and the ratio of CP to CS that of 9 to 8; the spheroid is therefore produced by the revolution of an ellipse round its smaller diameter, 131. In order to determine the refraction of oblique rays, Huygens saw that they depended on the proportion between the velocity without and within the crystal; and he supposes this proportion to be such that while the light in the crystal describes the spheroid GSP, fig. 4, it describes without it a sphere whose semi-diameter is equal to the line N, which will be afterwards determined. Now, if RC is a ray incident at the surface C K, draw CO perpendicular to R C, and from O draw O K equal to N, and at right angles to CO. Having then drawn K I, touching the ellipse GSP, the line IC will be the refracted ray required. For the refraction of RC is nothing else than the progress of the point C of the curve CO, continued in the crystal; but the points H H of this wave, during the time that O comes to K, will arrive at the surface C K by the lines Hr, &c., and will also have produced in the crystal particular hemispheroidal waves from the centres x, x, &c., similarly situated with the hemispheroid GS Pg; and whose great and small diameters will have the same ratio to the lines av (which are continuations of Hr to K B parallel to CO) as the diameters of the spheroid GSP have to the line C B or N. The common tangent, therefore, of all the spheroids, which are here represented by ellipses, will be the line I K, which will be the propagation of the wave CO; and the point I that of C. 132. In order to find the point of contact, I, we must find a third proportional CD to the lines CK, CG; and having drawn DI parallel to CM, the diameter conjugate to CG, and previously determined, we have only to draw H I, which will touch the ellipse at I. 133. In like manner we may find the refracted ray Ci of any other ray r C, incident on the opposite side, by drawing C o perpendicular to Cr and following the rest of the preceding con struction. 134. Hence we see that, if Cr and CR are incident at equal angles, we shall have Cd CD, since CkCK and Cg CG. Consequently I will be bisected in E by CM, to which D I, Di, are parallel; and, because C M is the conjugate diameter to C G, it follows that Ii will be parallel to g G. If we then prolong the refracted rays, CI, Ci, till they meet the tangent M L in T and t, the distances M T, M t, will also be equal. 135. Huygens next found, by measuring the irregular refraction, that the ratio of N to CG was that of 156962 to 98779, or of 8 to 5, and he proceeds to explain an abridged method of finding the irregular refraction. 136. Let g G, fig. 5, be the surface of the crystal, G P g the ellipse, and C M the refraction of the perpendicular ray FC which deviates 6° 40', and let RC be any other ray, whose refraction is required. 137. From the centre C, and with the semidiameter C G, describe the circumference g RG, cutting the ray RC in R, and let fall RV perpendicular to C G. Then, since C D is known from the analogy, N: CGCV: CD; draw DI parallel to CM, cutting the ellipse g MG in I, and joining C and I, CI will be the refracted ray required. For let CO be perpendicular to ČR, and O K drawn equal to Ñ, and perpendicular to CO; then, if KI touches the ellipse in I, CI must be the refraction of RC. Since RCO is a right angle, the right-angled triangles RC V, K CO, are similar. Hence CK : KO = RC; CV; but KON, and RC = CG; Whence CK: N=CG:CV; but by con struction, N:CG=CV:CD; therefore, CK: CG CG; CD; and, because DI is parallel to C M, the diameter conjugate to CD, it follows that K I touches the ellipse in I. 138. It appears, therefore, that, as there is in ordinary refractions, a constant ratio between the sines of incidence and refraction, so in the present case there is such a proportion between CV and CD or I E, that is between the sine of the angle of incidence, and the line intercepted between the refracted ray and the diameter C M ; for the, ratio of CV to CD is always the same as that of N to the semi-diameter C G. 139. In comparing the regular and irregular refraction, Huygens observes, that if AP BS, fig. 6, is the spheroid by which the light propagates itself in the crystal in a certain space of time, and produces the irregular refraction, then the inscribed sphere BV ST is the propagation, in the same space of time, of the light which serves for the regular refraction. For, since N is the radius of a spherical wave of light in the air, while in the crystal it is propagated by the spheroid AP BS, the ratio of N to CS is that of 156962 to 93410. But the ratio of the regular. refraction is that of 5 to 3, that is, N being the radius of a spherical wave of light in air, its propagation in the crystal forms in the same time a sphere whose radius is to N as 3 to 5. But 156962 is to 93410 as 513 minus 140. Though there are two different propagations of light within the crystal, yet it is only in the direction of the perpendiculars to the axis BS of the spheroid, that one of these propagations is more rapid than the other, for they have the same velocity in another direction, namely, in that of lines parallel to the same axis BS, which is also the axis of the obtuse angle of the crystal. 