposes that for this it requires only one
+minute of time, then it is manifest that the angle CEG will only be 33
+minutes; and if it requires only ten seconds of time, the angle will
+be less than six minutes. And then it will not be easy to perceive
+anything of it in observations of the Eclipse; nor, consequently, will
+it be permissible to deduce from it that the movement of light is
+instantaneous.
+
+It is true that we are here supposing a strange velocity that would be
+a hundred thousand times greater than that of Sound. For Sound,
+according to what I have observed, travels about 180 Toises in the
+time of one Second, or in about one beat of the pulse. But this
+supposition ought not to seem to be an impossibility; since it is not
+a question of the transport of a body with so great a speed, but of a
+successive movement which is passed on from some bodies to others. I
+have then made no difficulty, in meditating on these things, in
+supposing that the emanation of light is accomplished with time,
+seeing that in this way all its phenomena can be explained, and that
+in following the contrary opinion everything is incomprehensible. For
+it has always seemed tome that even Mr. Des Cartes, whose aim has been
+to treat all the subjects of Physics intelligibly, and who assuredly
+has succeeded in this better than any one before him, has said nothing
+that is not full of difficulties, or even inconceivable, in dealing
+with Light and its properties.
+
+But that which I employed only as a hypothesis, has recently received
+great seemingness as an established truth by the ingenious proof of
+Mr. Roemer which I am going here to relate, expecting him himself to
+give all that is needed for its confirmation. It is founded as is the
+preceding argument upon celestial observations, and proves not only
+that Light takes time for its passage, but also demonstrates how much
+time it takes, and that its velocity is even at least six times
+greater than that which I have just stated.
+
+For this he makes use of the Eclipses suffered by the little planets
+which revolve around Jupiter, and which often enter his shadow: and
+see what is his reasoning. Let A be the Sun, BCDE the annual orbit of
+the Earth, F Jupiter, GN the orbit of the nearest of his Satellites,
+for it is this one which is more apt for this investigation than any
+of the other three, because of the quickness of its revolution. Let G
+be this Satellite entering into the shadow of Jupiter, H the same
+Satellite emerging from the shadow.
+
+[Illustration]
+
+Let it be then supposed, the Earth being at B some time before the
+last quadrature, that one has seen the said Satellite emerge from the
+shadow; it must needs be, if the Earth remains at the same place,
+that, after 42-1/2 hours, one would again see a similar emergence,
+because that is the time in which it makes the round of its orbit, and
+when it would come again into opposition to the Sun. And if the Earth,
+for instance, were to remain always at B during 30 revolutions of this
+Satellite, one would see it again emerge from the shadow after 30
+times 42-1/2 hours. But the Earth having been carried along during
+this time to C, increasing thus its distance from Jupiter, it follows
+that if Light requires time for its passage the illumination of the
+little planet will be perceived later at C than it would have been at
+B, and that there must be added to this time of 30 times 42-1/2 hours
+that which the Light has required to traverse the space MC, the
+difference of the spaces CH, BH. Similarly at the other quadrature
+when the earth has come to E from D while approaching toward Jupiter,
+the immersions of the Satellite ought to be observed at E earlier than
+they would have been seen if the Earth had remained at D.
+
+Now in quantities of observations of these Eclipses, made during ten
+consecutive years, these differences have been found to be very
+considerable, such as ten minutes and more; and from them it has been
+concluded that in order to traverse the whole diameter of the annual
+orbit KL, which is double the distance from here to the sun, Light
+requires about 22 minutes of time.
+
+The movement of Jupiter in his orbit while the Earth passed from B to
+C, or from D to E, is included in this calculation; and this makes it
+evident that one cannot attribute the retardation of these
+illuminations or the anticipation of the eclipses, either to any
+irregularity occurring in the movement of the little planet or to its
+eccentricity.
+
+If one considers the vast size of the diameter KL, which according to
+me is some 24 thousand diameters of the Earth, one will acknowledge
+the extreme velocity of Light. For, supposing that KL is no more than
+22 thousand of these diameters, it appears that being traversed in 22
+minutes this makes the speed a thousand diameters in one minute, that
+is 16-2/3 diameters in one second or in one beat of the pulse, which
+makes more than 11 hundred times a hundred thousand toises; since the
+diameter of the Earth contains 2,865 leagues, reckoned at 25 to the
+degree, and each each league is 2,282 Toises, according to the exact
+measurement which Mr. Picard made by order of the King in 1669. But
+Sound, as I have said above, only travels 180 toises in the same time
+of one second: hence the velocity of Light is more than six hundred
+thousand times greater than that of Sound. This, however, is quite
+another thing from being instantaneous, since there is all the
+difference between a finite thing and an infinite. Now the successive
+movement of Light being confirmed in this way, it follows, as I have
+said, that it spreads by spherical waves, like the movement of Sound.
+
+But if the one resembles the other in this respect, they differ in
+many other things; to wit, in the first production of the movement
+which causes them; in the matter in which the movement spreads; and in
+the manner in which it is propagated. As to that which occurs in the
+production of Sound, one knows that it is occasioned by the agitation
+undergone by an entire body, or by a considerable part of one, which
+shakes all the contiguous air. But the movement of the Light must
+originate as from each point of the luminous object, else we should
+not be able to perceive all the different parts of that object, as
+will be more evident in that which follows. And I do not believe that
+this movement can be better explained than by supposing that all those
+of the luminous bodies which are liquid, such as flames, and
+apparently the sun and the stars, are composed of particles which
+float in a much more subtle medium which agitates them with great
+rapidity, and makes them strike against the particles of the ether
+which surrounds them, and which are much smaller than they. But I hold
+also that in luminous solids such as charcoal or metal made red hot in
+the fire, this same movement is caused by the violent agitation of
+the particles of the metal or of the wood; those of them which are on
+the surface striking similarly against the ethereal matter. The
+agitation, moreover, of the particles which engender the light ought
+to be much more prompt and more rapid than is that of the bodies which
+cause sound, since we do not see that the tremors of a body which is
+giving out a sound are capable of giving rise to Light, even as the
+movement of the hand in the air is not capable of producing Sound.
+
+Now if one examines what this matter may be in which the movement
+coming from the luminous body is propagated, which I call Ethereal
+matter, one will see that it is not the same that serves for the
+propagation of Sound. For one finds that the latter is really that
+which we feel and which we breathe, and which being removed from any
+place still leaves there the other kind of matter that serves to
+convey Light. This may be proved by shutting up a sounding body in a
+glass vessel from which the air is withdrawn by the machine which Mr.
+Boyle has given us, and with which he has performed so many beautiful
+experiments. But in doing this of which I speak, care must be taken to
+place the sounding body on cotton or on feathers, in such a way that
+it cannot communicate its tremors either to the glass vessel which
+encloses it, or to the machine; a precaution which has hitherto been
+neglected. For then after having exhausted all the air one hears no
+Sound from the metal, though it is struck.
+
+One sees here not only that our air, which does not penetrate through
+glass, is the matter by which Sound spreads; but also that it is not
+the same air but another kind of matter in which Light spreads; since
+if the air is removed from the vessel the Light does not cease to
+traverse it as before.
+
+And this last point is demonstrated even more clearly by the
+celebrated experiment of Torricelli, in which the tube of glass from
+which the quicksilver has withdrawn itself, remaining void of air,
+transmits Light just the same as when air is in it. For this proves
+that a matter different from air exists in this tube, and that this
+matter must have penetrated the glass or the quicksilver, either one
+or the other, though they are both impenetrable to the air. And when,
+in the same experiment, one makes the vacuum after putting a little
+water above the quicksilver, one concludes equally that the said
+matter passes through glass or water, or through both.
+
+As regards the different modes in which I have said the movements of
+Sound and of Light are communicated, one may sufficiently comprehend
+how this occurs in the case of Sound if one considers that the air is
+of such a nature that it can be compressed and reduced to a much
+smaller space than that which it ordinarily occupies. And in
+proportion as it is compressed the more does it exert an effort to
+regain its volume; for this property along with its penetrability,
+which remains notwithstanding its compression, seems to prove that it
+is made up of small bodies which float about and which are agitated
+very rapidly in the ethereal matter composed of much smaller parts. So
+that the cause of the spreading of Sound is the effort which these
+little bodies make in collisions with one another, to regain freedom
+when they are a little more squeezed together in the circuit of these
+waves than elsewhere.
+
+But the extreme velocity of Light, and other properties which it has,
+cannot admit of such a propagation of motion, and I am about to show
+here the way in which I conceive it must occur. For this, it is
+needful to explain the property which hard bodies must possess to
+transmit movement from one to another.
+
+When one takes a number of spheres of equal size, made of some very
+hard substance, and arranges them in a straight line, so that they
+touch one another, one finds, on striking with a similar sphere
+against the first of these spheres, that the motion passes as in an
+instant to the last of them, which separates itself from the row,
+without one's being able to perceive that the others have been
+stirred. And even that one which was used to strike remains motionless
+with them. Whence one sees that the movement passes with an extreme
+velocity which is the greater, the greater the hardness of the
+substance of the spheres.
+
+But it is still certain that this progression of motion is not
+instantaneous, but successive, and therefore must take time. For if
+the movement, or the disposition to movement, if you will have it so,
+did not pass successively through all these spheres, they would all
+acquire the movement at the same time, and hence would all advance
+together; which does not happen. For the last one leaves the whole row
+and acquires the speed of the one which was pushed. Moreover there are
+experiments which demonstrate that all the bodies which we reckon of
+the hardest kind, such as quenched steel, glass, and agate, act as
+springs and bend somehow, not only when extended as rods but also when
+they are in the form of spheres or of other shapes. That is to say
+they yield a little in themselves at the place where they are struck,
+and immediately regain their former figure. For I have found that on
+striking with a ball of glass or of agate against a large and quite
+thick thick piece of the same substance which had a flat surface,
+slightly soiled with breath or in some other way, there remained round
+marks, of smaller or larger size according as the blow had been weak
+or strong. This makes it evident that these substances yield where
+they meet, and spring back: and for this time must be required.
+
+Now in applying this kind of movement to that which produces Light
+there is nothing to hinder us from estimating the particles of the
+ether to be of a substance as nearly approaching to perfect hardness
+and possessing a springiness as prompt as we choose. It is not
+necessary to examine here the causes of this hardness, or of that
+springiness, the consideration of which would lead us too far from our
+subject. I will say, however, in passing that we may conceive that the
+particles of the ether, notwithstanding their smallness, are in turn
+composed of other parts and that their springiness consists in the
+very rapid movement of a subtle matter which penetrates them from
+every side and constrains their structure to assume such a disposition
+as to give to this fluid matter the most overt and easy passage
+possible. This accords with the explanation which Mr. Des Cartes gives
+for the spring, though I do not, like him, suppose the pores to be in
+the form of round hollow canals. And it must not be thought that in
+this there is anything absurd or impossible, it being on the contrary
+quite credible that it is this infinite series of different sizes of
+corpuscles, having different degrees of velocity, of which Nature
+makes use to produce so many marvellous effects.
+
+But though we shall ignore the true cause of springiness we still see
+that there are many bodies which possess this property; and thus there
+is nothing strange in supposing that it exists also in little
+invisible bodies like the particles of the Ether. Also if one wishes
+to seek for any other way in which the movement of Light is
+successively communicated, one will find none which agrees better,
+with uniform progression, as seems to be necessary, than the property
+of springiness; because if this movement should grow slower in
+proportion as it is shared over a greater quantity of matter, in
+moving away from the source of the light, it could not conserve this
+great velocity over great distances. But by supposing springiness in
+the ethereal matter, its particles will have the property of equally
+rapid restitution whether they are pushed strongly or feebly; and thus
+the propagation of Light will always go on with an equal velocity.
+
+[Illustration]
+
+And it must be known that although the particles of the ether are not
+ranged thus in straight lines, as in our row of spheres, but
+confusedly, so that one of them touches several others, this does not
+hinder them from transmitting their movement and from spreading it
+always forward. As to this it is to be remarked that there is a law of
+motion serving for this propagation, and verifiable by experiment. It
+is that when a sphere, such as A here, touches several other similar
+spheres CCC, if it is struck by another sphere B in such a way as to
+exert an impulse against all the spheres CCC which touch it, it
+transmits to them the whole of its movement, and remains after that
+motionless like the sphere B. And without supposing that the ethereal
+particles are of spherical form (for I see indeed no need to suppose
+them so) one may well understand that this property of communicating
+an impulse does not fail to contribute to the aforesaid propagation
+of movement.
+
+Equality of size seems to be more necessary, because otherwise there
+ought to be some reflexion of movement backwards when it passes from a
+smaller particle to a larger one, according to the Laws of Percussion
+which I published some years ago.
+
+However, one will see hereafter that we have to suppose such an
+equality not so much as a necessity for the propagation of light as
+for rendering that propagation easier and more powerful; for it is not
+beyond the limits of probability that the particles of the ether have
+been made equal for a purpose so important as that of light, at least
+in that vast space which is beyond the region of atmosphere and which
+seems to serve only to transmit the light of the Sun and the Stars.
+
+I have then shown in what manner one may conceive Light to spread
+successively, by spherical waves, and how it is possible that this
+spreading is accomplished with as great a velocity as that which
+experiments and celestial observations demand. Whence it may be
+further remarked that although the particles are supposed to be in
+continual movement (for there are many reasons for this) the
+successive propagation of the waves cannot be hindered by this;
+because the propagation consists nowise in the transport of those
+particles but merely in a small agitation which they cannot help
+communicating to those surrounding, notwithstanding any movement which
+may act on them causing them to be changing positions amongst
+themselves.
+
+But we must consider still more particularly the origin of these
+waves, and the manner in which they spread. And, first, it follows
+from what has been said on the production of Light, that each little
+region of a luminous body, such as the Sun, a candle, or a burning
+coal, generates its own waves of which that region is the centre. Thus
+in the flame of a candle, having distinguished the points A, B, C,
+concentric circles described about each of these points represent the
+waves which come from them. And one must imagine the same about every
+point of the surface and of the part within the flame.
+
+[Illustration]
+
+But as the percussions at the centres of these waves possess no
+regular succession, it must not be supposed that the waves themselves
+follow one another at equal distances: and if the distances marked in
+the figure appear to be such, it is rather to mark the progression of
+one and the same wave at equal intervals of time than to represent
+several of them issuing from one and the same centre.
+
+After all, this prodigious quantity of waves which traverse one
+another without confusion and without effacing one another must not be
+deemed inconceivable; it being certain that one and the same particle
+of matter can serve for many waves coming from different sides or even
+from contrary directions, not only if it is struck by blows which
+follow one another closely but even for those which act on it at the
+same instant. It can do so because the spreading of the movement is
+successive. This may be proved by the row of equal spheres of hard
+matter, spoken of above. If against this row there are pushed from two
+opposite sides at the same time two similar spheres A and D, one will
+see each of them rebound with the same velocity which it had in
+striking, yet the whole row will remain in its place, although the
+movement has passed along its whole length twice over. And if these
+contrary movements happen to meet one another at the middle sphere, B,
+or at some other such as C, that sphere will yield and act as a spring
+at both sides, and so will serve at the same instant to transmit these
+two movements.
+
+[Illustration]
+
+But what may at first appear full strange and even incredible is that
+the undulations produced by such small movements and corpuscles,
+should spread to such immense distances; as for example from the Sun
+or from the Stars to us. For the force of these waves must grow feeble
+in proportion as they move away from their origin, so that the action
+of each one in particular will without doubt become incapable of
+making itself felt to our sight. But one will cease to be astonished
+by considering how at a great distance from the luminous body an
+infinitude of waves, though they have issued from different points of
+this body, unite together in such a way that they sensibly compose one
+single wave only, which, consequently, ought to have enough force to
+make itself felt. Thus this infinite number of waves which originate
+at the same instant from all points of a fixed star, big it may be as
+the Sun, make practically only one single wave which may well have
+force enough to produce an impression on our eyes. Moreover from each
+luminous point there may come many thousands of waves in the smallest
+imaginable time, by the frequent percussion of the corpuscles which
+strike the Ether at these points: which further contributes to
+rendering their action more sensible.
+
+[Illustration]
+
+There is the further consideration in the emanation of these waves,
+that each particle of matter in which a wave spreads, ought not to
+communicate its motion only to the next particle which is in the
+straight line drawn from the luminous point, but that it also imparts
+some of it necessarily to all the others which touch it and which
+oppose themselves to its movement. So it arises that around each
+particle there is made a wave of which that particle is the centre.
+Thus if DCF is a wave emanating from the luminous point A, which is
+its centre, the particle B, one of those comprised within the sphere
+DCF, will have made its particular or partial wave KCL, which will
+touch the wave DCF at C at the same moment that the principal wave
+emanating from the point A has arrived at DCF; and it is clear that it
+will be only the region C of the wave KCL which will touch the wave
+DCF, to wit, that which is in the straight line drawn through AB.
+Similarly the other particles of the sphere DCF, such as _bb_, _dd_,
+etc., will each make its own wave. But each of these waves can be
+infinitely feeble only as compared with the wave DCF, to the
+composition of which all the others contribute by the part of their
+surface which is most distant from the centre A.
+
+One sees, in addition, that the wave DCF is determined by the
+distance attained in a certain space of time by the movement which
+started from the point A; there being no movement beyond this wave,
+though there will be in the space which it encloses, namely in parts
+of the particular waves, those parts which do not touch the sphere
+DCF. And all this ought not to seem fraught with too much minuteness
+or subtlety, since we shall see in the sequel that all the properties
+of Light, and everything pertaining to its reflexion and its
+refraction, can be explained in principle by this means. This is a
+matter which has been quite unknown to those who hitherto have begun
+to consider the waves of light, amongst whom are Mr. Hooke in his
+_Micrographia_, and Father Pardies, who, in a treatise of which he let
+me see a portion, and which he was unable to complete as he died
+shortly afterward, had undertaken to prove by these waves the effects
+of reflexion and refraction. But the chief foundation, which consists
+in the remark I have just made, was lacking in his demonstrations; and
+for the rest he had opinions very different from mine, as may be will
+appear some day if his writing has been preserved.
+
+To come to the properties of Light. We remark first that each portion
+of a wave ought to spread in such a way that its extremities lie
+always between the same straight lines drawn from the luminous point.
+Thus the portion BG of the wave, having the luminous point A as its
+centre, will spread into the arc CE bounded by the straight lines ABC,
+AGE. For although the particular waves produced by the particles
+comprised within the space CAE spread also outside this space, they
+yet do not concur at the same instant to compose a wave which
+terminates the movement, as they do precisely at the circumference
+CE, which is their common tangent.
+
+And hence one sees the reason why light, at least if its rays are not
+reflected or broken, spreads only by straight lines, so that it
+illuminates no object except when the path from its source to that
+object is open along such lines.
+
+For if, for example, there were an opening BG, limited by opaque
+bodies BH, GI, the wave of light which issues from the point A will
+always be terminated by the straight lines AC, AE, as has just been
+shown; the parts of the partial waves which spread outside the space
+ACE being too feeble to produce light there.
+
+Now, however small we make the opening BG, there is always the same
+reason causing the light there to pass between straight lines; since
+this opening is always large enough to contain a great number of
+particles of the ethereal matter, which are of an inconceivable
+smallness; so that it appears that each little portion of the wave
+necessarily advances following the straight line which comes from the
+luminous point. Thus then we may take the rays of light as if they
+were straight lines.
+
+It appears, moreover, by what has been remarked touching the
+feebleness of the particular waves, that it is not needful that all
+the particles of the Ether should be equal amongst themselves, though
+equality is more apt for the propagation of the movement. For it is
+true that inequality will cause a particle by pushing against another
+larger one to strive to recoil with a part of its movement; but it
+will thereby merely generate backwards towards the luminous point some
+partial waves incapable of causing light, and not a wave compounded of
+many as CE was.
