file which includes the original illustrations4 ,see #adgbe-;h4htm or #adgbe-;h4zip3 7https3_/_/www4gutenberg4org_/dirs_/#a_/ d_/g_/b_/adgbe_/adgbe-;h_/#adgbe-;h4htm7 or 7https3_/_/www4gutenberg4org_/dirs_/#a_/ d_/g_/b_/adgbe_/adgbe-;h4zip7 ,,treatise ,,on ,,light ,in which are explained ,the causes of that which occurs ,in ,,reflexion1 in ,,refraction ,and particularly ,in the strange ,,refraction ,,of ,,iceland ,,crystal by ,,christiaan ,,huygens ,rendered into ,english by ,,silvanus ,p4 ,,thompson ,university of ,chicago ,press ,,preface ,i wrote this ,treatise during my sojourn in ,france twelve years ago1 and ,i communicated it in the year #afgh to the learned persons who then composed the ,royal ,academy of ,science1 to the membership of which the ,king had done me the honour of calling1 me4 ,several of that body who are still alive will remember having been present when ,i read it1 and above the rest those amongst them who applied themselves particularly to the study of ,mathematics2 of whom ,i cannot cite more than the celebrated gentlemen ,cassini1 ,rmer1 and ,de la ,hire4 ,and1 although ,i have since corrected and changed some parts1 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 ,crystal1 and a novel observation on the refraction of ,rock ,crystal4 ,i have desired to relate these particulars to make known how long ,i have meditated the things which now ,i publish1 and not for the purpose of detracting from the merit of those who1 without having seen anything that ,i have written1 may be found to have treated of like matters3 as has in fact occurred to two eminent ,geometricians1 ,messieurs ,newton and ,leibnitz1 with respect to the ,problem of the figure of glasses for collecting rays when one of the surfaces is given4 ,one may ask why ,i have so long delayed to bring this work to the light4 ,the reason is that ,i wrote it rather carelessly in the ,language in which it appears1 with the intention of translating it into ,latin1 so doing in order to obtain greater attention to the thing4 ,after which ,i proposed to myself to give it out along with another ,treatise on ,dioptrics1 in which ,i explain the effects of ,telescopes and those things which belong more to that ,science4 ,but the pleasure of novelty being past1 ,i have put off from time to time the execution of this design1 and ,i know not when ,i shall ever come to an end if it1 being often turned aside either by business or by some new study4 ,considering which ,i have finally judged that it was better worth while to publish this writing1 such as it is1 than to let it run the risk1 by waiting longer1 of remaining lost4 ,there will be seen in it demonstrations of those kinds which do not produce as great a certitude as those of ,geometry1 and which even differ much therefrom1 since whereas the ,geometers prove their ,propositions by fixed and incontestable ,principles1 here the ,principles are verified by the conclusions to be drawn from them2 the nature of these things not allowing of this being done otherwise4 ,it is always possible to attain thereby to a degree of probability which very often is scarcely less than complete proof4 ,to wit1 when things which have been demonstrated by the ,principles that have been assumed correspond perfectly to the phenomena which experiment has brought under observation2 especially when there are a great number of them1 and further1 principally1 when one can imagine and foresee new phenomena which ought to follow from the hypotheses which one employs1 and when one finds that therein the fact corresponds to our prevision4 ,but if all these proofs of probability are met with in that which ,i propose to discuss1 as it seems to me they are1 this ought to be a very strong confirmation of the success of my inquiry2 and it must be ill if the facts are not pretty much as ,i represent them4 ,i would believe then that those who love to know the ,causes of things and who are able to admire the marvels of ,light1 will find some satisfaction in these various speculations regarding it1 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 them4 ,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 do1 since the subject must be far from being exhausted4 ,this appears from the passages which ,i have indicated where ,i leave certain difficulties without having resolved them1 and still more from matters which ,i have not touched at all1 such as ,luminous ,bodies of several sorts1 and all that concerns ,colours2 in which no one until now can boast of having succeeded4 ,finally1 there remains much more to be investigated touching the nature of ,light which ,i do not pretend to have disclosed1 and ,i shall owe much in return to him who shall be able to supplement that which is here lacking to me in knowledge4 ,the ,hague4 ,the #h ,january #afij4 ,,note ,,by ,,the ,,translator ,considering the great influence which this ,treatise has exercised in the development of the ,science of ,optics1 it seems strange that two centuries should have passed before an ,english edition of the work appeared4 ,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 followers4 ,the ,treatise on ,light of ,huygens has1 however1 withstood the test of time3 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 crystals1 and of the refraction of the atmosphere1 will excite the admiration of the student of ,optics4 ,it is true that his wave theory was far from the complete doctrine as subsequently developed by ,thomas ,young and ,augustin ,fresnel1 and belonged rather to geometrical than to physical ,optics4 ,if ,huygens had no conception of transverse vibrations1 of the principle of interference1 or of the existence of the ordered sequence of waves in trains1 he nevertheless attained to a remarkably clear understanding of the principles of wave-propagation2 and his exposition of the subject marks an epoch in the treatment of ,optical problems4 ,it has been needful in preparing this translation to exercise care lest one should import into the author's text ideas of subsequent date1 by using words that have come to imply modern conceptions4 ,hence the adoption of as literal a rendering as possible4 ,a few of the author's terms need explanation4 ,he uses the word refraction1 for example1 both for the phenomenon or process usually so denoted1 and for the result of that process3 thus the refracted ray he habitually terms the refraction of the incident ray4 ,when a wave-front1 or1 as he terms it1 a wave1 has passed from some initial position to a subsequent one1 he terms the wave-front in its subsequent position the continuation of the wave4 ,he also speaks of the envelope of a set of elementary waves1 formed by coalescence of those elementary wave-fronts1 as the termination of the wave2 and the elementary wave-fronts he terms particular waves4 ,owing to the circumstance that the ,french word rayon possesses the double signification of ray of light and radius of a circle1 he avoids its use in the latter sense and speaks always of the semi-diameter1 not of the radius4 ,his speculations as to the ether1 his suggestive views of the structure of crystalline bodies1 and his explanation of opacity1 slight as they are1 will possibly surprise the reader by their seeming modernness4 ,and none can read his investigation of the phenomena found in ,iceland spar without marvelling at his insight and sagacity4 ,s4,p4,t4 ,june1 #aiab4 ,,table ,,of ,,matters ,contained in this ,treatise ,,chapter ,i4 ,on ,rays ,propagated in ,straight ,lines4 ,that ,light is produced by a certain movement4 ,that no substance passes from the luminous object to the eyes4 ,that ,light spreads spherically1 almost as ,sound does4 ,whether ,light takes time to spread4 ,experience seeming to prove that it passes instantaneously4 ,experience proving that it takes time4 ,how much its speed is greater than that of ,sound4 ,in what the emission of ,light differs from that of ,sound4 ,that it is not the same medium which serves for ,light and ,sound4 ,how ,sound is propagated4 ,how ,light is propagated4 ,detailed ,remarks on the propagation of ,light4 ,why ,rays are propagated only in straight lines4 ,how ,light coming in different directions can cross itself4 ,,chapter ,,ii4 ,on ,reflexion4 ,demonstration of equality of angles of incidence and reflexion4 ,why the incident and reflected rays are in the same plane perpendicular to the reflecting surface4 ,that it is not needful for the reflecting surface to be perfectly flat to attain equality of the angles of incidence and reflexion4 ,,chapter ,,iii4 ,on ,refraction4 ,that bodies may be transparent without any substance passing through them4 ,proof that the ethereal matter passes through transparent bodies4 ,how this matter passing through can render them transparent4 ,that the most solid bodies in appearance are of a very loose texture4 ,that ,light spreads more slowly in water and in glass than in air4 ,third hypothesis to explain transparency1 and the retardation which ,light suffers4 ,on that which makes bodies opaque4 ,demonstration why ,refraction obeys the known proportion of ,sines4 ,why the incident and refracted ,rays produce one another reciprocally4 ,why ,reflexion within a triangular glass prism is suddenly augmented when the ,light can no longer penetrate4 ,that bodies which cause greater ,refraction also cause stronger ,reflexion4 ,demonstration of the ,theorem of ,mr4 ,fermat4 ,,chapter ,,iv4 ,on the ,refraction of the ,air4 ,that the emanations of ,light in the air are not spherical4 ,how consequently some objects appear higher than they are4 ,how the ,sun may appear on the ,horizon before he has risen4 ,that the rays of light become curved in the ,air of the ,atmosphere1 and what effects this produces4 ,,chapter ,v4 ,on the ,strange ,refraction of ,iceland ,crystal4 ,that this ,crystal grows also in other countries4 ,who first-wrote about it4 ,description of ,iceland ,crystal2 its substance1 shape1 and properties4 ,that it has two different ,refractions4 ,that the ray perpendicular to the surface suffers refraction1 and that some rays inclined to the surface pass without suffering refraction4 ,observation of the refractions in this ,crystal4 ,that there is a ,regular and an ,irregular ,refraction4 ,the way of measuring the two ,refractions of ,iceland ,crystal4 ,remarkable properties of the ,irregular ,refraction4 ,hypothesis to explain the double ,refraction4 ,that ,rock ,crystal has also a double ,refraction4 ,hypothesis of emanations of ,light1 within ,iceland ,crystal1 of spheroidal form1 for the ,irregular ,refraction4 ,how a perpendicular ray can suffer ,refraction4 ,how the position and form of the spheroidal emanations in this ,crystal can be defined4 ,explanation of the ,irregular ,refraction by these spheroidal emanations4 ,easy way to find the ,irregular ,refraction of each incident ray4 ,demonstration of the oblique ray which traverses the ,crystal without being refracted4 ,other irregularities of ,refraction explained4 ,that an object placed beneath the ,crystal appears double1 in two images of different heights4 ,why the apparent heights of one of the images change on changing the position of the eyes above the ,crystal4 ,of the different sections of this ,crystal which produce yet other refractions1 and confirm all this ,theory4 ,particular way of polishing the surfaces after it has been cut4 ,surprising phenomenon touching the rays which pass through two separated pieces2 the cause of which is not explained4 ,probable conjecture on the internal composition of ,iceland ,crystal1 and of what figure its particles are4 ,tests to confirm this conjecture4 ,calculations which have been supposed in this ,chapter4 ,,chapter ,,vi4 ,on the ,figures of transparent bodies which serve for ,refraction and for ,reflexion4 ,general and easy rule to find these ,figures4 ,invention of the ,ovals of ,mr4 ,des ,cartes for ,dioptrics4 ,how he was able to find these ,lines4 ,way of finding the surface of a glass for perfect refraction1 when the other surface is given4 ,remark on what happens to rays refracted at a spherical surface4 ,remark on the curved line which is formed by reflexion in a spherical concave mirror4 ,,chapter ,i ,,on ,,rays ,,propagated ,,in ,,straight ,,lines ,as happens in all the sciences in which ,geometry is applied to matter1 the demonstrations concerning ,optics are founded on truths drawn from experience4 ,such are that the rays of light are propagated in straight lines2 that the angles of reflexion and of incidence are equal2 and that in refraction the ray is bent according to the law of sines1 now so well known1 and which is no less certain than the preceding laws4 ,the majority of those who have written touching the various parts of ,optics have contented themselves with presuming these truths4 ,but some1 more inquiring1 have desired to investigate the origin and the causes1 considering these to be in themselves wonderful effects of ,nature4 ,in which they advanced some ingenious things1 but not however such that the most intelligent folk do not wish for better and more satisfactory explanations4 ,wherefore ,i here desire to propound what ,i have meditated on the subject1 so as to contribute as much as ,i can to the explanation of this department of ,natural ,science1 which1 not without reason1 is reputed to be one of its most difficult parts4 ,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 enveloped1 and to give us hope that they might be explained by intelligible reasoning4 ,but1 on the other hand ,i am astonished also that even here these have often been willing to offer1 as assured and demonstrative1 reasonings which were far from conclusive4 ,for ,i do not find that any one has yet given a probable explanation of the first and most notable phenomena of light1 namely why it is not propagated except in straight lines1 and how visible rays1 coming from an infinitude of diverse places1 cross one another without hindering one another in any way4 ,i shall therefore essay in this book1 to give1 in accordance with the principles accepted in the ,philosophy of the present day1 some clearer and more probable reasons1 firstly of these properties of light propagated rectilinearly2 secondly of light which is reflected on meeting other bodies4 ,then ,i shall explain the phenomena of those rays which are said to suffer refraction on passing through transparent bodies of different sorts2 and in this part ,i shall also explain the effects of the refraction of the air by the different densities of the ,atmosphere4 ,thereafter ,i shall examine the causes of the strange refraction of a certain kind of ,crystal which is brought from ,iceland4 ,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 ways4 ,from this it will be seen with what facility1 following our new ,theory1 we find not only the ,ellipses1 ,hyperbolas1 and other curves which ,mr4 ,des ,cartes has ingeniously invented for this purpose2 but also those which the surface of a glass lens ought to possess when its other surface is given as spherical or plane1 or of any other figure that may be4 ,it is inconceivable to doubt that light consists in the motion of some sort of matter4 ,for whether one considers its production1 one sees that here upon the ,earth it is chiefly engendered by fire and flame which contain without doubt bodies that are in rapid motion1 since they dissolve and melt many other bodies1 even the most solid2 or whether one considers its effects1 one sees that when light is collected1 as by concave mirrors1 it has the property of burning as a fire does1 that is to say it disunites the particles of bodies4 ,this is assuredly the mark of motion1 at least in the true ,philosophy1 in which one conceives the causes of all natural effects in terms of mechanical motions4 ,this1 in my opinion1 we must necessarily do1 or else renounce all hopes of ever comprehending anything in ,physics4 ,and as1 according to this ,philosophy1 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 eyes1 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 body4 ,further1 when one considers the extreme speed with which light spreads on every side1 and how1 when it comes from different regions1 even from those directly opposite1 the rays traverse one another without hindrance1 one may well understand that when we see a luminous object1 it cannot be by any transport of matter coming to us from this object1 in the way in which a shot or an arrow traverses the air2 for assuredly that would too greatly impugn these two properties of light1 especially the second of them4 ,it is then in some other way that light spreads2 and that which can lead us to comprehend it is the knowledge which we have of the spreading of ,sound in the air4 ,we know that by means of the air1 which is an invisible and impalpable body1 ,sound spreads around the spot where it has been produced1 by a movement which is passed on successively from one part of the air to another2 and that the spreading of this movement1 taking place equally rapidly on all sides1 ought to form spherical surfaces ever enlarging and which strike our ears4 ,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 two2 since1 as we have already seen1 it cannot be by the transport of a body which passes from one to the other4 ,if1 in addition1 light takes time for its passage--which we are now going to examine--it will follow that this movement1 impressed on the intervening matter1 is successive2 and consequently it spreads1 as ,sound does1 by spherical surfaces and waves3 for ,i call them waves from their resemblance to those which are seen to be formed in water when a stone is thrown into it1 and which present a successive spreading as circles1 though these arise from another cause1 and are only in a flat surface4 ,to see then whether the spreading of light takes time1 let us consider first whether there are any facts of experience which can convince us to the contrary4 ,as to those which can be made here on the ,earth1 by striking lights at great distances1 although they prove that light takes no sensible time to pass over these distances1 one may say with good reason that they are too small1 and that the only conclusion to be drawn from them is that the passage of light is extremely rapid4 ,mr4 ,des ,cartes1 who was of opinion that it is instantaneous1 founded his views1 not without reason1 upon a better basis of experience1 drawn from the ,eclipses of the ,moon2 which1 nevertheless1 as ,i shall show1 is not at all convincing4 ,i will set it forth1 in a way a little different from his1 in order to make the conclusion more comprehensible4 ,illustration ,let ,a be the place of the sun1 ,,bd a part of the orbit or annual path of the ,earth3 ,,abc a straight line which ,i suppose to meet the orbit of the ,moon1 which is represented by the circle ,,cd1 at ,c4 ,now if light requires time1 for example one hour1 to traverse the space which is between the ,earth and the ,moon1 it will follow that the ,earth having arrived at ,b1 the shadow which it casts1 or the interruption of the light1 will not yet have arrived at the point ,c1 but will only arrive there an hour after4 ,it will then be one hour after1 reckoning from the moment when the ,earth was at ,b1 that the ,moon1 arriving at ,c1 will be obscured3 but this obscuration or interruption of the light will not reach the ,earth till after another hour4 ,let us suppose that the ,earth in these two hours will have arrived at ,e4 ,the ,earth then1 being at ,e1 will see the ,eclipsed ,moon at ,c1 which it left an hour before1 and at the same time will see the sun at ,a4 ,for it being immovable1 as ,i suppose with ,copernicus1 and the light moving always in straight lines1 it must always appear where it is4 ,but one has always observed1 we are told1 that the eclipsed ,moon appears at the point of the ,ecliptic opposite to the ,sun2 and yet here it would appear in arrear of that point by an amount equal to the angle ,,gec1 the supplement of ,,aec4 ,this1 however1 is contrary to experience1 since the angle ,,gec would be very sensible1 and about #cc degrees4 ,now according to our computation1 which is given in the ,treatise on the causes of the phenomena of ,saturn1 the distance ,,ba between the ,earth and the ,sun is about twelve thousand diameters of the ,earth1 and hence four hundred times greater than ,,bc the distance of the ,moon1 which is #cj diameters4 ,then the angle ,,ecb will be nearly four hundred times greater than ,,bae1 which is five minutes2 namely1 the path which the earth travels in two hours along its orbit2 and thus the angle ,,bce will be nearly #cc degrees2 and likewise the angle ,,ceg1 which is greater by five minutes4 ,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 ,moon4 ,if one supposes that for this it requires only one minute of time1 then it is manifest that the angle ,,ceg will only be #cc minutes2 and if it requires only ten seconds of time1 the angle will be less than six minutes4 ,and then it will not be easy to perceive anything of it in observations of the ,eclipse2 nor1 consequently1 will it be permissible to deduce from it that the movement of light is instantaneous4 ,it is true that we are here supposing a strange velocity that would be a hundred thousand times greater than that of ,sound4 ,for ,sound1 according to what ,i have observed1 travels about #ahj ,toises in the time of one ,second1 or in about one beat of the pulse4 ,but this supposition ought not to seem to be an impossibility2 since it is not a question of the transport of a body with so great a speed1 but of a successive movement which is passed on from some bodies to others4 ,i have then made no difficulty1 in meditating on these things1 in supposing that the emanation of light is accomplished with time1 seeing that in this way all its phenomena can be explained1 and that in following the contrary opinion everything is incomprehensible4 ,for it has always seemed tome that even ,mr4 ,des ,cartes1 whose aim has been to treat all the subjects of ,physics intelligibly1 and who assuredly has succeeded in this better than any one before him1 has said nothing that is not full of difficulties1 or even inconceivable1 in dealing with ,light and its properties4 ,but that which ,i employed only as a hypothesis1 has recently received great seemingness as an established truth by the ingenious proof of ,mr4 ,rmer which ,i am going here to relate1 expecting him himself to give all that is needed for its confirmation4 ,it is founded as is the preceding argument upon celestial observations1 and proves not only that ,light takes time for its passage1 but also demonstrates how much time it takes1 and that its velocity is even at least six times greater than that which ,i have just stated4 ,for this he makes use of the ,eclipses suffered by the little planets which revolve around ,jupiter1 and which often enter his shadow3 and see what is his reasoning4 ,let ,a be the ,sun1 ,,bcde the annual orbit of the ,earth1 ,f ,jupiter1 ,,gn the orbit of the nearest of his ,satellites1 for it is this one which is more apt for this investigation than any of the other three1 because of the quickness of its revolution4 ,let ,g be this ,satellite entering into the shadow of ,jupiter1 ,h the same ,satellite emerging from the shadow4 ,illustration ,let it be then supposed1 the ,earth being at ,b some time before the last quadrature1 that one has seen the said ,satellite emerge from the shadow2 it must needs be1 if the ,earth remains at the same place1 that1 after #db-a_/b hours1 one would again see a similar emergence1 because that is the time in which it makes the round of its orbit1 and when it would come again into opposition to the ,sun4 ,and if the ,earth1 for instance1 were to remain always at ,b during #cj revolutions of this ,satellite1 one would see it again emerge from the shadow after #cj times #db-a_/b hours4 ,but the ,earth having been carried along during this time to ,c1 increasing thus its distance from ,jupiter1 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 ,b1 and that there must be added to this time of #cj times #db-a_/b hours that which the ,light has required to traverse the space ,,mc1 the difference of the spaces ,,ch1 ,,bh4 ,similarly at the other quadrature when the earth has come to ,e from ,d while approaching toward ,jupiter1 the immersions of the ,satellite ought to be observed at ,e earlier than they would have been seen if the ,earth had remained at ,d4 ,now in quantities of observations of these ,eclipses1 made during ten consecutive years1 these differences have been found to be very considerable1 such as ten minutes and more2 and from them it has been concluded that in order to traverse the whole diameter of the annual orbit ,,kl1 which is double the distance from here to the sun1 ,light requires about #bb minutes of time4 ,the movement of ,jupiter in his orbit while the ,earth passed from ,b to ,c1 or from ,d to ,e1 is included in this calculation2 and this makes it evident that one cannot attribute the retardation of these illuminations or the anticipation of the eclipses1 either to any irregularity occurring in the movement of the little planet or to its eccentricity4 ,if one considers the vast size of the diameter ,,kl1 which according to me is some #bd thousand diameters of the ,earth1 one will acknowledge the extreme velocity of ,light4 ,for1 supposing that ,,kl is no more than #bb thousand of these diameters1 it appears that being traversed in #bb minutes this makes the speed a thousand diameters in one minute1 that is #af-b_/c diameters in one second or in one beat of the pulse1 which makes more than #aa hundred times a hundred thousand toises2 since the diameter of the ,earth contains #b1hfe leagues1 reckoned at #be to the degree1 and each each league is #b1bhb ,toises1 according to the exact measurement which ,mr4 ,picard made by order of the ,king in #affi4 ,but ,sound1 as ,i have said above1 only travels #ahj toises in the same time of one second3 hence the velocity of ,light is more than six hundred thousand times greater than that of ,sound4 ,this1 however1 is quite another thing from being instantaneous1 since there is all the difference between a finite thing and an infinite4 ,now the successive movement of ,light being confirmed in this way1 it follows1 as ,i have said1 that it spreads by spherical waves1 like the movement of ,sound4 ,but if the one resembles the other in this respect1 they differ in many other things2 to wit1 in the first production of the movement which causes them2 in the matter in which the movement spreads2 and in the manner in which it is propagated4 ,as to that which occurs in the production of ,sound1 one knows that it is occasioned by the agitation undergone by an entire body1 or by a considerable part of one1 which shakes all the contiguous air4 ,but the movement of the ,light must originate as from each point of the luminous object1 else we should not be able to perceive all the different parts of that object1 as will be more evident in that which follows4 ,and ,i do not believe that this movement can be better explained than by supposing that all those of the luminous bodies which are liquid1 such as flames1 and apparently the sun and the stars1 are composed of particles which float in a much more subtle medium which agitates them with great rapidity1 and makes them strike against the particles of the ether which surrounds them1 and which are much smaller than they4 ,but ,i hold also that in luminous solids such as charcoal or metal made red hot in the fire1 this same movement is caused by the violent agitation of the particles of the metal or of the wood2 those of them which are on the surface striking similarly against the ethereal matter4 ,the agitation1 moreover1 of the particles which engender the light ought to be much more prompt and more rapid than is that of the bodies which cause sound1 since we do not see that the tremors of a body which is giving out a sound are capable of giving rise to ,light1 even as the movement of the hand in the air is not capable of producing ,sound4 ,now if one examines what this matter may be in which the movement coming from the luminous body is propagated1 which ,i call ,ethereal matter1 one will see that it is not the same that serves for the propagation of ,sound4 ,for one finds that the latter is really that which we feel and which we breathe1 and which being removed from any place still leaves there the other kind of matter that serves to convey ,light4 ,this may be proved by shutting up a sounding body in a glass vessel from which the air is withdrawn by the machine which ,mr4 ,boyle has given us1 and with which he has performed so many beautiful experiments4 ,but in doing this of which ,i speak1 care must be taken to place the sounding body on cotton or on feathers1 in such a way that it cannot communicate its tremors either to the glass vessel which encloses it1 or to the machine2 a precaution which has hitherto been neglected4 ,for then after having exhausted all the air one hears no ,sound from the metal1 though it is struck4 ,one sees here not only that our air1 which does not penetrate through glass1 is the matter by which ,sound spreads2 but also that it is not the same air but another kind of matter in which ,light spreads2 since if the air is removed from the vessel the ,light does not cease to traverse it as before4 ,and this last point is demonstrated even more clearly by the celebrated experiment of ,torricelli1 in which the tube of glass from which the quicksilver has withdrawn itself1 remaining void of air1 transmits ,light just the same as when air is in it4 ,for this proves that a matter different from air exists in this tube1 and that this matter must have penetrated the glass or the quicksilver1 either one or the other1 though they are both impenetrable to the air4 ,and when1 in the same experiment1 one makes the vacuum after putting a little water above the quicksilver1 one concludes equally that the said matter passes through glass or water1 or through both4 ,as regards the different modes in which ,i have said the movements of ,sound and of ,light are communicated1 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 occupies4 ,and in proportion as it is compressed the more does it exert an effort to regain its volume2 for this property along with its penetrability1 which remains notwithstanding its compression1 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 parts4 ,so that the cause of the spreading of ,sound is the effort which these little bodies make in collisions with one another1 to regain freedom when they are a little more squeezed together in the circuit of these waves than elsewhere4 ,but the extreme velocity of ,light1 and other properties which it has1 cannot admit of such a propagation of motion1 and ,i am about to show here the way in which ,i conceive it must occur4 ,for this1 it is needful to explain the property which hard bodies must possess to transmit movement from one to another4 ,when one takes a number of spheres of equal size1 made of some very hard substance1 and arranges them in a straight line1 so that they touch one another1 one finds1 on striking with a similar sphere against the first of these spheres1 that the motion passes as in an instant to the last of them1 which separates itself from the row1 without one's being able to perceive that the others have been stirred4 ,and even that one which was used to strike remains motionless with them4 ,whence one sees that the movement passes with an extreme velocity which is the greater1 the greater the hardness of the substance of the spheres4 ,but it is still certain that this progression of motion is not instantaneous1 but successive1 and therefore must take time4 ,for if the movement1 or the disposition to movement1 if you will have it so1 did not pass successively through all these spheres1 they would all acquire the movement at the same time1 and hence would all advance together2 which does not happen4 ,for the last one leaves the whole row and acquires the speed of the one which was pushed4 ,moreover there are experiments which demonstrate that all the bodies which we reckon of the hardest kind1 such as quenched steel1 glass1 and agate1 act as springs and bend somehow1 not only when extended as rods but also when they are in the form of spheres or of other shapes4 ,that is to say they yield a little in themselves at the place where they are struck1 and immediately regain their former figure4 ,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 surface1 slightly soiled with breath or in some other way1 there remained round marks1 of smaller or larger size according as the blow had been weak or strong4 ,this makes it evident that these substances yield where they meet1 and spring back3 and for this time must be required4 ,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 choose4 ,it is not necessary to examine here the causes of this hardness1 or of that springiness1 the consideration of which would lead us too far from our subject4 ,i will say1 however1 in passing that we may conceive that the particles of the ether1 notwithstanding their smallness1 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 possible4 ,this accords with the explanation which ,mr4 ,des ,cartes gives for the spring1 though ,i do not1 like him1 suppose the pores to be in the form of round hollow canals4 ,and it must not be thought that in this there is anything absurd or impossible1 it being on the contrary quite credible that it is this infinite series of different sizes of corpuscles1 having different degrees of velocity1 of which ,nature makes use to produce so many marvellous effects4 ,but though we shall ignore the true cause of springiness we still see that there are many bodies which possess this property2 and thus there is nothing strange in supposing that it exists also in little invisible bodies like the particles of the ,ether4 ,also if one wishes to seek for any other way in which the movement of ,light is successively communicated1 one will find none which agrees better1 with uniform progression1 as seems to be necessary1 than the property of springiness2 because if this movement should grow slower in proportion as it is shared over a greater quantity of matter1 in moving away from the source of the light1 it could not conserve this great velocity over great distances4 ,but by supposing springiness in the ethereal matter1 its particles will have the property of equally rapid restitution whether they are pushed strongly or feebly2 and thus the propagation of ,light will always go on with an equal velocity4 ,illustration ,and it must be known that although the particles of the ether are not ranged thus in straight lines1 as in our row of spheres1 but confusedly1 so that one of them touches several others1 this does not hinder them from transmitting their movement and from spreading it always forward4 ,as to this it is to be remarked that there is a law of motion serving for this propagation1 and verifiable by experiment4 ,it is that when a sphere1 such as ,a here1 touches several other similar spheres ,,ccc1 if it is struck by another sphere ,b in such a way as to exert an impulse against all the spheres ,,ccc which touch it1 it transmits to them the whole of its movement1 and remains after that motionless like the sphere ,b4 ,and without supposing that the ethereal particles are of spherical form 7for ,i see indeed no need to suppose them so7 one may well understand that this property of communicating an impulse does not fail to contribute to the aforesaid propagation of movement4 ,equality of size seems to be more necessary1 because otherwise there ought to be some reflexion of movement backwards when it passes from a smaller particle to a larger one1 according to the ,laws of ,percussion which ,i published some years ago4 ,however1 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 powerful2 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 light1 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 ,stars4 ,i have then shown in what manner one may conceive ,light to spread successively1 by spherical waves1 and how it is possible that this spreading is accomplished with as great a velocity as that which experiments and celestial observations demand4 ,whence it may be further remarked that although the particles are supposed to be in continual movement 7for there are many reasons for this7 the successive propagation of the waves cannot be hindered by this2 because the propagation consists nowise in the transport of those particles but merely in a small agitation which they cannot help communicating to those surrounding1 notwithstanding any movement which may act on them causing them to be changing positions amongst themselves4 ,but we must consider still more particularly the origin of these waves1 and the manner in which they spread4 ,and1 first1 it follows from what has been said on the production of ,light1 that each little region of a luminous body1 such as the ,sun1 a candle1 or a burning coal1 generates its own waves of which that region is the centre4 ,thus in the flame of a candle1 having distinguished the points ,a1 ,b1 ,c1 concentric circles described about each of these points represent the waves which come from them4 ,and one must imagine the same about every point of the surface and of the part within the flame4 ,illustration ,but as the percussions at the centres of these waves possess no regular succession1 it must not be supposed that the waves themselves follow one another at equal distances3 and if the distances marked in the figure appear to be such1 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 centre4 ,after all1 this prodigious quantity of waves which traverse one another without confusion and without effacing one another must not be deemed inconceivable2 it being certain that one and the same particle of matter can serve for many waves coming from different sides or even from contrary directions1 not only if it is struck by blows which follow one another closely but even for those which act on it at the same instant4 ,it can do so because the spreading of the movement is successive4 ,this may be proved by the row of equal spheres of hard matter1 spoken of above4 ,if against this row there are pushed from two opposite sides at the same time two similar spheres ,a and ,d1 one will see each of them rebound with the same velocity which it had in striking1 yet the whole row will remain in its place1 although the movement has passed along its whole length twice over4 ,and if these contrary movements happen to meet one another at the middle sphere1 ,b1 or at some other such as ,c1 that sphere will yield and act as a spring at both sides1 and so will serve at the same instant to transmit these two movements4 ,illustration ,but what may at first appear full strange and even incredible is that the undulations produced by such small movements and corpuscles1 should spread to such immense distances2 as for example from the ,sun or from the ,stars to us4 ,for the force of these waves must grow feeble in proportion as they move away from their origin1 so that the action of each one in particular will without doubt become incapable of making itself felt to our sight4 ,but one will cease to be astonished by considering how at a great distance from the luminous body an infinitude of waves1 though they have issued from different points of this body1 unite together in such a way that they sensibly compose one single wave only1 which1 consequently1 ought to have enough force to make itself felt4 ,thus this infinite number of waves which originate at the same instant from all points of a fixed star1 big it may be as the ,sun1 make practically only one single wave which may well have force enough to produce an impression on our eyes4 ,moreover from each luminous point there may come many thousands of waves in the smallest imaginable time1 by the frequent percussion of the corpuscles which strike the ,ether at these points3 which further contributes to rendering their action more sensible4 ,illustration ,there is the further consideration in the emanation of these waves1 that each particle of matter in which a wave spreads1 ought not to communicate its motion only to the next particle which is in the straight line drawn from the luminous point1 but that it also imparts some of it necessarily to all the others which touch it and which oppose themselves to its movement4 ,so it arises that around each particle there is made a wave of which that particle is the centre4 ,thus if ,,dcf is a wave emanating from the luminous point ,a1 which is its centre1 the particle ,b1 one of those comprised within the sphere ,,dcf1 will have made its particular or partial wave ,,kcl1 which will touch the wave ,,dcf at ,c at the same moment that the principal wave emanating from the point ,a has arrived at ,,dcf2 and it is clear that it will be only the region ,c of the wave ,,kcl which will touch the wave ,,dcf1 to wit1 that which is in the straight line drawn through ,,ab4 ,similarly the other particles of the sphere ,,dcf1 such as bb1 dd1 etc41 will each make its own wave4 ,but each of these waves can be infinitely feeble only as compared with the wave ,,dcf1 to the composition of which all the others contribute by the part of their surface which is most distant from the centre ,a4 ,one sees1 in addition1 that the wave ,,dcf is determined by the distance attained in a certain space of time by the movement which started from the point ,a2 there being no movement beyond this wave1 though there will be in the space which it encloses1 namely in parts of the particular waves1 those parts which do not touch the sphere ,,dcf4 ,and all this ought not to seem fraught with too much minuteness or subtlety1 since we shall see in the sequel that all the properties of ,light1 and everything pertaining to its reflexion and its refraction1 can be explained in principle by this means4 ,this is a matter which has been quite unknown to those who hitherto have begun to consider the waves of light1 amongst whom are ,mr4 ,hooke in his ,micrographia1 and ,father ,pardies1 who1 in a treatise of which he let me see a portion1 and which he was unable to complete as he died shortly afterward1 had undertaken to prove by these waves the effects of reflexion and refraction4 ,but the chief foundation1 which consists in the remark ,i have just made1 was lacking in his demonstrations2 and for the rest he had opinions very different from mine1 as may be will appear some day if his writing has been preserved4 ,to come to the properties of ,light4 ,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 point4 ,thus the portion ,,bg of the wave1 having the luminous point ,a as its centre1 will spread into the arc ,,ce bounded by the straight lines ,,abc1 ,,age4 ,for although the particular waves produced by the particles comprised within the space ,,cae spread also outside this space1 they yet do not concur at the same instant to compose a wave which terminates the movement1 as they do precisely at the circumference ,,ce1 which is their common tangent4 ,and hence one sees the reason why light1 at least if its rays are not reflected or broken1 spreads only by straight lines1 so that it illuminates no object except when the path from its source to that object is open along such lines4 ,for if1 for example1 there were an opening ,,bg1 limited by opaque bodies ,,bh1 ,,gi1 the wave of light which issues from the point ,a will always be terminated by the straight lines ,,ac1 ,,ae1 as has just been shown2 the parts of the partial waves which spread outside the space ,,ace being too feeble to produce light there4 ,now1 however small we make the opening ,,bg1 there is always the same reason causing the light there to pass between straight lines2 since this opening is always large enough to contain a great number of particles of the ethereal matter1 which are of an inconceivable smallness2 so that it appears that each little portion of the wave necessarily advances following the straight line which comes from the luminous point4 ,thus then we may take the rays of light as if they were straight lines4 ,it appears1 moreover1 by what has been remarked touching the feebleness of the particular waves1 that it is not needful that all the particles of the ,ether should be equal amongst themselves1 though equality is more apt for the propagation of the movement4 ,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 movement2 but it will thereby merely generate backwards towards the luminous point some partial waves incapable of causing light1 and not a wave compounded of many as ,,ce was4 ,another property of waves of light1 and one of the most marvellous1 is that when some of them come from different or even from opposing sides1 they produce their effect across one another without any hindrance4 ,whence also it comes about that a number of spectators may view different objects at the same time through the same opening1 and that two persons can at the same time see one another's eyes4 ,now according to the explanation which has been given of the action of light1 how the waves do not destroy nor interrupt one another when they cross one another1 these effects which ,i have just mentioned are easily conceived4 ,but in my judgement they are not at all easy to explain according to the views of ,mr4 ,des ,cartes1 who makes ,light to consist in a continuous pressure merely tending to movement4 ,for this pressure not being able to act from two opposite sides at the same time1 against bodies which have no inclination to approach one another1 it is impossible so to understand what ,i have been saying about two persons mutually seeing one another's eyes1 or how two torches can illuminate one another4 ,,chapter ,,ii ,,on ,,reflexion ,having explained the effects of waves of light which spread in a homogeneous matter1 we will examine next that which happens to them on encountering other bodies4 ,we will first make evident how the ,reflexion of light is explained by these same waves1 and why it preserves equality of angles4 ,let there be a surface ,,ab2 plane and polished1 of some metal1 glass1 or other body1 which at first ,i will consider as perfectly uniform 7reserving to myself to deal at the end of this demonstration with the inequalities from which it cannot be exempt71 and let a line ,,ac1 inclined to ,,ad1 represent a portion of a wave of light1 the centre of which is so distant that this portion ,,ac may be considered as a straight line2 for ,i consider all this as in one plane1 imagining to myself that the plane in which this figure is1 cuts the sphere of the wave through its centre and intersects the plane ,,ab at right angles4 ,this explanation will suffice once for all4 ,illustration ,the piece ,c of the wave ,,ac1 will in a certain space of time advance as far as the plane ,,ab at ,b1 following the straight line ,,cb1 which may be supposed to come from the luminous centre1 and which in consequence is perpendicular to ,,ac4 ,now in this same space of time the portion ,a of the same wave1 which has been hindered from communicating its movement beyond the plane ,,ab1 or at least partly so1 ought to have continued its movement in the matter which is above this plane1 and this along a distance equal to ,,cb1 making its own partial spherical wave1 according to what has been said above4 ,which wave is here represented by the circumference ,,snr1 the centre of which is ,a1 and its semi-diameter ,,an equal to ,,cb4 ,if one considers further the other pieces ,h of the wave ,,ac1 it appears that they will not only have reached the surface ,,ab by straight lines ,,hk parallel to ,,cb1 but that in addition they will have generated in the transparent air1 from the centres ,k1 ,k1 ,k1 particular spherical waves1 represented here by circumferences the semi-diameters of which are equal to ,,km1 that is to say to the continuations of ,,hk as far as the line ,,bg parallel to ,,ac4 ,but all these circumferences have as a common tangent the straight line ,,bn1 namely the same which is drawn from ,b as a tangent to the first of the circles1 of which ,a is the centre1 and ,,an the semi-diameter equal to ,,bc1 as is easy to see4 ,it is then the line ,,bn 7comprised between ,b and the point ,n where the perpendicular from the point ,a falls7 which is as it were formed by all these circumferences1 and which terminates the movement which is made by the reflexion of the wave ,,ac2 and it is also the place where the movement occurs in much greater quantity than anywhere else4 ,wherefore1 according to that which has been explained1 ,,bn is the propagation of the wave ,,ac at the moment when the piece ,c of it has arrived at ,b4 ,for there is no other line which like ,,bn is a common tangent to all the aforesaid circles1 except ,,bg below the plane ,,ab2 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 plane4 ,and if one wishes to see how the wave ,,ac has come successively to ,,bn1 one has only to draw in the same figure the straight lines ,,ko parallel to ,,bn1 and the straight lines ,,kl parallel to ,,ac4 ,thus one will see that the straight wave ,,ac has become broken up into all the ,,okl parts successively1 and that it has become straight again at ,,nb4 ,now it is apparent here that the angle of reflexion is made equal to the angle of incidence4 ,for the triangles ,,acb1 ,,bna being rectangular and having the side ,,ab common1 and the side ,,cb equal to ,,na1 it follows that the angles opposite to these sides will be equal1 and therefore also the angles ,,cba1 ,,nab4 ,but as ,,cb1 perpendicular to ,,ca1 marks the direction of the incident ray1 so ,,an1 perpendicular to the wave ,,bn1 marks the direction of the reflected ray2 hence these rays are equally inclined to the plane ,,ab4 ,but in considering the preceding demonstration1 one might aver that it is indeed true that ,,bn is the common tangent of the circular waves in the plane of this figure1 but that these waves1 being in truth spherical1 have still an infinitude of similar tangents1 namely all the straight lines which are drawn from the point ,b in the surface generated by the straight line ,,bn about the axis ,,ba4 ,it remains1 therefore1 to demonstrate that there is no difficulty herein3 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 plane4 ,i say then that the wave ,,ac1 being regarded only as a line1 produces no light4 ,for a visible ray of light1 however