141. The ratios of the refractions being such as have now been determined, it follows that a ray of light, RC fig. 7, incident at an angle of 73° 30′ with C G, should have its refraction CI in the same straight line with R C, or should pass through the crystal without refraction. For since CGCR 98779; C M = 100000 and RCV 73° 20′, CV will be 28330. But, because C I is the refraction of R C, CV: CD 156962: 98779=N: CG, and CD=17828. And since CGCM2 GD × Dg: DI' we have DICE 98353. But as CE:EI=CM; MT; MT=18127, which being added to M L=11609 (the sine of LCM =6°-40′) we have LT⇒ 27936, which is to LC 99324 as CV is to V R, that is as 29938, the cotangent of RC V, is to the radius. Whence it appears that RCIT is a straight line. 142. Huygens goes on to show that the ray CI, emerging at the opposite surface of the crystal, ought to pass straight on without refraction, by demonstrating in general, that the reciprocation of refractions takes place in this crystal as well as with transparent bodies; that is, if a ray RC, fig. 8, incident on the surface of the crystal CG, is refracted in CI, the ray C I, emerging at the opposite and parallel surface I B of the crystal, will have its refraction IA parallel to the ray RC. 143. Let CO, perpendicular to CR, represent, as formerly, a portion of a wave, whose continuation in the crystal is IK, so that the point C is continued by CI during the time that O arrives in K. If we now take a second space of time equal to the first, the point K of the wave IK will, in this second portion of time, have moved through the right line K B, equal and parallel to CI, because every point of the wave CO, in arriving at the surface C K, ought to continue in the crystal the same as the point C, and in the same time it will propagate from the point I in the air a spherical wave having a semidiameter IA = KO, since KO is described in the same time. If we consider any other point k of the wave I K, it will advance by hm parallel to CI, and reach the surface I B, while the point K describes K1 - h m, and during the time that K has completed the remainder, IB, there will have been propagated from the point m a sphenical wave, whose semi-diameter mn will have the same ratio to B as IA: K B. The waves mn and I A will therefore have the same tangent AB, and the same is true of all the other spherical waves that are propagated out of the crystal by the impulsion of all the points of the wave IK against the surface of the other I B. The tangent B A will therefore be the continuation of the wave I K out of the crystal, when the point K has come to B; and consequently I A, which is the perpendicular to BA, will be the refraction of the ray C I in going out of the crystal. But IA is parallel to RC since IB CK, and IA =KO and A and O right angles. 144. The only observations which Sir Isaac Newton appears to have published, on the subject of double refraction and polarisation, are contained in the queries printed at the end of the Third Book of his Optics. As they are written with great perspicuity, and easily understood, we shall lay them before our readers in his own words. 145.Query 25. Are there not other original properties of the rays of light, besides those already described? An instance of another original property we have in the refraction of Iceland crystal, described first by Erasmus Bartholine, and afterwards more exactly by Huygenius, in his book De la Lumiere. This crystal is a pellucid fossile store, clear as water or crystal of the rock, and without color; enduring a red heat without losing its transparency, and in a very strong heat calcining without fusion. Steeped a day or two in water it loses its natural polish. Being rubbed on cloth it attracts pieces of straws and other light things, like amber or glass; and with aquafortis it makes an ebullition. It seems to be a sort of talc, and is found in form of an oblique parallelopiped, with six paralellogram sides and eight solid angles. The obtuse angles of the parallelograms are each of them 101° 52'; the acute ones 78° 8'. Two of the solid angles opposite to one another, as C and E, fig. 9, are compassed each of them with three of these obtuse angles, and each of the other six with one obtuse and two acute ones. It cleaves easily in planes parallel to any of its sides; and not in any other planes. It cleaves with a glossy polished surface not perfectly plane, but with some little unevenness. It is easily scratched, and by means of its softness it takes a polish very difficultly. It polishes better upon polished looking glass than upon metal, and perhaps better upon pitch, leather, or parchment. Afterwards it must be rubbed with a little oil or white of an egg, to fill up its scratches; whereby it will become very transparent and polished. But for several experiments it is not necessary to polish it. If a piece of this crystalline stone be laid upon a book, every letter of the book seen through it will appear double, by means of a double refraction. And if any beam of light falls either perpendicularly, or in any oblique handle upon any surface of this crystal, it becomes divided into two beams, by means of the same double refraction. Which beams are of the same color with the incident beam of light, and seem equal to one another in the quantity of their light, or very nearly equal. One of these refractions is performed by the usual rule of optics, the sine of incidence out of air into this crystal being, to the sine of refraction as five to three. The other refraction, which may be called the unusual refraction, is performed by the following rule. 146. Let AD BC represent the refracting surface of the crystal, C the biggest solid angle at that surface, GE H F the opposite surface, and CK a perpendicular on that surface. This perpendicular makes, with the edge of the crystal CF, an angle of 19° 3'. Join K F, and in it take KL, so that the angle K CL be 6° 40′, and the angle LCF 12° 23'. And, if S T represent any beam of light incident at T in any angle upon the refracting surface A D B C, let TV be the refracted beam determined