+
+Another property of waves of light, and one of the most marvellous,
+is that when some of them come from different or even from opposing
+sides, they produce their effect across one another without any
+hindrance. Whence also it comes about that a number of spectators may
+view different objects at the same time through the same opening, and
+that two persons can at the same time see one another's eyes. Now
+according to the explanation which has been given of the action of
+light, how the waves do not destroy nor interrupt one another when
+they cross one another, these effects which I have just mentioned are
+easily conceived. But in my judgement they are not at all easy to
+explain according to the views of Mr. Des Cartes, who makes Light to
+consist in a continuous pressure merely tending to movement. For this
+pressure not being able to act from two opposite sides at the same
+time, against bodies which have no inclination to approach one
+another, it is impossible so to understand what I have been saying
+about two persons mutually seeing one another's eyes, or how two
+torches can illuminate one another.
+
+
+
+
+CHAPTER II
+
+ON REFLEXION
+
+
+Having explained the effects of waves of light which spread in a
+homogeneous matter, we will examine next that which happens to them on
+encountering other bodies. We will first make evident how the
+Reflexion of light is explained by these same waves, and why it
+preserves equality of angles.
+
+Let there be a surface AB; plane and polished, of some metal, glass,
+or other body, which at first I will consider as perfectly uniform
+(reserving to myself to deal at the end of this demonstration with the
+inequalities from which it cannot be exempt), and let a line AC,
+inclined to AD, represent a portion of a wave of light, the centre of
+which is so distant that this portion AC may be considered as a
+straight line; for I consider all this as in one plane, imagining to
+myself that the plane in which this figure is, cuts the sphere of the
+wave through its centre and intersects the plane AB at right angles.
+This explanation will suffice once for all.
+
+[Illustration]
+
+The piece C of the wave AC, will in a certain space of time advance as
+far as the plane AB at B, following the straight line CB, which may be
+supposed to come from the luminous centre, and which in consequence is
+perpendicular to AC. Now in this same space of time the portion A of
+the same wave, which has been hindered from communicating its movement
+beyond the plane AB, or at least partly so, ought to have continued
+its movement in the matter which is above this plane, and this along a
+distance equal to CB, making its own partial spherical wave,
+according to what has been said above. Which wave is here represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter AN equal to CB.
+
+If one considers further the other pieces H of the wave AC, it appears
+that they will not only have reached the surface AB by straight lines
+HK parallel to CB, but that in addition they will have generated in
+the transparent air, from the centres K, K, K, particular spherical
+waves, represented here by circumferences the semi-diameters of which
+are equal to KM, that is to say to the continuations of HK as far as
+the line BG parallel to AC. But all these circumferences have as a
+common tangent the straight line BN, namely the same which is drawn
+from B as a tangent to the first of the circles, of which A is the
+centre, and AN the semi-diameter equal to BC, as is easy to see.
+
+It is then the line BN (comprised between B and the point N where the
+perpendicular from the point A falls) which is as it were formed by
+all these circumferences, and which terminates the movement which is
+made by the reflexion of the wave AC; and it is also the place where
+the movement occurs in much greater quantity than anywhere else.
+Wherefore, according to that which has been explained, BN is the
+propagation of the wave AC at the moment when the piece C of it has
+arrived at B. For there is no other line which like BN is a common
+tangent to all the aforesaid circles, except BG below the plane AB;
+which line BG would be the propagation of the wave if the movement
+could have spread in a medium homogeneous with that which is above the
+plane. And if one wishes to see how the wave AC has come successively
+to BN, one has only to draw in the same figure the straight lines KO
+parallel to BN, and the straight lines KL parallel to AC. Thus one
+will see that the straight wave AC has become broken up into all the
+OKL parts successively, and that it has become straight again at NB.
+
+Now it is apparent here that the angle of reflexion is made equal to
+the angle of incidence. For the triangles ACB, BNA being rectangular
+and having the side AB common, and the side CB equal to NA, it follows
+that the angles opposite to these sides will be equal, and therefore
+also the angles CBA, NAB. But as CB, perpendicular to CA, marks the
+direction of the incident ray, so AN, perpendicular to the wave BN,
+marks the direction of the reflected ray; hence these rays are equally
+inclined to the plane AB.
+
+But in considering the preceding demonstration, one might aver that it
+is indeed true that BN is the common tangent of the circular waves in
+the plane of this figure, but that these waves, being in truth
+spherical, have still an infinitude of similar tangents, namely all
+the straight lines which are drawn from the point B in the surface
+generated by the straight line BN about the axis BA. It remains,
+therefore, to demonstrate that there is no difficulty herein: and by
+the same argument one will see why the incident ray and the reflected
+ray are always in one and the same plane perpendicular to the
+reflecting plane. I say then that the wave AC, being regarded only as
+a line, produces no light. For a visible ray of light, however narrow
+it may be, has always some width, and consequently it is necessary, in
+representing the wave whose progression constitutes the ray, to put
+instead of a line AC some plane figure such as the circle HC in the
+following figure, by supposing, as we have done, the luminous point to
+be infinitely distant. Now it is easy to see, following the preceding
+demonstration, that each small piece of this wave HC having arrived at
+the plane AB, and there generating each one its particular wave, these
+will all have, when C arrives at B, a common plane which will touch
+them, namely a circle BN similar to CH; and this will be intersected
+at its middle and at right angles by the same plane which likewise
+intersects the circle CH and the ellipse AB.
+
+[Illustration]
+
+One sees also that the said spheres of the partial waves cannot have
+any common tangent plane other than the circle BN; so that it will be
+this plane where there will be more reflected movement than anywhere
+else, and which will therefore carry on the light in continuance from
+the wave CH.
+
+I have also stated in the preceding demonstration that the movement of
+the piece A of the incident wave is not able to communicate itself
+beyond the plane AB, or at least not wholly. Whence it is to be
+remarked that though the movement of the ethereal matter might
+communicate itself partly to that of the reflecting body, this could
+in nothing alter the velocity of progression of the waves, on which
+the angle of reflexion depends. For a slight percussion ought to
+generate waves as rapid as strong percussion in the same matter. This
+comes about from the property of bodies which act as springs, of which
+we have spoken above; namely that whether compressed little or much
+they recoil in equal times. Equally so in every reflexion of the
+light, against whatever body it may be, the angles of reflexion and
+incidence ought to be equal notwithstanding that the body might be of
+such a nature that it takes away a portion of the movement made by the
+incident light. And experiment shows that in fact there is no polished
+body the reflexion of which does not follow this rule.
+
+
+But the thing to be above all remarked in our demonstration is that it
+does not require that the reflecting surface should be considered as a
+uniform plane, as has been supposed by all those who have tried to
+explain the effects of reflexion; but only an evenness such as may be
+attained by the particles of the matter of the reflecting body being
+set near to one another; which particles are larger than those of the
+ethereal matter, as will appear by what we shall say in treating of
+the transparency and opacity of bodies. For the surface consisting
+thus of particles put together, and the ethereal particles being
+above, and smaller, it is evident that one could not demonstrate the
+equality of the angles of incidence and reflexion by similitude to
+that which happens to a ball thrown against a wall, of which writers
+have always made use. In our way, on the other hand, the thing is
+explained without difficulty. For the smallness of the particles of
+quicksilver, for example, being such that one must conceive millions
+of them, in the smallest visible surface proposed, arranged like a
+heap of grains of sand which has been flattened as much as it is
+capable of being, this surface then becomes for our purpose as even
+as a polished glass is: and, although it always remains rough with
+respect to the particles of the Ether it is evident that the centres
+of all the particular spheres of reflexion, of which we have spoken,
+are almost in one uniform plane, and that thus the common tangent can
+fit to them as perfectly as is requisite for the production of light.
+And this alone is requisite, in our method of demonstration, to cause
+equality of the said angles without the remainder of the movement
+reflected from all parts being able to produce any contrary effect.
+
+
+
+
+CHAPTER III
+
+ON REFRACTION
+
+
+In the same way as the effects of Reflexion have been explained by
+waves of light reflected at the surface of polished bodies, we will
+explain transparency and the phenomena of refraction by waves which
+spread within and across diaphanous bodies, both solids, such as
+glass, and liquids, such as water, oils, etc. But in order that it may
+not seem strange to suppose this passage of waves in the interior of
+these bodies, I will first show that one may conceive it possible in
+more than one mode.
+
+First, then, if the ethereal matter cannot penetrate transparent
+bodies at all, their own particles would be able to communicate
+successively the movement of the waves, the same as do those of the
+Ether, supposing that, like those, they are of a nature to act as a
+spring. And this is easy to conceive as regards water and other
+transparent liquids, they being composed of detached particles. But it
+may seem more difficult as regards glass and other transparent and
+hard bodies, because their solidity does not seem to permit them to
+receive movement except in their whole mass at the same time. This,
+however, is not necessary because this solidity is not such as it
+appears to us, it being probable rather that these bodies are composed
+of particles merely placed close to one another and held together by
+some pressure from without of some other matter, and by the
+irregularity of their shapes. For primarily their rarity is shown by
+the facility with which there passes through them the matter of the
+vortices of the magnet, and that which causes gravity. Further, one
+cannot say that these bodies are of a texture similar to that of a
+sponge or of light bread, because the heat of the fire makes them flow
+and thereby changes the situation of the particles amongst themselves.
+It remains then that they are, as has been said, assemblages of
+particles which touch one another without constituting a continuous
+solid. This being so, the movement which these particles receive to
+carry on the waves of light, being merely communicated from some of
+them to others, without their going for that purpose out of their
+places or without derangement, it may very well produce its effect
+without prejudicing in any way the apparent solidity of the compound.
+
+By pressure from without, of which I have spoken, must not be
+understood that of the air, which would not be sufficient, but that of
+some other more subtle matter, a pressure which I chanced upon by
+experiment long ago, namely in the case of water freed from air, which
+remains suspended in a tube open at its lower end, notwithstanding
+that the air has been removed from the vessel in which this tube is
+enclosed.
+
+One can then in this way conceive of transparency in a solid without
+any necessity that the ethereal matter which serves for light should
+pass through it, or that it should find pores in which to insinuate
+itself. But the truth is that this matter not only passes through
+solids, but does so even with great facility; of which the experiment
+of Torricelli, above cited, is already a proof. Because on the
+quicksilver and the water quitting the upper part of the glass tube,
+it appears that it is immediately filled with ethereal matter, since
+light passes across it. But here is another argument which proves this
+ready penetrability, not only in transparent bodies but also in all
+others.
+
+When light passes across a hollow sphere of glass, closed on all
+sides, it is certain that it is full of ethereal matter, as much as
+the spaces outside the sphere. And this ethereal matter, as has been
+shown above, consists of particles which just touch one another. If
+then it were enclosed in the sphere in such a way that it could not
+get out through the pores of the glass, it would be obliged to follow
+the movement of the sphere when one changes its place: and it would
+require consequently almost the same force to impress a certain
+velocity on this sphere, when placed on a horizontal plane, as if it
+were full of water or perhaps of quicksilver: because every body
+resists the velocity of the motion which one would give to it, in
+proportion to the quantity of matter which it contains, and which is
+obliged to follow this motion. But on the contrary one finds that the
+sphere resists the impress of movement only in proportion to the
+quantity of matter of the glass of which it is made. Then it must be
+that the ethereal matter which is inside is not shut up, but flows
+through it with very great freedom. We shall demonstrate hereafter
+that by this process the same penetrability may be inferred also as
+relating to opaque bodies.
+
+The second mode then of explaining transparency, and one which appears
+more probably true, is by saying that the waves of light are carried
+on in the ethereal matter, which continuously occupies the interstices
+or pores of transparent bodies. For since it passes through them
+continuously and freely, it follows that they are always full of it.
+And one may even show that these interstices occupy much more space
+than the coherent particles which constitute the bodies. For if what
+we have just said is true: that force is required to impress a certain
+horizontal velocity on bodies in proportion as they contain coherent
+matter; and if the proportion of this force follows the law of
+weights, as is confirmed by experiment, then the quantity of the
+constituent matter of bodies also follows the proportion of their
+weights. Now we see that water weighs only one fourteenth part as much
+as an equal portion of quicksilver: therefore the matter of the water
+does not occupy the fourteenth part of the space which its mass
+obtains. It must even occupy much less of it, since quicksilver is
+less heavy than gold, and the matter of gold is by no means dense, as
+follows from the fact that the matter of the vortices of the magnet
+and of that which is the cause of gravity pass very freely through it.
+
+But it may be objected here that if water is a body of so great
+rarity, and if its particles occupy so small a portion of the space of
+its apparent bulk, it is very strange how it yet resists Compression
+so strongly without permitting itself to be condensed by any force
+which one has hitherto essayed to employ, preserving even its entire
+liquidity while subjected to this pressure.
+
+This is no small difficulty. It may, however, be resolved by saying
+that the very violent and rapid motion of the subtle matter which
+renders water liquid, by agitating the particles of which it is
+composed, maintains this liquidity in spite of the pressure which
+hitherto any one has been minded to apply to it.
+
+The rarity of transparent bodies being then such as we have said, one
+easily conceives that the waves might be carried on in the ethereal
+matter which fills the interstices of the particles. And, moreover,
+one may believe that the progression of these waves ought to be a
+little slower in the interior of bodies, by reason of the small
+detours which the same particles cause. In which different velocity of
+light I shall show the cause of refraction to consist.
+
+Before doing so, I will indicate the third and last mode in which
+transparency may be conceived; which is by supposing that the movement
+of the waves of light is transmitted indifferently both in the
+particles of the ethereal matter which occupy the interstices of
+bodies, and in the particles which compose them, so that the movement
+passes from one to the other. And it will be seen hereafter that this
+hypothesis serves excellently to explain the double refraction of
+certain transparent bodies.
+
+Should it be objected that if the particles of the ether are smaller
+than those of transparent bodies (since they pass through their
+intervals), it would follow that they can communicate to them but
+little of their movement, it may be replied that the particles of
+these bodies are in turn composed of still smaller particles, and so
+it will be these secondary particles which will receive the movement
+from those of the ether.
+
+Furthermore, if the particles of transparent bodies have a recoil a
+little less prompt than that of the ethereal particles, which nothing
+hinders us from supposing, it will again follow that the progression
+of the waves of light will be slower in the interior of such bodies
+than it is outside in the ethereal matter.
+
+All this I have found as most probable for the mode in which the waves
+of light pass across transparent bodies. To which it must further be
+added in what respect these bodies differ from those which are opaque;
+and the more so since it might seem because of the easy penetration of
+bodies by the ethereal matter, of which mention has been made, that
+there would not be any body that was not transparent. For by the same
+reasoning about the hollow sphere which I have employed to prove the
+smallness of the density of glass and its easy penetrability by the
+ethereal matter, one might also prove that the same penetrability
+obtains for metals and for every other sort of body. For this sphere
+being for example of silver, it is certain that it contains some of
+the ethereal matter which serves for light, since this was there as
+well as in the air when the opening of the sphere was closed. Yet,
+being closed and placed upon a horizontal plane, it resists the
+movement which one wishes to give to it, merely according to the
+quantity of silver of which it is made; so that one must conclude, as
+above, that the ethereal matter which is enclosed does not follow the
+movement of the sphere; and that therefore silver, as well as glass,
+is very easily penetrated by this matter. Some of it is therefore
+present continuously and in quantities between the particles of silver
+and of all other opaque bodies: and since it serves for the
+propagation of light it would seem that these bodies ought also to be
+transparent, which however is not the case.
+
+Whence then, one will say, does their opacity come? Is it because the
+particles which compose them are soft; that is to say, these particles
+being composed of others that are smaller, are they capable of
+changing their figure on receiving the pressure of the ethereal
+particles, the motion of which they thereby damp, and so hinder the
+continuance of the waves of light? That cannot be: for if the
+particles of the metals are soft, how is it that polished silver and
+mercury reflect light so strongly? What I find to be most probable
+herein, is to say that metallic bodies, which are almost the only
+really opaque ones, have mixed amongst their hard particles some soft
+ones; so that some serve to cause reflexion and the others to hinder
+transparency; while, on the other hand, transparent bodies contain
+only hard particles which have the faculty of recoil, and serve
+together with those of the ethereal matter for the propagation of the
+waves of light, as has been said.
+
+[Illustration]
+
+Let us pass now to the explanation of the effects of Refraction,
+assuming, as we have done, the passage of waves of light through
+transparent bodies, and the diminution of velocity which these same
+waves suffer in them.
+
+The chief property of Refraction is that a ray of light, such as AB,
+being in the air, and falling obliquely upon the polished surface of a
+transparent body, such as FG, is broken at the point of incidence B,
+in such a way that with the straight line DBE which cuts the surface
+perpendicularly it makes an angle CBE less than ABD which it made with
+the same perpendicular when in the air. And the measure of these
+angles is found by describing, about the point B, a circle which cuts
+the radii AB, BC. For the perpendiculars AD, CE, let fall from the
+points of intersection upon the straight line DE, which are called the
+Sines of the angles ABD, CBE, have a certain ratio between themselves;
+which ratio is always the same for all inclinations of the incident
+ray, at least for a given transparent body. This ratio is, in glass,
+very nearly as 3 to 2; and in water very nearly as 4 to 3; and is
+likewise different in other diaphanous bodies.
+
+Another property, similar to this, is that the refractions are
+reciprocal between the rays entering into a transparent body and those
+which are leaving it. That is to say that if the ray AB in entering
+the transparent body is refracted into BC, then likewise CB being
+taken as a ray in the interior of this body will be refracted, on
+passing out, into BA.
+
+[Illustration]
+
+To explain then the reasons of these phenomena according to our
+principles, let AB be the straight line which represents a plane
+surface bounding the transparent substances which lie towards C and
+towards N. When I say plane, that does not signify a perfect evenness,
+but such as has been understood in treating of reflexion, and for the
+same reason. Let the line AC represent a portion of a wave of light,
+the centre of which is supposed so distant that this portion may be
+considered as a straight line. The piece C, then, of the wave AC, will
+in a certain space of time have advanced as far as the plane AB
+following the straight line CB, which may be imagined as coming from
+the luminous centre, and which consequently will cut AC at right
+angles. Now in the same time the piece A would have come to G along
+the straight line AG, equal and parallel to CB; and all the portion of
+wave AC would be at GB if the matter of the transparent body
+transmitted the movement of the wave as quickly as the matter of the
+Ether. But let us suppose that it transmits this movement less
+quickly, by one-third, for instance. Movement will then be spread from
+the point A, in the matter of the transparent body through a distance
+equal to two-thirds of CB, making its own particular spherical wave
+according to what has been said before. This wave is then represented
+by the circumference SNR, the centre of which is A, and its
+semi-diameter equal to two-thirds of CB. Then if one considers in
+order the other pieces H of the wave AC, it appears that in the same
+time that the piece C reaches B they will not only have arrived at the
+surface AB along the straight lines HK parallel to CB, but that, in
+addition, they will have generated in the diaphanous substance from
+the centres K, partial waves, represented here by circumferences the
+semi-diameters of which are equal to two-thirds of the lines KM, that
+is to say, to two-thirds of the prolongations of HK down to the
+straight line BG; for these semi-diameters would have been equal to
+entire lengths of KM if the two transparent substances had been of the
+same penetrability.
+
+Now all these circumferences have for a common tangent the straight
+line BN; namely the same line which is drawn as a tangent from the
+point B to the circumference SNR which we considered first. For it is
+easy to see that all the other circumferences will touch the same BN,
+from B up to the point of contact N, which is the same point where AN
+falls perpendicularly on BN.
