be as the ordinates BD, FG, &c. to the line DGE, the areas BFGD, BHIKD, &c. will be as the charges made on the fquare of the velocity at B, when the particle arrives at the points F, H, &c. For let BC be divided into innumerable Imall portions, of which let FH be one, and let the force be fuppofed to act uniformly, or to be of invariable intenfity during the motion along FH; draw GI perpendicular to HK; it is evident that the rectangle FHIG will be as the product of the accelerating force by the space along which it acts, and will therefore exprefs the momentary increment of the fquare of the velocity. (Lemma 1.) The fame may be faid of every fuch rectangle And if the number of the portions, fuch as FH, be increased, and their magnitude diminished without end, the rectangles will ultimately occupy the whole curvili neal area, and the force will be continually varying in its intenfity. The curvilineal areas will therefore be as the finite changes made on the fquares of the velocity, and the proportion is demonftrated.

237. Carol. The whole change made on the fquare of the velocity is equal to the fquare of that velocity which the accelerating force would communicate to the particle, by impelling it along BC from a ftate of reft in B. For the area BCED will ftill exprefs the fquare of this velocity, and it equally expreffes the change made on the fquare of any velocity wherewith the particle may pafs through the point B, and is independent on the magnitude of that velocity.''

238. Remark. The figure is adapted to the cafe where the forces all confpire with the initial motion of the particle, or all oppofe it; and the area expreffes an augmentation or a diminution of the fquare of the initial velocity. But the reafoning would have been the fame, although, in fome parts of the line BC, the forces had confpired with the initial motion, and in other parts had oppofed it. In fuch a cafe, the ordinates which exprefs the intenfity of the forces muft lie on different fides of the abfciffa BC, and that part of the area which lies on one fide muft be confidered as negative with respect to the other, and be fubtracted from it. Thus, if the forces are reprefented by the ordinates of the dotted curve-line DHe, which croffes the abfciffa in H, the figure will correfpond to the motion of a particle, which, after moving uniformly along AB, is fubjected to the action of a variable accelerating force during its motion along BH, and the fquare of its initial velocity is increased by the quantity BHD; after which it is retarded during its motion along HC, and the fquare of its velocity in His diminished by a quantity HCe. Therefore the fquare of the initial velocity is changed by a quantity BHD-HC e, or HC e-BHD.

239. This propofition is perhaps the most im portant in the whole fcience of mechanics, being the foundation of every application of mechanical theory to the explanation of natural phenomena. No traces of it are to be found in the writings of philofophers before the publication of Newton's Principia, although it is affumed by John Bernoulli and other detractors from Newton's greatnefs, as an elementary truth, without any acknowledgment of their obligations to its author. It is ufually expreffed by the equation ƒv and

ƒƒs=v2, i. e. the sum of the momentary actions is equal to the whole or finite increment of the fquare of the velocity.

240. PROPOSITION. When light paffes obliquely into or out of a tranfparent fubftance, it is refracted fo that the fine of the angle of incidence is to the fine of the angle of refraction in the conftant ratio of the velocity of the refracted light to that of the incident light.

241. Let ST, KR (fig. 7.), reprefent two planes (paralle! to, and equidiftant from, the refracting furface XY) which bound the fpace in which the light, during its paffage, is acted on by the refracting forces, as explained in § 225. The in tenfity of the refracting forces, being fuppofed equal at equal diftances from the bounding planes, though any how different at different diftances from them, may be reprefented by the ordinates Ta, nq, pr, cR, &c. of the curve a bnpe, of which the form must be determined from obfervation, or may remain for ever unknown. The phenomena of inflected light fhew us that it is attracted by the refracting fubftance at fome diftances, and repelled at others.

242. Let the light, moving uniformly in the direction AB, enter the refracting ftratum at B. It will not proceed in that direction, but its path will be incurvated upwards, while acted on by a repullive force, and downwards, while impelled by an attractive force. It will defcribe fome curvilineal path B do CDE, which AB touches in B, and will finally emerge from the refracting ftratum at E, and move uniformly in a ftraight line EF, which touches the curve in E. If, through b, the interfection of the curve of forces with its abfcifla, we draw bo, cutting the path of the light in o, it is evident that this path will be concave upwards between B and o, and concave downwards between

and E. Alfo, if the initial velocity of the light has been fufficiently fmall, its path may be fo much bent upwards, that in fome point. dits direction may be parallel to the bounding planes. In this cafe it is evident, that being under the influence of a repulfive force, it will be more bent upwards, and it will defcribe df, equal and fimilar to d B, and emerge in an angle gƒS equal to ABG. In this cafe it is reflected, making the angle of reflection equal to that of incidence. By which it appears how reflection, refriction, and inflection, are produced by the fame forces, and performed by the fame laws.