narrow it may be1 has always some width1 and consequently it is necessary1 in representing the wave whose progression constitutes the ray1 to put instead of a line ,,ac some plane figure such as the circle ,,hc in the following figure1 by supposing1 as we have done1 the luminous point to be infinitely distant4 ,now it is easy to see1 following the preceding demonstration1 that each small piece of this wave ,,hc having arrived at the plane ,,ab1 and there generating each one its particular wave1 these will all have1 when ,c arrives at ,b1 a common plane which will touch them1 namely a circle ,,bn similar to ,,ch2 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 ,,ab4 ,illustration ,one sees also that the said spheres of the partial waves cannot have any common tangent plane other than the circle ,,bn2 so that it will be this plane where there will be more reflected movement than anywhere else1 and which will therefore carry on the light in continuance from the wave ,,ch4 ,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 ,,ab1 or at least not wholly4 ,whence it is to be remarked that though the movement of the ethereal matter might communicate itself partly to that of the reflecting body1 this could in nothing alter the velocity of progression of the waves1 on which the angle of reflexion depends4 ,for a slight percussion ought to generate waves as rapid as strong percussion in the same matter4 ,this comes about from the property of bodies which act as springs1 of which we have spoken above2 namely that whether compressed little or much they recoil in equal times4 ,equally so in every reflexion of the light1 against whatever body it may be1 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 light4 ,and experiment shows that in fact there is no polished body the reflexion of which does not follow this rule4 ,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 plane1 as has been supposed by all those who have tried to explain the effects of reflexion2 but only an evenness such as may be attained by the particles of the matter of the reflecting body being set near to one another2 which particles are larger than those of the ethereal matter1 as will appear by what we shall say in treating of the transparency and opacity of bodies4 ,for the surface consisting thus of particles put together1 and the ethereal particles being above1 and smaller1 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 wall1 of which writers have always made use4 ,in our way1 on the other hand1 the thing is explained without difficulty4 ,for the smallness of the particles of quicksilver1 for example1 being such that one must conceive millions of them1 in the smallest visible surface proposed1 arranged like a heap of grains of sand which has been flattened as much as it is capable of being1 this surface then becomes for our purpose as even as a polished glass is3 and1 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 reflexion1 of which we have spoken1 are almost in one uniform plane1 and that thus the common tangent can fit to them as perfectly as is requisite for the production of light4 ,and this alone is requisite1 in our method of demonstration1 to cause equality of the said angles without the remainder of the movement reflected from all parts being able to produce any contrary effect4 ,,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 bodies1 we will explain transparency and the phenomena of refraction by waves which spread within and across diaphanous bodies1 both solids1 such as glass1 and liquids1 such as water1 oils1 etc4 ,but in order that it may not seem strange to suppose this passage of waves in the interior of these bodies1 ,i will first show that one may conceive it possible in more than one mode4 ,first1 then1 if the ethereal matter cannot penetrate transparent bodies at all1 their own particles would be able to communicate successively the movement of the waves1 the same as do those of the ,ether1 supposing that1 like those1 they are of a nature to act as a spring4 ,and this is easy to conceive as regards water and other transparent liquids1 they being composed of detached particles4 ,but it may seem more difficult as regards glass and other transparent and hard bodies1 because their solidity does not seem to permit them to receive movement except in their whole mass at the same time4 ,this1 however1 is not necessary because this solidity is not such as it appears to us1 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 matter1 and by the irregularity of their shapes4 ,for primarily their rarity is shown by the facility with which there passes through them the matter of the vortices of the magnet1 and that which causes gravity4 ,further1 one cannot say that these bodies are of a texture similar to that of a sponge or of light bread1 because the heat of the fire makes them flow and thereby changes the situation of the particles amongst themselves4 ,it remains then that they are1 as has been said1 assemblages of particles which touch one another without constituting a continuous solid4 ,this being so1 the movement which these particles receive to carry on the waves of light1 being merely communicated from some of them to others1 without their going for that purpose out of their places or without derangement1 it may very well produce its effect without prejudicing in any way the apparent solidity of the compound4 ,by pressure from without1 of which ,i have spoken1 must not be understood that of the air1 which would not be sufficient1 but that of some other more subtle matter1 a pressure which ,i chanced upon by experiment long ago1 namely in the case of water freed from air1 which remains suspended in a tube open at its lower end1 notwithstanding that the air has been removed from the vessel in which this tube is enclosed4 ,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 it1 or that it should find pores in which to insinuate itself4 ,but the truth is that this matter not only passes through solids1 but does so even with great facility2 of which the experiment of ,torricelli1 above cited1 is already a proof4 ,because on the quicksilver and the water quitting the upper part of the glass tube1 it appears that it is immediately filled with ethereal matter1 since light passes across it4 ,but here is another argument which proves this ready penetrability1 not only in transparent bodies but also in all others4 ,when light passes across a hollow sphere of glass1 closed on all sides1 it is certain that it is full of ethereal matter1 as much as the spaces outside the sphere4 ,and this ethereal matter1 as has been shown above1 consists of particles which just touch one another4 ,if then it were enclosed in the sphere in such a way that it could not get out through the pores of the glass1 it would be obliged to follow the movement of the sphere when one changes its place3 and it would require consequently almost the same force to impress a certain velocity on this sphere1 when placed on a horizontal plane1 as if it were full of water or perhaps of quicksilver3 because every body resists the velocity of the motion which one would give to it1 in proportion to the quantity of matter which it contains1 and which is obliged to follow this motion4 ,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 made4 ,then it must be that the ethereal matter which is inside is not shut up1 but flows through it with very great freedom4 ,we shall demonstrate hereafter that by this process the same penetrability may be inferred also as relating to opaque bodies4 ,the second mode then of explaining transparency1 and one which appears more probably true1 is by saying that the waves of light are carried on in the ethereal matter1 which continuously occupies the interstices or pores of transparent bodies4 ,for since it passes through them continuously and freely1 it follows that they are always full of it4 ,and one may even show that these interstices occupy much more space than the coherent particles which constitute the bodies4 ,for if what we have just said is true3 that force is required to impress a certain horizontal velocity on bodies in proportion as they contain coherent matter2 and if the proportion of this force follows the law of weights1 as is confirmed by experiment1 then the quantity of the constituent matter of bodies also follows the proportion of their weights4 ,now we see that water weighs only one fourteenth part as much as an equal portion of quicksilver3 therefore the matter of the water does not occupy the fourteenth part of the space which its mass obtains4 ,it must even occupy much less of it1 since quicksilver is less heavy than gold1 and the matter of gold is by no means dense1 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 it4 ,but it may be objected here that if water is a body of so great rarity1 and if its particles occupy so small a portion of the space of its apparent bulk1 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 employ1 preserving even its entire liquidity while subjected to this pressure4 ,this is no small difficulty4 ,it may1 however1 be resolved by saying that the very violent and rapid motion of the subtle matter which renders water liquid1 by agitating the particles of which it is composed1 maintains this liquidity in spite of the pressure which hitherto any one has been minded to apply to it4 ,the rarity of transparent bodies being then such as we have said1 one easily conceives that the waves might be carried on in the ethereal matter which fills the interstices of the particles4 ,and1 moreover1 one may believe that the progression of these waves ought to be a little slower in the interior of bodies1 by reason of the small detours which the same particles cause4 ,in which different velocity of light ,i shall show the cause of refraction to consist4 ,before doing so1 ,i will indicate the third and last mode in which transparency may be conceived2 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 bodies1 and in the particles which compose them1 so that the movement passes from one to the other4 ,and it will be seen hereafter that this hypothesis serves excellently to explain the double refraction of certain transparent bodies4 ,should it be objected that if the particles of the ether are smaller than those of transparent bodies 7since they pass through their intervals71 it would follow that they can communicate to them but little of their movement1 it may be replied that the particles of these bodies are in turn composed of still smaller particles1 and so it will be these secondary particles which will receive the movement from those of the ether4 ,furthermore1 if the particles of transparent bodies have a recoil a little less prompt than that of the ethereal particles1 which nothing hinders us from supposing1 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 matter4 ,all this ,i have found as most probable for the mode in which the waves of light pass across transparent bodies4 ,to which it must further be added in what respect these bodies differ from those which are opaque2 and the more so since it might seem because of the easy penetration of bodies by the ethereal matter1 of which mention has been made1 that there would not be any body that was not transparent4 ,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 matter1 one might also prove that the same penetrability obtains for metals and for every other sort of body4 ,for this sphere being for example of silver1 it is certain that it contains some of the ethereal matter which serves for light1 since this was there as well as in the air when the opening of the sphere was closed4 ,yet1 being closed and placed upon a horizontal plane1 it resists the movement which one wishes to give to it1 merely according to the quantity of silver of which it is made2 so that one must conclude1 as above1 that the ethereal matter which is enclosed does not follow the movement of the sphere2 and that therefore silver1 as well as glass1 is very easily penetrated by this matter4 ,some of it is therefore present continuously and in quantities between the particles of silver and of all other opaque bodies3 and since it serves for the propagation of light it would seem that these bodies ought also to be transparent1 which however is not the case4 ,whence then1 one will say1 does their opacity come8 ,is it because the particles which compose them are soft2 that is to say1 these particles being composed of others that are smaller1 are they capable of changing their figure on receiving the pressure of the ethereal particles1 the motion of which they thereby damp1 and so hinder the continuance of the waves of light8 ,that cannot be3 for if the particles of the metals are soft1 how is it that polished silver and mercury reflect light so strongly8 ,what ,i find to be most probable herein1 is to say that metallic bodies1 which are almost the only really opaque ones1 have mixed amongst their hard particles some soft ones2 so that some serve to cause reflexion and the others to hinder transparency2 while1 on the other hand1 transparent bodies contain only hard particles which have the faculty of recoil1 and serve together with those of the ethereal matter for the propagation of the waves of light1 as has been said4 ,illustration ,let us pass now to the explanation of the effects of ,refraction1 assuming1 as we have done1 the passage of waves of light through transparent bodies1 and the diminution of velocity which these same waves suffer in them4 ,the chief property of ,refraction is that a ray of light1 such as ,,ab1 being in the air1 and falling obliquely upon the polished surface of a transparent body1 such as ,,fg1 is broken at the point of incidence ,b1 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 air4 ,and the measure of these angles is found by describing1 about the point ,b1 a circle which cuts the radii ,,ab1 ,,bc4 ,for the perpendiculars ,,ad1 ,,ce1 let fall from the points of intersection upon the straight line ,,de1 which are called the ,sines of the angles ,,abd1 ,,cbe1 have a certain ratio between themselves2 which ratio is always the same for all inclinations of the incident ray1 at least for a given transparent body4 ,this ratio is1 in glass1 very nearly as #c to #b2 and in water very nearly as #d to #c2 and is likewise different in other diaphanous bodies4 ,another property1 similar to this1 is that the refractions are reciprocal between the rays entering into a transparent body and those which are leaving it4 ,that is to say that if the ray ,,ab in entering the transparent body is refracted into ,,bc1 then likewise ,,cb being taken as a ray in the interior of this body will be refracted1 on passing out1 into ,,ba4 ,illustration ,to explain then the reasons of these phenomena according to our principles1 let ,,ab be the straight line which represents a plane surface bounding the transparent substances which lie towards ,c and towards ,n4 ,when ,i say plane1 that does not signify a perfect evenness1 but such as has been understood in treating of reflexion1 and for the same reason4 ,let the line ,,ac represent a portion of a wave of light1 the centre of which is supposed so distant that this portion may be considered as a straight line4 ,the piece ,c1 then1 of the wave ,,ac1 will in a certain space of time have advanced as far as the plane ,,ab following the straight line ,,cb1 which may be imagined as coming from the luminous centre1 and which consequently will cut ,,ac at right angles4 ,now in the same time the piece ,a would have come to ,g along the straight line ,,ag1 equal and parallel to ,,cb2 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 ,ether4 ,but let us suppose that it transmits this movement less quickly1 by one-third1 for instance4 ,movement will then be spread from the point ,a1 in the matter of the transparent body through a distance equal to two-thirds of ,,cb1 making its own particular spherical wave according to what has been said before4 ,this wave is then represented by the circumference ,,snr1 the centre of which is ,a1 and its semi-diameter equal to two-thirds of ,,cb4 ,then if one considers in order the other pieces ,h of the wave ,,ac1 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 ,,cb1 but that1 in addition1 they will have generated in the diaphanous substance from the centres ,k1 partial waves1 represented here by circumferences the semi-diameters of which are equal to two-thirds of the lines ,,km1 that is to say1 to two-thirds of the prolongations of ,,hk down to the straight line ,,bg2 for these semi-diameters would have been equal to entire lengths of ,,km if the two transparent substances had been of the same penetrability4 ,now all these circumferences have for a common tangent the straight line ,,bn2 namely the same line which is drawn as a tangent from the point ,b to the circumference ,,snr which we considered first4 ,for it is easy to see that all the other circumferences will touch the same ,,bn1 from ,b up to the point of contact ,n1 which is the same point where ,,an falls perpendicularly on ,,bn4 ,it is then ,,bn1 which is formed by small arcs of these circumferences1 which terminates the movement that the wave ,,ac has communicated within the transparent body1 and where this movement occurs in much greater amount than anywhere else4 ,and for that reason this line1 in accordance with what has been said more than once1 is the propagation of the wave ,,ac at the moment when its piece ,c has reached ,b4 ,for there is no other line below the plane ,,ab which is1 like ,,bn1 a common tangent to all these partial waves4 ,and if one would know how the wave ,,ac has come progressively to ,,bn1 it is necessary only to draw in the same figure the straight lines ,,ko parallel to ,,bn1 and all the lines ,,kl parallel to ,,ac4 ,thus one will see that the wave ,,ca1 from being a straight line1 has become broken in all the positions ,,lko successively1 and that it has again become a straight line at ,,bn4 ,this being evident by what has already been demonstrated1 there is no need to explain it further4 ,now1 in the same figure1 if one draws ,,eaf1 which cuts the plane ,,ab at right angles at the point ,a1 since ,,ad is perpendicular to the wave ,,ac1 it will be ,,da which will mark the ray of incident light1 and ,,an which was perpendicular to ,,bn1 the refracted ray3 since the rays are nothing else than the straight lines along which the portions of the waves advance4 ,whence it is easy to recognize this chief property of refraction1 namely that the ,sine of the angle ,,dae has always the same ratio to the ,sine of the angle ,,naf1 whatever be the inclination of the ray ,,da3 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 ,,af4 ,for1 considering ,,ab as the radius of a circle1 the ,sine of the angle ,,bac is ,,bc1 and the ,sine of the angle ,,abn is ,,an4 ,but the angle ,,bac is equal to ,,dae1 since each of them added to ,,cae makes a right angle4 ,and the angle ,,abn is equal to ,,naf1 since each of them with ,,ban makes a right angle4 ,then also the ,sine of the angle ,,dae is to the ,sine of ,,naf as ,,bc is to ,,an4 ,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 ,,af2 therefore also the ,sine of the angle ,,dae will be to the ,sine of the angle ,,naf the same as the said velocities of light4 ,to see1 consequently1 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 7let us again assume the ratio of #c to #b71 it is only necessary to repeat all the same construction and demonstration which we have just used1 merely substituting everywhere #c_/b instead of #b_/c4 ,and it will be found by the same reasoning1 in this other figure1 that when the piece ,c of the wave ,,ac shall have reached the surface ,,ab at ,b1 all the portions of the wave ,,ac will have advanced as far as ,,bn1 so that ,,bc the perpendicular on ,,ac is to ,,an the perpendicular on ,,bn as #b to #c4 ,and there will finally be this same ratio of #b to #c between the ,sine of the angle ,,bad and the ,sine of the angle ,,fan4 ,hence one sees the reciprocal relation of the refractions of the ray on entering and on leaving one and the same transparent body3 namely that if ,,na falling on the external surface ,,ab is refracted into the direction ,,ad1 so the ray ,,ad will be refracted on leaving the transparent body into the direction ,,an4 ,illustration ,one sees also the reason for a noteworthy accident which happens in this refraction3 which is this1 that after a certain obliquity of the incident ray ,,da1 it begins to be quite unable to penetrate into the other transparent substance4 ,for if the angle ,,daq or ,,cba is such that in the triangle ,,acb1 ,,cb is equal to #b_/c of ,,ab1 or is greater1 then ,,an cannot form one side of the triangle ,,anb1 since it becomes equal to or greater than ,,ab3 so that the portion of wave ,,bn cannot be found anywhere1 neither consequently can ,,an1 which ought to be perpendicular to it4 ,and thus the incident ray ,,da does not then pierce the surface ,,ab4 ,when the ratio of the velocities of the waves is as two to three1 as in our example1 which is that which obtains for glass and air1 the angle ,,daq must be more than #dh degrees #aa minutes in order that the ray ,,da may be able to pass by refraction4 ,and when the ratio of the velocities is as #c to #d1 as it is very nearly in water and air1 this angle ,,daq must exceed #da degrees #bd minutes4 ,and this accords perfectly with experiment4 ,but it might here be asked3 since the meeting of the wave ,,ac against the surface ,,ab ought to produce movement in the matter which is on the other side1 why does no light pass there8 ,to which the reply is easy if one remembers what has been said before4 ,for although it generates an infinitude of partial waves in the matter which is at the other side of ,,ab1 these waves never have a common tangent line 7either straight or curved7 at the same moment2 and so there is no line terminating the propagation of the wave ,,ac beyond the plane ,,ab1 nor any place where the movement is gathered together in sufficiently great quantity to produce light4 ,and one will easily see the truth of this1 namely that ,,cb being larger than #b_/c of ,,ab1 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 #c_/b of the lengths ,,lb to which they correspond4 ,for all these circles will be enclosed in one another and will all pass beyond the point ,b4 ,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 substance1 one finds that the interior reflexion which occurs at the surface ,,ab is much augmented in brightness1 as is easy to realize by experiment with a triangular prism2 and for this our theory can afford this reason4 ,when the angle ,,daq is still large enough to enable the ray ,,da to pass1 it is evident that the light from the portion ,,ac of the wave is collected in a minimum space when it reaches ,,bn4 ,it appears also that the wave ,,bn becomes so much the smaller as the angle ,,cba or ,,daq is made less2 until when the latter is diminished to the limit indicated a little previously1 this wave ,,bn is collected together always at one point4 ,that is to say1 that when the piece ,c of the wave ,,ac has then arrived at ,b1 the wave ,,bn which is the propagation of ,,ac is entirely reduced to the same point ,b4 ,similarly when the piece ,h has reached ,k1 the part ,,ah is entirely reduced to the same point ,k4 ,this makes it evident that in proportion as the wave ,,ca comes to meet the surface ,,ab1 there occurs a great quantity of movement along that surface2 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 ,,ab1 according to the laws of reflexion previously explained4 ,and because a slight diminution of the angle of incidence ,,daq causes the wave ,,bn1 however great it was1 to be reduced to zero1 7for this angle being #di degrees #aa minutes in the glass1 the angle ,,ban is still #aa degrees #ba minutes1 and the same angle being reduced by one degree only the angle ,,ban is reduced to zero1 and so the wave ,,bn reduced to a point7 thence it comes about that the interior reflexion from being obscure becomes suddenly bright1 so soon as the angle of incidence is such that it no longer gives passage to the refraction4 ,now as concerns ordinary external reflexion1 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 ,,ab1 this reflexion should occur against the particles of the substance which touches the transparent body on its outside4 ,and it apparently occurs against the particles of the air or others mingled with the ethereal particles and larger than they4 ,so on the other hand the external reflexion of these bodies occurs against the particles which compose them1 and which are also larger than those of the ethereal matter1 since the latter flows in their interstices4 ,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 it1 as in vessels or tubes from which the air has been extracted4 ,experience1 moreover1 teaches us that these two reflexions are of nearly equal force1 and that in different transparent bodies they are so much the stronger as the refraction of these bodies is the greater4 ,thus one sees manifestly that the reflexion of glass is stronger than that of water1 and that of diamond stronger than that of glass4 ,i will finish this