angles? For it has not, that we know of, been proved, that all the rays of light iffuing from the fun, proceed in the directions of radii, drawn from the center through the points in the furface, or rather, in a direction perpendicular to the tangent of the furface of the fun. But even fuppofe that they did, might not the particles of our atmofphere refract some, fo as to meet others in the fame point, in the refracting furface? This however, is only a conjecture, and does not affect the author's theory, which fhews what effect thefe rays produce, when refracted by different mediums. He proceeds; Every ray of light paffing from a rarer into a denfer medium, is refracted towards the perpendicular; but from a denfer into a rarer one, from the perpendicular; and the fines of the angles of incidence and refraction are in a given ratio. But light confifting of parts, ⚫ which are differently refrangible, each part of an original, or compound ray, has a ratio peculiar to itfelf; and there fore, the more a heterogene ray is refracted, the more will the colours diverge, fince the ratios of the fines of the homogene rays are conftant; and equal refractions produce equal divergencies.' The first part of this paragraph, is demonftrated from the general law of attraction; fince all particles of matter attract each other in proportion to their quantity directly, and as the fquares of their diftances inverfely; fo that a ray of light falling obliquely upon a furface, will be bent more or lefs, as the denfity of the fubftance of which this furface, or rather folid, is greater or lefs and as to the latter part, it has been demonftrated by all optical writers. The Author obferves, that it has been hitherto fuppofed, that the divergency of the colours is the fame under equal refractions;. which he thinks is not always true. This he endeavours to prove by fome experiments, which we fhall mention hereafter, and then he proceeds, as no medium is known, which will refract light without diverging the colours, and as difference of refrangibility feems thence to be a property inherent in light itfelf, opticians have, upon that confideration, concluded, that equal refractions must produce equal divergencies in every fort of medium: whence it should alfo follow, that equal and contrary refractions must not only deftroy each other, but that the divergency of the colours from ⚫ one refraction would likewife be corrected by the other; and there could be no poffibility of producing any fuch thing as refraction which would not be affected by the different refrangibility of light; or, in other words, that however a REVIEW, Aug. 1759. K • ray ray of light might be refracted backwards and forwards by different mediums, as water, glafs, &c. provided it was fo done, that the emergent ray fhould be parallel to the inci⚫dent one, it would ever after be white; and conversely, if it fhould come out inclined to the incident, it would diverge, and ever after be coloured. From which it was natural to infer, that all spherical object glaffes of telescopes • must be equally affected by the different refrangibility of light, in proportion to their apertures, whatever material they be formed of. may But it seems worthy of confideration, that notwithstanding this notion has been generally adopted as an incontestible truth, yet it does not feem to have been hitherto so confirmed by evident experiment, as the nature of fo important a matter juftly demands; and this it was that determined me to attempt putting the thing to iffue by experiment.' If we mistake not, opticians did not think that the different refrangibility of light was in proportion to the apertures; but rather that it was owing to the deviation from the true figure which the object-glafs fhould have to refract all the rays to the fame focus; and those rays which did fall short of, or furpaffed the focus, caufed the colouring round the edge of the image. Befides it is well known, that the smaller the part of the fpheric surface is, in refpect to the radius, the less colour is to be feen through the image; and it is for this feason, that refracters are made fo long, in order to get a larger field with lefs error from the true figure. After this the Author enumerates feveral experiments he made in a glass prifmatic veffel filled with water, with a glass prifm in it; but as this is the fame with the eighth experiment of Sir Ifaac Newton's optics, book I. part ii. after prop. 8; it would be needless to repeat here, any more of it, than that the refult was quite contrary to the prefent: for the object, though not at all refracted, was yet as much infected with prifmatic colours, as if it had been seen through a glafs prifm, whose refracting angle was near thirty degrees. From whence he concludes, that the divergency of the colours, by different fubftances, was by no means in proportion to the refractions; and, that there was a poffibility of refraction, without any divergency of the light at all. In confequence of thefe experiments, he made fome objest glafies for telescopes, of two fpheric forms with water between them; which, he fays, were free from the errors ari fing fing from the different refrangibility of light: but not so distinct as might have been expected, because the radii of the • spherical surfaces of thofe glaffes were required fo fhort, in < order to make the refractions in the required proportions, that they must produce as great, or greater errors in the image, than those from the different refrangibility of • light. ་ As thefe experiments clearly proved, that different fub• ftances diverged the light very differently, in proportion to the refraction; I began to fufpect, that fuch variety might poffibly