243. But let the velocity be fuppofed fufficiently great to enable the light to penetrate through the refracting ftratum, and emerge from it in the direction EF; let AB and EF be fuppofed to be defcribed in equal times: They will be proportional to the initial and final velocities of the light. Now, because the refracting forces must act in a direction perpendicular to the refracting furface (fince they arife from the joint action of · all the particles of a homogeneous fubftance which are within the fphere of mutual action), they' cannot affect the motion of the light eftimated in the direction of the refracting furface. If, therefore, AG be drawn perpendicular to ST, and FK to KR, the lines GB, EK, must be equal, because they are the motions AB, EF, eftimated in the direction of the planes. Draw now EL parallel to

222

A

AB. It is alfo equal to it. Therefore EL, EF, are as the initial and final velocities of the light. But EF is to EL as the fine of the angle ELK to the fine of the angle EFK; that is, as the fine of the angle ABH to the fine of the angle FEI; that is, as the fine of the angle of incidence to the fine of the angle of refraction.

244. By the fame reasoning it will appear that light, moving in the direction and with the velocity FE, will describe the path EDB, and will emerge in the direction and with the velocity BA. Let another ray enter the refracting ftratum perpendicularly at B, and emerge at Q. Take two points N, P, in the line BQ, extremely near to each other, fo that the refracting forces may be supposed to act uniformly along the space NP; draw NC, PD, parallel to ST, CM perpendicular to DP, and MO perpendicular to CD, which may be taken for a ftraight line; then, becaufe the forces at C and N are equal, by fuppofition they may be reprefented by the equal lines CM and NP. The force NP is wholly employed in accelerating the light along NP; but the force CM being tranfverfe to the motion BD, is but partly fo employed, and may be conceived as arifing from the joint action of the forces CO, OM, of which CO only is employed in accelerating the motion of the light, while OM is employed in incurvating its path. Now, it is evident, from the fimilarity of the triangles DCM, MCO, that DC, CM CM:CO, and that DC X CO=CMX CM NPX NP. But DCX CO and NPX NP are as the products of the spaces by the accelerating forces, and exprefs the momentary increments of the fquares of the velocities at C and N. (Lemma 1.) These increments, therefore, are equal. And as this must be faid of every portion of the paths BCE and BNQ, it follows that the whole increment of the fquare of the initial velocity produced in the motion along BCE, is equal to the increment produced in the motion along BNQ. And, because the initial velocities were equal in both paths, their fquares were equal. Therefore the fquares of the final velocities are alfo equal in both paths, and the final velocities themselves are equal. The initial and final velocities are therefore in a conftant ratio, whatever are the directions; and the ratio of the fines of the angles of incidence and refraction being the ratio of the velocities of the refracted and incident light, by the former cafe of Prop. 1. is also conftant.

245. Remark. The augmentation of the fquare of the initial velocity is equal to the fquare of the velocity, which a particle of light would have acquired, if impelled from a ftate of reft at B along the line BQ (Corol. of Lemma 2.), and is therefore independent on the initial velocity. As this augmentation is expreffed by the curvilineal arca aTbnpc R, it depends both on the intenfity of the refracting forces, expreffed by the ordinates, and on the pace through which they act, viz. TR. Thefe circumftances arife from the nature of the tranfparent fubftance, and are characteristic of that fubftance. Therefore, to abbreviate language, we fhall call this the specific velocity. This Specific velocity is eafily determined for any fubfrance in which the refraction is obferved, by drawing Li perpendicular to EL, meeting in i,

the circle described with the radius EF. For Ei being equal to EF, will reprefent the velocity of the refracted light, and EL reprefent the velocity of the incident light, and E-EL2XL2, and therefore Li is the augmentation of the square of the initial velocity, and Li is the specific velocity.

246. It is now proper to deduce fome corollaries from thefe propofitions, tending to explain the chief phenomena of refraction.