theory of refraction by demonstrating a remarkable proposition which depends on it2 namely1 that a ray of light in order to go from one point to another1 when these points are in different media1 is refracted in such wise at the plane surface which joins these two media that it employs the least possible time3 and exactly the same happens in the case of reflexion against a plane surface4 ,mr4 ,fermat was the first to propound this property of refraction1 holding with us1 and directly counter to the opinion of ,mr4 ,des ,cartes1 that light passes more slowly through glass and water than through air4 ,but he assumed besides this a constant ratio of ,sines1 which we have just proved by these different degrees of velocity alone3 or rather1 what is equivalent1 he assumed not only that the velocities were different but that the light took the least time possible for its passage1 and thence deduced the constant ratio of the ,sines4 ,his demonstration1 which may be seen in his printed works1 and in the volume of letters of ,mr4 ,des ,cartes1 is very long2 wherefore ,i give here another which is simpler and easier4 ,illustration ,let ,,kf be the plane surface2 ,a the point in the medium which the light traverses more easily1 as the air2 ,c the point in the other which is more difficult to penetrate1 as water4 ,and suppose that a ray has come from ,a1 by ,b1 to ,c1 having been refracted at ,b according to the law demonstrated a little before2 that is to say that1 having drawn ,,pbq1 which cuts the plane at right angles1 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 is4 ,it is to be shown that the time of passage of light along ,,ab and ,,bc taken together1 is the shortest that can be4 ,let us assume that it may have come by other lines1 and1 in the first place1 along ,,af1 ,,fc1 so that the point of refraction ,f may be further from ,b than the point ,a2 and let ,,ao be a line perpendicular to ,,ab1 and ,,fo parallel to ,,ab2 ,,bh perpendicular to ,,fo1 and ,,fg to ,,bc4 ,since then the angle ,,hbf is equal to ,,pba1 and the angle ,,bfg equal to ,,qbc1 it follows that the sine of the angle ,,hbf will also have the same ratio to the sine of ,,bfg1 as the velocity of light in the medium ,a is to its velocity in the medium ,c4 ,but these sines are the straight lines ,,hf1 ,,bg1 if we take ,,bf as the semi-diameter of a circle4 ,then these lines ,,hf1 ,,bg1 will bear to one another the said ratio of the velocities4 ,and1 therefore1 the time of the light along ,,hf1 supposing that the ray had been ,,of1 would be equal to the time along ,,bg in the interior of the medium ,c4 ,but the time along ,,ab is equal to the time along ,,oh2 therefore the time along ,,of is equal to the time along ,,ab1 ,,bg4 ,again the time along ,,fc is greater than that along ,,gc2 then the time along ,,ofc will be longer than that along ,,abc4 ,but ,,af is longer than ,,of1 then the time along ,,afc will by just so much more exceed the time along ,,abc4 ,now let us assume that the ray has come from ,a to ,c along ,,ak1 ,,kc2 the point of refraction ,k being nearer to ,a than the point ,b is2 and let ,,cn be the perpendicular upon ,,bc1 ,,kn parallel to ,,bc3 ,,bm perpendicular upon ,,kn1 and ,,kl upon ,,ba4 ,here ,,bl and ,,km are the sines of angles ,,bkl1 ,,kbm2 that is to say1 of the angles ,,pba1 ,,qbc2 and therefore they are to one another as the velocity of light in the medium ,a is to the velocity in the medium ,c4 ,then the time along ,,lb is equal to the time along ,,km2 and since the time along ,,bc is equal to the time along ,,mn1 the time along ,,lbc will be equal to the time along ,,kmn4 ,but the time along ,,ak is longer than that along ,,al3 hence the time along ,,akn is longer than that along ,,abc4 ,and ,,kc being longer than ,,kn1 the time along ,,akc will exceed1 by as much more1 the time along ,,abc4 ,hence it appears that the time along ,,abc is the shortest possible2 which was to be proven4 ,,chapter ,,iv ,,on ,,the ,,refraction ,,of ,,the ,,air ,we have shown how the movement which constitutes light spreads by spherical waves in any homogeneous matter4 ,and it is evident that when the matter is not homogeneous1 but of such a constitution that the movement is communicated in it more rapidly toward one side than toward another1 these waves cannot be spherical3 but that they must acquire their figure according to the different distances over which the successive movement passes in equal times4 ,it is thus that we shall in the first place explain the refractions which occur in the air1 which extends from here to the clouds and beyond4 ,the effects of which refractions are very remarkable2 for by them we often see objects which the rotundity of the ,earth ought otherwise to hide2 such as ,islands1 and the tops of mountains when one is at sea4 ,because also of them the ,sun and the ,moon appear as risen before in fact they have1 and appear to set later3 so that at times the ,moon has been seen eclipsed while the ,sun appeared still above the horizon4 ,and so also the heights of the ,sun and of the ,moon1 and those of all the ,stars always appear a little greater than they are in reality1 because of these same refractions1 as ,astronomers know4 ,but there is one experiment which renders this refraction very evident2 which is that of fixing a telescope on some spot so that it views an object1 such as a steeple or a house1 at a distance of half a league or more4 ,if then you look through it at different hours of the day1 leaving it always fixed in the same way1 you will see that the same spots of the object will not always appear at the middle of the aperture of the telescope1 but that generally in the morning and in the evening1 when there are more vapours near the ,earth1 these objects seem to rise higher1 so that the half or more of them will no longer be visible2 and so that they seem lower toward mid-day when these vapours are dissipated4 ,those who consider refraction to occur only in the surfaces which separate transparent bodies of different nature1 would find it difficult to give a reason for all that ,i have just related2 but according to our ,theory the thing is quite easy4 ,it is known that the air which surrounds us1 besides the particles which are proper to it and which float in the ethereal matter as has been explained1 is full also of particles of water which are raised by the action of heat2 and it has been ascertained further by some very definite experiments that as one mounts up higher the density of air diminishes in proportion4 ,now whether the particles of water and those of air take part1 by means of the particles of ethereal matter1 in the movement which constitutes light1 but have a less prompt recoil than these1 or whether the encounter and hindrance which these particles of air and water offer to the propagation of movement of the ethereal progress1 retard the progression1 it follows that both kinds of particles flying amidst the ethereal particles1 must render the air1 from a great height down to the ,earth1 gradually less easy for the spreading of the waves of light4 ,illustration ,whence the configuration of the waves ought to become nearly such as this figure represents3 namely1 if ,a is a light1 or the visible point of a steeple1 the waves which start from it ought to spread more widely upwards and less widely downwards1 but in other directions more or less as they approximate to these two extremes4 ,this being so1 it necessarily follows that every line intersecting one of these waves at right angles will pass above the point ,a1 always excepting the one line which is perpendicular to the horizon4 ,illustration ,let ,,bc be the wave which brings the light to the spectator who is at ,b1 and let ,,bd be the straight line which intersects this wave at right angles4 ,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 eye1 as will be understood by what was said above1 it is manifest that the point ,a will be perceived as being in the line ,,bd1 and therefore higher than in fact it is4 ,similarly if the ,earth be ,,ab1 and the top of the ,atmosphere ,,cd1 which probably is not a well defined spherical surface 7since we know that the air becomes rare in proportion as one ascends1 for above there is so much less of it to press down upon it71 the waves of light from the sun coming1 for instance1 in such a way that so long as they have not reached the ,atmosphere ,,cd the straight line ,,ae intersects them perpendicularly1 they ought1 when they enter the ,atmosphere1 to advance more quickly in elevated regions than in regions nearer to the ,earth4 ,so that if ,,ca is the wave which brings the light to the spectator at ,a1 its region ,c will be the furthest advanced2 and the straight line ,,af1 which intersects this wave at right angles1 and which determines the apparent place of the ,sun1 will pass above the real ,sun1 which will be seen along the line ,,ae4 ,and so it may occur that when it ought not to be visible in the absence of vapours1 because the line ,,ae encounters the rotundity of the ,earth1 it will be perceived in the line ,,af by refraction4 ,but this angle ,,eaf is scarcely ever more than half a degree because the attenuation of the vapours alters the waves of light but little4 ,furthermore these refractions are not altogether constant in all weathers1 particularly at small elevations of #b or #c degrees2 which results from the different quantity of aqueous vapours rising above the ,earth4 ,and this same thing is the cause why at certain times a distant object will be hidden behind another less distant one1 and yet may at another time be able to be seen1 although the spot from which it is viewed is always the same4 ,but the reason for this effect will be still more evident from what we are going to remark touching the curvature of rays4 ,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 ray4 ,now these rays1 instead of being straight as they are in homogeneous media1 ought to be curved in an atmosphere of unequal penetrability4 ,for they necessarily follow from the object to the eye the line which intersects at right angles all the progressions of the waves1 as in the first figure the line ,,aeb does1 as will be shown hereafter2 and it is this line which determines what interposed bodies would or would not hinder us from seeing the object4 ,for although the point of the steeple ,a appears raised to ,d1 it would yet not appear to the eye ,b if the tower ,h was between the two1 because it crosses the curve ,,aeb4 ,but the tower ,e1 which is beneath this curve1 does not hinder the point ,a from being seen4 ,now according as the air near the ,earth exceeds in density that which is higher1 the curvature of the ray ,,aeb becomes greater3 so that at certain times it passes above the summit ,e1 which allows the point ,a to be perceived by the eye at ,b2 and at other times it is intercepted by the same tower ,e which hides ,a from this same eye4 ,illustration ,but to demonstrate this curvature of the rays conformably to all our preceding ,theory1 let us imagine that ,,ab is a small portion of a wave of light coming from the side ,c1 which we may consider as a straight line4 ,let us also suppose that it is perpendicular to the ,horizon1 the portion ,b being nearer to the ,earth than the portion ,a2 and that because the vapours are less hindering at ,a than at ,b1 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 ,,be2 ,,ad and ,,be being parallel to the ,horizon4 ,further1 supposing the straight lines ,,fg1 ,,hi1 etc41 to be drawn from an infinitude of points in the straight line ,,ab and to terminate on the line ,,de 7which is straight or may be considered as such71 let the different penetrabilities at the different heights in the air between ,a and ,b be represented by all these lines2 so that the particular wave1 originating from the point ,f1 will spread across the space ,,fg1 and that from the point ,h across the space ,,hi1 while that from the point ,a spreads across the space ,,ad4 ,now if about the centres ,a1 ,b1 one describes the circles ,,dk1 ,,el1 which represent the spreading of the waves which originate from these two points1 and if one draws the straight line ,,kl which touches these two circles1 it is easy to see that this same line will be the common tangent to all the other circles drawn about the centres ,f1 ,h1 etc42 and that all the points of contact will fall within that part of this line which is comprised between the perpendiculars ,,ak1 ,,bl4 ,then it will be the line ,,kl which will terminate the movement of the particular waves originating from the points of the wave ,,ab2 and this movement will be stronger between the points ,,kl1 than anywhere else at the same instant1 since an infinitude of circumferences concur to form this straight line2 and consequently ,,kl will be the propagation of the portion of wave ,,ab1 as has been said in explaining reflexion and ordinary refraction4 ,now it appears that ,,ak and ,,bl dip down toward the side where the air is less easy to penetrate3 for ,,ak being longer than ,,bl1 and parallel to it1 it follows that the lines ,,ab and ,,kl1 being prolonged1 would meet at the side ,l4 ,but the angle ,k is a right angle3 hence ,,kab is necessarily acute1 and consequently less than ,,dab4 ,if one investigates in the same way the progression of the portion of the wave ,,kl1 one will find that after a further time it has arrived at ,,mn in such a manner that the perpendiculars ,,km1 ,,ln1 dip down even more than do ,,ak1 ,,bl4 ,and this suffices to show that the ray will continue along the curved line which intersects all the waves at right angles1 as has been said4 ,,chapter ,v ,,on ,,the ,,strange ,,refraction ,,of ,,iceland ,,crystal #a4 ,there is brought from ,iceland1 which is an ,island in the ,north ,sea1 in the latitude of #ff degrees1 a kind of ,crystal or transparent stone1 very remarkable for its figure and other qualities1 but above all for its strange refractions4 ,the causes of this have seemed to me to be worthy of being carefully investigated1 the more so because amongst transparent bodies this one alone does not follow the ordinary rules with respect to rays of light4 ,i have even been under some necessity to make this research1 because the refractions of this ,crystal seemed to overturn our preceding explanation of regular refraction2 which explanation1 on the contrary1 they strongly confirm1 as will be seen after they have been brought under the same principle4 ,in ,iceland are found great lumps of this ,crystal1 some of which ,i have seen of #d or #e pounds4 ,but it occurs also in other countries1 for ,i have had some of the same sort which had been found in ,france near the town of ,troyes in ,champagne1 and some others which came from the ,island of ,corsica1 though both were less clear and only in little bits1 scarcely capable of letting any effect of refraction be observed4 #b4 ,the first knowledge which the public has had about it is due to ,mr4 ,erasmus ,bartholinus1 who has given a description of ,iceland ,crystal and of its chief phenomena4 ,but here ,i shall not desist from giving my own1 both for the instruction of those who may not have seen his book1 and because as respects some of these phenomena there is a slight difference between his observations and those which ,i have made3 for ,i have applied myself with great exactitude to examine these properties of refraction1 in order to be quite sure before undertaking to explain the causes of them4 #c4 ,as regards the hardness of this stone1 and the property which it has of being easily split1 it must be considered rather as a species of ,talc than of ,crystal4 ,for an iron spike effects an entrance into it as easily as into any other ,talc or ,alabaster1 to which it is equal in gravity4 ,illustration #d4 ,the pieces of it which are found have the figure of an oblique parallelepiped2 each of the six faces being a parallelogram2 and it admits of being split in three directions parallel to two of these opposed faces4 ,even in such wise1 if you will1 that all the six faces are equal and similar rhombuses4 ,the figure here added represents a piece of this ,crystal4 ,the obtuse angles of all the parallelograms1 as ,c1 ,d1 here1 are angles of #aja degrees #eb minutes1 and consequently the acute angles1 such as ,a and ,b1 are of #gh degrees #h minutes4 #e4 ,of the solid angles there are two opposite to one another1 such as ,c and ,e1 which are each composed of three equal obtuse plane angles4 ,the other six are composed of two acute angles and one obtuse4 ,all that ,i have just said has been likewise remarked by ,mr4 ,bartholinus in the aforesaid treatise2 if we differ it is only slightly about the values of the angles4 ,he recounts moreover some other properties of this ,crystal2 to wit1 that when rubbed against cloth it attracts straws and other light things as do amber1 diamond1 glass1 and ,spanish wax4 ,let a piece be covered with water for a day or more1 the surface loses its natural polish4 ,when aquafortis is poured on it it produces ebullition1 especially1 as ,i have found1 if the ,crystal has been pulverized4 ,i have also found by experiment that it may be heated to redness in the fire without being in anywise altered or rendered less transparent2 but a very violent fire calcines it nevertheless4 ,its transparency is scarcely less than that of water or of ,rock ,crystal1 and devoid of colour4 ,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 explain2 reserving for the end of this ,treatise the statement of my conjectures touching the formation and extraordinary configuration of this ,crystal4 #f4 ,in all other transparent bodies that we know there is but one sole and simple refraction2 but in this substance there are two different ones4 ,the effect is that objects seen through it1 especially such as are placed right against it1 appear double2 and that a ray of sunlight1 falling on one of its surfaces1 parts itself into two rays and traverses the ,crystal thus4 #g4 ,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 refraction1 and that an oblique ray is always refracted4 ,but in this ,crystal the perpendicular ray suffers refraction1 and there are oblique rays which pass through it quite straight4 ,illustration #h4 ,but in order to explain these phenomena more particularly1 let there be1 in the first place1 a piece ,,abfe of the same ,crystal1 and let the obtuse angle ,,acb1 one of the three which constitute the equilateral solid angle ,c1 be divided into two equal parts by the straight line ,,cg1 and let it be conceived that the ,crystal is intersected by a plane which passes through this line and through the side ,,cf1 which plane will necessarily be perpendicular to the surface ,,ab2 and its section in the ,crystal will form a parallelogram ,,gcfh4 ,we will call this section the principal section of the ,crystal4 #i4 ,now if one covers the surface ,,ab1 leaving there only a small aperture at the point ,k1 situated in the straight line ,,cg1 and if one exposes it to the sun1 so that his rays face it perpendicularly above1 then the ray ,,ik will divide itself at the point ,k into two1 one of which will continue to go on straight by ,,kl1 and the other will separate itself along the straight line ,,km1 which is in the plane ,,gcfh1 and which makes with ,,kl an angle of about #f degrees #dj minutes1 tending from the side of the solid angle ,c2 and on emerging from the other side of the ,crystal it will turn again parallel to ,,jk1 along ,,mz4 ,and as1 in this extraordinary refraction1 the point ,m is seen by the refracted ray ,,mki1 which ,i consider as going to the eye at ,i1 it necessarily follows that the point ,l1 by virtue of the same refraction1 will be seen by the refracted ray ,,lri1 so that ,,lr will be parallel to ,,mk if the distance from the eye ,,ki is supposed very great4 ,the point ,l appears then as being in the straight line ,,irs2 but the same point appears also1 by ordinary refraction1 to be in the straight line ,,ik1 hence it is necessarily judged to be double4 ,and similarly if ,l be a small hole in a sheet of paper or other substance which is laid against the ,crystal1 it will appear when turned towards daylight as if there were two holes1 which will seem the wider apart from one another the greater the thickness of the ,crystal4 #aj4 ,again1 if one turns the ,crystal in such wise that an incident ray ,,no1 of sunlight1 which ,i suppose to be in the plane continued from ,,gcfh1 makes with ,,gc an angle of #gc degrees and #bj minutes1 and is consequently nearly parallel to the edge ,,cf1 which makes with ,,fh an angle of #gj degrees #eg minutes1 according to the calculation which ,i shall put at the end1 it will divide itself at the point ,o into two rays1 one of which will continue along ,,op in a straight line with ,,no1 and will similarly pass out of the other side of the crystal without any refraction2 but the other will be refracted and will go along ,,oq4 ,and it must be noted that it is special to the plane through ,,gcf and to those which are parallel to it1 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 double2 for it is quite otherwise for rays in all other planes which intersect the ,crystal1 as we shall see afterwards4 #aa4 ,i recognized at first by these experiments and by some others that of the two refractions which the ray suffers in this ,crystal1 there is one which follows the ordinary rules2 and it is this to which the rays ,,kl and ,,oq belong4 ,this is why ,i have distinguished this ordinary refraction from the other2 and having measured it by exact observation1 ,i found that its proportion1 considered as to the ,sines of the angles which the incident and refracted rays make with the perpendicular1 was very precisely that of #e to #c1 as was found also by ,mr4 ,bartholinus1 and consequently much greater than that of ,rock ,crystal1 or of glass1 which is nearly #c to #b4 ,illustration #ab4 ,the mode of making these observations exactly is as follows4 ,upon a leaf of paper fixed on a thoroughly flat table there is traced a black line ,,ab1 and two others1 ,,ced and ,,kml1 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 oblique4 ,then place the ,crystal upon the intersection ,e so that the line ,,ab concurs with that which bisects the obtuse angle of the lower surface1 or with some line parallel to it4 ,then by placing the eye directly above the line ,,ab it will appear single only2 and one will see that the portion viewed through the ,crystal and the portions which appear outside it1 meet together in a straight line3 but the line ,,cd will appear double1 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 other1 or again by the circumstance that1 when the ,crystal is turned around on the paper1 this image remains stationary1 whereas the other image shifts and moves entirely around4 ,afterwards let the eye be placed at ,i 7remaining always in the plane perpendicular through ,,ab7 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 ,crystal4 ,and then1 marking on the surface of the ,crystal the point ,h where the intersection ,e appears1 this point will be directly above ,e4 ,then draw back the eye towards ,o1 keeping always in the plane perpendicular through ,,ab1 so that the image of the line ,,cd1 which is formed by ordinary refraction1 may appear in a straight line with the line ,,kl viewed without refraction2 and then mark on the ,crystal the point ,n where the point of intersection ,e appears4 #ac4 ,then one will know the length and position of the lines ,,nh1 ,,em1 and of ,,he1 which is the thickness of the ,crystal3 which lines being traced separately upon a plan1 and then joining ,,ne and ,,nm which cuts ,,he at ,p1 the proportion of the refraction will be that of ,,en to ,,np1 because these lines are to one another as the sines of the angles ,,nph1 ,,nep1 which are equal to those which the incident ray ,,on and its refraction ,,ne make with the perpendicular to the surface4 ,this proportion1 as ,i have said1 is sufficiently precisely as #e to #c1 and is always the same for all inclinations of the incident ray4 #ad4 ,the same mode of observation has also served me for examining the extraordinary or irregular refraction of this ,crystal4 ,for1 the point ,h having been found and marked1 as aforesaid1 directly above the point ,e1 ,i observed the appearance of the line ,,cd1 which is made by the extraordinary refraction2 and having placed the eye at ,q1 so that this appearance made a straight line with the line ,,kl viewed without refraction1 ,i ascertained the triangles ,,reh1 ,,res1 and consequently the angles ,,rsh1 ,,res1 which the incident and the refracted ray make with the perpendicular4 #ae4 ,but ,i found in this refraction that the ratio of ,,fr to ,,rs was not constant1 like the ordinary refraction1 but that it varied with the varying obliquity of the incident ray4 #af4 ,i found also that when ,,qre made a straight line1 that is1 when the incident ray entered the ,crystal without being refracted 7as ,i ascertained by the circumstance that then the point ,e viewed by the extraordinary refraction appeared in the line ,,cd1 as seen without refraction7 ,i found1 ,i say1 then that the angle ,,qrg was #gc degrees #bj minutes1 as has been already remarked2 and so it