+
+It is then BN, which is formed by small arcs of these circumferences,
+which terminates the movement that the wave AC has communicated within
+the transparent body, and where this movement occurs in much greater
+amount than anywhere else. And for that reason this line, in
+accordance with what has been said more than once, is the propagation
+of the wave AC at the moment when its piece C has reached B. For there
+is no other line below the plane AB which is, like BN, a common
+tangent to all these partial waves. And if one would know how the wave
+AC has come progressively to BN, it is necessary only to draw in the
+same figure the straight lines KO parallel to BN, and all the lines KL
+parallel to AC. Thus one will see that the wave CA, from being a
+straight line, has become broken in all the positions LKO
+successively, and that it has again become a straight line at BN. This
+being evident by what has already been demonstrated, there is no need
+to explain it further.
+
+Now, in the same figure, if one draws EAF, which cuts the plane AB at
+right angles at the point A, since AD is perpendicular to the wave AC,
+it will be DA which will mark the ray of incident light, and AN which
+was perpendicular to BN, the refracted ray: since the rays are nothing
+else than the straight lines along which the portions of the waves
+advance.
+
+Whence it is easy to recognize this chief property of refraction,
+namely that the Sine of the angle DAE has always the same ratio to the
+Sine of the angle NAF, whatever be the inclination of the ray DA: and
+that this ratio is the same as that of the velocity of the waves in
+the transparent substance which is towards AE to their velocity in the
+transparent substance towards AF. For, considering AB as the radius of
+a circle, the Sine of the angle BAC is BC, and the Sine of the angle
+ABN is AN. But the angle BAC is equal to DAE, since each of them added
+to CAE makes a right angle. And the angle ABN is equal to NAF, since
+each of them with BAN makes a right angle. Then also the Sine of the
+angle DAE is to the Sine of NAF as BC is to AN. But the ratio of BC to
+AN was the same as that of the velocities of light in the substance
+which is towards AE and in that which is towards AF; therefore also
+the Sine of the angle DAE will be to the Sine of the angle NAF the
+same as the said velocities of light.
+
+To see, consequently, what the refraction will be when the waves of
+light pass into a substance in which the movement travels more quickly
+than in that from which they emerge (let us again assume the ratio of
+3 to 2), it is only necessary to repeat all the same construction and
+demonstration which we have just used, merely substituting everywhere
+3/2 instead of 2/3. And it will be found by the same reasoning, in
+this other figure, that when the piece C of the wave AC shall have
+reached the surface AB at B, all the portions of the wave AC will
+have advanced as far as BN, so that BC the perpendicular on AC is to
+AN the perpendicular on BN as 2 to 3. And there will finally be this
+same ratio of 2 to 3 between the Sine of the angle BAD and the Sine of
+the angle FAN.
+
+Hence one sees the reciprocal relation of the refractions of the ray
+on entering and on leaving one and the same transparent body: namely
+that if NA falling on the external surface AB is refracted into the
+direction AD, so the ray AD will be refracted on leaving the
+transparent body into the direction AN.
+
+[Illustration]
+
+One sees also the reason for a noteworthy accident which happens in
+this refraction: which is this, that after a certain obliquity of the
+incident ray DA, it begins to be quite unable to penetrate into the
+other transparent substance. For if the angle DAQ or CBA is such that
+in the triangle ACB, CB is equal to 2/3 of AB, or is greater, then AN
+cannot form one side of the triangle ANB, since it becomes equal to or
+greater than AB: so that the portion of wave BN cannot be found
+anywhere, neither consequently can AN, which ought to be perpendicular
+to it. And thus the incident ray DA does not then pierce the surface
+AB.
+
+When the ratio of the velocities of the waves is as two to three, as
+in our example, which is that which obtains for glass and air, the
+angle DAQ must be more than 48 degrees 11 minutes in order that the
+ray DA may be able to pass by refraction. And when the ratio of the
+velocities is as 3 to 4, as it is very nearly in water and air, this
+angle DAQ must exceed 41 degrees 24 minutes. And this accords
+perfectly with experiment.
+
+But it might here be asked: since the meeting of the wave AC against
+the surface AB ought to produce movement in the matter which is on the
+other side, why does no light pass there? To which the reply is easy
+if one remembers what has been said before. For although it generates
+an infinitude of partial waves in the matter which is at the other
+side of AB, these waves never have a common tangent line (either
+straight or curved) at the same moment; and so there is no line
+terminating the propagation of the wave AC beyond the plane AB, nor
+any place where the movement is gathered together in sufficiently
+great quantity to produce light. And one will easily see the truth of
+this, namely that CB being larger than 2/3 of AB, the waves excited
+beyond the plane AB will have no common tangent if about the centres K
+one then draws circles having radii equal to 3/2 of the lengths LB to
+which they correspond. For all these circles will be enclosed in one
+another and will all pass beyond the point B.
+
+Now it is to be remarked that from the moment when the angle DAQ is
+smaller than is requisite to permit the refracted ray DA to pass into
+the other transparent substance, one finds that the interior reflexion
+which occurs at the surface AB is much augmented in brightness, as is
+easy to realize by experiment with a triangular prism; and for this
+our theory can afford this reason. When the angle DAQ is still large
+enough to enable the ray DA to pass, it is evident that the light from
+the portion AC of the wave is collected in a minimum space when it
+reaches BN. It appears also that the wave BN becomes so much the
+smaller as the angle CBA or DAQ is made less; until when the latter is
+diminished to the limit indicated a little previously, this wave BN is
+collected together always at one point. That is to say, that when the
+piece C of the wave AC has then arrived at B, the wave BN which is the
+propagation of AC is entirely reduced to the same point B. Similarly
+when the piece H has reached K, the part AH is entirely reduced to the
+same point K. This makes it evident that in proportion as the wave CA
+comes to meet the surface AB, there occurs a great quantity of
+movement along that surface; which movement ought also to spread
+within the transparent body and ought to have much re-enforced the
+partial waves which produce the interior reflexion against the surface
+AB, according to the laws of reflexion previously explained.
+
+And because a slight diminution of the angle of incidence DAQ causes
+the wave BN, however great it was, to be reduced to zero, (for this
+angle being 49 degrees 11 minutes in the glass, the angle BAN is still
+11 degrees 21 minutes, and the same angle being reduced by one degree
+only the angle BAN is reduced to zero, and so the wave BN reduced to a
+point) thence it comes about that the interior reflexion from being
+obscure becomes suddenly bright, so soon as the angle of incidence is
+such that it no longer gives passage to the refraction.
+
+Now as concerns ordinary external reflexion, that is to say which
+occurs when the angle of incidence DAQ is still large enough to enable
+the refracted ray to penetrate beyond the surface AB, this reflexion
+should occur against the particles of the substance which touches the
+transparent body on its outside. And it apparently occurs against the
+particles of the air or others mingled with the ethereal particles and
+larger than they. So on the other hand the external reflexion of these
+bodies occurs against the particles which compose them, and which are
+also larger than those of the ethereal matter, since the latter flows
+in their interstices. It is true that there remains here some
+difficulty in those experiments in which this interior reflexion
+occurs without the particles of air being able to contribute to it, as
+in vessels or tubes from which the air has been extracted.
+
+Experience, moreover, teaches us that these two reflexions are of
+nearly equal force, and that in different transparent bodies they are
+so much the stronger as the refraction of these bodies is the greater.
+Thus one sees manifestly that the reflexion of glass is stronger than
+that of water, and that of diamond stronger than that of glass.
+
+I will finish this theory of refraction by demonstrating a remarkable
+proposition which depends on it; namely, that a ray of light in order
+to go from one point to another, when these points are in different
+media, is refracted in such wise at the plane surface which joins
+these two media that it employs the least possible time: and exactly
+the same happens in the case of reflexion against a plane surface. Mr.
+Fermat was the first to propound this property of refraction, holding
+with us, and directly counter to the opinion of Mr. Des Cartes, that
+light passes more slowly through glass and water than through air.
+But he assumed besides this a constant ratio of Sines, which we have
+just proved by these different degrees of velocity alone: or rather,
+what is equivalent, he assumed not only that the velocities were
+different but that the light took the least time possible for its
+passage, and thence deduced the constant ratio of the Sines. His
+demonstration, which may be seen in his printed works, and in the
+volume of letters of Mr. Des Cartes, is very long; wherefore I give
+here another which is simpler and easier.
+
+[Illustration]
+
+Let KF be the plane surface; A the point in the medium which the light
+traverses more easily, as the air; C the point in the other which is
+more difficult to penetrate, as water. And suppose that a ray has come
+from A, by B, to C, having been refracted at B according to the law
+demonstrated a little before; that is to say that, having drawn PBQ,
+which cuts the plane at right angles, let the sine of the angle ABP
+have to the sine of the angle CBQ the same ratio as the velocity of
+light in the medium where A is to the velocity of light in the medium
+where C is. It is to be shown that the time of passage of light along
+AB and BC taken together, is the shortest that can be. Let us assume
+that it may have come by other lines, and, in the first place, along
+AF, FC, so that the point of refraction F may be further from B than
+the point A; and let AO be a line perpendicular to AB, and FO parallel
+to AB; BH perpendicular to FO, and FG to BC.
+
+Since then the angle HBF is equal to PBA, and the angle BFG equal to
+QBC, it follows that the sine of the angle HBF will also have the same
+ratio to the sine of BFG, as the velocity of light in the medium A is
+to its velocity in the medium C. But these sines are the straight
+lines HF, BG, if we take BF as the semi-diameter of a circle. Then
+these lines HF, BG, will bear to one another the said ratio of the
+velocities. And, therefore, the time of the light along HF, supposing
+that the ray had been OF, would be equal to the time along BG in the
+interior of the medium C. But the time along AB is equal to the time
+along OH; therefore the time along OF is equal to the time along AB,
+BG. Again the time along FC is greater than that along GC; then the
+time along OFC will be longer than that along ABC. But AF is longer
+than OF, then the time along AFC will by just so much more exceed the
+time along ABC.
+
+Now let us assume that the ray has come from A to C along AK, KC; the
+point of refraction K being nearer to A than the point B is; and let
+CN be the perpendicular upon BC, KN parallel to BC: BM perpendicular
+upon KN, and KL upon BA.
+
+Here BL and KM are the sines of angles BKL, KBM; that is to say, of
+the angles PBA, QBC; and therefore they are to one another as the
+velocity of light in the medium A is to the velocity in the medium C.
+Then the time along LB is equal to the time along KM; and since the
+time along BC is equal to the time along MN, the time along LBC will
+be equal to the time along KMN. But the time along AK is longer than
+that along AL: hence the time along AKN is longer than that along ABC.
+And KC being longer than KN, the time along AKC will exceed, by as
+much more, the time along ABC. Hence it appears that the time along
+ABC is the shortest possible; which was to be proven.
+
+
+
+
+CHAPTER IV
+
+ON THE REFRACTION OF THE AIR
+
+
+We have shown how the movement which constitutes light spreads by
+spherical waves in any homogeneous matter. And it is evident that when
+the matter is not homogeneous, but of such a constitution that the
+movement is communicated in it more rapidly toward one side than
+toward another, these waves cannot be spherical: but that they must
+acquire their figure according to the different distances over which
+the successive movement passes in equal times.
+
+It is thus that we shall in the first place explain the refractions
+which occur in the air, which extends from here to the clouds and
+beyond. The effects of which refractions are very remarkable; for by
+them we often see objects which the rotundity of the Earth ought
+otherwise to hide; such as Islands, and the tops of mountains when one
+is at sea. Because also of them the Sun and the Moon appear as risen
+before in fact they have, and appear to set later: so that at times
+the Moon has been seen eclipsed while the Sun appeared still above the
+horizon. And so also the heights of the Sun and of the Moon, and those
+of all the Stars always appear a little greater than they are in
+reality, because of these same refractions, as Astronomers know. But
+there is one experiment which renders this refraction very evident;
+which is that of fixing a telescope on some spot so that it views an
+object, such as a steeple or a house, at a distance of half a league
+or more. If then you look through it at different hours of the day,
+leaving it always fixed in the same way, you will see that the same
+spots of the object will not always appear at the middle of the
+aperture of the telescope, but that generally in the morning and in
+the evening, when there are more vapours near the Earth, these objects
+seem to rise higher, so that the half or more of them will no longer
+be visible; and so that they seem lower toward mid-day when these
+vapours are dissipated.
+
+Those who consider refraction to occur only in the surfaces which
+separate transparent bodies of different nature, would find it
+difficult to give a reason for all that I have just related; but
+according to our Theory the thing is quite easy. It is known that the
+air which surrounds us, besides the particles which are proper to it
+and which float in the ethereal matter as has been explained, is full
+also of particles of water which are raised by the action of heat; and
+it has been ascertained further by some very definite experiments that
+as one mounts up higher the density of air diminishes in proportion.
+Now whether the particles of water and those of air take part, by
+means of the particles of ethereal matter, in the movement which
+constitutes light, but have a less prompt recoil than these, or
+whether the encounter and hindrance which these particles of air and
+water offer to the propagation of movement of the ethereal progress,
+retard the progression, it follows that both kinds of particles flying
+amidst the ethereal particles, must render the air, from a great
+height down to the Earth, gradually less easy for the spreading of the
+waves of light.
+
+[Illustration]
+
+Whence the configuration of the waves ought to become nearly such as
+this figure represents: namely, if A is a light, or the visible point
+of a steeple, the waves which start from it ought to spread more
+widely upwards and less widely downwards, but in other directions more
+or less as they approximate to these two extremes. This being so, it
+necessarily follows that every line intersecting one of these waves at
+right angles will pass above the point A, always excepting the one
+line which is perpendicular to the horizon.
+
+[Illustration]
+
+Let BC be the wave which brings the light to the spectator who is at
+B, and let BD be the straight line which intersects this wave at right
+angles. Now because the ray or straight line by which we judge the
+spot where the object appears to us is nothing else than the
+perpendicular to the wave that reaches our eye, as will be understood
+by what was said above, it is manifest that the point A will be
+perceived as being in the line BD, and therefore higher than in fact it
+is.
+
+Similarly if the Earth be AB, and the top of the Atmosphere CD, which
+probably is not a well defined spherical surface (since we know that
+the air becomes rare in proportion as one ascends, for above there is
+so much less of it to press down upon it), the waves of light from the
+sun coming, for instance, in such a way that so long as they have not
+reached the Atmosphere CD the straight line AE intersects them
+perpendicularly, they ought, when they enter the Atmosphere, to
+advance more quickly in elevated regions than in regions nearer to the
+Earth. So that if CA is the wave which brings the light to the
+spectator at A, its region C will be the furthest advanced; and the
+straight line AF, which intersects this wave at right angles, and
+which determines the apparent place of the Sun, will pass above the
+real Sun, which will be seen along the line AE. And so it may occur
+that when it ought not to be visible in the absence of vapours,
+because the line AE encounters the rotundity of the Earth, it will be
+perceived in the line AF by refraction. But this angle EAF is scarcely
+ever more than half a degree because the attenuation of the vapours
+alters the waves of light but little. Furthermore these refractions
+are not altogether constant in all weathers, particularly at small
+elevations of 2 or 3 degrees; which results from the different
+quantity of aqueous vapours rising above the Earth.
+
+And this same thing is the cause why at certain times a distant object
+will be hidden behind another less distant one, and yet may at another
+time be able to be seen, although the spot from which it is viewed is
+always the same. But the reason for this effect will be still more
+evident from what we are going to remark touching the curvature of
+rays. It appears from the things explained above that the progression
+or propagation of a small part of a wave of light is properly what one
+calls a ray. Now these rays, instead of being straight as they are in
+homogeneous media, ought to be curved in an atmosphere of unequal
+penetrability. For they necessarily follow from the object to the eye
+the line which intersects at right angles all the progressions of the
+waves, as in the first figure the line AEB does, as will be shown
+hereafter; and it is this line which determines what interposed bodies
+would or would not hinder us from seeing the object. For although the
+point of the steeple A appears raised to D, it would yet not appear to
+the eye B if the tower H was between the two, because it crosses the
+curve AEB. But the tower E, which is beneath this curve, does not
+hinder the point A from being seen. Now according as the air near the
+Earth exceeds in density that which is higher, the curvature of the
+ray AEB becomes greater: so that at certain times it passes above the
+summit E, which allows the point A to be perceived by the eye at B;
+and at other times it is intercepted by the same tower E which hides A
+from this same eye.
+
+[Illustration]
+
+But to demonstrate this curvature of the rays conformably to all our
+preceding Theory, let us imagine that AB is a small portion of a wave
+of light coming from the side C, which we may consider as a straight
+line. Let us also suppose that it is perpendicular to the Horizon, the
+portion B being nearer to the Earth than the portion A; and that
+because the vapours are less hindering at A than at B, the particular
+wave which comes from the point A spreads through a certain space AD
+while the particular wave which starts from the point B spreads
+through a shorter space BE; AD and BE being parallel to the Horizon.
+Further, supposing the straight lines FG, HI, etc., to be drawn from
+an infinitude of points in the straight line AB and to terminate on
+the line DE (which is straight or may be considered as such), let the
+different penetrabilities at the different heights in the air between
+A and B be represented by all these lines; so that the particular
+wave, originating from the point F, will spread across the space FG,
+and that from the point H across the space HI, while that from the
+point A spreads across the space AD.
+
+Now if about the centres A, B, one describes the circles DK, EL, which
+represent the spreading of the waves which originate from these two
+points, and if one draws the straight line KL which touches these two
+circles, it is easy to see that this same line will be the common
+tangent to all the other circles drawn about the centres F, H, etc.;
+and that all the points of contact will fall within that part of this
+line which is comprised between the perpendiculars AK, BL. Then it
+will be the line KL which will terminate the movement of the
+particular waves originating from the points of the wave AB; and this
+movement will be stronger between the points KL, than anywhere else at
+the same instant, since an infinitude of circumferences concur to form
+this straight line; and consequently KL will be the propagation of the
+portion of wave AB, as has been said in explaining reflexion and
+ordinary refraction. Now it appears that AK and BL dip down toward the
+side where the air is less easy to penetrate: for AK being longer than
+BL, and parallel to it, it follows that the lines AB and KL, being
+prolonged, would meet at the side L. But the angle K is a right angle:
+hence KAB is necessarily acute, and consequently less than DAB. If one
+investigates in the same way the progression of the portion of the
+wave KL, one will find that after a further time it has arrived at MN
+in such a manner that the perpendiculars KM, LN, dip down even more
+than do AK, BL. And this suffices to show that the ray will continue
+along the curved line which intersects all the waves at right angles,
+as has been said.
+
+
+
+
+CHAPTER V
+
+ON THE STRANGE REFRACTION OF ICELAND CRYSTAL
+
+
+1.
+
+There is brought from Iceland, which is an Island in the North Sea, in
+the latitude of 66 degrees, a kind of Crystal or transparent stone,
+very remarkable for its figure and other qualities, but above all for
+its strange refractions. The causes of this have seemed to me to be
+worthy of being carefully investigated, the more so because amongst
+transparent bodies this one alone does not follow the ordinary rules
+with respect to rays of light. I have even been under some necessity
+to make this research, because the refractions of this Crystal seemed
+to overturn our preceding explanation of regular refraction; which
+explanation, on the contrary, they strongly confirm, as will be seen
+after they have been brought under the same principle. In Iceland are
+found great lumps of this Crystal, some of which I have seen of 4 or 5
+pounds. But it occurs also in other countries, for I have had some of
+the same sort which had been found in France near the town of Troyes
+in Champagne, and some others which came from the Island of Corsica,
+though both were less clear and only in little bits, scarcely capable
+of letting any effect of refraction be observed.
+
+2. The first knowledge which the public has had about it is due to Mr.