247. I. When light is refracted towards the perpendicular to the refracting surface, it is accelerated; and it is retarded when it is refracted from the perpendicular. In the firft cafe, therefore, it must be confidered as having been acted on by forces confpiring (in part at leaft) with its motion, and vice versa. Therefore, because we see that it is always refracted towards the perpendicular, when paffing from a void into any transparent fubftance, we conclude that it is, on the whole, attracted by that fubftance. We draw the fame conclufion from obferving, that it is refracted from the perpendicular in its paffage out of any tranf parent fubftance whatever into a void. It has been attracted backwards by that substance. This acceleration of light in refraction is contrary to the opinion of those who maintain, that illumination is produced by the undulation of an elastic medium. EULER attempts to prove, by mechanical laws, that the velocities of the incident and refracted light are proportional to the fines of incidence and refraction, while our principles make them in this ratio inverfely. BoscovicH proposed a fine experiment for deciding this queftion. The aberration of the fixed ftars arifes from the combination of the motion of light with the motion of the telescope by which it is obferved; therefore it should be greater or lefs when obferved by a telescope filled with water, according as light moves flower or fwifter through water than through air. The experiment has not yet been made in a convincing manner; because no fluid has been found of fufficient tranfparency to admit of the neceffary magnifying power. It is an experiment of the greateft importance to optical science.

248. II. If the light be moving within the tranfparent fubftance, and if its velocity (eftimated in a direction perpendicular to the furface) do not exceed the fpecific velocity of that fubftance, it will not emerge from it, but will be reflected backwards in an angle equal to that of its incidence. For it must be obferved, that in the figure of laft propofition, the excess of the fquare of EF above the fquare of EL is the fame with the excess of the fquare of KF above the fquare of KL. Therefore the fquare of the specific velocity is equal to the augmentation or diminution of the fquare of the perpendicular velocity. If therefore the initial perpendicular velocity FK (Fig. 8. Pl. CCLI.) be precifely equal to the fpecific velocity, the fight will just reach the farther fide of the attracting ftratum, as at B, where its perpendicular velocity will be completely extinguished, and its motion will be in the direction BT. But it is here under the influence of forces tending towards the plane KR, and its motion will therefore be ftill incur vated towards it; and it will defcribe a curve BD equal and fimilar to EB, and finally emerge back from the refracting ftratum into the transparent

fubftance

fubftance in an angle RDA equal to KEF. If the direction of the light be still more oblique, fo that. its perpendicular velocity is lefs than the specific velocity, it will not reach the plane ST, but be reflected as foon as it has penetrated fo far that the specific velocity of the part penetrated (eftimated by the compounding part of the area of forces), is equal to its perpendicular velocity. Thus the ray ƒE will defcribe the path EdDa penetrating to bd, fo that the correfponding area of forces abce is equal to the fquare of fk, its perpendicular velocity.

249. The extreme brilliancy of DEW DROPS and of JEWELS had often excited the attention of philofophers, and it always appeared a difficulty, how light was reflected at all from the pofterior furface of transparent bodies. It afforded Sir. Haac Newton his strongest argument against the ufual theory of reflection, viz. that it was produced by impact on folid elaftic matter. He was the first who took notice of the total reflection in great obliquities; and very properly afked how it can be faid that there is any impact in this cafe, or that the reflecting impact should ceafe at a particular obliquity?

250. It is a very curious circumstance, that a body which is perfectly tranfparent fhould ceafe to be fo at a certain OBLIQUITY; that a great obliquity fhould not hinder light from paffing from a void into a piece of glass; but that the fame obliquity fhould prevent it from paffing, from the glafs into a void. The finest experiment for illuftrating the fact is, to take two pieces of mirror glass, not filvered, and put them together with a piece of paper between them, forming a narrow margin all round to keep them apart. Plunge this apparatus into water. When held nearly parallel to the furface of the water, every thing at the bottom of the veffel will be feen clearly through the glaffes; but when turned fo as to be inclined about 50°, they will intercept the light as much as if they were plates of iron. It will be proper to foak the paper in varnish, to prevent water from getting between the glaffes.