is not the ray parallel to the edge of the ,crystal1 which crosses it in a straight line without being refracted1 as ,mr4 ,bartholinus believed1 since that inclination is only #gj degrees #eg minutes1 as was stated above4 ,and this is to be noted1 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 mentioned4 ,illustration #ag4 ,finally1 continuing my observations to discover the nature of this refraction1 ,i learned that it obeyed the following remarkable rule4 ,let the parallelogram ,,gcfh1 made by the principal section of the ,crystal1 as previously determined1 be traced separately4 ,i found then that always1 when the inclinations of two rays which come from opposite sides1 as ,,vk1 ,,sk here1 are equal1 their refractions ,,kx and ,,kt meet the bottom line ,,hf in such wise that points ,x and ,t are equally distant from the point ,m1 where the refraction of the perpendicular ray ,,ik falls2 and this occurs also for refractions in other sections of this ,crystal4 ,but before speaking of those1 which have also other particular properties1 we will investigate the causes of the phenomena which ,i have already reported4 ,it was after having explained the refraction of ordinary transparent bodies by means of the spherical emanations of light1 as above1 that ,i resumed my examination of the nature of this ,crystal1 wherein ,i had previously been unable to discover anything4 #ah4 ,as there were two different refractions1 ,i conceived that there were also two different emanations of waves of light1 and that one could occur in the ethereal matter extending through the body of the ,crystal4 ,which matter1 being present in much larger quantity than is that of the particles which compose it1 was alone capable of causing transparency1 according to what has been explained heretofore4 ,i attributed to this emanation of waves the regular refraction which is observed in this stone1 by supposing these waves to be ordinarily of spherical form1 and having a slower progression within the ,crystal than they have outside it2 whence proceeds refraction as ,i have demonstrated4 #ai4 ,as to the other emanation which should produce the irregular refraction1 ,i wished to try what ,elliptical waves1 or rather spheroidal waves1 would do2 and these ,i supposed would spread indifferently both in the ethereal matter diffused throughout the crystal and in the particles of which it is composed1 according to the last mode in which ,i have explained transparency4 ,it seemed to me that the disposition or regular arrangement of these particles could contribute to form spheroidal waves 7nothing more being required for this than that the successive movement of light should spread a little more quickly in one direction than in the other7 and ,i scarcely doubted that there were in this crystal such an arrangement of equal and similar particles1 because of its figure and of its angles with their determinate and invariable measure4 ,touching which particles1 and their form and disposition1 ,i shall1 at the end of this ,treatise1 propound my conjectures and some experiments which confirm them4 #bj4 ,the double emission of waves of light1 which ,i had imagined1 became more probable to me after ,i had observed a certain phenomenon in the ordinary ,rock ,crystal1 which occurs in hexagonal form1 and which1 because of this regularity1 seems also to be composed of particles1 of definite figure1 and ranged in order4 ,this was1 that this crystal1 as well as that from ,iceland1 has a double refraction1 though less evident4 ,for having had cut from it some well polished ,prisms of different sections1 ,i remarked in all1 in viewing through them the flame of a candle or the lead of window panes1 that everything appeared double1 though with images not very distant from one another4 ,whence ,i understood the reason why this substance1 though so transparent1 is useless for ,telescopes1 when they have ever so little length4 #ba4 ,now this double refraction1 according to my ,theory hereinbefore established1 seemed to demand a double emission of waves of light1 both of them spherical 7for both the refractions are regular7 and those of one series a little slower only than the others4 ,for thus the phenomenon is quite naturally explained1 by postulating substances which serve as vehicle for these waves1 as ,i have done in the case of ,iceland ,crystal4 ,i had then less trouble after that in admitting two emissions of waves in one and the same body4 ,and since it might have been objected that in composing these two kinds of crystal of equal particles of a certain figure1 regularly piled1 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 there1 ,i removed this difficulty by regarding these particles as being of a very rare texture1 or rather as composed of other much smaller particles1 between which the ethereal matter passes quite freely4 ,this1 moreover1 necessarily follows from that which has been already demonstrated touching the small quantity of matter of which the bodies are built up4 #bb4 ,supposing then these spheroidal waves besides the spherical ones1 ,i began to examine whether they could serve to explain the phenomena of the irregular refraction1 and how by these same phenomena ,i could determine the figure and position of the spheroids3 as to which ,i obtained at last the desired success1 by proceeding as follows4 ,illustration #bc4 ,i considered first the effect of waves so formed1 as respects the ray which falls perpendicularly on the flat surface of a transparent body in which they should spread in this manner4 ,i took ,,ab for the exposed region of the surface4 ,and1 since a ray perpendicular to a plane1 and coming from a very distant source of light1 is nothing else1 according to the precedent ,theory1 than the incidence of a portion of the wave parallel to that plane1 ,i supposed the straight line ,,rc1 parallel and equal to ,,ab1 to be a portion of a wave of light1 in which an infinitude of points such as ,,rhh,c come to meet the surface ,,ab at the points ,,akk,b4 ,then instead of the hemispherical partial waves which in a body of ordinary refraction would spread from each of these last points1 as we have above explained in treating of refraction1 these must here be hemi-spheroids4 ,the axes 7or rather the major diameters7 of these ,i supposed to be oblique to the plane ,,ab1 as is ,,av the semi-axis or semi-major diameter of the spheroid ,,svt1 which represents the partial wave coming from the point ,a1 after the wave ,,rc has reached ,,ab4 ,i say axis or major diameter1 because the same ellipse ,,svt may be considered as the section of a spheroid of which the axis is ,,az perpendicular to ,,av4 ,but1 for the present1 without yet deciding one or other1 we will consider these spheroids only in those sections of them which make ellipses in the plane of this figure4 ,now taking a certain space of time during which the wave ,,svt has spread from ,a1 it would needs be that from all the other points ,kk,b there should proceed1 in the same time1 waves similar to ,,svt and similarly situated4 ,and the common tangent ,,nq of all these semi-ellipses would be the propagation of the wave ,,rc which fell on ,,ab1 and would be the place where this movement occurs in much greater amount than anywhere else1 being made up of arcs of an infinity of ellipses1 the centres of which are along the line ,,ab4 #bd4 ,now it appeared that this common tangent ,,nq was parallel to ,,ab1 and of the same length1 but that it was not directly opposite to it1 since it was comprised between the lines ,,an1 ,,bq1 which are diameters of ellipses having ,a and ,b for centres1 conjugate with respect to diameters which are not in the straight line ,,ab4 ,and in this way ,i comprehended1 a matter which had seemed to me very difficult1 how a ray perpendicular to a surface could suffer refraction on entering a transparent body2 seeing that the wave ,,rc1 having come to the aperture ,,ab1 went on forward thence1 spreading between the parallel lines ,,an1 ,,bq1 yet itself remaining always parallel to ,,ab1 so that here the light does not spread along lines perpendicular to its waves1 as in ordinary refraction1 but along lines cutting the waves obliquely4 ,illustration #be4 ,inquiring subsequently what might be the position and form of these spheroids in the crystal1 ,i considered that all the six faces produced precisely the same refractions4 ,taking1 then1 the parallelopiped ,,afb1 of which the obtuse solid angle ,c is contained between the three equal plane angles1 and imagining in it the three principal sections1 one of which is perpendicular to the face ,,dc and passes through the edge ,,cf1 another perpendicular to the face ,,bf passing through the edge ,,ca1 and the third perpendicular to the face ,,af passing through the edge ,,bc2 ,i knew that the refractions of the incident rays belonging to these three planes were all similar4 ,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 ,c4 ,consequently ,i saw that the axis of this angle1 that is to say the straight line which traversed the crystal from the point ,c with equal inclination to the edges ,,cf1 ,,ca1 ,,cb was the line which determined the position of the axis of all the spheroidal waves which one imagined to originate from some point1 taken within or on the surface of the crystal1 since all these spheroids ought to be alike1 and have their axes parallel to one another4 #bf4 ,considering after this the plane of one of these three sections1 namely that through ,,gcf1 the angle of which is #aji degrees #c minutes1 since the angle ,f was shown above to be #gj degrees #eg minutes2 and1 imagining a spheroidal wave about the centre ,c1 ,i knew1 because ,i have just explained it1 that its axis must be in the same plane1 the half of which axis ,i have marked ,,cs in the next figure3 and seeking by calculation 7which will be given with others at the end of this discourse7 the value of the angle ,,cgs1 ,i found it #de degrees #bj minutes4 ,illustration #bg4 ,to know from this the form of this spheroid1 that is to say the proportion of the semi-diameters ,,cs1 ,,cp1 of its elliptical section1 which are perpendicular to one another1 ,i considered that the point ,m where the ellipse is touched by the straight line ,,fh1 parallel to ,,cg1 ought to be so situated that ,,cm makes with the perpendicular ,,cl an angle of #f degrees #dj minutes2 since1 this being so1 this ellipse satisfies what has been said about the refraction of the ray perpendicular to the surface ,,cg1 which is inclined to the perpendicular ,,cl by the same angle4 ,this1 then1 being thus disposed1 and taking ,,cm at #ajj1jjj parts1 ,i found by the calculation which will be given at the end1 the semi-major diameter ,,cp to be #aje1jcb1 and the semi-axis ,,cs to be #ic1daj1 the ratio of which numbers is very nearly #i to #h2 so that the spheroid was of the kind which resembles a compressed sphere1 being generated by the revolution of an ellipse about its smaller diameter4 ,i found also the value of ,,cg the semi-diameter parallel to the tangent ,,ml to be #ih1ggi4 ,illustration #bh4 ,now passing to the investigation of the refractions which obliquely incident rays must undergo1 according to our hypothesis of spheroidal waves1 ,i saw that these refractions depended on the ratio between the velocity of movement of the light outside the crystal in the ether1 and that within the crystal4 ,for supposing1 for example1 this proportion to be such that while the light in the crystal forms the spheroid ,,gsp1 as ,i have just said1 it forms outside a sphere the semi-diameter of which is equal to the line ,n which will be determined hereafter1 the following is the way of finding the refraction of the incident rays4 ,let there be such a ray ,,rc falling upon the surface ,,ck4 ,make ,,co perpendicular to ,,rc1 and across the angle ,,kco adjust ,,ok1 equal to ,n and perpendicular to ,,co2 then draw ,,ki1 which touches the ,ellipse ,,gsp1 and from the point of contact ,i join ,,ic1 which will be the required refraction of the ray ,,rc4 ,the demonstration of this is1 it will be seen1 entirely similar to that of which we made use in explaining ordinary refraction4 ,for the refraction of the ray ,,rc is nothing else than the progression of the portion ,c of the wave ,,co1 continued in the crystal4 ,now the portions ,h of this wave1 during the time that ,o came to ,k1 will have arrived at the surface ,,ck along the straight lines ,hx1 and will moreover have produced in the crystal around the centres x some hemi-spheroidal partial waves similar to the hemi-spheroidal ,,gspg1 and similarly disposed1 and of which the major and minor diameters will bear the same proportions to the lines xv 7the continuations of the lines ,hx up to ,,kb parallel to ,,co7 that the diameters of the spheroid ,,gspg bear to the line ,,cb1 or ,n4 ,and it is quite easy to see that the common tangent of all these spheroids1 which are here represented by ,ellipses1 will be the straight line ,,ik1 which consequently will be the propagation of the wave ,,co2 and the point ,i will be that of the point ,c1 conformably with that which has been demonstrated in ordinary refraction4 ,now as to finding the point of contact ,i1 it is known that one must find ,,cd a third proportional to the lines ,,ck1 ,,cg1 and draw ,,di parallel to ,,cm1 previously determined1 which is the conjugate diameter to ,,cg2 for then1 by drawing ,,ki it touches the ,ellipse at ,i4 #bi4 ,now as we have found ,,ci the refraction of the ray ,,rc1 similarly one will find ,ci the refraction of the ray r,c1 which comes from the opposite side1 by making ,co perpendicular to r,c and following out the rest of the construction as before4 ,whence one sees that if the ray r,c is inclined equally with ,,rc1 the line ,cd will necessarily be equal to ,,cd1 because ,ck is equal to ,,ck1 and ,cg to ,,cg4 ,and in consequence ,ii will be cut at ,e into equal parts by the line ,,cm1 to which ,,di and di are parallel4 ,and because ,,cm is the conjugate diameter to ,,cg1 it follows that i,i will be parallel to g,g4 ,therefore if one prolongs the refracted rays ,,ci1 ,ci1 until they meet the tangent ,,ml at ,t and t1 the distances ,,mt1 ,mt1 will also be equal4 ,and so1 by our hypothesis1 we explain perfectly the phenomenon mentioned above2 to wit1 that when there are two rays equally inclined1 but coming from opposite sides1 as here the rays ,,rc1 rc1 their refractions diverge equally from the line followed by the refraction of the ray perpendicular to the surface1 by considering these divergences in the direction parallel to the surface of the crystal4 #cj4 ,to find the length of the line ,n1 in proportion to ,,cp1 ,,cs1 ,,cg1 it must be determined by observations of the irregular refraction which occurs in this section of the crystal2 and ,i find thus that the ratio of ,n to ,,gc is just a little less than #h to #e4 ,and having regard to some other observations and phenomena of which ,i shall speak afterwards1 ,i put ,n at #aef1ifb parts1 of which the semi-diameter ,,cg is found to contain #ih1ggi1 making this ratio #h to #e-a_/bi4 ,now this proportion1 which there is between the line ,n and ,,cg1 may be called the ,proportion of the ,refraction2 similarly as in glass that of #c to #b1 as will be manifest when ,i shall have explained a short process in the preceding way to find the irregular refractions4 #ca4 ,supposing then1 in the next figure1 as previously1 the surface of the crystal g,g1 the ,ellipse ,,gpg1 and the line ,n2 and ,,cm the refraction of the perpendicular ray ,,fc1 from which it diverges by #f degrees #dj minutes4 ,now let there be some other ray ,,rc1 the refraction of which must be found4 ,about the centre ,c1 with semi-diameter ,,cg1 let the circumference g,,rg be described1 cutting the ray ,,rc at ,r2 and let ,,rv be the perpendicular on ,,cg4 ,then as the line ,n is to ,,cg let ,,cv be to ,,cd1 and let ,,di be drawn parallel to ,,cm1 cutting the ,ellipse g,,mg at ,i2 then joining ,,ci1 this will be the required refraction of the ray ,,rc4 ,which is demonstrated thus4 ,illustration ,let ,,co be perpendicular to ,,cr1 and across the angle ,,ocg let ,,ok be adjusted1 equal to ,n and perpendicular to ,,co1 and let there be drawn the straight line ,,ki1 which if it is demonstrated to be a tangent to the ,ellipse at ,i1 it will be evident by the things heretofore explained that ,,ci is the refraction of the ray ,,rc4 ,now since the angle ,,rco is a right angle1 it is easy to see that the right-angled triangles ,,rcv1 ,,kco1 are similar4 ,as then1 ,,ck is to ,,ko1 so also is ,,rc to ,,cv4 ,but ,,ko is equal to ,n1 and ,,rc to ,,cg3 then as ,,ck is to ,n so will ,,cg be to ,,cv4 ,but as ,n is to ,,cg1 so1 by construction1 is ,,cv to ,,cd4 ,then as ,,ck is to ,,cg so is ,,cg to ,,cd4 ,and because ,,di is parallel to ,,cm1 the conjugate diameter to ,,cg1 it follows that ,,ki touches the ,ellipse at ,i2 which remained to be shown4 #cb4 ,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 perpendicular1 so here there is such a proportion between ,,cv and ,,cd or ,,ie2 that is to say between the ,sine of the angle which the incident ray makes with the perpendicular1 and the horizontal intercept1 in the ,ellipse1 between the refraction of this ray and the diameter ,,cm4 ,for the ratio of ,,cv to ,,cd is1 as has been said1 the same as that of ,n to the semi-diameter ,,cg4 #cc4 ,i will add here1 before passing away1 that in comparing together the regular and irregular refraction of this crystal1 there is this remarkable fact1 that if ,,abps be the spheroid by which light spreads in the ,crystal in a certain space of time 7which spreading1 as has been said1 serves for the irregular refraction71 then the inscribed sphere ,,bvst is the extension in the same space of time of the light which serves for the regular refraction4 ,illustration ,for we have stated before this1 that the line ,n being the radius of a spherical wave of light in air1 while in the crystal it spread through the spheroid ,,abps1 the ratio of ,n to ,,cs will be #aef1ifb to #ic1daj4 ,but it has also been stated that the proportion of the regular refraction was #e to #c2 that is to say1 that ,n being the radius of a spherical wave of light in air1 its extension in the crystal would1 in the same space of time1 form a sphere the radius of which would be to ,n as #c to #e4 ,now #aef1ifb is to #ic1daj as #e to #c less #a_/da4 ,so that it is sufficiently nearly1 and may be exactly1 the sphere ,,bvst1 which the light describes for the regular refraction in the crystal1 while it describes the spheroid ,,bpsa for the irregular refraction1 and while it describes the sphere of radius ,n in air outside the crystal4 ,although then there are1 according to what we have supposed1 two different propagations of light within the crystal1 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 other2 but that they have an equal velocity in the other direction1 namely1 in that parallel to the same axis ,,bs1 which is also the axis of the obtuse angle of the crystal4 ,illustration #cd4 ,the proportion of the refraction being what we have just seen1 ,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 refraction4 ,for supposing the same things as before1 and that the ray makes with the same surface g,g the angle ,,rcg of #gc degrees #bj minutes1 inclining to the same side as the crystal 7of which ray mention has been made above72 if one investigates1 by the process above explained1 the refraction ,,ci1 one will find that it makes exactly a straight line with ,,rc1 and that thus this ray is not deviated at all1 conformably with experiment4 ,this is proved as follows by calculation4 ,,cg or ,,cr being1 as precedently1 #ih1ggi2 ,,cm being #ajj1jjj2 and the angle ,,rcv #gc degrees #bj minutes1 ,,cv will be #bh1ccj4 ,but because ,,ci is the refraction of the ray ,,rc1 the proportion of ,,cv to ,,cd is #aef1ifb to #ih1ggi1 namely1 that of ,n to ,,cg2 then ,,cd is #ag1hbh4 ,now the rectangle g,,dc is to the square of ,,di as the square of ,,cg is to the square of ,,cm2 hence ,,di or ,,ce will be #ih1cec4 ,but as ,,ce is to ,,ei1 so will ,,cm be to ,,mt1 which will then be #ah1abg4 ,and being added to ,,ml1 which is #aa1fji 7namely the sine of the angle ,,lcm1 which is #f degrees #dj minutes1 taking ,,cm #ajj1jjj as radius7 we get ,,lt #bg1icf2 and this is to ,,lc #ii1cbd as ,,cv to ,,vr1 that is to say1 as #bi1ich1 the tangent of the complement of the angle ,,rcv1 which is #gc degrees #bj minutes1 is to the radius of the ,tables4 ,whence it appears that ,,rcit is a straight line2 which was to be proved4 #ce4 ,further it will be seen that the ray ,,ci in emerging through the opposite surface of the crystal1 ought to pass out quite straight1 according to the following demonstration1 which proves that the reciprocal relation of refraction obtains in this crystal the same as in other transparent bodies2 that is to say1 that if a ray ,,rc in meeting the surface of the crystal ,,cg is refracted as ,,ci1 the ray ,,ci emerging through the opposite parallel surface of the crystal1 which ,i suppose to be ,,ib1 will have its refraction ,,ia parallel to the ray ,,rc4 ,illustration ,let the same things be supposed as before2 that is to say1 let ,,co1 perpendicular to ,,cr1 represent a portion of a wave the continuation of which in the crystal is ,,ik1 so that the piece ,c will be continued on along the straight line ,,ci1 while ,o comes to ,k4 ,now if one takes a second period of time equal to the first1 the piece ,k of the wave ,,ik will1 in this second period1 have advanced along the straight line ,,kb1 equal and parallel to ,,ci1 because every piece of the wave ,,co1 on arriving at the surface ,,ck1 ought to go on in the crystal the same as the piece ,c2 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 ,,ko1 since ,,ko has been traversed in an equal time4 ,similarly1 if one considers some other point of the wave ,,ik1 such as h1 it will go along hm1 parallel to ,,ci1 to meet the surface ,,ib1 while the point ,k traverses ,kl equal to hm2 and while this accomplishes the remainder l,b1 there will start from the point m a partial wave the semi-diameter of which1 mn1 will have the same ratio to l,b as ,,ia to ,,kb4 ,whence it is evident that this wave of semi-diameter mn1 and the other of semi-diameter ,,ia will have the same tangent ,,ba4 ,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 ,,ib4 ,it is then precisely the tangent ,,ba which will be the continuation of the wave ,,ik1 outside the crystal1 when the piece ,k has reached ,b4 ,and in consequence ,,ia1 which is perpendicular to ,,ba1 will be the refraction of the ray ,,ci on emerging from the crystal4 ,now it is clear that ,,ia is parallel to the incident ray ,,rc1 since ,,ib is equal to ,,ck1 and ,,ia equal to ,,ko1 and the angles ,a and ,o are right angles4 ,it is seen then that1 according to our hypothesis1 the reciprocal relation of refraction holds good in this crystal as well as in ordinary transparent bodies2 as is thus in fact found by observation4 #cf4 ,i pass now to the consideration of other sections of the crystal1 and of the refractions there produced1 on which1 as will be seen1 some other very remarkable phenomena depend4 ,let ,,abh be a parallelepiped of crystal1 and let the top surface ,,aehf be a perfect rhombus1 the obtuse angles of which are equally divided by the straight line ,,ef1 and the acute angles by the straight line ,,ah perpendicular to ,,fe4 ,the section which we have hitherto considered is that which passes through the lines ,,ef1 ,,eb1 and which at the same time cuts the plane ,,aehf at right angles4 ,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 angles1 is that in which the refracted ray also is found4 ,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 surface1 and turns away towards the side of the slope of the crystal4 ,for which fact we shall show the reason1 in the first place1 for the section through ,,ah2 and we shall show at the same time how one can determine the refraction1 according to our hypothesis4 ,let there be1 then1 in the plane which passes through ,,ah1 and which is perpendicular to the plane ,,afhe1 the incident ray ,,rc2 it is required to find its refraction in the crystal4 ,illustration #cg4 ,about the centre ,c1 which ,i suppose to be in the intersection of ,,ah and ,,fe1 let there be imagined a hemi-spheroid ,,qgqg,m1 such as the light would form in spreading in the crystal1 and let its section by the plane ,,aehf form the ,ellipse ,,qgqg1 the major diameter of which ,qq1 which is in the line ,,ah1 will necessarily be one of the major diameters of the spheroid2 because the axis of the spheroid being in the plane through ,,feb1 to which ,,qc is perpendicular1 it follows that ,,qc is also perpendicular to the axis of the spheroid1 and consequently ,,qcq one of its major diameters4 ,but the minor diameter of this ,ellipse1 ,gg1 will bear to ,qq the proportion which has been defined previously1 ,article #bg1 between ,,cg and the major semi-diameter of the spheroid1 ,,cp1 namely1 that of #ih1ggi to #aje1jcb4 ,let the line ,n be the length of the travel of light in air during the time in which1 within the crystal1 it makes1 from the centre ,c1 the spheroid ,,qcqg,m4 ,then having drawn ,,co