by the given proportion of the sines five to three, according to the usual rule of optics, Draw V X parallel and equal to K L. Draw it the same way from V in which L lieth from K; and, joining TX, this line T X shall be the other refracted beam carried from T to X, by the unusual refraction. perpendicular to the refracting surface, the two beams TV and TX, into which it shall become divided, shall be parallel to the lines C K and CL; one of these beams going through the crystal perpendicularly, as it ought to do by the usual laws of optics, and the other TX by an unusual refraction diverging from the perpendicular, and making with it an angle VTX of about 63°, as is found by experience. And hence the plane V TX, and such like planes which are parallel to the plane CF K, may be called the planes of perpendicular refraction. And the coast towards which the lines K L and VX are drawn, may be called the coast of an unusual refraction. 148. In like manner crystal of the rock has a double refraction: but the differences of the two refractions is not so great and manifest as in Iceland crystal. 149. When the beam ST, incident on Iceland crystal, is divided into two beams TV and TX, and these two beams arrive at the farther surface of the glass; the beam TV, which was refracted at the first surface after the usual manner, shall be again refracted entirely after the usual manner at the second surface; and the beam T X, which was refracted after the unusual manner in the first surface, shall be again refracted entirely after the unusual manner in the second surface; so that both these beams shall emerge out of the second surface in lines parallel to the first incident beam S T. 150. And if two pieces of Iceland crystal be placed one after another, in such a manner that all the surfaces of the latter be parallel to all the corresponding surfaces of the former, the rays which are refracted after the usual manner in the first surface of the first crystal shall be refracted after the usual manner in all the following surfaces, and the rays which are refracted after the unusual manner in the first surface, shall be refracted after the unusual manner in all the following surfaces. And the same thing happens though the surfaces of the crystal be any ways inclined to one another, provided that their planes of perpendicular refraction be parallel to one another. 151. And, therefore, there is an original difference in the rays of light, by means of which some rays are, in this experiment, constantly refracted after the usual manner, and others constantly after the unusual manner. For if the difference be not original, but arises from new modifications impressed on the rays at their first refraction, it would be altered by new modifications in the three following refractions; whereas it suffers no alteration, but is constant, and has the same effect upon the rays in all the refractions. The unusual refraction is, therefore, performed by an original property of the rays. And it remains to be enquired, whether the rays have not more original properties than are yet discovered? all of them refracted after the usual manner in passing through the second crystal, and the rays which are refracted after the unusual manner in passing through the first crystal will all of them be refracted after the usual manner in passing through the second crystal. And, therefore, there are not two sorts of rays differing in their nature from one another, one of which is constantly and in all positions refracted after the usual manner, and the other constantly and in all positions after the unusual manner. The difference between the two sorts of rays, in the experiment mentioned in the twenty-fifth question, was only in the positions of the sides of the rays to the planes of perpendicular refraction. For one and the same ray is here refracted sometimes after the usual, and sometimes after the unusual manner, according to the position which its sides have to the crystals. If the sides of the ray are posited the same way to both crystals, it is refracted after the same manner in them both. But, if that side of the ray which looks towards the coast of the unusual refraction of the first crystal be 90° from that side of the same ray which looks towards the coast of the unusual refraction of the second crystal (which may be effected by varying the position of the second crystal to the first, and by consequence to the rays of light), the ray shall be refracted after several manners in the several crystals. There is nothing more required to determine whether the rays of light which fall upon the second crystal shall be refracted after the usual, or after the unusual manner, but to turn about this crystal, so that the coast of this crystal's unusual refraction may be on this or on that side of the ray. And, therefore, every ray may be considered as having four sides or quarters; two of which, opposite to one another, incline the ray to be refracted after the unusual manner, as often as either of them are turned towards the coast of unusual refraction, and the other two, whenever either of them are turned towards the coast of unusual refraction, do not incline it to be otherwise refracted than after the usual manner. The two first may, therefore, be called the sides of unusual refraction. And since these dispositions were in the rays before their incidence on the second, third, and fourth surfaces of the two crystals, and suffered no alteration (so far as appears) by the refraction of the rays in their passage through those surfaces, and the rays were refracted by the same laws in all the four surfaces: it appears that those dispositions were in the rays originally, and suffered no alteration by the first refraction, and that by means of those dispositions the rays were refracted at their incidence on the first surface of the first crystal, some of them after the usual, and some of them after the unusual manner, according as their sides of unusual refraction were then turned towards the coast of the unusual refraction from that crystal or sideways from it. which the sides of the rays differ, and are distinguished from one another. 