251. What is called the brilliant cut in DIAMONDS, is fuch a difpofition of the pofterior facets of the diamond, that the light is made to fall upon them fo obliquely, that none of it can go through, but all is reflected. To produce this effect in the greatest poffible degree, is a matter of calculation, and merits the attention of the lapidary. When diamonds are too thin to admit of this form, they are cut in what is called the rofe form. This has a plain back, and the facets are all on the front, and fo difpofed as to refract the rays into fufficient obliquities, to be ftrongly reflected from the posterior plane: Doublets are made by cutting one thin diamond rofe-fashion, and another fimilar one is put behind it, with their plane furfaces joined. Or, more frequently, the outfide diamond has the anterior facets of the brilliant, and the inner has the form of the inner part of a brilliant. If they be joined with very pure and ftrongly refracting varnish, little light is reflected from the feparating plane, and their brilliancy is very confiderable, though inferior to that of a true and deep brilliant. If no varnish be used, much of the light is reflected from the flat

fide, and the effect of the pofterior facets is much diminished. But doublets might be conftructed, by making the touching furfaces of a spherical form (of which the curvature should have a due proportion to the fize of the ftone), that would produce an effect nearly equal to that of the moft perfect brilliant.

252. III. Since the change made on the fquare of the velocity of the incident light is a conftant quantity, it follows, that the refraction will di minith as the velocity of the incident light increafes. For if Li in fig. 7. be a conftant quantity, and EL be increased, it is evident that the ratio of Ei, or its equal EF, to EL will be diminish. ed, and the angle LEF, which conflitutes the refraction, will be diminished. The physical caute of this is eafily feen: When the velocity of the incident light is increafed, it employs lets time in paffing through the refracting ftratum or space between the planes ST and KR, and is therefore lefs influenced by the refracting forces. A fimilar effect would follow if the transparent body were moving with great velocity towards the luminous body.

253. The refraction of a ftar which is in our meridian at fix o'clock P. M. fhould be greater than that of a ftar which comes on the meridian at fix A. M.; because we are moving away from the firft, and approaching to the laft. But the difference is but of the whole, and cannot be observed with fufficient accuracy in any way yet practifed. A form of obfervation has been propofed by Dr BLAIR, profeffor of aftronomy, in the university of Edinburgh, which promifes a very fenfible difference of refraction. It is alfo to be expected, that a difference will be obferved iu the refraction of the light from the E. and W. ends of Saturn's ring. Its diameter is about 26 times that of the earth, and it revolves in oh. 32'; fo that the velocity of its edge is about roooo of the velocity of the fun's light. If therefore the light be reflected from it according to the laws of perfect elafticity, or in the manner here explained, that which comes to us from the western extremity will move more flowly than that which comes from the caftern extremity in the proportion of 2500 to 2501. And if Saturn can be seen distinctly after a refraction of 30° through a prifm, the diameter of the ring will be increased one half in one pofition of the telescope, and will be as much diminished by turning the telescope half round its axis; and an intermediate pofition will exhibit the ring of a distorted shape. This experiment is moft interefting to optical fcience, as its refult will be a, fevere touchstone of the theories which have been attempted for explaining the phenomena on mechanical principles.

254. If the tail of a comet be impelled by the rays of the fun, as is with great probability fuppofed by EULER and others, the light by which its extreme parts are feen by us must have its velocity greatly diminished, being reflected by particles which are moving away from the fun with immenfe rapidity. This may perhaps be difcovered by its greater aberration and refrangibility..

255. As common DAY-LIGHT is nothing but

the

the fun's light reflected from terrestrial bodies, it is reasonable to expect that it will fuffer the fame refraction. But nothing but obfervation could affure us that this would be the cafe with the light of the ftars; and it is rather furprifing that the velocity of their light is the fame with that of the fun's light. It is a circumftance of connection between the folar fyftem and the reft of the universe. It was as little to be looked for on the light of terreftrial luminaries. If light be conceived as fmall particles of matter emitted from bodies by the action of accelerating forces of any kind, the vast diverfity which we obferve in the conftitution of fublunary bodies fhould make us expect differences in this particular. Yet it is found, that the light of a candle, of a glow-worm, &c. fuffers the fame refraction, and confifts of the fame colours.