+Erasmus Bartholinus, who has given a description of Iceland Crystal
+and of its chief phenomena. But here I shall not desist from giving my
+own, both for the instruction of those who may not have seen his book,
+and because as respects some of these phenomena there is a slight
+difference between his observations and those which I have made: for I
+have applied myself with great exactitude to examine these properties
+of refraction, in order to be quite sure before undertaking to explain
+the causes of them.
+
+3. As regards the hardness of this stone, and the property which it
+has of being easily split, it must be considered rather as a species
+of Talc than of Crystal. For an iron spike effects an entrance into it
+as easily as into any other Talc or Alabaster, to which it is equal in
+gravity.
+
+[Illustration]
+
+4. The pieces of it which are found have the figure of an oblique
+parallelepiped; each of the six faces being a parallelogram; and it
+admits of being split in three directions parallel to two of these
+opposed faces. Even in such wise, if you will, that all the six faces
+are equal and similar rhombuses. The figure here added represents a
+piece of this Crystal. The obtuse angles of all the parallelograms, as
+C, D, here, are angles of 101 degrees 52 minutes, and consequently
+the acute angles, such as A and B, are of 78 degrees 8 minutes.
+
+5. Of the solid angles there are two opposite to one another, such as
+C and E, which are each composed of three equal obtuse plane angles.
+The other six are composed of two acute angles and one obtuse. All
+that I have just said has been likewise remarked by Mr. Bartholinus in
+the aforesaid treatise; if we differ it is only slightly about the
+values of the angles. He recounts moreover some other properties of
+this Crystal; to wit, that when rubbed against cloth it attracts
+straws and other light things as do amber, diamond, glass, and Spanish
+wax. Let a piece be covered with water for a day or more, the surface
+loses its natural polish. When aquafortis is poured on it it produces
+ebullition, especially, as I have found, if the Crystal has been
+pulverized. I have also found by experiment that it may be heated to
+redness in the fire without being in anywise altered or rendered less
+transparent; but a very violent fire calcines it nevertheless. Its
+transparency is scarcely less than that of water or of Rock Crystal,
+and devoid of colour. But rays of light pass through it in another
+fashion and produce those marvellous refractions the causes of which I
+am now going to try to explain; reserving for the end of this Treatise
+the statement of my conjectures touching the formation and
+extraordinary configuration of this Crystal.
+
+6. In all other transparent bodies that we know there is but one sole
+and simple refraction; but in this substance there are two different
+ones. The effect is that objects seen through it, especially such as
+are placed right against it, appear double; and that a ray of
+sunlight, falling on one of its surfaces, parts itself into two rays
+and traverses the Crystal thus.
+
+7. It is again a general law in all other transparent bodies that the
+ray which falls perpendicularly on their surface passes straight on
+without suffering refraction, and that an oblique ray is always
+refracted. But in this Crystal the perpendicular ray suffers
+refraction, and there are oblique rays which pass through it quite
+straight.
+
+[Illustration]
+
+8. But in order to explain these phenomena more particularly, let
+there be, in the first place, a piece ABFE of the same Crystal, and
+let the obtuse angle ACB, one of the three which constitute the
+equilateral solid angle C, be divided into two equal parts by the
+straight line CG, and let it be conceived that the Crystal is
+intersected by a plane which passes through this line and through the
+side CF, which plane will necessarily be perpendicular to the surface
+AB; and its section in the Crystal will form a parallelogram GCFH. We
+will call this section the principal section of the Crystal.
+
+9. Now if one covers the surface AB, leaving there only a small
+aperture at the point K, situated in the straight line CG, and if one
+exposes it to the sun, so that his rays face it perpendicularly above,
+then the ray IK will divide itself at the point K into two, one of
+which will continue to go on straight by KL, and the other will
+separate itself along the straight line KM, which is in the plane
+GCFH, and which makes with KL an angle of about 6 degrees 40 minutes,
+tending from the side of the solid angle C; and on emerging from the
+other side of the Crystal it will turn again parallel to JK, along MZ.
+And as, in this extraordinary refraction, the point M is seen by the
+refracted ray MKI, which I consider as going to the eye at I, it
+necessarily follows that the point L, by virtue of the same
+refraction, will be seen by the refracted ray LRI, so that LR will be
+parallel to MK if the distance from the eye KI is supposed very great.
+The point L appears then as being in the straight line IRS; but the
+same point appears also, by ordinary refraction, to be in the straight
+line IK, hence it is necessarily judged to be double. And similarly if
+L be a small hole in a sheet of paper or other substance which is laid
+against the Crystal, it will appear when turned towards daylight as if
+there were two holes, which will seem the wider apart from one another
+the greater the thickness of the Crystal.
+
+10. Again, if one turns the Crystal in such wise that an incident ray
+NO, of sunlight, which I suppose to be in the plane continued from
+GCFH, makes with GC an angle of 73 degrees and 20 minutes, and is
+consequently nearly parallel to the edge CF, which makes with FH an
+angle of 70 degrees 57 minutes, according to the calculation which I
+shall put at the end, it will divide itself at the point O into two
+rays, one of which will continue along OP in a straight line with NO,
+and will similarly pass out of the other side of the crystal without
+any refraction; but the other will be refracted and will go along OQ.
+And it must be noted that it is special to the plane through GCF and
+to those which are parallel to it, that all incident rays which are in
+one of these planes continue to be in it after they have entered the
+Crystal and have become double; for it is quite otherwise for rays in
+all other planes which intersect the Crystal, as we shall see
+afterwards.
+
+11. I recognized at first by these experiments and by some others that
+of the two refractions which the ray suffers in this Crystal, there is
+one which follows the ordinary rules; and it is this to which the rays
+KL and OQ belong. This is why I have distinguished this ordinary
+refraction from the other; and having measured it by exact
+observation, I found that its proportion, considered as to the Sines
+of the angles which the incident and refracted rays make with the
+perpendicular, was very precisely that of 5 to 3, as was found also by
+Mr. Bartholinus, and consequently much greater than that of Rock
+Crystal, or of glass, which is nearly 3 to 2.
+
+[Illustration]
+
+12. The mode of making these observations exactly is as follows. Upon
+a leaf of paper fixed on a thoroughly flat table there is traced a
+black line AB, and two others, CED and KML, which cut it at right
+angles and are more or less distant from one another according as it
+is desired to examine a ray that is more or less oblique. Then place
+the Crystal upon the intersection E so that the line AB concurs with
+that which bisects the obtuse angle of the lower surface, or with some
+line parallel to it. Then by placing the eye directly above the line
+AB it will appear single only; and one will see that the portion
+viewed through the Crystal and the portions which appear outside it,
+meet together in a straight line: but the line CD will appear double,
+and one can distinguish the image which is due to regular refraction
+by the circumstance that when one views it with both eyes it seems
+raised up more than the other, or again by the circumstance that, when
+the Crystal is turned around on the paper, this image remains
+stationary, whereas the other image shifts and moves entirely around.
+Afterwards let the eye be placed at I (remaining always in the plane
+perpendicular through AB) so that it views the image which is formed
+by regular refraction of the line CD making a straight line with the
+remainder of that line which is outside the Crystal. And then, marking
+on the surface of the Crystal the point H where the intersection E
+appears, this point will be directly above E. Then draw back the eye
+towards O, keeping always in the plane perpendicular through AB, so
+that the image of the line CD, which is formed by ordinary refraction,
+may appear in a straight line with the line KL viewed without
+refraction; and then mark on the Crystal the point N where the point
+of intersection E appears.
+
+13. Then one will know the length and position of the lines NH, EM,
+and of HE, which is the thickness of the Crystal: which lines being
+traced separately upon a plan, and then joining NE and NM which cuts
+HE at P, the proportion of the refraction will be that of EN to NP,
+because these lines are to one another as the sines of the angles NPH,
+NEP, which are equal to those which the incident ray ON and its
+refraction NE make with the perpendicular to the surface. This
+proportion, as I have said, is sufficiently precisely as 5 to 3, and
+is always the same for all inclinations of the incident ray.
+
+14. The same mode of observation has also served me for examining the
+extraordinary or irregular refraction of this Crystal. For, the point
+H having been found and marked, as aforesaid, directly above the point
+E, I observed the appearance of the line CD, which is made by the
+extraordinary refraction; and having placed the eye at Q, so that this
+appearance made a straight line with the line KL viewed without
+refraction, I ascertained the triangles REH, RES, and consequently the
+angles RSH, RES, which the incident and the refracted ray make with
+the perpendicular.
+
+15. But I found in this refraction that the ratio of FR to RS was not
+constant, like the ordinary refraction, but that it varied with the
+varying obliquity of the incident ray.
+
+16. I found also that when QRE made a straight line, that is, when the
+incident ray entered the Crystal without being refracted (as I
+ascertained by the circumstance that then the point E viewed by the
+extraordinary refraction appeared in the line CD, as seen without
+refraction) I found, I say, then that the angle QRG was 73 degrees 20
+minutes, as has been already remarked; and so it is not the ray
+parallel to the edge of the Crystal, which crosses it in a straight
+line without being refracted, as Mr. Bartholinus believed, since that
+inclination is only 70 degrees 57 minutes, as was stated above. And
+this is to be noted, in order that no one may search in vain for the
+cause of the singular property of this ray in its parallelism to the
+edges mentioned.
+
+[Illustration]
+
+17. Finally, continuing my observations to discover the nature of
+this refraction, I learned that it obeyed the following remarkable
+rule. Let the parallelogram GCFH, made by the principal section of the
+Crystal, as previously determined, be traced separately. I found then
+that always, when the inclinations of two rays which come from
+opposite sides, as VK, SK here, are equal, their refractions KX and KT
+meet the bottom line HF in such wise that points X and T are equally
+distant from the point M, where the refraction of the perpendicular
+ray IK falls; and this occurs also for refractions in other sections
+of this Crystal. But before speaking of those, which have also other
+particular properties, we will investigate the causes of the phenomena
+which I have already reported.
+
+It was after having explained the refraction of ordinary transparent
+bodies by means of the spherical emanations of light, as above, that I
+resumed my examination of the nature of this Crystal, wherein I had
+previously been unable to discover anything.
+
+18. As there were two different refractions, I conceived that there
+were also two different emanations of waves of light, and that one
+could occur in the ethereal matter extending through the body of the
+Crystal. Which matter, being present in much larger quantity than is
+that of the particles which compose it, was alone capable of causing
+transparency, according to what has been explained heretofore. I
+attributed to this emanation of waves the regular refraction which is
+observed in this stone, by supposing these waves to be ordinarily of
+spherical form, and having a slower progression within the Crystal
+than they have outside it; whence proceeds refraction as I have
+demonstrated.
+
+19. As to the other emanation which should produce the irregular
+refraction, I wished to try what Elliptical waves, or rather
+spheroidal waves, would do; and these I supposed would spread
+indifferently both in the ethereal matter diffused throughout the
+crystal and in the particles of which it is composed, according to the
+last mode in which I have explained transparency. It seemed to me that
+the disposition or regular arrangement of these particles could
+contribute to form spheroidal waves (nothing more being required for
+this than that the successive movement of light should spread a little
+more quickly in one direction than in the other) and I scarcely
+doubted that there were in this crystal such an arrangement of equal
+and similar particles, because of its figure and of its angles with
+their determinate and invariable measure. Touching which particles,
+and their form and disposition, I shall, at the end of this Treatise,
+propound my conjectures and some experiments which confirm them.
+
+20. The double emission of waves of light, which I had imagined,
+became more probable to me after I had observed a certain phenomenon
+in the ordinary [Rock] Crystal, which occurs in hexagonal form, and
+which, because of this regularity, seems also to be composed of
+particles, of definite figure, and ranged in order. This was, that
+this crystal, as well as that from Iceland, has a double refraction,
+though less evident. For having had cut from it some well polished
+Prisms of different sections, I remarked in all, in viewing through
+them the flame of a candle or the lead of window panes, that
+everything appeared double, though with images not very distant from
+one another. Whence I understood the reason why this substance, though
+so transparent, is useless for Telescopes, when they have ever so
+little length.
+
+21. Now this double refraction, according to my Theory hereinbefore
+established, seemed to demand a double emission of waves of light,
+both of them spherical (for both the refractions are regular) and
+those of one series a little slower only than the others. For thus the
+phenomenon is quite naturally explained, by postulating substances
+which serve as vehicle for these waves, as I have done in the case of
+Iceland Crystal. I had then less trouble after that in admitting two
+emissions of waves in one and the same body. And since it might have
+been objected that in composing these two kinds of crystal of equal
+particles of a certain figure, regularly piled, the interstices which
+these particles leave and which contain the ethereal matter would
+scarcely suffice to transmit the waves of light which I have localized
+there, I removed this difficulty by regarding these particles as being
+of a very rare texture, or rather as composed of other much smaller
+particles, between which the ethereal matter passes quite freely.
+This, moreover, necessarily follows from that which has been already
+demonstrated touching the small quantity of matter of which the bodies
+are built up.
+
+22. Supposing then these spheroidal waves besides the spherical ones,
+I began to examine whether they could serve to explain the phenomena
+of the irregular refraction, and how by these same phenomena I could
+determine the figure and position of the spheroids: as to which I
+obtained at last the desired success, by proceeding as follows.
+
+[Illustration]
+
+23. I considered first the effect of waves so formed, as respects the
+ray which falls perpendicularly on the flat surface of a transparent
+body in which they should spread in this manner. I took AB for the
+exposed region of the surface. And, since a ray perpendicular to a
+plane, and coming from a very distant source of light, is nothing
+else, according to the precedent Theory, than the incidence of a
+portion of the wave parallel to that plane, I supposed the straight
+line RC, parallel and equal to AB, to be a portion of a wave of light,
+in which an infinitude of points such as RH_h_C come to meet the
+surface AB at the points AK_k_B. Then instead of the hemispherical
+partial waves which in a body of ordinary refraction would spread from
+each of these last points, as we have above explained in treating of
+refraction, these must here be hemi-spheroids. The axes (or rather the
+major diameters) of these I supposed to be oblique to the plane AB, as
+is AV the semi-axis or semi-major diameter of the spheroid SVT, which
+represents the partial wave coming from the point A, after the wave RC
+has reached AB. I say axis or major diameter, because the same ellipse
+SVT may be considered as the section of a spheroid of which the axis
+is AZ perpendicular to AV. But, for the present, without yet deciding
+one or other, we will consider these spheroids only in those sections
+of them which make ellipses in the plane of this figure. Now taking a
+certain space of time during which the wave SVT has spread from A, it
+would needs be that from all the other points K_k_B there should
+proceed, in the same time, waves similar to SVT and similarly
+situated. And the common tangent NQ of all these semi-ellipses would
+be the propagation of the wave RC which fell on AB, and would be the
+place where this movement occurs in much greater amount than anywhere
+else, being made up of arcs of an infinity of ellipses, the centres of
+which are along the line AB.
+
+24. Now it appeared that this common tangent NQ was parallel to AB,
+and of the same length, but that it was not directly opposite to it,
+since it was comprised between the lines AN, BQ, which are diameters
+of ellipses having A and B for centres, conjugate with respect to
+diameters which are not in the straight line AB. And in this way I
+comprehended, a matter which had seemed to me very difficult, how a
+ray perpendicular to a surface could suffer refraction on entering a
+transparent body; seeing that the wave RC, having come to the aperture
+AB, went on forward thence, spreading between the parallel lines AN,
+BQ, yet itself remaining always parallel to AB, so that here the light
+does not spread along lines perpendicular to its waves, as in ordinary
+refraction, but along lines cutting the waves obliquely.
+
+[Illustration]
+
+25. Inquiring subsequently what might be the position and form of
+these spheroids in the crystal, I considered that all the six faces
+produced precisely the same refractions. Taking, then, the
+parallelopiped AFB, of which the obtuse solid angle C is contained
+between the three equal plane angles, and imagining in it the three
+principal sections, one of which is perpendicular to the face DC and
+passes through the edge CF, another perpendicular to the face BF
+passing through the edge CA, and the third perpendicular to the face
+AF passing through the edge BC; I knew that the refractions of the
+incident rays belonging to these three planes were all similar. But
+there could be no position of the spheroid which would have the same
+relation to these three sections except that in which the axis was
+also the axis of the solid angle C. Consequently I saw that the axis
+of this angle, that is to say the straight line which traversed the
+crystal from the point C with equal inclination to the edges CF, CA,
+CB was the line which determined the position of the axis of all the
+spheroidal waves which one imagined to originate from some point,
+taken within or on the surface of the crystal, since all these
+spheroids ought to be alike, and have their axes parallel to one
+another.
+
+26. Considering after this the plane of one of these three sections,
+namely that through GCF, the angle of which is 109 degrees 3 minutes,
+since the angle F was shown above to be 70 degrees 57 minutes; and,
+imagining a spheroidal wave about the centre C, I knew, because I have
+just explained it, that its axis must be in the same plane, the half
+of which axis I have marked CS in the next figure: and seeking by
+calculation (which will be given with others at the end of this
+discourse) the value of the angle CGS, I found it 45 degrees 20
+minutes.
+
+[Illustration]
+
+27. To know from this the form of this spheroid, that is to say the
+proportion of the semi-diameters CS, CP, of its elliptical section,
+which are perpendicular to one another, I considered that the point M
+where the ellipse is touched by the straight line FH, parallel to CG,
+ought to be so situated that CM makes with the perpendicular CL an
+angle of 6 degrees 40 minutes; since, this being so, this ellipse
+satisfies what has been said about the refraction of the ray
+perpendicular to the surface CG, which is inclined to the
+perpendicular CL by the same angle. This, then, being thus disposed,
+and taking CM at 100,000 parts, I found by the calculation which will
+be given at the end, the semi-major diameter CP to be 105,032, and the
+semi-axis CS to be 93,410, the ratio of which numbers is very nearly 9
+to 8; so that the spheroid was of the kind which resembles a
+compressed sphere, being generated by the revolution of an ellipse
+about its smaller diameter. I found also the value of CG the
+semi-diameter parallel to the tangent ML to be 98,779.
+
+[Illustration]
+
+28. Now passing to the investigation of the refractions which
+obliquely incident rays must undergo, according to our hypothesis of
+spheroidal waves, I saw that these refractions depended on the ratio
+between the velocity of movement of the light outside the crystal in
+the ether, and that within the crystal. For supposing, for example,
+this proportion to be such that while the light in the crystal forms
+the spheroid GSP, as I have just said, it forms outside a sphere the
+semi-diameter of which is equal to the line N which will be determined
+hereafter, the following is the way of finding the refraction of the
+incident rays. Let there be such a ray RC falling upon the surface
+CK. Make CO perpendicular to RC, and across the angle KCO adjust OK,
+equal to N and perpendicular to CO; then draw KI, which touches the
+Ellipse GSP, and from the point of contact I join IC, which will be
+the required refraction of the ray RC. The demonstration of this is,
+it will be seen, entirely similar to that of which we made use in
+explaining ordinary refraction. For the refraction of the ray RC is
+nothing else than the progression of the portion C of the wave CO,
+continued in the crystal. Now the portions H of this wave, during the
+time that O came to K, will have arrived at the surface CK along the
+straight lines H_x_, and will moreover have produced in the crystal
+around the centres _x_ some hemi-spheroidal partial waves similar to
+the hemi-spheroidal GSP_g_, and similarly disposed, and of which the
+major and minor diameters will bear the same proportions to the lines
+_xv_ (the continuations of the lines H_x_ up to KB parallel to CO)
+that the diameters of the spheroid GSP_g_ bear to the line CB, or N.
+And it is quite easy to see that the common tangent of all these
+spheroids, which are here represented by Ellipses, will be the
+straight line IK, which consequently will be the propagation of the
+wave CO; and the point I will be that of the point C, conformably with
+that which has been demonstrated in ordinary refraction.