154. In explaining the difference of the sides of the rays above mentioned, I have supposed that the rays fall perpendicularly on the first crystal. But, if they fall obliquely on it, the success is the same. Those rays which are refracted after the usual manner in the first crystal will be refracted after the unusual manner in the second crystal, supposing the planes of perpendicular refraction to be at right angles with one another, as above, and on the contrary. 155. If the planes of the perpendicular refraction of the two crystals be neither parallel nor perpendicular to one another, but contain an acute angle, the two beams of light which emerge out of the first crystal will be each of them divided into two more at their incidence on the second crystal. For in this case the rays in each of the two beams will some of them have their sides of unusual refraction, and some of them their other sides turned towards the coast of the unusual refraction of the second crystal. 156. To explain the unusual refraction of Iceland crystal by pressure or motion propagated, has not hitherto been attempted (to my knowledge) except by Huygens, who, for that end, supposed two several vibrating mediums within that crystal. But when he tried the refractions in two successive pieces of that crystal, and found them such as is mentioned above, he confessed himself at a loss for explaining them. For pressions or motions, propagated from a shining body through a uniform medium, must be on all sides alike; whereas by those experiments it appears, that the rays of light have different properties in their different sides. He suspected that the pulses of ether, in passing through the first crystal, might receive certain new modifications, which might determine them to be propagated in this or that medium within the second crystal, according to the position of that crystal. But what modifications those might be he could not say, nor think of any thing satisfactory on that point. And if he had known that the unusual refraction depends, not on new modifications, but on the original and unchangeable dispositions of the rays, he would have found it as difficult to explain how those dispositions which he supposed to be impressed on the rays by the first crystal could be in them before their incidence on that crystal; and, in general, how all rays emitted by shining bodies can have those dispositions in them from the beginning. To me, at least, this seems inexplicable, if light be nothing else than pression or motion propagated through ether.' 157. Those who have already examined the law of double refraction, as given by Huygens, and its agreement with observations made in all sections of the crystals of Iceland-spar, must experience no small degree of surprize, when they find that Sir Isaac Newton has proposed another law, different from his, and absolutely incompatible with observation. As Sir Isaac remarks that Huygens has described the phenomena more exactly than Bartholinus, there is reason to believe that he made some experiments on the subject, which confirmed those of Huygens; and yet it is strange that, without assigning any reasons he should reject Huygens's law, and substitute another, entirely inconsistent with the very experiments he has praised. In his speculations respecting the cause of the disappearance and reappearance of the pencil, when light is transmitted through two rhombs of calcareous spar, Newton has been more, fortunate; and he has undoubtedly the merit of having first' suggested the idea of the polarity of light, and of having described the phenomena of the polarisation of the pencils in Iceland-spar, to original properties possessed by different sides of the rays. 158. A paper entitled An Account of the Double Refractions in Crystals, by Father John Beccaria, professor of experimental philosophy at Turin, was read before the Royal Society of London on the 18th of March 1762, and printed in the Transactions for that year, vol. lii, p. 486. The principal result of these experiments is, that the double refraction in rock crystal is greatest when the ray is perpendicular to the axis of the crystal, and that the images approached to coincidence as the ray approached to that axis. This conclusion must be considered as of some importance, as it overturns the peculiar law of double refraction, which Huygens had devised for rock crystal alone. According to this law, the double refraction of rock crystal should be the same in every direction; whereas Beccaria has proved that it diminishes as the ray approaches to the axis. 159. Beccaria's paper is concluded with some unimportant queries, in one of which he conjectures that the examination of the double refraction of different crystals may lead to the determination of their structure, and mode of formation. 160. Towards the end of the last century the abbé Hauy began to direct his attention to the kindred subjects of crystallography and mineralogy, and to lay the foundation of that beautiful system which has immortalised his name. 161. Not content with the external examination of mineral bodies, he directed his notice to all their physical properties, and thus availed himself of the lights of natural philosophy, in giving a scientific form to the science of minerals. 