256. IV. When two transparent bodies are contiguous, the light in its paffage out of the one into the other will be refracted towards or from the perpendicular, according as the refracting forces of the fecond are greater or lefs than thofe of the first, or rather according as the area expreffing the fquare of the fpecific velocity is greater, or lefs. And as the difference of thefe areas is a determined quantity, the difference between the velocity in the medium of incidence and the ve locity in the medium of refraction, will also be a determined quantity. Therefore the fime of the angle of incidence will be in a conftant ratio to the fine of the angle of refraction; and this ratio will be compounded of the ratio of the line of incidence in the first medium to the fine of refraction in a void; and the ratio of the fine of incidence in a void to the fine of refraction in the fe cond medium. If therefore a ray of light, moving through a void in any direction, fhall pafs through any number of media bounded by parallel planes, its direction in the laft medium will be the fame as if it had come into it from a void.

257. V. It alfo follows from thefe propofitions, that if the obliquity of incidence on the pofterior furface of a tranfparent body be fuch, that the light should be reflected back again, the placing a mafs of the fame or of another medium in contact with this furface, will caufe it to be tranfmitted, and this the more completely, as the added medium is more denfe or more refractive; and the reflection from the feparating furface will be the more vivid, in proportion as the pofterior fubftance is lefs denfe, or of a fmaller refractive power. It is not even neceffary that the other, body be in contact; it is enough if it be so near that thofe parts of the refracting ftrata which are beyond the bodies interfere with or coincide with each other. All these confequences are agreeable to experience. The brilliant reflection from a dew-drop ceafes when it touches the leaf on which it refts: The brilliancy of a diamond is greatly, damaged by moisture getting behind it: The opa city of the combined mirror plates, mentioned in the II. corollary, is removed by letting water get between them: A piece of GLASS is diftinctly or clearly seen in air, more faintly when immerfed in water, ftill more faintly amidst oil of olives, and it is hardly perceived in fpirits of turpentine.

Thefe phenomena are incompatible with the notion that reflection is occafioned by impact on folid matter, whether of the tranfparent body, or of any æther or other fancied fluid behind it; and their perfect coincidence with the legitimate confequences of the affumed principles, is a ftrong argument in favour of the truth of those principles.

258. A fact, taken notice of by Mr BEGUELIN, has been objected as a great difficulty in the Newtonian theory of refraction. To get the greateft poffible refraction, and the fimpleft measure of the refracting power at the anterior surface of any tranfparent fubftance, Sir Ifaac Newton enjoins us to employ a ray of light falling on the furface quam obliquiffime. But Mr Beguelin found, that when the obiquity of incidence in glafs was about 89° 50, no light was refracted, but it was wholly reflected. He also observed, that when he gradually increafed the obliquity of incidence on the pofterior furface of the glafs, the fight which emerged laft of all did not skim along the furface, making an angle of 90° with the perpendicular, as it thould do by the Newtonian theory, but made an angle of more than ten minutes with the pofterior farface Alfo, when he began with very great obliquities, fo that all the light was reflected back into the glafs, and gradually diminifhed the obliquity of incidence, the first ray of light which emerged did not fkim along the furface, but was railed about 10 or 15 minutes.

259. But all thefe phenomena are neceffary confequences of our principles, combined with what obfervation teaches us concerning the forces which bodies exert on the rays of light. It is evident,” from the experiments of GRIMALDI and NEWTON, that light is both attracted and repelled by folid bodies. Newton's analyfis of thefe experiments difcovered feveral alternations of actual inflection and deflection; and he gives us the precife distance from the body when fome of thefe attractions end, 'and repulfion commences; and the most remote action to be observed in his experiments is repulfion. Suppofe that the forces are reprefented by the ordinates of a curve a b'n cp (see fig. 7. pl. 251.) which croffes the abfciffa in b. Draw bo parallel to the refracting surface. When the obliquity of incidence of the ray AB has become fo great, that its path in the glass, or in the refracting ftratum, does not cut, but only touches the line o b, it can penetrate no further, but is totally reflected: and this must happen in all greater obliquities. On the other hand, when the ray LE, moving within the glass, has but a very small perpendicular velocity, it will penetrate the refracting ftratum no further than till this perpendicular velocity is extinguished, and its path becomes parallel to the furface, and it will be reflected back. As the perpendicular velocity increases by diminishing the obliquity of incidence, it will penetrate farther; and the last reflection will happen when it penetrates fo far that its path touches the line ob. Now diminish the obliquity by a single second; the light will get over the line o b, will defcribe an arch o dB concave upwards, and will emerge in a direction BA, which does not fkim the furface, but is fenfibly raised above it. And thus the facts obferved by M. Be

guelin,