+
+Now as to finding the point of contact I, it is known that one must
+find CD a third proportional to the lines CK, CG, and draw DI parallel
+to CM, previously determined, which is the conjugate diameter to CG;
+for then, by drawing KI it touches the Ellipse at I.
+
+29. Now as we have found CI the refraction of the ray RC, similarly
+one will find C_i_ the refraction of the ray _r_C, which comes from
+the opposite side, by making C_o_ perpendicular to _r_C and following
+out the rest of the construction as before. Whence one sees that if
+the ray _r_C is inclined equally with RC, the line C_d_ will
+necessarily be equal to CD, because C_k_ is equal to CK, and C_g_ to
+CG. And in consequence I_i_ will be cut at E into equal parts by the
+line CM, to which DI and _di_ are parallel. And because CM is the
+conjugate diameter to CG, it follows that _i_I will be parallel to
+_g_G. Therefore if one prolongs the refracted rays CI, C_i_, until
+they meet the tangent ML at T and _t_, the distances MT, M_t_, will
+also be equal. And so, by our hypothesis, we explain perfectly the
+phenomenon mentioned above; to wit, that when there are two rays
+equally inclined, but coming from opposite sides, as here the rays RC,
+_rc_, their refractions diverge equally from the line followed by the
+refraction of the ray perpendicular to the surface, by considering
+these divergences in the direction parallel to the surface of the
+crystal.
+
+30. To find the length of the line N, in proportion to CP, CS, CG, it
+must be determined by observations of the irregular refraction which
+occurs in this section of the crystal; and I find thus that the ratio
+of N to GC is just a little less than 8 to 5. And having regard to
+some other observations and phenomena of which I shall speak
+afterwards, I put N at 156,962 parts, of which the semi-diameter CG is
+found to contain 98,779, making this ratio 8 to 5-1/29. Now this
+proportion, which there is between the line N and CG, may be called
+the Proportion of the Refraction; similarly as in glass that of 3 to
+2, as will be manifest when I shall have explained a short process in
+the preceding way to find the irregular refractions.
+
+31. Supposing then, in the next figure, as previously, the surface of
+the crystal _g_G, the Ellipse GP_g_, and the line N; and CM the
+refraction of the perpendicular ray FC, from which it diverges by 6
+degrees 40 minutes. Now let there be some other ray RC, the refraction
+of which must be found.
+
+About the centre C, with semi-diameter CG, let the circumference _g_RG
+be described, cutting the ray RC at R; and let RV be the perpendicular
+on CG. Then as the line N is to CG let CV be to CD, and let DI be
+drawn parallel to CM, cutting the Ellipse _g_MG at I; then joining CI,
+this will be the required refraction of the ray RC. Which is
+demonstrated thus.
+
+[Illustration]
+
+Let CO be perpendicular to CR, and across the angle OCG let OK be
+adjusted, equal to N and perpendicular to CO, and let there be drawn
+the straight line KI, which if it is demonstrated to be a tangent to
+the Ellipse at I, it will be evident by the things heretofore
+explained that CI is the refraction of the ray RC. Now since the angle
+RCO is a right angle, it is easy to see that the right-angled
+triangles RCV, KCO, are similar. As then, CK is to KO, so also is RC
+to CV. But KO is equal to N, and RC to CG: then as CK is to N so will
+CG be to CV. But as N is to CG, so, by construction, is CV to CD. Then
+as CK is to CG so is CG to CD. And because DI is parallel to CM, the
+conjugate diameter to CG, it follows that KI touches the Ellipse at I;
+which remained to be shown.
+
+32. One sees then that as there is in the refraction of ordinary
+media a certain constant proportion between the sines of the angles
+which the incident ray and the refracted ray make with the
+perpendicular, so here there is such a proportion between CV and CD or
+IE; that is to say between the Sine of the angle which the incident
+ray makes with the perpendicular, and the horizontal intercept, in the
+Ellipse, between the refraction of this ray and the diameter CM. For
+the ratio of CV to CD is, as has been said, the same as that of N to
+the semi-diameter CG.
+
+33. I will add here, before passing away, that in comparing together
+the regular and irregular refraction of this crystal, there is this
+remarkable fact, that if ABPS be the spheroid by which light spreads
+in the Crystal in a certain space of time (which spreading, as has
+been said, serves for the irregular refraction), then the inscribed
+sphere BVST is the extension in the same space of time of the light
+which serves for the regular refraction.
+
+[Illustration]
+
+For we have stated before this, that the line N being the radius of a
+spherical wave of light in air, while in the crystal it spread through
+the spheroid ABPS, the ratio of N to CS will be 156,962 to 93,410. But
+it has also been stated that the proportion of the regular refraction
+was 5 to 3; that is to say, that N being the radius of a spherical
+wave of light in air, its extension in the crystal would, in the same
+space of time, form a sphere the radius of which would be to N as 3 to
+5. Now 156,962 is to 93,410 as 5 to 3 less 1/41. So that it is
+sufficiently nearly, and may be exactly, the sphere BVST, which the
+light describes for the regular refraction in the crystal, while it
+describes the spheroid BPSA for the irregular refraction, and while it
+describes the sphere of radius N in air outside the crystal.
+
+Although then there are, according to what we have supposed, two
+different propagations of light within the crystal, it appears that it
+is only in directions perpendicular to the axis BS of the spheroid
+that one of these propagations occurs more rapidly than the other; but
+that they have an equal velocity in the other direction, namely, in
+that parallel to the same axis BS, which is also the axis of the
+obtuse angle of the crystal.
+
+[Illustration]
+
+34. The proportion of the refraction being what we have just seen, I
+will now show that there necessarily follows thence that notable
+property of the ray which falling obliquely on the surface of the
+crystal enters it without suffering refraction. For supposing the same
+things as before, and that the ray makes with the same surface _g_G
+the angle RCG of 73 degrees 20 minutes, inclining to the same side as
+the crystal (of which ray mention has been made above); if one
+investigates, by the process above explained, the refraction CI, one
+will find that it makes exactly a straight line with RC, and that thus
+this ray is not deviated at all, conformably with experiment. This is
+proved as follows by calculation.
+
+CG or CR being, as precedently, 98,779; CM being 100,000; and the
+angle RCV 73 degrees 20 minutes, CV will be 28,330. But because CI is
+the refraction of the ray RC, the proportion of CV to CD is 156,962 to
+98,779, namely, that of N to CG; then CD is 17,828.
+
+Now the rectangle _g_DC is to the square of DI as the square of CG is
+to the square of CM; hence DI or CE will be 98,353. But as CE is to
+EI, so will CM be to MT, which will then be 18,127. And being added to
+ML, which is 11,609 (namely the sine of the angle LCM, which is 6
+degrees 40 minutes, taking CM 100,000 as radius) we get LT 27,936; and
+this is to LC 99,324 as CV to VR, that is to say, as 29,938, the
+tangent of the complement of the angle RCV, which is 73 degrees 20
+minutes, is to the radius of the Tables. Whence it appears that RCIT
+is a straight line; which was to be proved.
+
+35. Further it will be seen that the ray CI in emerging through the
+opposite surface of the crystal, ought to pass out quite straight,
+according to the following demonstration, which proves that the
+reciprocal relation of refraction obtains in this crystal the same as
+in other transparent bodies; that is to say, that if a ray RC in
+meeting the surface of the crystal CG is refracted as CI, the ray CI
+emerging through the opposite parallel surface of the crystal, which
+I suppose to be IB, will have its refraction IA parallel to the ray
+RC.
+
+[Illustration]
+
+Let the same things be supposed as before; that is to say, let CO,
+perpendicular to CR, represent a portion of a wave the continuation of
+which in the crystal is IK, so that the piece C will be continued on
+along the straight line CI, while O comes to K. Now if one takes a
+second period of time equal to the first, the piece K of the wave IK
+will, in this second period, have advanced along the straight line KB,
+equal and parallel to CI, because every piece of the wave CO, on
+arriving at the surface CK, ought to go on in the crystal the same as
+the piece C; and in this same time there will be formed in the air
+from the point I a partial spherical wave having a semi-diameter IA
+equal to KO, since KO has been traversed in an equal time. Similarly,
+if one considers some other point of the wave IK, such as _h_, it will
+go along _hm_, parallel to CI, to meet the surface IB, while the point
+K traverses K_l_ equal to _hm_; and while this accomplishes the
+remainder _l_B, there will start from the point _m_ a partial wave the
+semi-diameter of which, _mn_, will have the same ratio to _l_B as IA
+to KB. Whence it is evident that this wave of semi-diameter _mn_, and
+the other of semi-diameter IA will have the same tangent BA. And
+similarly for all the partial spherical waves which will be formed
+outside the crystal by the impact of all the points of the wave IK
+against the surface of the Ether IB. It is then precisely the tangent
+BA which will be the continuation of the wave IK, outside the crystal,
+when the piece K has reached B. And in consequence IA, which is
+perpendicular to BA, will be the refraction of the ray CI on emerging
+from the crystal. Now it is clear that IA is parallel to the incident
+ray RC, since IB is equal to CK, and IA equal to KO, and the angles A
+and O are right angles.
+
+It is seen then that, according to our hypothesis, the reciprocal
+relation of refraction holds good in this crystal as well as in
+ordinary transparent bodies; as is thus in fact found by observation.
+
+36. I pass now to the consideration of other sections of the crystal,
+and of the refractions there produced, on which, as will be seen, some
+other very remarkable phenomena depend.
+
+Let ABH be a parallelepiped of crystal, and let the top surface AEHF
+be a perfect rhombus, the obtuse angles of which are equally divided
+by the straight line EF, and the acute angles by the straight line AH
+perpendicular to FE.
+
+The section which we have hitherto considered is that which passes
+through the lines EF, EB, and which at the same time cuts the plane
+AEHF at right angles. Refractions in this section have this in common
+with the refractions in ordinary media that the plane which is drawn
+through the incident ray and which also intersects the surface of the
+crystal at right angles, is that in which the refracted ray also is
+found. But the refractions which appertain to every other section of
+this crystal have this strange property that the refracted ray always
+quits the plane of the incident ray perpendicular to the surface, and
+turns away towards the side of the slope of the crystal. For which
+fact we shall show the reason, in the first place, for the section
+through AH; and we shall show at the same time how one can determine
+the refraction, according to our hypothesis. Let there be, then, in
+the plane which passes through AH, and which is perpendicular to the
+plane AFHE, the incident ray RC; it is required to find its refraction
+in the crystal.
+
+[Illustration]
+
+37. About the centre C, which I suppose to be in the intersection of
+AH and FE, let there be imagined a hemi-spheroid QG_qg_M, such as the
+light would form in spreading in the crystal, and let its section by
+the plane AEHF form the Ellipse QG_qg_, the major diameter of which
+Q_q_, which is in the line AH, will necessarily be one of the major
+diameters of the spheroid; because the axis of the spheroid being in
+the plane through FEB, to which QC is perpendicular, it follows that
+QC is also perpendicular to the axis of the spheroid, and consequently
+QC_q_ one of its major diameters. But the minor diameter of this
+Ellipse, G_g_, will bear to Q_q_ the proportion which has been defined
+previously, Article 27, between CG and the major semi-diameter of the
+spheroid, CP, namely, that of 98,779 to 105,032.
+
+Let the line N be the length of the travel of light in air during the
+time in which, within the crystal, it makes, from the centre C, the
+spheroid QC_qg_M. Then having drawn CO perpendicular to the ray CR and
+situate in the plane through CR and AH, let there be adjusted, across
+the angle ACO, the straight line OK equal to N and perpendicular to
+CO, and let it meet the straight line AH at K. Supposing consequently
+that CL is perpendicular to the surface of the crystal AEHF, and that
+CM is the refraction of the ray which falls perpendicularly on this
+same surface, let there be drawn a plane through the line CM and
+through KCH, making in the spheroid the semi-ellipse QM_q_, which will
+be given, since the angle MCL is given of value 6 degrees 40 minutes.
+And it is certain, according to what has been explained above, Article
+27, that a plane which would touch the spheroid at the point M, where
+I suppose the straight line CM to meet the surface, would be parallel
+to the plane QG_q_. If then through the point K one now draws KS
+parallel to G_g_, which will be parallel also to QX, the tangent to
+the Ellipse QG_q_ at Q; and if one conceives a plane passing through
+KS and touching the spheroid, the point of contact will necessarily be
+in the Ellipse QM_q_, because this plane through KS, as well as the
+plane which touches the spheroid at the point M, are parallel to QX,
+the tangent of the spheroid: for this consequence will be demonstrated
+at the end of this Treatise. Let this point of contact be at I, then
+making KC, QC, DC proportionals, draw DI parallel to CM; also join CI.
+I say that CI will be the required refraction of the ray RC. This will
+be manifest if, in considering CO, which is perpendicular to the ray
+RC, as a portion of the wave of light, we can demonstrate that the
+continuation of its piece C will be found in the crystal at I, when O
+has arrived at K.
+
+38. Now as in the Chapter on Reflexion, in demonstrating that the
+incident and reflected rays are always in the same plane perpendicular
+to the reflecting surface, we considered the breadth of the wave of
+light, so, similarly, we must here consider the breadth of the wave CO
+in the diameter G_g_. Taking then the breadth C_c_ on the side toward
+the angle E, let the parallelogram CO_oc_ be taken as a portion of a
+wave, and let us complete the parallelograms CK_kc_, CI_ic_, Kl_ik_,
+OK_ko_. In the time then that the line O_o_ arrives at the surface of
+the crystal at K_k_, all the points of the wave CO_oc_ will have
+arrived at the rectangle K_c_ along lines parallel to OK; and from the
+points of their incidences there will originate, beyond that, in the
+crystal partial hemi-spheroids, similar to the hemi-spheroid QM_q_,
+and similarly disposed. These hemi-spheroids will necessarily all
+touch the plane of the parallelogram KI_ik_ at the same instant that
+O_o_ has reached K_k_. Which is easy to comprehend, since, of these
+hemi-spheroids, all those which have their centres along the line CK,
+touch this plane in the line KI (for this is to be shown in the same
+way as we have demonstrated the refraction of the oblique ray in the
+principal section through EF) and all those which have their centres
+in the line C_c_ will touch the same plane KI in the line I_i_; all
+these being similar to the hemi-spheroid QM_q_. Since then the
+parallelogram K_i_ is that which touches all these spheroids, this
+same parallelogram will be precisely the continuation of the wave
+CO_oc_ in the crystal, when O_o_ has arrived at K_k_, because it forms
+the termination of the movement and because of the quantity of
+movement which occurs more there than anywhere else: and thus it
+appears that the piece C of the wave CO_oc_ has its continuation at I;
+that is to say, that the ray RC is refracted as CI.
+
+From this it is to be noted that the proportion of the refraction for
+this section of the crystal is that of the line N to the semi-diameter
+CQ; by which one will easily find the refractions of all incident
+rays, in the same way as we have shown previously for the case of the
+section through FE; and the demonstration will be the same. But it
+appears that the said proportion of the refraction is less here than
+in the section through FEB; for it was there the same as the ratio of
+N to CG, that is to say, as 156,962 to 98,779, very nearly as 8 to 5;
+and here it is the ratio of N to CQ the major semi-diameter of the
+spheroid, that is to say, as 156,962 to 105,032, very nearly as 3 to
+2, but just a little less. Which still agrees perfectly with what one
+finds by observation.
+
+39. For the rest, this diversity of proportion of refraction produces
+a very singular effect in this Crystal; which is that when it is
+placed upon a sheet of paper on which there are letters or anything
+else marked, if one views it from above with the two eyes situated in
+the plane of the section through EF, one sees the letters raised up by
+this irregular refraction more than when one puts one's eyes in the
+plane of section through AH: and the difference of these elevations
+appears by comparison with the other ordinary refraction of the
+crystal, the proportion of which is as 5 to 3, and which always raises
+the letters equally, and higher than the irregular refraction does.
+For one sees the letters and the paper on which they are written, as
+on two different stages at the same time; and in the first position of
+the eyes, namely, when they are in the plane through AH these two
+stages are four times more distant from one another than when the eyes
+are in the plane through EF.
+
+We will show that this effect follows from the refractions; and it
+will enable us at the same time to ascertain the apparent place of a
+point of an object placed immediately under the crystal, according to
+the different situation of the eyes.
+
+40. Let us see first by how much the irregular refraction of the plane
+through AH ought to lift the bottom of the crystal. Let the plane of
+this figure represent separately the section through Q_q_ and CL, in
+which section there is also the ray RC, and let the semi-elliptic
+plane through Q_q_ and CM be inclined to the former, as previously, by
+an angle of 6 degrees 40 minutes; and in this plane CI is then the
+refraction of the ray RC.
+
+[Illustration]
+
+If now one considers the point I as at the bottom of the crystal, and
+that it is viewed by the rays ICR, _Icr_, refracted equally at the
+points C_c_, which should be equally distant from D, and that these
+rays meet the two eyes at R_r_; it is certain that the point I will
+appear raised to S where the straight lines RC, _rc_, meet; which
+point S is in DP, perpendicular to Q_q_. And if upon DP there is drawn
+the perpendicular IP, which will lie at the bottom of the crystal, the
+length SP will be the apparent elevation of the point I above the
+bottom.
+
+Let there be described on Q_q_ a semicircle cutting the ray CR at B,
+from which BV is drawn perpendicular to Q_q_; and let the proportion
+of the refraction for this section be, as before, that of the line N
+to the semi-diameter CQ.
+
+Then as N is to CQ so is VC to CD, as appears by the method of finding
+the refraction which we have shown above, Article 31; but as VC is to
+CD, so is VB to DS. Then as N is to CQ, so is VB to DS. Let ML be
+perpendicular to CL. And because I suppose the eyes R_r_ to be distant
+about a foot or so from the crystal, and consequently the angle RS_r_
+very small, VB may be considered as equal to the semi-diameter CQ, and
+DP as equal to CL; then as N is to CQ so is CQ to DS. But N is valued
+at 156,962 parts, of which CM contains 100,000 and CQ 105,032. Then DS
+will have 70,283. But CL is 99,324, being the sine of the complement
+of the angle MCL which is 6 degrees 40 minutes; CM being supposed as
+radius. Then DP, considered as equal to CL, will be to DS as 99,324 to
+70,283. And so the elevation of the point I by the refraction of this
+section is known.
+
+[Illustration]
+
+41. Now let there be represented the other section through EF in the
+figure before the preceding one; and let CM_g_ be the semi-ellipse,
+considered in Articles 27 and 28, which is made by cutting a
+spheroidal wave having centre C. Let the point I, taken in this
+ellipse, be imagined again at the bottom of the Crystal; and let it be
+viewed by the refracted rays ICR, I_cr_, which go to the two eyes; CR
+and _cr_ being equally inclined to the surface of the crystal G_g_.
+This being so, if one draws ID parallel to CM, which I suppose to be
+the refraction of the perpendicular ray incident at the point C, the
+distances DC, D_c_, will be equal, as is easy to see by that which has
+been demonstrated in Article 28. Now it is certain that the point I
+should appear at S where the straight lines RC, _rc_, meet when
+prolonged; and that this point will fall in the line DP perpendicular
+to G_g_. If one draws IP perpendicular to this DP, it will be the
+distance PS which will mark the apparent elevation of the point I. Let
+there be described on G_g_ a semicircle cutting CR at B, from which
+let BV be drawn perpendicular to G_g_; and let N to GC be the
+proportion of the refraction in this section, as in Article 28. Since
+then CI is the refraction of the radius BC, and DI is parallel to CM,
+VC must be to CD as N to GC, according to what has been demonstrated
+in Article 31. But as VC is to CD so is BV to DS. Let ML be drawn
+perpendicular to CL. And because I consider, again, the eyes to be
+distant above the crystal, BV is deemed equal to the semi-diameter CG;
+and hence DS will be a third proportional to the lines N and CG: also
+DP will be deemed equal to CL. Now CG consisting of 98,778 parts, of
+which CM contains 100,000, N is taken as 156,962. Then DS will be
+62,163. But CL is also determined, and contains 99,324 parts, as has
+been said in Articles 34 and 40. Then the ratio of PD to DS will be as
+99,324 to 62,163. And thus one knows the elevation of the point at the
+bottom I by the refraction of this section; and it appears that this
+elevation is greater than that by the refraction of the preceding
+section, since the ratio of PD to DS was there as 99,324 to 70,283.