perpendicular to the ray ,,cr and situate in the plane through ,,cr and ,,ah1 let there be adjusted1 across the angle ,,aco1 the straight line ,,ok equal to ,n and perpendicular to ,,co1 and let it meet the straight line ,,ah at ,k4 ,supposing consequently that ,,cl is perpendicular to the surface of the crystal ,,aehf1 and that ,,cm is the refraction of the ray which falls perpendicularly on this same surface1 let there be drawn a plane through the line ,,cm and through ,,kch1 making in the spheroid the semi-ellipse ,,qmq1 which will be given1 since the angle ,,mcl is given of value #f degrees #dj minutes4 ,and it is certain1 according to what has been explained above1 ,article #bg1 that a plane which would touch the spheroid at the point ,m1 where ,i suppose the straight line ,,cm to meet the surface1 would be parallel to the plane ,,qgq4 ,if then through the point ,k one now draws ,,ks parallel to ,gg1 which will be parallel also to ,,qx1 the tangent to the ,ellipse ,,qgq at ,q2 and if one conceives a plane passing through ,,ks and touching the spheroid1 the point of contact will necessarily be in the ,ellipse ,,qmq1 because this plane through ,,ks1 as well as the plane which touches the spheroid at the point ,m1 are parallel to ,,qx1 the tangent of the spheroid3 for this consequence will be demonstrated at the end of this ,treatise4 ,let this point of contact be at ,i1 then making ,,kc1 ,,qc1 ,,dc proportionals1 draw ,,di parallel to ,,cm2 also join ,,ci4 ,i say that ,,ci will be the required refraction of the ray ,,rc4 ,this will be manifest if1 in considering ,,co1 which is perpendicular to the ray ,,rc1 as a portion of the wave of light1 we can demonstrate that the continuation of its piece ,c will be found in the crystal at ,i1 when ,o has arrived at ,k4 #ch4 ,now as in the ,chapter on ,reflexion1 in demonstrating that the incident and reflected rays are always in the same plane perpendicular to the reflecting surface1 we considered the breadth of the wave of light1 so1 similarly1 we must here consider the breadth of the wave ,,co in the diameter ,gg4 ,taking then the breadth ,cc on the side toward the angle ,e1 let the parallelogram ,,cooc be taken as a portion of a wave1 and let us complete the parallelograms ,,ckkc1 ,,ciic1 ,klik1 ,,okko4 ,in the time then that the line ,oo arrives at the surface of the crystal at ,kk1 all the points of the wave ,,cooc will have arrived at the rectangle ,kc along lines parallel to ,,ok2 and from the points of their incidences there will originate1 beyond that1 in the crystal partial hemi-spheroids1 similar to the hemi-spheroid ,,qmq1 and similarly disposed4 ,these hemi-spheroids will necessarily all touch the plane of the parallelogram ,,kiik at the same instant that ,oo has reached ,kk4 ,which is easy to comprehend1 since1 of these hemi-spheroids1 all those which have their centres along the line ,,ck1 touch this plane in the line ,,ki 7for this is to be shown in the same way as we have demonstrated the refraction of the oblique ray in the principal section through ,,ef7 and all those which have their centres in the line ,cc will touch the same plane ,,ki in the line ,ii2 all these being similar to the hemi-spheroid ,,qmq4 ,since then the parallelogram ,ki is that which touches all these spheroids1 this same parallelogram will be precisely the continuation of the wave ,,cooc in the crystal1 when ,oo has arrived at ,kk1 because it forms the termination of the movement and because of the quantity of movement which occurs more there than anywhere else3 and thus it appears that the piece ,c of the wave ,,cooc has its continuation at ,i2 that is to say1 that the ray ,,rc is refracted as ,,ci4 ,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 ,,cq2 by which one will easily find the refractions of all incident rays1 in the same way as we have shown previously for the case of the section through ,,fe2 and the demonstration will be the same4 ,but it appears that the said proportion of the refraction is less here than in the section through ,,feb2 for it was there the same as the ratio of ,n to ,,cg1 that is to say1 as #aef1ifb to #ih1ggi1 very nearly as #h to #e2 and here it is the ratio of ,n to ,,cq the major semi-diameter of the spheroid1 that is to say1 as #aef1ifb to #aje1jcb1 very nearly as #c to #b1 but just a little less4 ,which still agrees perfectly with what one finds by observation4 #ci4 ,for the rest1 this diversity of proportion of refraction produces a very singular effect in this ,crystal2 which is that when it is placed upon a sheet of paper on which there are letters or anything else marked1 if one views it from above with the two eyes situated in the plane of the section through ,,ef1 one sees the letters raised up by this irregular refraction more than when one puts one's eyes in the plane of section through ,,ah3 and the difference of these elevations appears by comparison with the other ordinary refraction of the crystal1 the proportion of which is as #e to #c1 and which always raises the letters equally1 and higher than the irregular refraction does4 ,for one sees the letters and the paper on which they are written1 as on two different stages at the same time2 and in the first position of the eyes1 namely1 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 ,,ef4 ,we will show that this effect follows from the refractions2 and it will enable us at the same time to ascertain the apparent place of a point of an object placed immediately under the crystal1 according to the different situation of the eyes4 #dj4 ,let us see first by how much the irregular refraction of the plane through ,,ah ought to lift the bottom of the crystal4 ,let the plane of this figure represent separately the section through ,qq and ,,cl1 in which section there is also the ray ,,rc1 and let the semi-elliptic plane through ,qq and ,,cm be inclined to the former1 as previously1 by an angle of #f degrees #dj minutes2 and in this plane ,,ci is then the refraction of the ray ,,rc4 ,illustration ,if now one considers the point ,i as at the bottom of the crystal1 and that it is viewed by the rays ,,icr1 ,icr1 refracted equally at the points ,cc1 which should be equally distant from ,d1 and that these rays meet the two eyes at ,rr2 it is certain that the point ,i will appear raised to ,s where the straight lines ,,rc1 rc1 meet2 which point ,s is in ,,dp1 perpendicular to ,qq4 ,and if upon ,,dp there is drawn the perpendicular ,,ip1 which will lie at the bottom of the crystal1 the length ,,sp will be the apparent elevation of the point ,i above the bottom4 ,let there be described on ,qq a semicircle cutting the ray ,,cr at ,b1 from which ,,bv is drawn perpendicular to ,qq2 and let the proportion of the refraction for this section be1 as before1 that of the line ,n to the semi-diameter ,,cq4 ,then as ,n is to ,,cq so is ,,vc to ,,cd1 as appears by the method of finding the refraction which we have shown above1 ,article #ca2 but as ,,vc is to ,,cd1 so is ,,vb to ,,ds4 ,then as ,n is to ,,cq1 so is ,,vb to ,,ds4 ,let ,,ml be perpendicular to ,,cl4 ,and because ,i suppose the eyes ,rr to be distant about a foot or so from the crystal1 and consequently the angle ,,rsr very small1 ,,vb may be considered as equal to the semi-diameter ,,cq1 and ,,dp as equal to ,,cl2 then as ,n is to ,,cq so is ,,cq to ,,ds4 ,but ,n is valued at #aef1ifb parts1 of which ,,cm contains #ajj1jjj and ,,cq #aje1jcb4 ,then ,,ds will have #gj1bhc4 ,but ,,cl is #ii1cbd1 being the sine of the complement of the angle ,,mcl which is #f degrees #dj minutes2 ,,cm being supposed as radius4 ,then ,,dp1 considered as equal to ,,cl1 will be to ,,ds as #ii1cbd to #gj1bhc4 ,and so the elevation of the point ,i by the refraction of this section is known4 ,illustration #da4 ,now let there be represented the other section through ,,ef in the figure before the preceding one2 and let ,,cmg be the semi-ellipse1 considered in ,articles #bg and #bh1 which is made by cutting a spheroidal wave having centre ,c4 ,let the point ,i1 taken in this ellipse1 be imagined again at the bottom of the ,crystal2 and let it be viewed by the refracted rays ,,icr1 ,icr1 which go to the two eyes2 ,,cr and cr being equally inclined to the surface of the crystal ,gg4 ,this being so1 if one draws ,,id parallel to ,,cm1 which ,i suppose to be the refraction of the perpendicular ray incident at the point ,c1 the distances ,,dc1 ,dc1 will be equal1 as is easy to see by that which has been demonstrated in ,article #bh4 ,now it is certain that the point ,i should appear at ,s where the straight lines ,,rc1 rc1 meet when prolonged2 and that this point will fall in the line ,,dp perpendicular to ,gg4 ,if one draws ,,ip perpendicular to this ,,dp1 it will be the distance ,,ps which will mark the apparent elevation of the point ,i4 ,let there be described on ,gg a semicircle cutting ,,cr at ,b1 from which let ,,bv be drawn perpendicular to ,gg2 and let ,n to ,,gc be the proportion of the refraction in this section1 as in ,article #bh4 ,since then ,,ci is the refraction of the radius ,,bc1 and ,,di is parallel to ,,cm1 ,,vc must be to ,,cd as ,n to ,,gc1 according to what has been demonstrated in ,article #ca4 ,but as ,,vc is to ,,cd so is ,,bv to ,,ds4 ,let ,,ml be drawn perpendicular to ,,cl4 ,and because ,i consider1 again1 the eyes to be distant above the crystal1 ,,bv is deemed equal to the semi-diameter ,,cg2 and hence ,,ds will be a third proportional to the lines ,n and ,,cg3 also ,,dp will be deemed equal to ,,cl4 ,now ,,cg consisting of #ih1ggh parts1 of which ,,cm contains #ajj1jjj1 ,n is taken as #aef1ifb4 ,then ,,ds will be #fb1afc4 ,but ,,cl is also determined1 and contains #ii1cbd parts1 as has been said in ,articles #cd and #dj4 ,then the ratio of ,,pd to ,,ds will be as #ii1cbd to #fb1afc4 ,and thus one knows the elevation of the point at the bottom ,i by the refraction of this section2 and it appears that this elevation is greater than that by the refraction of the preceding section1 since the ratio of ,,pd to ,,ds was there as #ii1cbd to #gj1bhc4 ,illustration ,but by the regular refraction of the crystal1 of which we have above said that the proportion is #e to #c1 the elevation of the point ,i1 or ,p1 from the bottom1 will be #b_/e of the height ,,dp2 as appears by this figure1 where the point ,p being viewed by the rays ,,pcr1 ,pcr1 refracted equally at the surface ,cc1 this point must needs appear to be at ,s1 in the perpendicular ,,pd where the lines ,,rc1 rc1 meet when prolonged3 and one knows that the line ,,pc is to ,,cs as #e to #c1 since they are to one another as the sine of the angle ,,csp or ,,dsc is to the sine of the angle ,,spc4 ,and because the ratio of ,,pd to ,,ds is deemed the same as that of ,,pc to ,,cs1 the two eyes ,rr being supposed very far above the crystal1 the elevation ,,ps will thus be #b_/e of ,,pd4 ,illustration #db4 ,if one takes a straight line ,,ab for the thickness of the crystal1 its point ,b being at the bottom1 and if one divides it at the points ,c1 ,d1 ,e1 according to the proportions of the elevations found1 making ,,ae #c_/e of ,,ab1 ,,ab to ,,ac as #ii1cbd to #gj1bhc1 and ,,ab to ,,ad as #ii1cbd to #fb1afc1 these points will divide ,,ab as in this figure4 ,and it will be found that this agrees perfectly with experiment2 that is to say by placing the eyes above in the plane which cuts the crystal according to the shorter diameter of the rhombus1 the regular refraction will lift up the letters to ,e2 and one will see the bottom1 and the letters over which it is placed1 lifted up to ,d by the irregular refraction4 ,but by placing the eyes above in the plane which cuts the crystal according to the longer diameter of the rhombus1 the regular refraction will lift the letters to ,e as before2 but the irregular refraction will make them1 at the same time1 appear lifted up only to ,c2 and in such a way that the interval ,,ce will be quadruple the interval ,,ed1 which one previously saw4 #dc4 ,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 refraction1 but they are separated from them by being more distant from the equilateral solid angle of the ,crystal4 ,that follows1 indeed1 from all that has been hitherto demonstrated about the irregular refraction2 and it is particularly shown by these last demonstrations1 from which one sees that the point ,i appears by irregular refraction at ,s in the perpendicular line ,,dp1 in which line also the image of the point ,p ought to appear by regular refraction1 but not the image of the point ,i1 which will be almost directly above the same point1 and higher than ,s4 ,but as to the apparent elevation of the point ,i in other positions of the eyes above the crystal1 besides the two positions which we have just examined1 the image of that point by the irregular refraction will always appear between the two heights of ,d and ,c1 passing from one to the other as one turns one's self around about the immovable crystal1 while looking down from above4 ,and all this is still found conformable to our hypothesis1 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 crystal1 besides the two which we have considered4 ,let us suppose one of the faces of the crystal1 in which let there be the ,ellipse ,,hde1 the centre ,c of which is also the centre of the spheroid ,,hme in which the light spreads1 and of which the said ,ellipse is the section4 ,and let the incident ray be ,,rc1 the refraction of which it is required to find4 ,let there be taken a plane passing through the ray ,,rc and which is perpendicular to the plane of the ellipse ,,hde1 cutting it along the straight line ,,bck2 and having in the same plane through ,,rc made ,,co perpendicular to ,,cr1 let ,,ok be adjusted across the angle ,,ock1 so as to be perpendicular to ,,oc and equal to the line ,n1 which ,i suppose to measure the travel of the light in air during the time that it spreads in the crystal through the spheroid ,,hdem4 ,then in the plane of the ,ellipse ,,hde let ,,kt be drawn1 through the point ,k1 perpendicular to ,,bck4 ,now if one conceives a plane drawn through the straight line ,,kt and touching the spheroid ,,hme at ,i1 the straight line ,,ci will be the refraction of the ray ,,rc1 as is easy to deduce from that which has been demonstrated in ,article #cf4 ,illustration ,but it must be shown how one can determine the point of contact ,i4 ,let there be drawn parallel to the line ,,kt a line ,,hf which touches the ,ellipse ,,hde1 and let this point of contact be at ,h4 ,and having drawn a straight line along ,,ch to meet ,,kt at ,t1 let there be imagined a plane passing through the same ,,ch and through ,,cm 7which ,i suppose to be the refraction of the perpendicular ray71 which makes in the spheroid the elliptical section ,,hme4 ,it is certain that the plane which will pass through the straight line ,,kt1 and which will touch the spheroid1 will touch it at a point in the ,ellipse ,,hme1 according to the ,lemma which will be demonstrated at the end of the ,chapter4 ,now this point is necessarily the point ,i which is sought1 since the plane drawn through ,,tk can touch the spheroid at one point only4 ,and this point ,i is easy to determine1 since it is needful only to draw from the point ,t1 which is in the plane of this ,ellipse1 the tangent ,,ti1 in the way shown previously4 ,for the ,ellipse ,,hme is given1 and its conjugate semi-diameters are ,,ch and ,,cm2 because a straight line drawn through ,m1 parallel to ,,he1 touches the ,ellipse ,,hme1 as follows from the fact that a plane taken through ,m1 and parallel to the plane ,,hde1 touches the spheroid at that point ,m1 as is seen from ,articles #bg and #bc4 ,for the rest1 the position of this ellipse1 with respect to the plane through the ray ,,rc and through ,,ck1 is also given2 from which it will be easy to find the position of ,,ci1 the refraction corresponding to the ray ,,rc4 ,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 ,,ck4 ,because every plane1 parallel to the straight line ,,hf1 or ,,tk1 which will touch the spheroid1 will touch it in this ellipse1 according to the ,lemma quoted a little before4 ,i have investigated thus1 in minute detail1 the properties of the irregular refraction of this ,crystal1 in order to see whether each phenomenon that is deduced from our hypothesis accords with that which is observed in fact4 ,and this being so it affords no slight proof of the truth of our suppositions and principles4 ,but what ,i am going to add here confirms them again marvellously4 ,it is this3 that there are different sections of this ,crystal1 the surfaces of which1 thereby produced1 give rise to refractions precisely such as they ought to be1 and as ,i had foreseen them1 according to the preceding ,theory4 ,in order to explain what these sections are1 let ,,abkf be the principal section through the axis of the crystal ,,ack1 in which there will also be the axis ,,ss of a spheroidal wave of light spreading in the crystal from the centre ,c2 and the straight line which cuts ,,ss through the middle and at right angles1 namely ,,pp1 will be one of the major diameters4 ,illustration3 ,section ,,abkf ,now as in the natural section of the crystal1 made by a plane parallel to two opposite faces1 which plane is here represented by the line ,,gg1 the refraction of the surfaces which are produced by it will be governed by the hemi-spheroids ,,gng1 according to what has been explained in the preceding ,theory4 ,similarly1 cutting the ,crystal through ,,nn1 by a plane perpendicular to the parallelogram ,,abkf1 the refraction of the surfaces will be governed by the hemi-spheroids ,,ngn4 ,and if one cuts it through ,,pp1 perpendicularly to the said parallelogram1 the refraction of the surfaces ought to be governed by the hemi-spheroids ,,psp1 and so for others4 ,but ,i saw that if the plane ,,nn was almost perpendicular to the plane ,,gg1 making the angle ,,ncg1 which is on the side ,a1 an angle of #ij degrees #dj minutes1 the hemi-spheroids ,,ngn would become similar to the hemi-spheroids ,,gng1 since the planes ,,nn and ,,gg were equally inclined by an angle of #de degrees #bj minutes to the axis ,,ss4 ,in consequence it must needs be1 if our theory is true1 that the surfaces which the section through ,,nn produces should effect the same refractions as the surfaces of the section through ,,gg4 ,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 #de degrees #bj minutes4 ,so that there are an infinitude of planes which ought to produce precisely the same refractions as the natural surfaces of the crystal1 or as the section parallel to any one of those surfaces which are made by cleavage4 ,i saw also that by cutting it by a plane taken through ,,pp1 and perpendicular to the axis ,,ss1 the refraction of the surfaces ought to be such that the perpendicular ray should suffer thereby no deviation2 and that for oblique rays there would always be an irregular refraction1 differing from the regular1 and by which objects placed beneath the crystal would be less elevated than by that other refraction4 ,that1 similarly1 by cutting the crystal by any plane through the axis ,,ss1 such as the plane of the figure is1 the perpendicular ray ought to suffer no refraction2 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 was4 ,now these things were found in fact so2 and1 after that1 ,i could not doubt that a similar success could be met with everywhere4 ,whence ,i concluded that one might form from this crystal solids similar to those which are its natural forms1 which should produce1 at all their surfaces1 the same regular and irregular refractions as the natural surfaces1 and which nevertheless would cleave in quite other ways1 and not in directions parallel to any of their faces4 ,that out of it one would be able to fashion pyramids1 having their base square1 pentagonal1 hexagonal1 or with as many sides as one desired1 all the surfaces of which should have the same refractions as the natural surfaces of the crystal1 except the base1 which will not refract the perpendicular ray4 ,these surfaces will each make an angle of #de degrees #bj minutes with the axis of the crystal1 and the base will be the section perpendicular to the axis4 ,that1 finally1 one could also fashion out of it triangular prisms1 or prisms with as many sides as one would1 of which neither the sides nor the bases would refract the perpendicular ray1 although they would yet all cause double refraction for oblique rays4 ,the cube is included amongst these prisms1 the bases of which are sections perpendicular to the axis of the crystal1 and the sides are sections parallel to the same axis4 ,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 composed1 and according to which it splits in three different senses1 that the cause resides of its irregular refraction2 and that it would be in vain to wish to seek it there4 ,but in order that any one who has some of this stone may be able to find1 by his own experience1 the truth of what ,i have just advanced1 ,i will state here the process of which ,i have made use to cut it1 and to polish it4 ,cutting is easy by the slicing wheels of lapidaries1 or in the way in which marble is sawn3 but polishing is very difficult1 and by employing the ordinary means one more often depolishes the surfaces than makes them lucent4 ,after many trials1 ,i have at last found that for this service no plate of metal must be used1 but a piece of mirror glass made matt and depolished4 ,upon this1 with fine sand and water1 one smoothes the crystal little by little1 in the same way as spectacle glasses1 and polishes it simply by continuing the work1 but ever reducing the material4 ,i have not1 however1 been able to give it perfect clarity and transparency2 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 stone1 which always have some inequality4 ,even when the surface is only moderately smoothed1 if one rubs it over with a little oil or white of egg1 it becomes quite transparent1 so that the refraction is discerned in it quite distinctly4 ,and this aid is specially necessary when it is wished to polish the natural surfaces to remove the inequalities2 because one cannot render them lucent equally with the surfaces of other sections1 which take a polish so much the better the less nearly they approximate to these natural planes4 ,before finishing the treatise on this ,crystal1 ,i will add one more marvellous phenomenon which ,i discovered after having written all the foregoing4 ,for though ,i have not been able till now to find its cause1 ,i do not for that reason wish to desist from describing it1 in order to give opportunity to others to investigate it4 ,it seems that it will be necessary to make still further suppositions besides those which ,i have made2 but these will not for all that cease to keep their probability after having been confirmed by so many tests4 ,illustration ,the phenomenon is1 that by taking two pieces of this crystal and applying them one over the other1 or rather holding them with a space between the two1 if all the sides of one are parallel to those of the other1 then a ray of light1 such as ,,ab1 is divided into two in the first piece1 namely into ,,bd and ,,bc1 following the two refractions1 regular and irregular4 ,on penetrating thence into the other piece each ray will pass there without further dividing itself in two2 but that one which underwent the regular refraction1 as here ,,dg1 will undergo again only a regular refraction at ,,gh2 and the other1 ,,ce1 an irregular refraction at ,,ef4 ,and the same thing occurs not only in this disposition1 but also in all those cases in which the principal section of each of the pieces is situated in one and the same plane1 without it being needful for the two neighbouring surfaces to be parallel4 ,now it is marvellous why the rays ,,ce and ,,dg1 incident from the air on the lower crystal1 do not divide themselves the same as the first ray ,,ab4 ,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 refraction2 and that likewise ,,ce has lost that which was necessary to move the matter which serves for regular refraction3 but there is yet another thing which upsets this reasoning4 ,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 angles1 whether the neighbouring surfaces are parallel or not1 then the ray which has come by the regular refraction1 as ,,dg1 undergoes only an irregular refraction in the lower piece2 and on the contrary the ray which has come by the irregular refraction1 as ,,ce1 undergoes only a regular refraction4 ,but in all the infinite other positions1 besides those which ,i have just stated1 the rays ,,dg1 ,,ce1 divide themselves anew each one into two1 by refraction in the lower crystal so that from the single ray ,,ab there are four1 sometimes of equal brightness1 sometimes some much less bright than others1 according to the varying agreement in the positions of the crystals3 but they do not appear to have all together more light than the single ray ,,ab4 ,when one considers here how1 while the rays ,,ce1 ,,dg1 remain the same1 it depends on the position that one gives to the lower piece1 whether it divides them both in two1 or whether it does not divide them1 and yet how the ray ,,ab above is always divided1 it seems that one is obliged to conclude that