162. In accomplishing this great event, he was naturally led not only to observe the double refraction of minerals, but to employ it as an essential character in his descriptions; and it was from this cause, more than from any other, that the attention of philosophers was recalled to a subject which had almost disappeared from modern physics. 163. The following are the general views given by this eminent author in his Traité de Mineralogie, vol. i. p. 230, which appeared in the year 1801. stances themselves. In zircon, for example, the double refraction is very strong, while it is much less sensible in emerald. Besides, this quantity varies in each substance from different causes. In general it increases or diminishes with the refracting angle, or that which is formed by the two faces through which we look at the object. But there is another cause of variation, which combines itself, with the preceding, and which depends on the position of the refracting surfaces, relative to the faces of the primitive form; and such is the influence of this cause that with two equal refracting angles, differently situated, we may have distances perceptibly unequal between the images of the same object, and there is even a limit where the effect of double refraction becomes nothing, that is, when the two images are combined into one. 165. This limit takes place in quartz and in emerald where one of the faces containing the refracting angle is perpendicular to the axis. It takes place in sulphate of barytes when one of the same faces, being parallel to the axis is at the same time parallel to a plane passing through the long diagonals of the base of its primitive form. 166. On this subject I possess only a small number of observations; but it is probable that all the substances which have double refraction are included in one or other of the two preceding cases, which give the limits of all the positions that the refracting surfaces can have, relative to the primitive form. But, as the position parallel to the axis is variable in its turn, between several limits which correspond to the diagonals and to the sides of the base of the primitive form, it is necessary to know which of these last limits is that which belongs to each sub stance. 167. I shall show, under the article Emerald, how a mistake conducted me to these results; and I confess that there remains still an uncertainty respecting the refraction of some substances, as I have not had leisure to multiply my researches sufficiently, in order to ascertain if a crystal, which gives only a single image of objects, would produce two after being cut in a particular manner. 168. Another observation, which will be of some use in generalising the theory of these phenomena, consists in this, that all the substances whose integrant molecule is remarkable by its symmetry have single refraction. Such are those which have for their primitive form the cube, the regular octahedron, and the rhomboidal dodecahedron. 169. Experiments on this object have hitherto been made only on those bodies which are called stones. I have extended these experiments to several called salts, and also to inflammable substances, and to metallic substances, oxidated and united to other substances, and I have found that there is no class of minerals which does not present bodies endowed with double refraction.' 170. After mentioning the methods which he employed for observing the double refraction, but which are entirely superseded by methods equally simple and much more certain, our author remarks, That it would be difficult to find a character more prominent than that which is drawn from double refraction, since it belongs to the very essence of the minerals in which it exists; and that it is fortunate that we are able, in this way, to supply the disappearance of crystalline forms by a physical observation, and thus to read in a certain degree in the interior of the stone, when its exterior no longer speaks to the eye.' 171. The following is the list of substances which have double refraction, as given by the abbé Haüy : Hauy's Table of Doubly Refracting Crystals. 1. Carbonate of lime strong. 2. Sulphate of lime. 7. Zircon, very strong. 10. Emerald. 14. Peridot, strong. 18. Carbonate of lead. 172. In his article on the Double Refraction of Iceland Spar, Haüy gives a detailed account of various experiments, that possess no very particular interest. He demonstrates by experiment the error of Newton's law, which we have already mentioned. He establishes the truth of the Huygenian law; but erroneously remarks, that it accords with observations only within certain limits, and he adopts the physical hypothesis of Newton, that the particles of light have two kinds of poles, on which the Iceland spar exerts a particular action, whose centre is placed in the region of the small solid angle of the crystal. 173. The only observations which we have to offer on the preceding articles, relate to the remark, that the cubical octahedral, and rhomhoido-dodecahedral crystals have single refraction. The only meaning which can be attached to this remark is, that some of these bodies have single refraction; for, out of the ten crystals which Hauy gives as having only single refraction, there are no fewer than six which belong to none of the above primitive forms, viz. those marked in italics. 174. The absolute incompatibility of Hauy's conclusions with his own facts, will appear still more strikingly from his list of transparent crys |