+
+[Illustration]
+
+But by the regular refraction of the crystal, of which we have above
+said that the proportion is 5 to 3, the elevation of the point I, or
+P, from the bottom, will be 2/5 of the height DP; as appears by this
+figure, where the point P being viewed by the rays PCR, P_cr_,
+refracted equally at the surface C_c_, this point must needs appear
+to be at S, in the perpendicular PD where the lines RC, _rc_, meet
+when prolonged: and one knows that the line PC is to CS as 5 to 3,
+since they are to one another as the sine of the angle CSP or DSC is
+to the sine of the angle SPC. And because the ratio of PD to DS is
+deemed the same as that of PC to CS, the two eyes Rr being supposed
+very far above the crystal, the elevation PS will thus be 2/5 of PD.
+
+[Illustration]
+
+42. If one takes a straight line AB for the thickness of the crystal,
+its point B being at the bottom, and if one divides it at the points
+C, D, E, according to the proportions of the elevations found, making
+AE 3/5 of AB, AB to AC as 99,324 to 70,283, and AB to AD as 99,324 to
+62,163, these points will divide AB as in this figure. And it will be
+found that this agrees perfectly with experiment; that is to say by
+placing the eyes above in the plane which cuts the crystal according
+to the shorter diameter of the rhombus, the regular refraction will
+lift up the letters to E; and one will see the bottom, and the letters
+over which it is placed, lifted up to D by the irregular refraction.
+But by placing the eyes above in the plane which cuts the crystal
+according to the longer diameter of the rhombus, the regular
+refraction will lift the letters to E as before; but the irregular
+refraction will make them, at the same time, appear lifted up only to
+C; and in such a way that the interval CE will be quadruple the
+interval ED, which one previously saw.
+
+
+43. I have only to make the remark here that in both the positions of
+the eyes the images caused by the irregular refraction do not appear
+directly below those which proceed from the regular refraction, but
+they are separated from them by being more distant from the
+equilateral solid angle of the Crystal. That follows, indeed, from all
+that has been hitherto demonstrated about the irregular refraction;
+and it is particularly shown by these last demonstrations, from which
+one sees that the point I appears by irregular refraction at S in the
+perpendicular line DP, in which line also the image of the point P
+ought to appear by regular refraction, but not the image of the point
+I, which will be almost directly above the same point, and higher than
+S.
+
+But as to the apparent elevation of the point I in other positions of
+the eyes above the crystal, besides the two positions which we have
+just examined, the image of that point by the irregular refraction
+will always appear between the two heights of D and C, passing from
+one to the other as one turns one's self around about the immovable
+crystal, while looking down from above. And all this is still found
+conformable to our hypothesis, as any one can assure himself after I
+shall have shown here the way of finding the irregular refractions
+which appear in all other sections of the crystal, besides the two
+which we have considered. Let us suppose one of the faces of the
+crystal, in which let there be the Ellipse HDE, the centre C of which
+is also the centre of the spheroid HME in which the light spreads, and
+of which the said Ellipse is the section. And let the incident ray be
+RC, the refraction of which it is required to find.
+
+Let there be taken a plane passing through the ray RC and which is
+perpendicular to the plane of the ellipse HDE, cutting it along the
+straight line BCK; and having in the same plane through RC made CO
+perpendicular to CR, let OK be adjusted across the angle OCK, so as
+to be perpendicular to OC and equal to the line N, which I suppose to
+measure the travel of the light in air during the time that it spreads
+in the crystal through the spheroid HDEM. Then in the plane of the
+Ellipse HDE let KT be drawn, through the point K, perpendicular to
+BCK. Now if one conceives a plane drawn through the straight line KT
+and touching the spheroid HME at I, the straight line CI will be the
+refraction of the ray RC, as is easy to deduce from that which has
+been demonstrated in Article 36.
+
+[Illustration]
+
+But it must be shown how one can determine the point of contact I. Let
+there be drawn parallel to the line KT a line HF which touches the
+Ellipse HDE, and let this point of contact be at H. And having drawn a
+straight line along CH to meet KT at T, let there be imagined a plane
+passing through the same CH and through CM (which I suppose to be the
+refraction of the perpendicular ray), which makes in the spheroid the
+elliptical section HME. It is certain that the plane which will pass
+through the straight line KT, and which will touch the spheroid, will
+touch it at a point in the Ellipse HME, according to the Lemma which
+will be demonstrated at the end of the Chapter. Now this point is
+necessarily the point I which is sought, since the plane drawn through
+TK can touch the spheroid at one point only. And this point I is easy
+to determine, since it is needful only to draw from the point T, which
+is in the plane of this Ellipse, the tangent TI, in the way shown
+previously. For the Ellipse HME is given, and its conjugate
+semi-diameters are CH and CM; because a straight line drawn through M,
+parallel to HE, touches the Ellipse HME, as follows from the fact that
+a plane taken through M, and parallel to the plane HDE, touches the
+spheroid at that point M, as is seen from Articles 27 and 23. For the
+rest, the position of this ellipse, with respect to the plane through
+the ray RC and through CK, is also given; from which it will be easy
+to find the position of CI, the refraction corresponding to the ray
+RC.
+
+Now it must be noted that the same ellipse HME serves to find the
+refractions of any other ray which may be in the plane through RC and
+CK. Because every plane, parallel to the straight line HF, or TK,
+which will touch the spheroid, will touch it in this ellipse,
+according to the Lemma quoted a little before.
+
+I have investigated thus, in minute detail, the properties of the
+irregular refraction of this Crystal, in order to see whether each
+phenomenon that is deduced from our hypothesis accords with that which
+is observed in fact. And this being so it affords no slight proof of
+the truth of our suppositions and principles. But what I am going to
+add here confirms them again marvellously. It is this: that there are
+different sections of this Crystal, the surfaces of which, thereby
+produced, give rise to refractions precisely such as they ought to be,
+and as I had foreseen them, according to the preceding Theory.
+
+In order to explain what these sections are, let ABKF _be_ the
+principal section through the axis of the crystal ACK, in which there
+will also be the axis SS of a spheroidal wave of light spreading in
+the crystal from the centre C; and the straight line which cuts SS
+through the middle and at right angles, namely PP, will be one of the
+major diameters.
+
+[Illustration: {Section ABKF}]
+
+Now as in the natural section of the crystal, made by a plane parallel
+to two opposite faces, which plane is here represented by the line GG,
+the refraction of the surfaces which are produced by it will be
+governed by the hemi-spheroids GNG, according to what has been
+explained in the preceding Theory. Similarly, cutting the Crystal
+through NN, by a plane perpendicular to the parallelogram ABKF, the
+refraction of the surfaces will be governed by the hemi-spheroids NGN.
+And if one cuts it through PP, perpendicularly to the said
+parallelogram, the refraction of the surfaces ought to be governed by
+the hemi-spheroids PSP, and so for others. But I saw that if the plane
+NN was almost perpendicular to the plane GG, making the angle NCG,
+which is on the side A, an angle of 90 degrees 40 minutes, the
+hemi-spheroids NGN would become similar to the hemi-spheroids GNG,
+since the planes NN and GG were equally inclined by an angle of 45
+degrees 20 minutes to the axis SS. In consequence it must needs be, if
+our theory is true, that the surfaces which the section through NN
+produces should effect the same refractions as the surfaces of the
+section through GG. And not only the surfaces of the section NN but
+all other sections produced by planes which might be inclined to the
+axis at an angle equal to 45 degrees 20 minutes. So that there are an
+infinitude of planes which ought to produce precisely the same
+refractions as the natural surfaces of the crystal, or as the section
+parallel to any one of those surfaces which are made by cleavage.
+
+I saw also that by cutting it by a plane taken through PP, and
+perpendicular to the axis SS, the refraction of the surfaces ought to
+be such that the perpendicular ray should suffer thereby no deviation;
+and that for oblique rays there would always be an irregular
+refraction, differing from the regular, and by which objects placed
+beneath the crystal would be less elevated than by that other
+refraction.
+
+That, similarly, by cutting the crystal by any plane through the axis
+SS, such as the plane of the figure is, the perpendicular ray ought to
+suffer no refraction; and that for oblique rays there were different
+measures for the irregular refraction according to the situation of
+the plane in which the incident ray was.
+
+Now these things were found in fact so; and, after that, I could not
+doubt that a similar success could be met with everywhere. Whence I
+concluded that one might form from this crystal solids similar to
+those which are its natural forms, which should produce, at all their
+surfaces, the same regular and irregular refractions as the natural
+surfaces, and which nevertheless would cleave in quite other ways, and
+not in directions parallel to any of their faces. That out of it one
+would be able to fashion pyramids, having their base square,
+pentagonal, hexagonal, or with as many sides as one desired, all the
+surfaces of which should have the same refractions as the natural
+surfaces of the crystal, except the base, which will not refract the
+perpendicular ray. These surfaces will each make an angle of 45
+degrees 20 minutes with the axis of the crystal, and the base will be
+the section perpendicular to the axis.
+
+That, finally, one could also fashion out of it triangular prisms, or
+prisms with as many sides as one would, of which neither the sides nor
+the bases would refract the perpendicular ray, although they would yet
+all cause double refraction for oblique rays. The cube is included
+amongst these prisms, the bases of which are sections perpendicular to
+the axis of the crystal, and the sides are sections parallel to the
+same axis.
+
+From all this it further appears that it is not at all in the
+disposition of the layers of which this crystal seems to be composed,
+and according to which it splits in three different senses, that the
+cause resides of its irregular refraction; and that it would be in
+vain to wish to seek it there.
+
+But in order that any one who has some of this stone may be able to
+find, by his own experience, the truth of what I have just advanced, I
+will state here the process of which I have made use to cut it, and to
+polish it. Cutting is easy by the slicing wheels of lapidaries, or in
+the way in which marble is sawn: but polishing is very difficult, and
+by employing the ordinary means one more often depolishes the surfaces
+than makes them lucent.
+
+After many trials, I have at last found that for this service no plate
+of metal must be used, but a piece of mirror glass made matt and
+depolished. Upon this, with fine sand and water, one smoothes the
+crystal little by little, in the same way as spectacle glasses, and
+polishes it simply by continuing the work, but ever reducing the
+material. I have not, however, been able to give it perfect clarity
+and transparency; but the evenness which the surfaces acquire enables
+one to observe in them the effects of refraction better than in those
+made by cleaving the stone, which always have some inequality.
+
+Even when the surface is only moderately smoothed, if one rubs it over
+with a little oil or white of egg, it becomes quite transparent, so
+that the refraction is discerned in it quite distinctly. And this aid
+is specially necessary when it is wished to polish the natural
+surfaces to remove the inequalities; because one cannot render them
+lucent equally with the surfaces of other sections, which take a
+polish so much the better the less nearly they approximate to these
+natural planes.
+
+Before finishing the treatise on this Crystal, I will add one more
+marvellous phenomenon which I discovered after having written all the
+foregoing. For though I have not been able till now to find its cause,
+I do not for that reason wish to desist from describing it, in order
+to give opportunity to others to investigate it. It seems that it will
+be necessary to make still further suppositions besides those which I
+have made; but these will not for all that cease to keep their
+probability after having been confirmed by so many tests.
+
+[Illustration]
+
+The phenomenon is, that by taking two pieces of this crystal and
+applying them one over the other, or rather holding them with a space
+between the two, if all the sides of one are parallel to those of the
+other, then a ray of light, such as AB, is divided into two in the
+first piece, namely into BD and BC, following the two refractions,
+regular and irregular. On penetrating thence into the other piece
+each ray will pass there without further dividing itself in two; but
+that one which underwent the regular refraction, as here DG, will
+undergo again only a regular refraction at GH; and the other, CE, an
+irregular refraction at EF. And the same thing occurs not only in this
+disposition, but also in all those cases in which the principal
+section of each of the pieces is situated in one and the same plane,
+without it being needful for the two neighbouring surfaces to be
+parallel. Now it is marvellous why the rays CE and DG, incident from
+the air on the lower crystal, do not divide themselves the same as the
+first ray AB. One would say that it must be that the ray DG in passing
+through the upper piece has lost something which is necessary to move
+the matter which serves for the irregular refraction; and that
+likewise CE has lost that which was necessary to move the matter
+which serves for regular refraction: but there is yet another thing
+which upsets this reasoning. It is that when one disposes the two
+crystals in such a way that the planes which constitute the principal
+sections intersect one another at right angles, whether the
+neighbouring surfaces are parallel or not, then the ray which has come
+by the regular refraction, as DG, undergoes only an irregular
+refraction in the lower piece; and on the contrary the ray which has
+come by the irregular refraction, as CE, undergoes only a regular
+refraction.
+
+But in all the infinite other positions, besides those which I have
+just stated, the rays DG, CE, divide themselves anew each one into
+two, by refraction in the lower crystal so that from the single ray AB
+there are four, sometimes of equal brightness, sometimes some much
+less bright than others, according to the varying agreement in the
+positions of the crystals: but they do not appear to have all together
+more light than the single ray AB.
+
+When one considers here how, while the rays CE, DG, remain the same,
+it depends on the position that one gives to the lower piece, whether
+it divides them both in two, or whether it does not divide them, and
+yet how the ray AB above is always divided, it seems that one is
+obliged to conclude that the waves of light, after having passed
+through the first crystal, acquire a certain form or disposition in
+virtue of which, when meeting the texture of the second crystal, in
+certain positions, they can move the two different kinds of matter
+which serve for the two species of refraction; and when meeting the
+second crystal in another position are able to move only one of these
+kinds of matter. But to tell how this occurs, I have hitherto found
+nothing which satisfies me.
+
+Leaving then to others this research, I pass to what I have to say
+touching the cause of the extraordinary figure of this crystal, and
+why it cleaves easily in three different senses, parallel to any one
+of its surfaces.
+
+There are many bodies, vegetable, mineral, and congealed salts, which
+are formed with certain regular angles and figures. Thus among flowers
+there are many which have their leaves disposed in ordered polygons,
+to the number of 3, 4, 5, or 6 sides, but not more. This well deserves
+to be investigated, both as to the polygonal figure, and as to why it
+does not exceed the number 6.
+
+Rock Crystal grows ordinarily in hexagonal bars, and diamonds are
+found which occur with a square point and polished surfaces. There is
+a species of small flat stones, piled up directly upon one another,
+which are all of pentagonal figure with rounded angles, and the sides
+a little folded inwards. The grains of gray salt which are formed from
+sea water affect the figure, or at least the angle, of the cube; and
+in the congelations of other salts, and in that of sugar, there are
+found other solid angles with perfectly flat faces. Small snowflakes
+almost always fall in little stars with 6 points, and sometimes in
+hexagons with straight sides. And I have often observed, in water
+which is beginning to freeze, a kind of flat and thin foliage of ice,
+the middle ray of which throws out branches inclined at an angle of 60
+degrees. All these things are worthy of being carefully investigated
+to ascertain how and by what artifice nature there operates. But it is
+not now my intention to treat fully of this matter. It seems that in
+general the regularity which occurs in these productions comes from
+the arrangement of the small invisible equal particles of which they
+are composed. And, coming to our Iceland Crystal, I say that if there
+were a pyramid such as ABCD, composed of small rounded corpuscles, not
+spherical but flattened spheroids, such as would be made by the
+rotation of the ellipse GH around its lesser diameter EF (of which the
+ratio to the greater diameter is very nearly that of 1 to the square
+root of 8)--I say that then the solid angle of the point D would be
+equal to the obtuse and equilateral angle of this Crystal. I say,
+further, that if these corpuscles were lightly stuck together, on
+breaking this pyramid it would break along faces parallel to those
+that make its point: and by this means, as it is easy to see, it would
+produce prisms similar to those of the same crystal as this other
+figure represents. The reason is that when broken in this fashion a
+whole layer separates easily from its neighbouring layer since each
+spheroid has to be detached only from the three spheroids of the next
+layer; of which three there is but one which touches it on its
+flattened surface, and the other two at the edges. And the reason why
+the surfaces separate sharp and polished is that if any spheroid of
+the neighbouring surface would come out by attaching itself to the
+surface which is being separated, it would be needful for it to detach
+itself from six other spheroids which hold it locked, and four of
+which press it by these flattened surfaces. Since then not only the
+angles of our crystal but also the manner in which it splits agree
+precisely with what is observed in the assemblage composed of such
+spheroids, there is great reason to believe that the particles are
+shaped and ranged in the same way.
+
+[Illustration: {Pyramid and section of spheroids}]
+
+There is even probability enough that the prisms of this crystal are
+produced by the breaking up of pyramids, since Mr. Bartholinus relates
+that he occasionally found some pieces of triangularly pyramidal
+figure. But when a mass is composed interiorly only of these little
+spheroids thus piled up, whatever form it may have exteriorly, it is
+certain, by the same reasoning which I have just explained, that if
+broken it would produce similar prisms. It remains to be seen whether
+there are other reasons which confirm our conjecture, and whether
+there are none which are repugnant to it.
+
+[Illustration: {paralleloid arrangement of spheroids with planes of
+potential cleavage}]
+
+It may be objected that this crystal, being so composed, might be
+capable of cleavage in yet two more fashions; one of which would be
+along planes parallel to the base of the pyramid, that is to say to
+the triangle ABC; the other would be parallel to a plane the trace of
+which is marked by the lines GH, HK, KL. To which I say that both the
+one and the other, though practicable, are more difficult than those
+which were parallel to any one of the three planes of the pyramid; and
+that therefore, when striking on the crystal in order to break it, it
+ought always to split rather along these three planes than along the
+two others. When one has a number of spheroids of the form above
+described, and ranges them in a pyramid, one sees why the two methods
+of division are more difficult. For in the case of that division which
+would be parallel to the base, each spheroid would be obliged to
+detach itself from three others which it touches upon their flattened
+surfaces, which hold more strongly than the contacts at the edges. And
+besides that, this division will not occur along entire layers,
+because each of the spheroids of a layer is scarcely held at all by
+the 6 of the same layer that surround it, since they only touch it at
+the edges; so that it adheres readily to the neighbouring layer, and
+the others to it, for the same reason; and this causes uneven
+surfaces. Also one sees by experiment that when grinding down the
+crystal on a rather rough stone, directly on the equilateral solid
+angle, one verily finds much facility in reducing it in this
+direction, but much difficulty afterwards in polishing the surface
+which has been flattened in this manner.
+
+As for the other method of division along the plane GHKL, it will be
+seen that each spheroid would have to detach itself from four of the
+neighbouring layer, two of which touch it on the flattened surfaces,
+and two at the edges. So that this division is likewise more difficult
+than that which is made parallel to one of the surfaces of the
+crystal; where, as we have said, each spheroid is detached from only
+three of the neighbouring layer: of which three there is one only
+which touches it on the flattened surface, and the other two at the
+edges only.