the waves of light1 after having passed through the first crystal1 acquire a certain form or disposition in virtue of which1 when meeting the texture of the second crystal1 in certain positions1 they can move the two different kinds of matter which serve for the two species of refraction2 and when meeting the second crystal in another position are able to move only one of these kinds of matter4 ,but to tell how this occurs1 ,i have hitherto found nothing which satisfies me4 ,leaving then to others this research1 ,i pass to what ,i have to say touching the cause of the extraordinary figure of this crystal1 and why it cleaves easily in three different senses1 parallel to any one of its surfaces4 ,there are many bodies1 vegetable1 mineral1 and congealed salts1 which are formed with certain regular angles and figures4 ,thus among flowers there are many which have their leaves disposed in ordered polygons1 to the number of #c1 #d1 #e1 or #f sides1 but not more4 ,this well deserves to be investigated1 both as to the polygonal figure1 and as to why it does not exceed the number #f4 ,rock ,crystal grows ordinarily in hexagonal bars1 and diamonds are found which occur with a square point and polished surfaces4 ,there is a species of small flat stones1 piled up directly upon one another1 which are all of pentagonal figure with rounded angles1 and the sides a little folded inwards4 ,the grains of gray salt which are formed from sea water affect the figure1 or at least the angle1 of the cube2 and in the congelations of other salts1 and in that of sugar1 there are found other solid angles with perfectly flat faces4 ,small snowflakes almost always fall in little stars with #f points1 and sometimes in hexagons with straight sides4 ,and ,i have often observed1 in water which is beginning to freeze1 a kind of flat and thin foliage of ice1 the middle ray of which throws out branches inclined at an angle of #fj degrees4 ,all these things are worthy of being carefully investigated to ascertain how and by what artifice nature there operates4 ,but it is not now my intention to treat fully of this matter4 ,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 composed4 ,and1 coming to our ,iceland ,crystal1 ,i say that if there were a pyramid such as ,,abcd1 composed of small rounded corpuscles1 not spherical but flattened spheroids1 such as would be made by the rotation of the ellipse ,,gh around its lesser diameter ,,ef 7of which the ratio to the greater diameter is very nearly that of #a to the square root of #h7--;,i say that then the solid angle of the point ,d would be equal to the obtuse and equilateral angle of this ,crystal4 ,i say1 further1 that if these corpuscles were lightly stuck together1 on breaking this pyramid it would break along faces parallel to those that make its point3 and by this means1 as it is easy to see1 it would produce prisms similar to those of the same crystal as this other figure represents4 ,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 layer2 of which three there is but one which touches it on its flattened surface1 and the other two at the edges4 ,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 separated1 it would be needful for it to detach itself from six other spheroids which hold it locked1 and four of which press it by these flattened surfaces4 ,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 spheroids1 there is great reason to believe that the particles are shaped and ranged in the same way4 ,illustration3 ,pyramid and section of spheroids ,there is even probability enough that the prisms of this crystal are produced by the breaking up of pyramids1 since ,mr4 ,bartholinus relates that he occasionally found some pieces of triangularly pyramidal figure4 ,but when a mass is composed interiorly only of these little spheroids thus piled up1 whatever form it may have exteriorly1 it is certain1 by the same reasoning which ,i have just explained1 that if broken it would produce similar prisms4 ,it remains to be seen whether there are other reasons which confirm our conjecture1 and whether there are none which are repugnant to it4 ,illustration3 paralleloid arrangement of spheroids with planes of potential cleavage ,it may be objected that this crystal1 being so composed1 might be capable of cleavage in yet two more fashions2 one of which would be along planes parallel to the base of the pyramid1 that is to say to the triangle ,,abc2 the other would be parallel to a plane the trace of which is marked by the lines ,,gh1 ,,hk1 ,,kl4 ,to which ,i say that both the one and the other1 though practicable1 are more difficult than those which were parallel to any one of the three planes of the pyramid2 and that therefore1 when striking on the crystal in order to break it1 it ought always to split rather along these three planes than along the two others4 ,when one has a number of spheroids of the form above described1 and ranges them in a pyramid1 one sees why the two methods of division are more difficult4 ,for in the case of that division which would be parallel to the base1 each spheroid would be obliged to detach itself from three others which it touches upon their flattened surfaces1 which hold more strongly than the contacts at the edges4 ,and besides that1 this division will not occur along entire layers1 because each of the spheroids of a layer is scarcely held at all by the #f of the same layer that surround it1 since they only touch it at the edges2 so that it adheres readily to the neighbouring layer1 and the others to it1 for the same reason2 and this causes uneven surfaces4 ,also one sees by experiment that when grinding down the crystal on a rather rough stone1 directly on the equilateral solid angle1 one verily finds much facility in reducing it in this direction1 but much difficulty afterwards in polishing the surface which has been flattened in this manner4 ,as for the other method of division along the plane ,,ghkl1 it will be seen that each spheroid would have to detach itself from four of the neighbouring layer1 two of which touch it on the flattened surfaces1 and two at the edges4 ,so that this division is likewise more difficult than that which is made parallel to one of the surfaces of the crystal2 where1 as we have said1 each spheroid is detached from only three of the neighbouring layer3 of which three there is one only which touches it on the flattened surface1 and the other two at the edges only4 ,however1 that which has made me know that in the crystal there are layers in this last fashion1 is that in a piece weighing half a pound which ,i possess1 one sees that it is split along its length1 as is the above-mentioned prism by the plane ,,ghkl2 as appears by colours of the ,iris extending throughout this whole plane although the two pieces still hold together4 ,all this proves then that the composition of the crystal is such as we have stated4 ,to which ,i again add this experiment2 that if one passes a knife scraping along any one of the natural surfaces1 and downwards as it were from the equilateral obtuse angle1 that is to say from the apex of the pyramid1 one finds it quite hard2 but by scraping in the opposite sense an incision is easily made4 ,this follows manifestly from the situation of the small spheroids2 over which1 in the first manner1 the knife glides2 but in the other manner it seizes them from beneath almost as if they were the scales of a fish4 ,i will not undertake to say anything touching the way in which so many corpuscles all equal and similar are generated1 nor how they are set in such beautiful order2 whether they are formed first and then assembled1 or whether they arrange themselves thus in coming into being and as fast as they are produced1 which seems to me more probable4 ,to develop truths so recondite there would be needed a knowledge of nature much greater than that which we have4 ,i will add only that these little spheroids could well contribute to form the spheroids of the waves of light1 here above supposed1 these as well as those being similarly situated1 and with their axes parallel4 ,calculations which have been supposed in this ,chapter4 ,mr4 ,bartholinus1 in his treatise of this ,crystal1 puts at #aja degrees the obtuse angles of the faces1 which ,i have stated to be #aja degrees #eb minutes4 ,he states that he measured these angles directly on the crystal1 which is difficult to do with ultimate exactitude1 because the edges such as ,,ca1 ,,cb1 in this figure1 are generally worn1 and not quite straight4 ,for more certainty1 therefore1 ,i preferred to measure actually the obtuse angle by which the faces ,,cbda1 ,,cbvf1 are inclined to one another1 namely the angle ,,ocn formed by drawing ,,cn perpendicular to ,,fv1 and ,,co perpendicular to ,,da4 ,this angle ,,ocn ,i found to be #aje degrees2 and its supplement ,,cnp1 to be #ge degrees1 as it should be4 ,illustration ,to find from this the obtuse angle ,,bca1 ,i imagined a sphere having its centre at ,c1 and on its surface a spherical triangle1 formed by the intersection of three planes which enclose the solid angle ,c4 ,in this equilateral triangle1 which is ,,abf in this other figure1 ,i see that each of the angles should be #aje degrees1 namely equal to the angle ,,ocn2 and that each of the sides should be of as many degrees as the angle ,,acb1 or ,,acf1 or ,,bcf4 ,having then drawn the arc ,,fq perpendicular to the side ,,ab1 which it divides equally at ,q1 the triangle ,,fqa has a right angle at ,q1 the angle ,a #aje degrees1 and ,f half as much1 namely #eb degrees #cj minutes2 whence the hypotenuse ,,af is found to be #aja degrees #eb minutes4 ,and this arc ,,af is the measure of the angle ,,acf in the figure of the crystal4 ,illustration ,in the same figure1 if the plane ,,cghf cuts the crystal so that it divides the obtuse angles ,,acb1 ,,mhv1 in the middle1 it is stated1 in ,article #aj1 that the angle ,,cfh is #gj degrees #eg minutes4 ,this again is easily shown in the same spherical triangle ,,abf1 in which it appears that the arc ,,fq is as many degrees as the angle ,,gcf in the crystal1 the supplement of which is the angle ,,cfh4 ,now the arc ,,fq is found to be #aji degrees #c minutes4 ,then its supplement1 #gj degrees #eg minutes1 is the angle ,,cfh4 ,it was stated1 in ,article #bf1 that the straight line ,,cs1 which in the preceding figure is ,,ch1 being the axis of the crystal1 that is to say being equally inclined to the three sides ,,ca1 ,,cb1 ,,cf1 the angle ,,gch is #de degrees #bj minutes4 ,this is also easily calculated by the same spherical triangle4 ,for by drawing the other arc ,,ad which cuts ,,bf equally1 and intersects ,,fq at ,s1 this point will be the centre of the triangle4 ,and it is easy to see that the arc ,,sq is the measure of the angle ,,gch in the figure which represents the crystal4 ,now in the triangle ,,qas1 which is right-angled1 one knows also the angle ,a1 which is #eb degrees #cj minutes1 and the side ,,aq #ej degrees #ef minutes2 whence the side ,,sq is found to be #de degrees #bj minutes4 ,in ,article #bg it was required to show that ,,pms being an ellipse the centre of which is ,c1 and which touches the straight line ,,md at ,m so that the angle ,,mcl which ,,cm makes with ,,cl1 perpendicular on ,,dm1 is #f degrees #dj minutes1 and its semi-minor axis ,,cs making with ,,cg 7which is parallel to ,,md7 an angle ,,gcs of #de degrees #bj minutes1 it was required to show1 ,i say1 that1 ,,cm being #ajj1jjj parts1 ,,pc the semi-major diameter of this ellipse is #aje1jcb parts1 and ,,cs1 the semi-minor diameter1 #ic1daj4 ,let ,,cp and ,,cs be prolonged and meet the tangent ,,dm at ,d and ,z2 and from the point of contact ,m let ,,mn and ,,mo be drawn as perpendiculars to ,,cp and ,,cs4 ,now because the angles ,,scp1 ,,gcl1 are right angles1 the angle ,,pcl will be equal to ,,gcs which was #de degrees #bj minutes4 ,and deducting the angle ,,lcm1 which is #f degrees #dj minutes1 from ,,lcp1 which is #de degrees #bj minutes1 there remains ,,mcp1 #ch degrees #dj minutes4 ,considering then ,,cm as a radius of #ajj1jjj parts1 ,,mn1 the sine of #ch degrees #dj minutes1 will be #fb1dgi4 ,and in the right-angled triangle ,,mnd1 ,,mn will be to ,,nd as the radius of the ,tables is to the tangent of #de degrees #bj minutes 7because the angle ,,nmd is equal to ,,dcl1 or ,,gcs72 that is to say as #ajj1jjj to #aja1agj3 whence results ,,nd #fc1baj4 ,but ,,nc is #gh1jgi of the same parts1 ,,cm being #ajj1jjj1 because ,,nc is the sine of the complement of the angle ,,mcp1 which was #ch degrees #dj minutes4 ,then the whole line ,,dc is #ada1bhi2 and ,,cp1 which is a mean proportional between ,,dc and ,,cn1 since ,,md touches the ,ellipse1 will be #aje1jcb4 ,illustration ,similarly1 because the angle ,,omz is equal to ,,cdz1 or ,,lcz1 which is #dd degrees #dj minutes1 being the complement of ,,gcs1 it follows that1 as the radius of the ,tables is to the tangent of #dd degrees #dj minutes1 so will ,,om #gh1jgi be to ,,oz #gg1agf4 ,but ,,oc is #fb1dgi of these same parts of which ,,cm is #ajj1jjj1 because it is equal to ,,mn1 the sine of the angle ,,mcp1 which is #ch degrees #dj minutes4 ,then the whole line ,,cz is #aci1fee2 and ,,cs1 which is a mean proportional between ,,cz and ,,co will be #ic1daj4 ,at the same place it was stated that ,,gc was found to be #ih1ggi parts4 ,to prove this1 let ,,pe be drawn in the same figure parallel to ,,dm1 and meeting ,,cm at ,e4 ,in the right-angled triangle ,,cld the side ,,cl is #ii1cbd 7,,cm being #ajj1jjj71 because ,,cl is the sine of the complement of the angle ,,lcm1 which is #f degrees #dj minutes4 ,and since the angle ,,lcd is #de degrees #bj minutes1 being equal to ,,gcs1 the side ,,ld is found to be #ajj1dhf3 whence deducting ,,ml #aa1fji there will remain ,,md #hh1hgg4 ,now as ,,cd 7which was #ada1bhi7 is to ,,dm #hh1hgg1 so will ,,cp #aje1jcb be to ,,pe #ff1jgj4 ,but as the rectangle ,,meh 7or rather the difference of the squares on ,,cm and ,,ce7 is to the square on ,,mc1 so is the square on ,,pe to the square on ,cg2 then also as the difference of the squares on ,,dc and ,,cp to the square on ,,cd1 so also is the square on ,,pe to the square on g,c4 ,but ,,dp1 ,,cp1 and ,,pe are known2 hence also one knows ,,gc1 which is #ih1ggi4 ,lemma which has been supposed4 ,if a spheroid is touched by a straight line1 and also by two or more planes which are parallel to this line1 though not parallel to one another1 all the points of contact of the line1 as well as of the planes1 will be in one and the same ellipse made by a plane which passes through the centre of the spheroid4 ,let ,,led be the spheroid touched by the line ,,bm at the point ,b1 and also by the planes parallel to this line at the points ,o and ,a4 ,it is required to demonstrate that the points ,b1 ,o1 and ,a are in one and the same ,ellipse made in the spheroid by a plane which passes through its centre4 ,illustration ,through the line ,,bm1 and through the points ,o and ,a1 let there be drawn planes parallel to one another1 which1 in cutting the spheroid make the ellipses ,,lbd1 ,,pop1 ,,qaq2 which will all be similar and similarly disposed1 and will have their centres ,k1 ,n1 ,r1 in one and the same diameter of the spheroid1 which will also be the diameter of the ellipse made by the section of the plane that passes through the centre of the spheroid1 and which cuts the planes of the three said ,ellipses at right angles3 for all this is manifest by proposition #ae of the book of ,conoids and ,spheroids of ,archimedes4 ,further1 the two latter planes1 which are drawn through the points ,o and ,a1 will also1 by cutting the planes which touch the spheroid in these same points1 generate straight lines1 as ,,oh and ,,as1 which will1 as is easy to see1 be parallel to ,,bm2 and all three1 ,,bm1 ,,oh1 ,,as1 will touch the ,ellipses ,,lbd1 ,,pop1 ,,qaq in these points1 ,b1 ,o1 ,a2 since they are in the planes of these ellipses1 and at the same time in the planes which touch the spheroid4 ,if now from these points ,b1 ,o1 ,a1 there are drawn the straight lines ,,bk1 ,,on1 ,,ar1 through the centres of the same ellipses1 and if through these centres there are drawn also the diameters ,,ld1 ,,pp1 ,,qq1 parallel to the tangents ,,bm1 ,,oh1 ,,as2 these will be conjugate to the aforesaid ,,bk1 ,,on1 ,,ar4 ,and because the three ellipses are similar and similarly disposed1 and have their diameters ,,ld1 ,,pp1 ,,qq parallel1 it is certain that their conjugate diameters ,,bk1 ,,on1 ,,ar1 will also be parallel4 ,and the centres ,k1 ,n1 ,r being1 as has been stated1 in one and the same diameter of the spheroid1 these parallels ,,bk1 ,,on1 ,,ar will necessarily be in one and the same plane1 which passes through this diameter of the spheroid1 and1 in consequence1 the points ,r1 ,o1 ,a are in one and the same ellipse made by the intersection of this plane4 ,which was to be proved4 ,and it is manifest that the demonstration would be the same if1 besides the points ,o1 ,a1 there had been others in which the spheroid had been touched by planes parallel to the straight line ,,bm4 ,,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 light1 and of opaque bodies1 and of transparent media1 ,i will here set forth a very easy and natural way of deducing1 from the same principles1 the true figures which serve1 either by reflexion or by refraction1 to collect or disperse the rays of light1 as may be desired4 ,for though ,i do not see yet that there are means of making use of these figures1 so far as relates to ,refraction1 not only because of the difficulty of shaping the glasses of ,telescopes with the requisite exactitude according to these figures1 but also because there exists in refraction itself a property which hinders the perfect concurrence of the rays1 as ,mr4 ,newton has very well proved by experiment1 ,i will yet not desist from relating the invention1 since it offers itself1 so to speak1 of itself1 and because it further confirms our ,theory of refraction1 by the agreement which here is found between the refracted ray and the reflected ray4 ,besides1 it may occur that some one in the future will discover in it utilities which at present are not seen4 ,illustration ,to proceed then to these figures1 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 ,a2 and that the summit of the surface shall be the given point ,d in the straight line ,,ab4 ,i say that1 whether by reflexion or by refraction1 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 ,,cde1 and from these to the point of concurrence 7as here the path along the straight lines ,,ac1 ,,cb1 along ,,al1 ,,lb1 and along ,,ad1 ,,db71 shall be everywhere traversed in equal times3 by which principle the finding of these curves becomes very easy4 ,illustration ,so far as relates to the reflecting surface1 since the sum of the lines ,,ac1 ,,cb ought to be equal to that of ,,ad1 ,,db1 it appears that ,,dce ought to be an ellipse2 and for refraction1 the ratio of the velocities of waves of light in the media ,a and ,b being supposed to be known1 for example that of #c to #b 7which is the same1 as we have shown1 as the ratio of the ,sines in the refraction71 it is only necessary to make ,,dh equal to #c_/b of ,,db2 and having after that described from the centre ,a some arc ,,fc1 cutting ,,db at ,f1 then describe another from centre ,b with its semi-diameter ,,bx equal to #b_/c of ,,fh2 and the point of intersection of the two arcs will be one of the points required1 through which the curve should pass4 ,for this point1 having been found in this fashion1 it is easy forthwith to demonstrate that the time along ,,ac1 ,,cb1 will be equal to the time along ,,ad1 ,,db4 ,for assuming that the line ,,ad represents the time which the light takes to traverse this same distance ,,ad in air1 it is evident that ,,dh1 equal to #c_/b of ,,db1 will represent the time of the light along ,,db in the medium1 because it needs here more time in proportion as its speed is slower4 ,therefore the whole line ,,ah will represent the time along ,,ad1 ,,db4 ,similarly the line ,,ac or ,,af will represent the time along ,,ac2 and ,,fh being by construction equal to #c_/b of ,,cb1 it will represent the time along ,,cb in the medium2 and in consequence the whole line ,,ah will represent also the time along ,,ac1 ,,cb4 ,whence it appears that the time along ,,ac1 ,,cb1 is equal to the time along ,,ad1 ,,db4 ,and similarly it can be shown if ,l and ,k are other points in the curve ,,cde1 that the times along ,,al1 ,,lb1 and along ,,ak1 ,,kb1 are always represented by the line ,,ah1 and therefore equal to the said time along ,,ad1 ,,db4 ,in order to show further that the surfaces1 which these curves will generate by revolution1 will direct all the rays which reach them from the point ,a in such wise that they tend towards ,b1 let there be supposed a point ,k in the curve1 farther from ,d than ,c is1 but such that the straight line ,,ak falls from outside upon the curve which serves for the refraction2 and from the centre ,b let the arc ,,ks be described1 cutting ,,bd at ,s1 and the straight line ,,cb at ,r2 and from the centre ,a describe the arc ,,dn meeting ,,ak at ,n4 ,since the sums of the times along ,,ak1 ,,kb1 and along ,,ac1 ,,cb are equal1 if from the former sum one deducts the time along ,,kb1 and if from the other one deducts the time along ,,rb1 there will remain the time along ,,ak as equal to the time along the two parts ,,ac1 ,,cr4 ,consequently in the time that the light has come along ,,ak it will also have come along ,,ac and will in addition have made1 in the medium from the centre ,c1 a partial spherical wave1 having a semi-diameter equal to ,,cr4 ,and this wave will necessarily touch the circumference ,,ks at ,r1 since ,,cb cuts this circumference at right angles4 ,similarly1 having taken any other point ,l in the curve1 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 wave1 from the centre ,l1 which will touch the same circumference ,,ks4 ,and so with all other points of the curve ,,cde4 ,then at the moment that the light reaches ,k the arc ,,krs will be the termination of the movement1 which has spread from ,a through ,,dck4 ,and thus this same arc will constitute in the medium the propagation of the wave emanating from ,a2 which wave may be represented by the arc ,,dn1 or by any other nearer the centre ,a4 ,but all the pieces of the arc ,,krs are propagated successively along straight lines which are perpendicular to them1 that is to say1 which tend to the centre ,b 7for 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 centre71 and these progressions of the pieces of the waves constitute the rays themselves of light4 ,it appears then that all these rays tend here towards the point ,b4 ,one might also determine the point ,c1 and all the others1 in this curve which serves for the refraction1 by dividing ,,da at ,g in such a way that ,,dg is #b_/c of ,,da1 and describing from the centre ,b any arc ,,cx which cuts ,,bd at ,n1 and another from the centre ,a with its semi-diameter ,,af equal to #c_/b of ,,gx2 or rather1 having described1 as before1 the arc ,,cx1 it is only necessary to make ,,df equal to #c_/b of ,,dx1 and from-the centre ,a to strike the arc ,,fc2 for these two constructions1 as may be easily known1 come back to the first one which was shown before4 ,and it is manifest by the last method that this curve is the same that ,mr4 ,des ,cartes has given in his ,geometry1 and which he calls the first of his ,ovals4 ,it is only a part of this oval which serves for the refraction1 namely1 the part ,,dk1 ending at ,k1 if ,,ak is the tangent4 ,as to the1 other part1 ,des ,cartes has remarked that it could serve for reflexions1 if there were some material of a mirror of such a nature that by its means the force of the rays 7or1 as we should say1 the velocity of the light1 which he could not say1 since he held that the movement of light was instantaneous7 could be augmented in the proportion of #c to #b4 ,but we have shown that in our way of explaining reflexion1 such a thing could not arise from the matter of the mirror1 and it is entirely impossible4 ,illustration ,illustration ,from what has been demonstrated about this oval1 it will be easy to find the figure which serves to collect to a point incident parallel rays4 ,for by supposing just the same construction1 but the point ,a infinitely distant1 giving parallel rays1 our oval becomes a true ,ellipse1 the construction of which differs in no way from that of the oval1 except that ,,fc1 which previously was an arc of a circle1 is here a straight line1 perpendicular to ,,db4 ,for the wave of light ,,dn1 being likewise represented by a straight line1 it will be seen that all the points of this wave1 travelling as far as the surface ,,kd along lines parallel to ,,db1 will advance subsequently towards the point ,b1 and will arrive there at the same time4 ,as for the ,ellipse which served for reflexion1 it is evident that it will here become a parabola1 since its focus ,a may be regarded as infinitely distant from the other1 ,b1 which is here the focus of the parabola1 towards which all the reflexions of rays parallel to ,,ab tend4 ,and the demonstration of these effects is just the same as the preceding4 ,but that this curved line ,,cde which serves