+
+However, that which has made me know that in the crystal there are
+layers in this last fashion, is that in a piece weighing half a pound
+which I possess, one sees that it is split along its length, as is the
+above-mentioned prism by the plane GHKL; as appears by colours of the
+Iris extending throughout this whole plane although the two pieces
+still hold together. All this proves then that the composition of the
+crystal is such as we have stated. To which I again add this
+experiment; that if one passes a knife scraping along any one of the
+natural surfaces, and downwards as it were from the equilateral obtuse
+angle, that is to say from the apex of the pyramid, one finds it quite
+hard; but by scraping in the opposite sense an incision is easily
+made. This follows manifestly from the situation of the small
+spheroids; over which, in the first manner, the knife glides; but in
+the other manner it seizes them from beneath almost as if they were
+the scales of a fish.
+
+I will not undertake to say anything touching the way in which so many
+corpuscles all equal and similar are generated, nor how they are set
+in such beautiful order; whether they are formed first and then
+assembled, or whether they arrange themselves thus in coming into
+being and as fast as they are produced, which seems to me more
+probable. To develop truths so recondite there would be needed a
+knowledge of nature much greater than that which we have. I will add
+only that these little spheroids could well contribute to form the
+spheroids of the waves of light, here above supposed, these as well as
+those being similarly situated, and with their axes parallel.
+
+
+_Calculations which have been supposed in this Chapter_.
+
+Mr. Bartholinus, in his treatise of this Crystal, puts at 101 degrees
+the obtuse angles of the faces, which I have stated to be 101 degrees
+52 minutes. He states that he measured these angles directly on the
+crystal, which is difficult to do with ultimate exactitude, because
+the edges such as CA, CB, in this figure, are generally worn, and not
+quite straight. For more certainty, therefore, I preferred to measure
+actually the obtuse angle by which the faces CBDA, CBVF, are inclined
+to one another, namely the angle OCN formed by drawing CN
+perpendicular to FV, and CO perpendicular to DA. This angle OCN I
+found to be 105 degrees; and its supplement CNP, to be 75 degrees, as
+it should be.
+
+[Illustration]
+
+To find from this the obtuse angle BCA, I imagined a sphere having its
+centre at C, and on its surface a spherical triangle, formed by the
+intersection of three planes which enclose the solid angle C. In this
+equilateral triangle, which is ABF in this other figure, I see that
+each of the angles should be 105 degrees, namely equal to the angle
+OCN; and that each of the sides should be of as many degrees as the
+angle ACB, or ACF, or BCF. Having then drawn the arc FQ perpendicular
+to the side AB, which it divides equally at Q, the triangle FQA has a
+right angle at Q, the angle A 105 degrees, and F half as much, namely
+52 degrees 30 minutes; whence the hypotenuse AF is found to be 101
+degrees 52 minutes. And this arc AF is the measure of the angle ACF in
+the figure of the crystal.
+
+[Illustration]
+
+In the same figure, if the plane CGHF cuts the crystal so that it
+divides the obtuse angles ACB, MHV, in the middle, it is stated, in
+Article 10, that the angle CFH is 70 degrees 57 minutes. This again is
+easily shown in the same spherical triangle ABF, in which it appears
+that the arc FQ is as many degrees as the angle GCF in the crystal,
+the supplement of which is the angle CFH. Now the arc FQ is found to
+be 109 degrees 3 minutes. Then its supplement, 70 degrees 57 minutes,
+is the angle CFH.
+
+It was stated, in Article 26, that the straight line CS, which in the
+preceding figure is CH, being the axis of the crystal, that is to say
+being equally inclined to the three sides CA, CB, CF, the angle GCH is
+45 degrees 20 minutes. This is also easily calculated by the same
+spherical triangle. For by drawing the other arc AD which cuts BF
+equally, and intersects FQ at S, this point will be the centre of the
+triangle. And it is easy to see that the arc SQ is the measure of the
+angle GCH in the figure which represents the crystal. Now in the
+triangle QAS, which is right-angled, one knows also the angle A, which
+is 52 degrees 30 minutes, and the side AQ 50 degrees 56 minutes;
+whence the side SQ is found to be 45 degrees 20 minutes.
+
+In Article 27 it was required to show that PMS being an ellipse the
+centre of which is C, and which touches the straight line MD at M so
+that the angle MCL which CM makes with CL, perpendicular on DM, is 6
+degrees 40 minutes, and its semi-minor axis CS making with CG (which
+is parallel to MD) an angle GCS of 45 degrees 20 minutes, it was
+required to show, I say, that, CM being 100,000 parts, PC the
+semi-major diameter of this ellipse is 105,032 parts, and CS, the
+semi-minor diameter, 93,410.
+
+Let CP and CS be prolonged and meet the tangent DM at D and Z; and
+from the point of contact M let MN and MO be drawn as perpendiculars
+to CP and CS. Now because the angles SCP, GCL, are right angles, the
+angle PCL will be equal to GCS which was 45 degrees 20 minutes. And
+deducting the angle LCM, which is 6 degrees 40 minutes, from LCP,
+which is 45 degrees 20 minutes, there remains MCP, 38 degrees 40
+minutes. Considering then CM as a radius of 100,000 parts, MN, the
+sine of 38 degrees 40 minutes, will be 62,479. And in the right-angled
+triangle MND, MN will be to ND as the radius of the Tables is to the
+tangent of 45 degrees 20 minutes (because the angle NMD is equal to
+DCL, or GCS); that is to say as 100,000 to 101,170: whence results ND
+63,210. But NC is 78,079 of the same parts, CM being 100,000, because
+NC is the sine of the complement of the angle MCP, which was 38
+degrees 40 minutes. Then the whole line DC is 141,289; and CP, which
+is a mean proportional between DC and CN, since MD touches the
+Ellipse, will be 105,032.
+
+[Illustration]
+
+Similarly, because the angle OMZ is equal to CDZ, or LCZ, which is 44
+degrees 40 minutes, being the complement of GCS, it follows that, as
+the radius of the Tables is to the tangent of 44 degrees 40 minutes,
+so will OM 78,079 be to OZ 77,176. But OC is 62,479 of these same
+parts of which CM is 100,000, because it is equal to MN, the sine of
+the angle MCP, which is 38 degrees 40 minutes. Then the whole line CZ
+is 139,655; and CS, which is a mean proportional between CZ and CO
+will be 93,410.
+
+At the same place it was stated that GC was found to be 98,779 parts.
+To prove this, let PE be drawn in the same figure parallel to DM, and
+meeting CM at E. In the right-angled triangle CLD the side CL is
+99,324 (CM being 100,000), because CL is the sine of the complement of
+the angle LCM, which is 6 degrees 40 minutes. And since the angle LCD
+is 45 degrees 20 minutes, being equal to GCS, the side LD is found to
+be 100,486: whence deducting ML 11,609 there will remain MD 88,877.
+Now as CD (which was 141,289) is to DM 88,877, so will CP 105,032 be
+to PE 66,070. But as the rectangle MEH (or rather the difference of
+the squares on CM and CE) is to the square on MC, so is the square on
+PE to the square on C_g_; then also as the difference of the squares
+on DC and CP to the square on CD, so also is the square on PE to the
+square on _g_C. But DP, CP, and PE are known; hence also one knows GC,
+which is 98,779.
+
+
+_Lemma which has been supposed_.
+
+If a spheroid is touched by a straight line, and also by two or more
+planes which are parallel to this line, though not parallel to one
+another, all the points of contact of the line, as well as of the
+planes, will be in one and the same ellipse made by a plane which
+passes through the centre of the spheroid.
+
+Let LED be the spheroid touched by the line BM at the point B, and
+also by the planes parallel to this line at the points O and A. It is
+required to demonstrate that the points B, O, and A are in one and the
+same Ellipse made in the spheroid by a plane which passes through its
+centre.
+
+[Illustration]
+
+Through the line BM, and through the points O and A, let there be
+drawn planes parallel to one another, which, in cutting the spheroid
+make the ellipses LBD, POP, QAQ; which will all be similar and
+similarly disposed, and will have their centres K, N, R, in one and
+the same diameter of the spheroid, which will also be the diameter of
+the ellipse made by the section of the plane that passes through the
+centre of the spheroid, and which cuts the planes of the three said
+Ellipses at right angles: for all this is manifest by proposition 15
+of the book of Conoids and Spheroids of Archimedes. Further, the two
+latter planes, which are drawn through the points O and A, will also,
+by cutting the planes which touch the spheroid in these same points,
+generate straight lines, as OH and AS, which will, as is easy to see,
+be parallel to BM; and all three, BM, OH, AS, will touch the Ellipses
+LBD, POP, QAQ in these points, B, O, A; since they are in the planes
+of these ellipses, and at the same time in the planes which touch the
+spheroid. If now from these points B, O, A, there are drawn the
+straight lines BK, ON, AR, through the centres of the same ellipses,
+and if through these centres there are drawn also the diameters LD,
+PP, QQ, parallel to the tangents BM, OH, AS; these will be conjugate
+to the aforesaid BK, ON, AR. And because the three ellipses are
+similar and similarly disposed, and have their diameters LD, PP, QQ
+parallel, it is certain that their conjugate diameters BK, ON, AR,
+will also be parallel. And the centres K, N, R being, as has been
+stated, in one and the same diameter of the spheroid, these parallels
+BK, ON, AR will necessarily be in one and the same plane, which passes
+through this diameter of the spheroid, and, in consequence, the points
+R, O, A are in one and the same ellipse made by the intersection of
+this plane. Which was to be proved. And it is manifest that the
+demonstration would be the same if, besides the points O, A, there had
+been others in which the spheroid had been touched by planes parallel
+to the straight line BM.
+
+
+
+
+CHAPTER VI
+
+ON THE FIGURES OF THE TRANSPARENT BODIES
+
+Which serve for Refraction and for Reflexion
+
+
+After having explained how the properties of reflexion and refraction
+follow from what we have supposed concerning the nature of light, and
+of opaque bodies, and of transparent media, I will here set forth a
+very easy and natural way of deducing, from the same principles, the
+true figures which serve, either by reflexion or by refraction, to
+collect or disperse the rays of light, as may be desired. For though I
+do not see yet that there are means of making use of these figures, so
+far as relates to Refraction, not only because of the difficulty of
+shaping the glasses of Telescopes with the requisite exactitude
+according to these figures, but also because there exists in
+refraction itself a property which hinders the perfect concurrence of
+the rays, as Mr. Newton has very well proved by experiment, I will yet
+not desist from relating the invention, since it offers itself, so to
+speak, of itself, and because it further confirms our Theory of
+refraction, by the agreement which here is found between the refracted
+ray and the reflected ray. Besides, it may occur that some one in the
+future will discover in it utilities which at present are not seen.
+
+[Illustration]
+
+To proceed then to these figures, let us suppose first that it is
+desired to find a surface CDE which shall reassemble at a point B rays
+coming from another point A; and that the summit of the surface shall
+be the given point D in the straight line AB. I say that, whether by
+reflexion or by refraction, it is only necessary to make this surface
+such that the path of the light from the point A to all points of the
+curved line CDE, and from these to the point of concurrence (as here
+the path along the straight lines AC, CB, along AL, LB, and along AD,
+DB), shall be everywhere traversed in equal times: by which principle
+the finding of these curves becomes very easy.
+
+[Illustration]
+
+So far as relates to the reflecting surface, since the sum of the
+lines AC, CB ought to be equal to that of AD, DB, it appears that DCE
+ought to be an ellipse; and for refraction, the ratio of the
+velocities of waves of light in the media A and B being supposed to be
+known, for example that of 3 to 2 (which is the same, as we have
+shown, as the ratio of the Sines in the refraction), it is only
+necessary to make DH equal to 3/2 of DB; and having after that
+described from the centre A some arc FC, cutting DB at F, then
+describe another from centre B with its semi-diameter BX equal to 2/3
+of FH; and the point of intersection of the two arcs will be one of
+the points required, through which the curve should pass. For this
+point, having been found in this fashion, it is easy forthwith to
+demonstrate that the time along AC, CB, will be equal to the time
+along AD, DB.
+
+For assuming that the line AD represents the time which the light
+takes to traverse this same distance AD in air, it is evident that DH,
+equal to 3/2 of DB, will represent the time of the light along DB in
+the medium, because it needs here more time in proportion as its speed
+is slower. Therefore the whole line AH will represent the time along
+AD, DB. Similarly the line AC or AF will represent the time along AC;
+and FH being by construction equal to 3/2 of CB, it will represent the
+time along CB in the medium; and in consequence the whole line AH will
+represent also the time along AC, CB. Whence it appears that the time
+along AC, CB, is equal to the time along AD, DB. And similarly it can
+be shown if L and K are other points in the curve CDE, that the times
+along AL, LB, and along AK, KB, are always represented by the line AH,
+and therefore equal to the said time along AD, DB.
+
+In order to show further that the surfaces, which these curves will
+generate by revolution, will direct all the rays which reach them from
+the point A in such wise that they tend towards B, let there be
+supposed a point K in the curve, farther from D than C is, but such
+that the straight line AK falls from outside upon the curve which
+serves for the refraction; and from the centre B let the arc KS be
+described, cutting BD at S, and the straight line CB at R; and from
+the centre A describe the arc DN meeting AK at N.
+
+Since the sums of the times along AK, KB, and along AC, CB are equal,
+if from the former sum one deducts the time along KB, and if from the
+other one deducts the time along RB, there will remain the time along
+AK as equal to the time along the two parts AC, CR. Consequently in
+the time that the light has come along AK it will also have come along
+AC and will in addition have made, in the medium from the centre C, a
+partial spherical wave, having a semi-diameter equal to CR. And this
+wave will necessarily touch the circumference KS at R, since CB cuts
+this circumference at right angles. Similarly, having taken any other
+point L in the curve, one can show that in the same time as the light
+passes along AL it will also have come along AL and in addition will
+have made a partial wave, from the centre L, which will touch the same
+circumference KS. And so with all other points of the curve CDE. Then
+at the moment that the light reaches K the arc KRS will be the
+termination of the movement, which has spread from A through DCK. And
+thus this same arc will constitute in the medium the propagation of
+the wave emanating from A; which wave may be represented by the arc
+DN, or by any other nearer the centre A. But all the pieces of the arc
+KRS are propagated successively along straight lines which are
+perpendicular to them, that is to say, which tend to the centre B (for
+that can be demonstrated in the same way as we have proved above that
+the pieces of spherical waves are propagated along the straight lines
+coming from their centre), and these progressions of the pieces of the
+waves constitute the rays themselves of light. It appears then that
+all these rays tend here towards the point B.
+
+One might also determine the point C, and all the others, in this
+curve which serves for the refraction, by dividing DA at G in such a
+way that DG is 2/3 of DA, and describing from the centre B any arc CX
+which cuts BD at N, and another from the centre A with its
+semi-diameter AF equal to 3/2 of GX; or rather, having described, as
+before, the arc CX, it is only necessary to make DF equal to 3/2 of
+DX, and from-the centre A to strike the arc FC; for these two
+constructions, as may be easily known, come back to the first one
+which was shown before. And it is manifest by the last method that
+this curve is the same that Mr. Des Cartes has given in his Geometry,
+and which he calls the first of his Ovals.
+
+It is only a part of this oval which serves for the refraction,
+namely, the part DK, ending at K, if AK is the tangent. As to the,
+other part, Des Cartes has remarked that it could serve for
+reflexions, if there were some material of a mirror of such a nature
+that by its means the force of the rays (or, as we should say, the
+velocity of the light, which he could not say, since he held that the
+movement of light was instantaneous) could be augmented in the
+proportion of 3 to 2. But we have shown that in our way of explaining
+reflexion, such a thing could not arise from the matter of the mirror,
+and it is entirely impossible.
+
+[Illustration]
+
+[Illustration]
+
+From what has been demonstrated about this oval, it will be easy to
+find the figure which serves to collect to a point incident parallel
+rays. For by supposing just the same construction, but the point A
+infinitely distant, giving parallel rays, our oval becomes a true
+Ellipse, the construction of which differs in no way from that of the
+oval, except that FC, which previously was an arc of a circle, is here
+a straight line, perpendicular to DB. For the wave of light DN, being
+likewise represented by a straight line, it will be seen that all the
+points of this wave, travelling as far as the surface KD along lines
+parallel to DB, will advance subsequently towards the point B, and
+will arrive there at the same time. As for the Ellipse which served
+for reflexion, it is evident that it will here become a parabola,
+since its focus A may be regarded as infinitely distant from the
+other, B, which is here the focus of the parabola, towards which all
+the reflexions of rays parallel to AB tend. And the demonstration of
+these effects is just the same as the preceding.
+
+But that this curved line CDE which serves for refraction is an
+Ellipse, and is such that its major diameter is to the distance
+between its foci as 3 to 2, which is the proportion of the refraction,
+can be easily found by the calculus of Algebra. For DB, which is
+given, being called _a_; its undetermined perpendicular DT being
+called _x_; and TC _y_; FB will be _a - y_; CB will be sqrt(_xx + aa
+-2ay + yy_). But the nature of the curve is such that 2/3 of TC
+together with CB is equal to DB, as was stated in the last
+construction: then the equation will be between _(2/3)y + sqrt(xx + aa
+- 2ay + yy)_ and _a_; which being reduced, gives _(6/5)ay - yy_ equal
+to _(9/5)xx_; that is to say that having made DO equal to 6/5 of DB,
+the rectangle DFO is equal to 9/5 of the square on FC. Whence it is
+seen that DC is an ellipse, of which the axis DO is to the parameter
+as 9 to 5; and therefore the square on DO is to the square of the
+distance between the foci as 9 to 9 - 5, that is to say 4; and finally
+the line DO will be to this distance as 3 to 2.
+
+[Illustration]
+
+Again, if one supposes the point B to be infinitely distant, in lieu
+of our first oval we shall find that CDE is a true Hyperbola; which
+will make those rays become parallel which come from the point A. And
+in consequence also those which are parallel within the transparent
+body will be collected outside at the point A. Now it must be remarked
+that CX and KS become straight lines perpendicular to BA, because they
+represent arcs of circles the centre of which is infinitely distant.
+And the intersection of the perpendicular CX with the arc FC will give
+the point C, one of those through which the curve ought to pass. And
+this operates so that all the parts of the wave of light DN, coming to
+meet the surface KDE, will advance thence along parallels to KS and
+will arrive at this straight line at the same time; of which the proof
+is again the same as that which served for the first oval. Besides one
+finds by a calculation as easy as the preceding one, that CDE is here
+a hyperbola of which the axis DO is 4/5 of AD, and the parameter
+equal to AD. Whence it is easily proved that DO is to the distance
+between the foci as 3 to 2.
+
+[Illustration]
+
+These are the two cases in which Conic sections serve for refraction,
+and are the same which are explained, in his _Dioptrique_, by Des
+Cartes, who first found out the use of these lines in relation to
+refraction, as also that of the Ovals the first of which we have
+already set forth. The second oval is that which serves for rays that
+tend to a given point; in which oval, if the apex of the surface which
+receives the rays is D, it will happen that the other apex will be
+situated between B and A, or beyond A, according as the ratio of AD to
+DB is given of greater or lesser value. And in this latter case it is
+the same as that which Des Cartes calls his 3rd oval.
+
+Now the finding and construction of this second oval is the same as
+that of the first, and the demonstration of its effect likewise. But
+it is worthy of remark that in one case this oval becomes a perfect
+circle, namely when the ratio of AD to DB is the same as the ratio of
+the refractions, here as 3 to 2, as I observed a long time ago. The
+4th oval, serving only for impossible reflexions, there is no need to
+set it forth.