for refraction is an ,ellipse1 and is such that its major diameter is to the distance between its foci as #c to #b1 which is the proportion of the refraction1 can be easily found by the calculus of ,algebra4 ,for ,,db1 which is given1 being called a2 its undetermined perpendicular ,,dt being called x2 and ,,tc y2 ,,fb will be a - y2 ,,cb will be sqrt7xx aa -#b;ay yy74 ,but the nature of the curve is such that #b_/c of ,,tc together with ,,cb is equal to ,,db1 as was stated in the last construction3 then the equation will be between 7#b_/c7y sqrt7xx aa - #b;ay yy7 and a2 which being reduced1 gives 7#f_/e7;ay - yy equal to 7#i_/e7xx2 that is to say that having made ,,do equal to #f_/e of ,,db1 the rectangle ,,dfo is equal to #i_/e of the square on ,,fc4 ,whence it is seen that ,,dc is an ellipse1 of which the axis ,,do is to the parameter as #i to #e2 and therefore the square on ,,do is to the square of the distance between the foci as #i to #i - #e1 that is to say #d2 and finally the line ,,do will be to this distance as #c to #b4 ,illustration ,again1 if one supposes the point ,b to be infinitely distant1 in lieu of our first oval we shall find that ,,cde is a true ,hyperbola2 which will make those rays become parallel which come from the point ,a4 ,and in consequence also those which are parallel within the transparent body will be collected outside at the point ,a4 ,now it must be remarked that ,,cx and ,,ks become straight lines perpendicular to ,,ba1 because they represent arcs of circles the centre of which is infinitely distant4 ,and the intersection of the perpendicular ,,cx with the arc ,,fc will give the point ,c1 one of those through which the curve ought to pass4 ,and this operates so that all the parts of the wave of light ,,dn1 coming to meet the surface ,,kde1 will advance thence along parallels to ,,ks and will arrive at this straight line at the same time2 of which the proof is again the same as that which served for the first oval4 ,besides one finds by a calculation as easy as the preceding one1 that ,,cde is here a hyperbola of which the axis ,,do is #d_/e of ,,ad1 and the parameter equal to ,,ad4 ,whence it is easily proved that ,,do is to the distance between the foci as #c to #b4 ,illustration ,these are the two cases in which ,conic sections serve for refraction1 and are the same which are explained1 in his ,dioptrique1 by ,des ,cartes1 who first found out the use of these lines in relation to refraction1 as also that of the ,ovals the first of which we have already set forth4 ,the second oval is that which serves for rays that tend to a given point2 in which oval1 if the apex of the surface which receives the rays is ,d1 it will happen that the other apex will be situated between ,b and ,a1 or beyond ,a1 according as the ratio of ,,ad to ,,db is given of greater or lesser value4 ,and in this latter case it is the same as that which ,des ,cartes calls his #crd oval4 ,now the finding and construction of this second oval is the same as that of the first1 and the demonstration of its effect likewise4 ,but it is worthy of remark that in one case this oval becomes a perfect circle1 namely when the ratio of ,,ad to ,,db is the same as the ratio of the refractions1 here as #c to #b1 as ,i observed a long time ago4 ,the #dth oval1 serving only for impossible reflexions1 there is no need to set it forth4 ,illustration ,as for the manner in which ,mr4 ,des ,cartes discovered these lines1 since he has given no explanation of it1 nor any one else since that ,i know of1 ,i will say here1 in passing1 what it seems to me it must have been4 ,let it be proposed to find the surface generated by the revolution of the curve ,,kde1 which1 receiving the incident rays coming to it from the point ,a1 shall deviate them toward the point ,b4 ,then considering this other curve as already known1 and that its apex ,d is in the straight line ,,ab1 let us divide it up into an infinitude of small pieces by the points ,g1 ,c1 ,f2 and having drawn from each of these points1 straight lines towards ,a to represent the incident rays1 and other straight lines towards ,b1 let there also be described with centre ,a the arcs ,,gl1 ,,cm1 ,,fn1 ,,do1 cutting the rays that come from ,a at ,l1 ,m1 ,n1 ,o2 and from the points ,k1 ,g1 ,c1 ,f1 let there be described the arcs ,,kq1 ,,gr1 ,,cs1 ,,ft cutting the rays towards ,b at ,q1 ,r1 ,s1 ,t2 and let us suppose that the straight line ,,hkz cuts the curve at ,k at right-angles4 ,illustration ,then ,,ak being an incident ray1 and ,,kb its refraction within the medium1 it needs must be1 according to the law of refraction which was known to ,mr4 ,des ,cartes1 that the sine of the angle ,,zka should be to the sine of the angle ,,hkb as #c to #b1 supposing that this is the proportion of the refraction of glass2 or rather1 that the sine of the angle ,,kgl should have this same ratio to the sine of the angle ,,gkq1 considering ,,kg1 ,,gl1 ,,kq as straight lines because of their smallness4 ,but these sines are the lines ,,kl and ,,gq1 if ,,gk is taken as the radius of the circle4 ,then ,,lk ought to be to ,,gq as #c to #b2 and in the same ratio ,,mg to ,,cr1 ,,nc to ,,fs1 ,,of to ,,dt4 ,then also the sum of all the antecedents to all the consequents would be as #c to #b4 ,now by prolonging the arc ,,do until it meets ,,ak at ,x1 ,,kx is the sum of the antecedents4 ,and by prolonging the arc ,,kq till it meets ,,ad at ,y1 the sum of the consequents is ,,dy4 ,then ,,kx ought to be to ,,dy as #c to #b4 ,whence it would appear that the curve ,,kde was of such a nature that having drawn from some point which had been assumed1 such as ,k1 the straight lines ,,ka1 ,,kb1 the excess by which ,,ak surpasses ,,ad should be to the excess of ,,db over ,,kb1 as #c to #b4 ,for it can similarly be demonstrated1 by taking any other point in the curve1 such as ,g1 that the excess of ,,ag over ,,ad1 namely ,,vg1 is to the excess of ,,bd over ,,dg1 namely ,,dp1 in this same ratio of #c to #b4 ,and following this principle ,mr4 ,des ,cartes constructed these curves in his ,geometric2 and he easily recognized that in the case of parallel rays1 these curves became ,hyperbolas and ,ellipses4 ,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 figure2 a figure not only plane or spherical1 or made by one of the conic sections 7which is the restriction with which ,des ,cartes proposed this problem1 leaving the solution to those who should come after him7 but generally any figure whatever3 that is to say1 one made by the revolution of any given curved line to which one must merely know how to draw straight lines as tangents4 ,let the given figure be that made by the revolution of some curve such as ,,ak about the axis ,,av1 and that this side of the glass receives rays coming from the point ,l4 ,furthermore1 let the thickness ,,ab of the middle of the glass be given1 and the point ,f at which one desires the rays to be all perfectly reunited1 whatever be the first refraction occurring at the surface ,,ak4 ,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 ,,ak1 and from thence to the surface ,,bdk1 and from thence to the point ,f1 shall be traversed everywhere in equal times1 and in each case in a time equal to that which the light employs1 to pass along the straight line ,,lf of which the part ,,ab is within the glass4 ,illustration ,let ,,lg be a ray falling on the arc ,,ak4 ,its refraction ,,gv will be given by means of the tangent which will be drawn at the point ,g4 ,now in ,,gv the point ,d must be found such that ,,fd together with #c_/b of ,,dg and the straight line ,,gl1 may be equal to ,,fb together with #c_/b of ,,ba and the straight line ,,al2 which1 as is clear1 make up a given length4 ,or rather1 by deducting from each the length of ,,lg1 which is also given1 it will merely be needful to adjust ,,fd up to the straight line ,,vg in such a way that ,,fd together with #c_/b of ,,dg is equal to a given straight line1 which is a quite easy plane problem3 and the point ,d will be one of those through which the curve ,,bdk ought to pass4 ,and similarly1 having drawn another ray ,,lm1 and found its refraction ,,mo1 the point ,n will be found in this line1 and so on as many times as one desires4 ,to demonstrate the effect of the curve1 let there be described about the centre ,l the circular arc ,,ah1 cutting ,,lg at ,h2 and about the centre ,f the arc ,,bp2 and in ,,ab let ,,as be taken equal to #b_/c of ,,hg2 and ,,se equal to ,,gd4 ,then considering ,,ah as a wave of light emanating from the point ,l1 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 ,,as2 for ,i suppose1 as above1 the proportion of the refraction to be as #c to #b4 ,now we know that the piece of wave which is incident on ,g1 advances thence along the line ,,gd1 since ,,gv is the refraction of the ray ,,lg4 ,then during the time that this piece of wave has taken from ,g to ,d1 the other piece which was at ,s has reached ,e1 since ,,gd1 ,,se are equal4 ,but while the latter will advance from ,e to ,b1 the piece of wave which was at ,d will have spread into the air its partial wave1 the semi-diameter of which1 ,,dc 7supposing this wave to cut the line ,,df at ,c71 will be #c_/b of ,,eb1 since the velocity of light outside the medium is to that inside as #c to #b4 ,now it is easy to show that this wave will touch the arc ,,bp at this point ,c4 ,for since1 by construction1 ,,fd #c_/b ,,dg ,,gl are equal to ,,fb #c_/b ,,ba ,,al2 on deducting the equals ,,lh1 ,,la1 there will remain ,,fd #c_/b ,,dg ,,gh equal to ,,fb #c_/b ,,ba4 ,and1 again1 deducting from one side ,,gh1 and from the other side #c_/b of ,,as1 which are equal1 there will remain ,,fd with #c_/b ,,dg equal to ,,fb with #c_/b of ,,bs4 ,but #c_/b of ,,dg are equal to #c_/b of ,,es2 then ,,fd is equal to ,,fb with #c_/b of ,,be4 ,but ,,dc was equal to #c_/b of ,,eb2 then deducting these equal lengths from one side and from the other1 there will remain ,,cf equal to ,,fb4 ,and thus it appears that the wave1 the semi-diameter of which is ,,dc1 touches the arc ,,bp at the moment when the light coming from the point ,l has arrived at ,b along the line ,,lb4 ,it can be demonstrated similarly that at this same moment the light that has come along any other ray1 such as ,,lm1 ,,mn1 will have propagated the movement which is terminated at the arc ,,bp4 ,whence it follows1 as has been often said1 that the propagation of the wave ,,ah1 after it has passed through the thickness of the glass1 will be the spherical wave ,,bp1 all the pieces of which ought to advance along straight lines1 which are the rays of light1 to the centre ,f4 ,which was to be proved4 ,similarly these curved lines can be found in all the cases which can be proposed1 as will be sufficiently shown by one or two examples which ,i will add4 ,let there be given the surface of the glass ,,ak1 made by the revolution about the axis ,,ba of the line ,,ak1 which may be straight or curved4 ,let there be also given in the axis the point ,l and the thickness ,,ba of the glass2 and let it be required to find the other surface ,,kdb1 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 ,l4 ,illustration ,from the point ,l let there be drawn to some point of the given line ,,ak the straight line ,,lg1 which1 being considered as a ray of light1 its refraction ,,gd will then be found4 ,and this line being then prolonged at one side or the other will meet the straight line ,,bl1 as here at ,v4 ,let there then be erected on ,,ab the perpendicular ,,bc1 which will represent a wave of light coming from the infinitely distant point ,f1 since we have supposed the rays to be parallel4 ,then all the parts of this wave ,,bc must arrive at the same time at the point ,l2 or rather all the parts of a wave emanating from the point ,l must arrive at the same time at the straight line ,,bc4 ,and for that1 it is necessary to find in the line ,,vgd the point ,d such that having drawn ,,dc parallel to ,,ab1 the sum of ,,cd1 plus #c_/b of ,,dg1 plus ,,gl may be equal to #c_/b of ,,ab1 plus ,,al3 or rather1 on deducting from both sides ,,gl1 which is given1 ,,cd plus #c_/b of ,,dg must be equal to a given length2 which is a still easier problem than the preceding construction4 ,the point ,d thus found will be one of those through which the curve ought to pass2 and the proof will be the same as before4 ,and by this it will be proved that the waves which come from the point ,l1 after having passed through the glass ,,kakb1 will take the form of straight lines1 as ,,bc2 which is the same thing as saying that the rays will become parallel4 ,whence it follows reciprocally that parallel rays falling on the surface ,,kdb will be reassembled at the point ,l4 ,illustration ,again1 let there be given the surface ,,ak1 of any desired form1 generated by revolution about the axis ,,ab1 and let the thickness of the glass at the middle be ,,ab4 ,also let the point ,l be given in the axis behind the glass2 and let it be supposed that the rays which fall on the surface ,,ak tend to this point1 and that it is required to find the surface ,,bd1 which on their emergence from the glass turns them as if they came from the point ,f in front of the glass4 ,having taken any point ,g in the line ,,ak1 and drawing the straight line ,,igl1 its part ,,gi will represent one of the incident rays1 the refraction of which1 ,,gv1 will then be found3 and it is in this line that we must find the point ,d1 one of those through which the curve ,,dg ought to pass4 ,let us suppose that it has been found3 and about ,l as centre let there be described ,,gt1 the arc of a circle cutting the straight line ,,ab at ,t1 in case the distance ,,lg is greater than ,,la2 for otherwise the arc ,,ah must be described about the same centre1 cutting the straight line ,,lg at ,h4 ,this arc ,,gt 7or ,,ah1 in the other case7 will represent an incident wave of light1 the rays of which tend towards ,l4 ,similarly1 about the centre ,f let there be described the circular arc ,,dq1 which will represent a wave emanating from the point ,f4 ,then the wave ,,tg1 after having passed through the glass1 must form the wave ,,qd2 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 three1 ,,ta1 ,,ab1 and ,,bq1 of which ,,ab alone is within the glass4 ,or rather1 having taken ,,as equal to #b_/c of ,,at1 ,i observe that #c_/b of ,,gd ought to be equal to #c_/b of ,,sb1 plus ,,bq2 and1 deducting both of them from ,,fd or ,,fq1 that ,,fd less #c_/b of ,,gd ought to be equal to ,,fb less #c_/b of ,,sb4 ,and this last difference is a given length3 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 thus4 ,which is a problem quite similar to that which served for the first of these constructions1 where ,,fd plus #c_/b of ,,gd had to be equal to a given length4 ,in the demonstration it is to be observed that1 since the arc ,,bc falls within the glass1 there must be conceived an arc ,,rx1 concentric with it and on the other side of ,,qd4 ,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 ,q1 which is easily deduced from the construction1 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 ,r1 and that thus this arc will at the same moment be the termination of the movement that comes from the wave ,,tg2 whence all the rest may be concluded4 ,having shown the method of finding these curved lines which serve for the perfect concurrence of the rays1 there remains to be explained a notable thing touching the uncoordinated refraction of spherical1 plane1 and other surfaces3 an effect which if ignored might cause some doubt concerning what we have several times said1 that rays of light are straight lines which intersect at right angles the waves which travel along them4 ,illustration ,for in the case of rays which1 for example1 fall parallel upon a spherical surface ,,afe1 intersecting one another1 after refraction1 at different points1 as this figure represents2 what can the waves of light be1 in this transparent body1 which are cut at right angles by the converging rays8 ,for they can not be spherical4 ,and what will these waves become after the said rays begin to intersect one another8 ,it will be seen in the solution of this difficulty that something very remarkable comes to pass herein1 and that the waves do not cease to persist though they do not continue entire1 as when they cross the glasses designed according to the construction we have seen4 ,according to what has been shown above1 the straight line ,,ad1 which has been drawn at the summit of the sphere1 at right angles to the axis parallel to which the rays come1 represents the wave of light2 and in the time taken by its piece ,d to reach the spherical surface ,,age at ,e1 its other parts will have met the same surface at ,f1 ,g1 ,h1 etc41 and will have also formed spherical partial waves of which these points are the centres4 ,and the surface ,,ek which all those waves will touch1 will be the continuation of the wave ,,ad in the sphere at the moment when the piece ,d has reached ,e4 ,now the line ,,ek is not an arc of a circle1 but is a curved line formed as the evolute of another curve ,,enc1 which touches all the rays ,,hl1 ,,gm1 ,,fo1 etc41 that are the refractions of the parallel rays1 if we imagine laid over the convexity ,,enc a thread which in unwinding describes at its end ,e the said curve ,,ek4 ,for1 supposing that this curve has been thus described1 we will show that the said waves formed from the centres ,f1 ,g1 ,h1 etc41 will all touch it4 ,it is certain that the curve ,,ek and all the others described by the evolution of the curve ,,enc1 with different lengths of thread1 will cut all the rays ,,hl1 ,,gm1 ,,fo1 etc41 at right angles1 and in such wise that the parts of them intercepted between two such curves will all be equal2 for this follows from what has been demonstrated in our treatise de ,motu ,pendulorum4 ,now imagining the incident rays as being infinitely near to one another1 if we consider two of them1 as ,,rg1 ,,tf1 and draw ,,gq perpendicular to ,,rg1 and if we suppose the curve ,,fs which intersects ,,gm at ,p to have been described by evolution from the curve ,,nc1 beginning at ,f1 as far as which the thread is supposed to extend1 we may assume the small piece ,,fp as a straight line perpendicular to the ray ,,gm1 and similarly the arc ,,gf as a straight line4 ,but ,,gm being the refraction of the ray ,,rg1 and ,,fp being perpendicular to it1 ,,qf must be to ,,gp as #c to #b1 that is to say in the proportion of the refraction2 as was shown above in explaining the discovery of ,des ,cartes4 ,and the same thing occurs in all the small arcs ,,gh1 ,,ha1 etc41 namely that in the quadrilaterals which enclose them the side parallel to the axis is to the opposite side as #c to #b4 ,then also as #c to #b will the sum of the one set be to the sum of the other2 that is to say1 ,,tf to ,,as1 and ,,de to ,,ak1 and ,,be to ,,sk or ,,dv1 supposing ,v to be the intersection of the curve ,,ek and the ray ,,fo4 ,but1 making ,,fb perpendicular to ,,de1 the ratio of #c to #b 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 ,,be4 ,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 ,,ek1 and consequently that the wave will touch this curve4 ,in the same way it can be proved that the same will apply to all the other waves above mentioned1 originating at the points ,g1 ,h1 etc42 to wit1 that they will touch the curve ,,ek at the moment when the piece ,d of the wave ,,ed shall have reached ,e4 ,now to say what these waves become after the rays have begun to cross one another3 it is that from thence they fold back and are composed of two contiguous parts1 one being a curve formed as evolute of the curve ,,enc in one sense1 and the other as evolute of the same curve in the opposite sense4 ,thus the wave ,,ke1 while advancing toward the meeting place becomes abc1 whereof the part ab is made by the evolute b,c1 a portion of the curve ,,enc1 while the end ,c remains attached2 and the part bc by the evolute of the portion b,e while the end ,e remains attached4 ,consequently the same wave becomes def1 then ghk1 and finally ,,cy1 from whence it subsequently spreads without any fold1 but always along curved lines which are evolutes of the curve ,,enc1 increased by some straight line at the end ,c4 ,there is even1 in this curve1 a part ,,en which is straight1 ,n being the point where the perpendicular from the centre ,x of the sphere falls upon the refraction of the ray ,,de1 which ,i now suppose to touch the sphere4 ,the folding of the waves of light begins from the point ,n up to the end of the curve ,c1 which point is formed by taking ,,ac to ,,cx in the proportion of the refraction1 as here #c to #b4 ,as many other points as may be desired in the curve ,,nc are found by a ,theorem which ,mr4 ,barrow has demonstrated in section #ab of his ,lectiones ,opticae1 though for another purpose4 ,and it is to be noted that a straight line equal in length to this curve can be given4 ,for since it together with the line ,,ne is equal to the line ,,ck1 which is known1 since ,,de is to ,,ak in the proportion of the refraction1 it appears that by deducting ,,en from ,,ck the remainder will be equal to the curve ,,nc4 ,similarly the waves that are folded back in reflexion by a concave spherical mirror can be found4 ,let ,,abc be the section1 through the axis1 of a hollow hemisphere1 the centre of which is ,d1 its axis being ,,db1 parallel to which ,i suppose the rays of light to come4 ,all the reflexions of those rays which fall upon the quarter-circle ,,ab will touch a curved line ,,afe1 of which line the end ,e is at the focus of the hemisphere1 that is to say1 at the point which divides the semi-diameter ,,bd into two equal parts4 ,the points through which this curve ought to pass are found by taking1 beyond ,a1 some arc ,,ao1 and making the arc ,,op double the length of it2 then dividing the chord ,,op at ,f in such wise that the part ,,fp is three times the part ,,fo2 for then ,f is one of the required points4 ,illustration ,and as the parallel rays are merely perpendiculars to the waves which fall on the concave surface1 which waves are parallel to ,,ad1 it will be found that as they come successively to encounter the surface ,,ab1 they form on reflexion folded waves composed of two curves which originate from two opposite evolutions of the parts of the curve ,,afe4 ,so1 taking ,,ad as an incident wave1 when the part ,,ag shall have met the surface ,,ai1 that is to say when the piece ,g shall have reached ,i1 it will be the curves ,,hf1 ,,fi1 generated as evolutes of the curves ,,fa1 ,,fe1 both beginning at ,f1 which together constitute the propagation of the part ,,ag4 ,and a little afterwards1 when the part ,,ak has met the surface ,,am1 the piece ,k having come to ,m1 then the curves ,,ln1 ,,nm1 will together constitute the propagation of that part4 ,and thus this folded wave will continue to advance until the point ,n has reached the focus ,e4 ,the curve ,,afe can be seen in smoke1 or in flying dust1 when a concave mirror is held opposite the sun4 ,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 ,,eb1 when that circle is made to roll within another whose semi-diameter is ,,ed and whose centre is ,d4 ,so that it is a kind of ,cycloid1 of which1 however1 the points can be found geometrically4 ,its length is exactly equal to #c_/d of the diameter of the sphere1 as can be found and demonstrated by means of these waves1 nearly in the same way as the mensuration of the preceding curve2 though it may also be demonstrated in other ways1 which ,i omit as outside the subject4 ,the area ,,aobefa1 comprised between the arc of the quarter-circle1 the straight line ,,be1 and the curve ,,efa1 is equal to the fourth part of the quadrant ,,dab4 ,,index ,archimedes1 #ajd4 ,atmospheric refraction1 #de4 ,barrow1 ,isaac1 #abf4 ,bartholinus1 ,erasmus1 #ec1 #ed1 #eg1 #fj1 #ig1 #ii4 ,boyle1 ,hon4 ,robert1 #aa4 ,cassini1 ,jacques1 iii4 ,caustic ,curves1 #abc4 ,crystals1 see ,iceland ,crystal1 ,rock ,crystal4 ,crystals1 configuration of1 #ie4 ,descartes1 ,rn1 #c1 #e1 #g1 #ad1 #bb1 #db1 #dc1 #aji1 #aac4 ,double ,refraction1 discovery of1 #ed1 #ha1 #ic4 ,elasticity1 #ab1 #ad4 ,ether1 the1 or ,ethereal matter1 #aa1 #ad1 #af1 #bh4 ,extraordinary refraction1 #ee1 #ef4 ,fermat1 principle of1 #db4 ,figures of transparent bodies1 #aje4 ,hooke1 ,robert1 #bj4 ,iceland ,crystal1 #b1 #eb sqq4 ,iceland ,crystal1 ,cutting and ,polishing of1 #ia1 #ib1 #ih4 ,leibnitz1 ,g4,w41 vi4 ,light1 nature of1 #c4 ,light1 velocity of1 #d1 #ae4 ,molecular texture of bodies1 #bg1 #ie4 ,newton1 ,sir ,isaac1 vi1 #ajf4 ,opacity1 #cd4 ,ovals1 ,cartesian1 #ajg1 #aac4 ,pardies1 ,rev4 ,father1 #bj4 ,rays1 definition of1 #ch1 #di4 ,reflexion1 #bb4 ,refraction1 #bh1 #cd4 ,rock ,crystal1 #ed1 #eg1 #fb1 #ie4 ,rmer1 ,olaf1 v1 #g4 ,roughness of surfaces1 #bg4 ,sines1 law of1 #a1 #ce1 #ch1 #dc4 ,spheres1 elasticity of1 #ae4 ,spheroidal waves in crystals1 #fc4 ,spheroids1 lemma about1 #ajc4 ,sound1 speed of1 #g1 #aj1 #ab4 ,telescopes1 lenses for1 #fb1 #aje4 ,torricelli's experiment1 #ab1 #cj4 ,transparency1 explanation of1 #bh1 #ca1 #cb4 ,waves1 no regular succession of1 #ag4 ,waves1 principle of wave envelopes1 #ai1 #bd4 ,waves1 principle of elementary wave fronts1 #ai4 ,waves1 propagation of light as1 #af1 #fc4