+
+[Illustration]
+
+As for the manner in which Mr. Des Cartes discovered these lines,
+since he has given no explanation of it, nor any one else since that I
+know of, I will say here, in passing, what it seems to me it must have
+been. Let it be proposed to find the surface generated by the
+revolution of the curve KDE, which, receiving the incident rays coming
+to it from the point A, shall deviate them toward the point B. Then
+considering this other curve as already known, and that its apex D is
+in the straight line AB, let us divide it up into an infinitude of
+small pieces by the points G, C, F; and having drawn from each of
+these points, straight lines towards A to represent the incident rays,
+and other straight lines towards B, let there also be described with
+centre A the arcs GL, CM, FN, DO, cutting the rays that come from A at
+L, M, N, O; and from the points K, G, C, F, let there be described
+the arcs KQ, GR, CS, FT cutting the rays towards B at Q, R, S, T; and
+let us suppose that the straight line HKZ cuts the curve at K at
+right-angles.
+
+[Illustration]
+
+Then AK being an incident ray, and KB its refraction within the
+medium, it needs must be, according to the law of refraction which was
+known to Mr. Des Cartes, that the sine of the angle ZKA should be to
+the sine of the angle HKB as 3 to 2, supposing that this is the
+proportion of the refraction of glass; or rather, that the sine of the
+angle KGL should have this same ratio to the sine of the angle GKQ,
+considering KG, GL, KQ as straight lines because of their smallness.
+But these sines are the lines KL and GQ, if GK is taken as the radius
+of the circle. Then LK ought to be to GQ as 3 to 2; and in the same
+ratio MG to CR, NC to FS, OF to DT. Then also the sum of all the
+antecedents to all the consequents would be as 3 to 2. Now by
+prolonging the arc DO until it meets AK at X, KX is the sum of the
+antecedents. And by prolonging the arc KQ till it meets AD at Y, the
+sum of the consequents is DY. Then KX ought to be to DY as 3 to 2.
+Whence it would appear that the curve KDE was of such a nature that
+having drawn from some point which had been assumed, such as K, the
+straight lines KA, KB, the excess by which AK surpasses AD should be
+to the excess of DB over KB, as 3 to 2. For it can similarly be
+demonstrated, by taking any other point in the curve, such as G, that
+the excess of AG over AD, namely VG, is to the excess of BD over DG,
+namely DP, in this same ratio of 3 to 2. And following this principle
+Mr. Des Cartes constructed these curves in his _Geometric_; and he
+easily recognized that in the case of parallel rays, these curves
+became Hyperbolas and Ellipses.
+
+Let us now return to our method and let us see how it leads without
+difficulty to the finding of the curves which one side of the glass
+requires when the other side is of a given figure; a figure not only
+plane or spherical, or made by one of the conic sections (which is the
+restriction with which Des Cartes proposed this problem, leaving the
+solution to those who should come after him) but generally any figure
+whatever: that is to say, one made by the revolution of any given
+curved line to which one must merely know how to draw straight lines
+as tangents.
+
+Let the given figure be that made by the revolution of some curve such
+as AK about the axis AV, and that this side of the glass receives rays
+coming from the point L. Furthermore, let the thickness AB of the
+middle of the glass be given, and the point F at which one desires the
+rays to be all perfectly reunited, whatever be the first refraction
+occurring at the surface AK.
+
+I say that for this the sole requirement is that the outline BDK which
+constitutes the other surface shall be such that the path of the
+light from the point L to the surface AK, and from thence to the
+surface BDK, and from thence to the point F, shall be traversed
+everywhere in equal times, and in each case in a time equal to that
+which the light employs, to pass along the straight line LF of which
+the part AB is within the glass.
+
+[Illustration]
+
+Let LG be a ray falling on the arc AK. Its refraction GV will be given
+by means of the tangent which will be drawn at the point G. Now in GV
+the point D must be found such that FD together with 3/2 of DG and the
+straight line GL, may be equal to FB together with 3/2 of BA and the
+straight line AL; which, as is clear, make up a given length. Or
+rather, by deducting from each the length of LG, which is also given,
+it will merely be needful to adjust FD up to the straight line VG in
+such a way that FD together with 3/2 of DG is equal to a given
+straight line, which is a quite easy plane problem: and the point D
+will be one of those through which the curve BDK ought to pass. And
+similarly, having drawn another ray LM, and found its refraction MO,
+the point N will be found in this line, and so on as many times as one
+desires.
+
+To demonstrate the effect of the curve, let there be described about
+the centre L the circular arc AH, cutting LG at H; and about the
+centre F the arc BP; and in AB let AS be taken equal to 2/3 of HG; and
+SE equal to GD. Then considering AH as a wave of light emanating from
+the point L, it is certain that during the time in which its piece H
+arrives at G the piece A will have advanced within the transparent
+body only along AS; for I suppose, as above, the proportion of the
+refraction to be as 3 to 2. Now we know that the piece of wave which
+is incident on G, advances thence along the line GD, since GV is the
+refraction of the ray LG. Then during the time that this piece of wave
+has taken from G to D, the other piece which was at S has reached E,
+since GD, SE are equal. But while the latter will advance from E to B,
+the piece of wave which was at D will have spread into the air its
+partial wave, the semi-diameter of which, DC (supposing this wave to
+cut the line DF at C), will be 3/2 of EB, since the velocity of light
+outside the medium is to that inside as 3 to 2. Now it is easy to show
+that this wave will touch the arc BP at this point C. For since, by
+construction, FD + 3/2 DG + GL are equal to FB + 3/2 BA + AL; on
+deducting the equals LH, LA, there will remain FD + 3/2 DG + GH equal
+to FB + 3/2 BA. And, again, deducting from one side GH, and from the
+other side 3/2 of AS, which are equal, there will remain FD with 3/2
+DG equal to FB with 3/2 of BS. But 3/2 of DG are equal to 3/2 of ES;
+then FD is equal to FB with 3/2 of BE. But DC was equal to 3/2 of EB;
+then deducting these equal lengths from one side and from the other,
+there will remain CF equal to FB. And thus it appears that the wave,
+the semi-diameter of which is DC, touches the arc BP at the moment
+when the light coming from the point L has arrived at B along the line
+LB. It can be demonstrated similarly that at this same moment the
+light that has come along any other ray, such as LM, MN, will have
+propagated the movement which is terminated at the arc BP. Whence it
+follows, as has been often said, that the propagation of the wave AH,
+after it has passed through the thickness of the glass, will be the
+spherical wave BP, all the pieces of which ought to advance along
+straight lines, which are the rays of light, to the centre F. Which
+was to be proved. Similarly these curved lines can be found in all the
+cases which can be proposed, as will be sufficiently shown by one or
+two examples which I will add.
+
+Let there be given the surface of the glass AK, made by the revolution
+about the axis BA of the line AK, which may be straight or curved. Let
+there be also given in the axis the point L and the thickness BA of
+the glass; and let it be required to find the other surface KDB, which
+receiving rays that are parallel to AB will direct them in such wise
+that after being again refracted at the given surface AK they will all
+be reassembled at the point L.
+
+[Illustration]
+
+From the point L let there be drawn to some point of the given line
+AK the straight line LG, which, being considered as a ray of light,
+its refraction GD will then be found. And this line being then
+prolonged at one side or the other will meet the straight line BL, as
+here at V. Let there then be erected on AB the perpendicular BC, which
+will represent a wave of light coming from the infinitely distant
+point F, since we have supposed the rays to be parallel. Then all the
+parts of this wave BC must arrive at the same time at the point L; or
+rather all the parts of a wave emanating from the point L must arrive
+at the same time at the straight line BC. And for that, it is
+necessary to find in the line VGD the point D such that having drawn
+DC parallel to AB, the sum of CD, plus 3/2 of DG, plus GL may be equal
+to 3/2 of AB, plus AL: or rather, on deducting from both sides GL,
+which is given, CD plus 3/2 of DG must be equal to a given length;
+which is a still easier problem than the preceding construction. The
+point D thus found will be one of those through which the curve ought
+to pass; and the proof will be the same as before. And by this it will
+be proved that the waves which come from the point L, after having
+passed through the glass KAKB, will take the form of straight lines,
+as BC; which is the same thing as saying that the rays will become
+parallel. Whence it follows reciprocally that parallel rays falling on
+the surface KDB will be reassembled at the point L.
+
+[Illustration]
+
+Again, let there be given the surface AK, of any desired form,
+generated by revolution about the axis AB, and let the thickness of
+the glass at the middle be AB. Also let the point L be given in the
+axis behind the glass; and let it be supposed that the rays which fall
+on the surface AK tend to this point, and that it is required to find
+the surface BD, which on their emergence from the glass turns them as
+if they came from the point F in front of the glass.
+
+Having taken any point G in the line AK, and drawing the straight line
+IGL, its part GI will represent one of the incident rays, the
+refraction of which, GV, will then be found: and it is in this line
+that we must find the point D, one of those through which the curve DG
+ought to pass. Let us suppose that it has been found: and about L as
+centre let there be described GT, the arc of a circle cutting the
+straight line AB at T, in case the distance LG is greater than LA; for
+otherwise the arc AH must be described about the same centre, cutting
+the straight line LG at H. This arc GT (or AH, in the other case) will
+represent an incident wave of light, the rays of which tend towards
+L. Similarly, about the centre F let there be described the circular
+arc DQ, which will represent a wave emanating from the point F.
+
+Then the wave TG, after having passed through the glass, must form the
+wave QD; and for this I observe that the time taken by the light along
+GD in the glass must be equal to that taken along the three, TA, AB,
+and BQ, of which AB alone is within the glass. Or rather, having taken
+AS equal to 2/3 of AT, I observe that 3/2 of GD ought to be equal to
+3/2 of SB, plus BQ; and, deducting both of them from FD or FQ, that FD
+less 3/2 of GD ought to be equal to FB less 3/2 of SB. And this last
+difference is a given length: and all that is required is to draw the
+straight line FD from the given point F to meet VG so that it may be
+thus. Which is a problem quite similar to that which served for the
+first of these constructions, where FD plus 3/2 of GD had to be equal
+to a given length.
+
+In the demonstration it is to be observed that, since the arc BC falls
+within the glass, there must be conceived an arc RX, concentric with
+it and on the other side of QD. Then after it shall have been shown
+that the piece G of the wave GT arrives at D at the same time that the
+piece T arrives at Q, which is easily deduced from the construction,
+it will be evident as a consequence that the partial wave generated at
+the point D will touch the arc RX at the moment when the piece Q shall
+have come to R, and that thus this arc will at the same moment be the
+termination of the movement that comes from the wave TG; whence all
+the rest may be concluded.
+
+Having shown the method of finding these curved lines which serve for
+the perfect concurrence of the rays, there remains to be explained a
+notable thing touching the uncoordinated refraction of spherical,
+plane, and other surfaces: an effect which if ignored might cause some
+doubt concerning what we have several times said, that rays of light
+are straight lines which intersect at right angles the waves which
+travel along them.
+
+[Illustration]
+
+For in the case of rays which, for example, fall parallel upon a
+spherical surface AFE, intersecting one another, after refraction, at
+different points, as this figure represents; what can the waves of
+light be, in this transparent body, which are cut at right angles by
+the converging rays? For they can not be spherical. And what will
+these waves become after the said rays begin to intersect one another?
+It will be seen in the solution of this difficulty that something very
+remarkable comes to pass herein, and that the waves do not cease to
+persist though they do not continue entire, as when they cross the
+glasses designed according to the construction we have seen.
+
+According to what has been shown above, the straight line AD, which
+has been drawn at the summit of the sphere, at right angles to the
+axis parallel to which the rays come, represents the wave of light;
+and in the time taken by its piece D to reach the spherical surface
+AGE at E, its other parts will have met the same surface at F, G, H,
+etc., and will have also formed spherical partial waves of which these
+points are the centres. And the surface EK which all those waves will
+touch, will be the continuation of the wave AD in the sphere at the
+moment when the piece D has reached E. Now the line EK is not an arc
+of a circle, but is a curved line formed as the evolute of another
+curve ENC, which touches all the rays HL, GM, FO, etc., that are the
+refractions of the parallel rays, if we imagine laid over the
+convexity ENC a thread which in unwinding describes at its end E the
+said curve EK. For, supposing that this curve has been thus described,
+we will show that the said waves formed from the centres F, G, H,
+etc., will all touch it.
+
+It is certain that the curve EK and all the others described by the
+evolution of the curve ENC, with different lengths of thread, will cut
+all the rays HL, GM, FO, etc., at right angles, and in such wise that
+the parts of them intercepted between two such curves will all be
+equal; for this follows from what has been demonstrated in our
+treatise _de Motu Pendulorum_. Now imagining the incident rays as
+being infinitely near to one another, if we consider two of them, as
+RG, TF, and draw GQ perpendicular to RG, and if we suppose the curve
+FS which intersects GM at P to have been described by evolution from
+the curve NC, beginning at F, as far as which the thread is supposed
+to extend, we may assume the small piece FP as a straight line
+perpendicular to the ray GM, and similarly the arc GF as a straight
+line. But GM being the refraction of the ray RG, and FP being
+perpendicular to it, QF must be to GP as 3 to 2, that is to say in the
+proportion of the refraction; as was shown above in explaining the
+discovery of Des Cartes. And the same thing occurs in all the small
+arcs GH, HA, etc., namely that in the quadrilaterals which enclose
+them the side parallel to the axis is to the opposite side as 3 to 2.
+Then also as 3 to 2 will the sum of the one set be to the sum of the
+other; that is to say, TF to AS, and DE to AK, and BE to SK or DV,
+supposing V to be the intersection of the curve EK and the ray FO.
+But, making FB perpendicular to DE, the ratio of 3 to 2 is also that
+of BE to the semi-diameter of the spherical wave which emanated from
+the point F while the light outside the transparent body traversed the
+space BE. Then it appears that this wave will intersect the ray FM at
+the same point V where it is intersected at right angles by the curve
+EK, and consequently that the wave will touch this curve. In the same
+way it can be proved that the same will apply to all the other waves
+above mentioned, originating at the points G, H, etc.; to wit, that
+they will touch the curve EK at the moment when the piece D of the
+wave ED shall have reached E.
+
+Now to say what these waves become after the rays have begun to cross
+one another: it is that from thence they fold back and are composed of
+two contiguous parts, one being a curve formed as evolute of the curve
+ENC in one sense, and the other as evolute of the same curve in the
+opposite sense. Thus the wave KE, while advancing toward the meeting
+place becomes _abc_, whereof the part _ab_ is made by the evolute
+_b_C, a portion of the curve ENC, while the end C remains attached;
+and the part _bc_ by the evolute of the portion _b_E while the end E
+remains attached. Consequently the same wave becomes _def_, then
+_ghk_, and finally CY, from whence it subsequently spreads without any
+fold, but always along curved lines which are evolutes of the curve
+ENC, increased by some straight line at the end C.
+
+There is even, in this curve, a part EN which is straight, N being the
+point where the perpendicular from the centre X of the sphere falls
+upon the refraction of the ray DE, which I now suppose to touch the
+sphere. The folding of the waves of light begins from the point N up
+to the end of the curve C, which point is formed by taking AC to CX in
+the proportion of the refraction, as here 3 to 2.
+
+As many other points as may be desired in the curve NC are found by a
+Theorem which Mr. Barrow has demonstrated in section 12 of his
+_Lectiones Opticae_, though for another purpose. And it is to be noted
+that a straight line equal in length to this curve can be given. For
+since it together with the line NE is equal to the line CK, which is
+known, since DE is to AK in the proportion of the refraction, it
+appears that by deducting EN from CK the remainder will be equal to
+the curve NC.
+
+Similarly the waves that are folded back in reflexion by a concave
+spherical mirror can be found. Let ABC be the section, through the
+axis, of a hollow hemisphere, the centre of which is D, its axis being
+DB, parallel to which I suppose the rays of light to come. All the
+reflexions of those rays which fall upon the quarter-circle AB will
+touch a curved line AFE, of which line the end E is at the focus of
+the hemisphere, that is to say, at the point which divides the
+semi-diameter BD into two equal parts. The points through which this
+curve ought to pass are found by taking, beyond A, some arc AO, and
+making the arc OP double the length of it; then dividing the chord OP
+at F in such wise that the part FP is three times the part FO; for
+then F is one of the required points.
+
+[Illustration]
+
+And as the parallel rays are merely perpendiculars to the waves which
+fall on the concave surface, which waves are parallel to AD, it will
+be found that as they come successively to encounter the surface AB,
+they form on reflexion folded waves composed of two curves which
+originate from two opposite evolutions of the parts of the curve AFE.
+So, taking AD as an incident wave, when the part AG shall have met the
+surface AI, that is to say when the piece G shall have reached I, it
+will be the curves HF, FI, generated as evolutes of the curves FA, FE,
+both beginning at F, which together constitute the propagation of the
+part AG. And a little afterwards, when the part AK has met the surface
+AM, the piece K having come to M, then the curves LN, NM, will
+together constitute the propagation of that part. And thus this folded
+wave will continue to advance until the point N has reached the focus
+E. The curve AFE can be seen in smoke, or in flying dust, when a
+concave mirror is held opposite the sun. And it should be known that
+it is none other than that curve which is described by the point E on
+the circumference of the circle EB, when that circle is made to roll
+within another whose semi-diameter is ED and whose centre is D. So
+that it is a kind of Cycloid, of which, however, the points can be
+found geometrically.
+
+Its length is exactly equal to 3/4 of the diameter of the sphere, as
+can be found and demonstrated by means of these waves, nearly in the
+same way as the mensuration of the preceding curve; though it may also
+be demonstrated in other ways, which I omit as outside the subject.
+The area AOBEFA, comprised between the arc of the quarter-circle, the
+straight line BE, and the curve EFA, is equal to the fourth part of
+the quadrant DAB.
+
+
+
+
+
+INDEX
+
+Archimedes, 104.
+
+Atmospheric refraction, 45.
+
+Barrow, Isaac, 126.
+
+Bartholinus, Erasmus, 53, 54, 57, 60, 97, 99.
+
+Boyle, Hon. Robert, 11.
+
+Cassini, Jacques, iii.
+
+Caustic Curves, 123.
+
+Crystals, see Iceland Crystal, Rock Crystal.
+
+Crystals, configuration of, 95.
+
+Descartes, Rene, 3, 5, 7, 14, 22, 42, 43, 109, 113.
+
+Double Refraction, discovery of, 54, 81, 93.
+
+Elasticity, 12, 14.
+
+Ether, the, or Ethereal matter, 11, 14, 16, 28.
+
+Extraordinary refraction, 55, 56.
+
+Fermat, principle of, 42.
+
+Figures of transparent bodies, 105.
+
+Hooke, Robert, 20.
+
+Iceland Crystal, 2, 52 sqq.
+
+Iceland Crystal, Cutting and Polishing of, 91, 92, 98.
+
+Leibnitz, G.W., vi.
+
+Light, nature of, 3.
+
+Light, velocity of, 4, 15.
+
+Molecular texture of bodies, 27, 95.
+
+Newton, Sir Isaac, vi, 106.
+
+Opacity, 34.
+
+Ovals, Cartesian, 107, 113.
+
+Pardies, Rev. Father, 20.
+
+Rays, definition of, 38, 49.
+
+Reflexion, 22.
+
+Refraction, 28, 34.
+
+Rock Crystal, 54, 57, 62, 95.
+
+Roemer, Olaf, v, 7.
+
+Roughness of surfaces, 27.
+
+Sines, law of, 1, 35, 38, 43.
+
+Spheres, elasticity of, 15.
+
+Spheroidal waves in crystals, 63.
+
+Spheroids, lemma about, 103.
+
+Sound, speed of, 7, 10, 12.
+
+Telescopes, lenses for, 62, 105.
+
+Torricelli's experiment, 12, 30.
+
+Transparency, explanation of, 28, 31, 32.
+
+Waves, no regular succession of, 17.
+
+Waves, principle of wave envelopes, 19, 24.
+
+Waves, principle of elementary wave fronts, 19.
+
+Waves, propagation of light as, 16, 63.
+
+
+
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