be found in different forts of glaffes, especially, as experience had already fhewn, that fome made much better object glaffes in the ufual way, than others and as no fatisfactory cause had as yet been affigned for fuch difference, there was great reafon to prefume, that it might be owing to the different divergency of the light by their refraction. I discovered a difference, far beyond my hopes, in the refractive qualities of different kinds of glafs, with respect to their divergency of colours. The yellow, or straw-coloured foreign fort, commonly called, Venice glafs, and the English crown glafs, are very near alike in that respect, though, in general, the crown glass feems to diverge the light, rather the least of the two. The common plate 'glafs made in England diverges more; and the white cryftal, or flint English glass, as it is called, the most of all.' Our author made feveral trials, in order to find two forts of glafs whofe difference was the greateft, which were the crown glafs, and the white flint, or cryftal; and he found, that when the refraction of the white glass, was to that of the crown glass, as two to three, the refracted light was intirely free from colours. Whence of two spherical glaffes which refract the light in contrary directions, the one muft be concave, and the other convex; and as the rays are to converge to a real focus, the excefs of refraction must be in the convex, which therefore, must be made of crown glass, and the concave with white flint glafs and as the refractions of fpherical glaffes are in an inverfe ratio to their focal distances, it is eafy to make these distances in the ratio given above. 'But it must be remembered, that the fpheric glaffes must have as large radii as they will admit of, although their focal diftances are limitted. He obferves, that the refracting powers of the fame fort of glafs, made at different times, vary; and that the two glaffes must be placed truly, on the common axis of the telescope, otherwife the defired effect will be in great meafure deftroyed, K 2 Article Article 85. A further attempt to facilitate the refolution of Isaperimetrical Problems. By Thomas Simpson, F. R. S. This branch of the mathematics, is of an old date, as we find by Apollonius Pergiae; but it received its greatest improvement from the method of fluxions: for general rules, are from thence formed, for all problems of that kind, which could not have been done by any other method. But as the progrefs of arts and fciences is gradual, moft writers upon fluxions, have added new problems to thofe already known: efpecially our learned author, who has treated this branch very copiously. He takes notice, in this paper, that about three years before, he had laid before the Royal Society, the inveftigation of a general rule, for the refolution of Ifoperimetrical problems, wherein only one of the two indeterminate quantities enters along with the fluxions into the expreffion, and which rule determines the greatest figures under given bounds; lines of the swifteft defcent; folids of the leaft refiftances; with innumerable other cafes. But as others may be proposed, fuch as actually may arife in inquiries into nature, wherein both the flowing quantities together with their fluxions, are jointly concerned; it is the investigation of a rule for the refolution of thefe, which he attempts in this paper. In order to this a general propofition is laid down, the purport of which is as follows: Let 2, R, &c. reprefent variable quantities, expreffed in terms of x and y, with proper co-cfficients, and let q, r, &c. denote as many others expreffed in terms of x and y; then it is propofed to find an equation for the relation of x and y, fo that the fluent of 29+Rr+&c. correfpondding to a given value of x (or y,) may be a maximum, or mininum. To folve this problem, he denotes the fluxions of 2 and R, by Qy, Ry, and the fluxions of q and r, by qj, ry; and after fuppofing y and j alone variable, and the two extreme ordinates conftant, he finds flux 27+R1r&c. =q1Q+r1R, &c. for his final equation That is, in words, the fluxion of the fluxion of 29+Rr, making y only variable, and divided by, is equal to the fluxion of the fame quantity 29+Rr, by making only y variable, and divided by j. Whereas the author gives the following GENERAL RULE. < Take the fluxion of the given expreffion, whofe fluent is required to be a maximum, or minimum, making alone variable * variable, and having divided by ÿ; let the quotient be • denoted by v: then take again the fluxion of the fame expreffion, making y alone variable, which divided by y; and then this last quotient will be.' The author obferves that, when y is not found in the quantity given, v will then be o; and confequently, the expreffion for equal to nothing alfo. But if y be abfent, then will, and confequently, the value of va conftant quantity: inftead of y and y, x and x may be made fucceffively variable. Morever, if the cafe to be refolved, fhould be confined to other restrictions, befides that of the maximum, or minimum; fuch as, having a certain number of other fluents, at the fame time equal to given quantities, the fame method may still be applied, with equal advantage, provided all thefe expreffions are connected together with proper co efficients. To exemplify by a particular cafe, the method of operation, he proposes the fluxionary quantity xn ym jp ., wherein the relation of x and y is fo required, that the fluent, correfponding to the given value of x and y, fhall be a maximum, or minimum: and proceeding according to the foregoing rule, he finds Þ m+p Xx p I for the 1. As this equa tion indicates that x and y increafe together from o to infinite, when their exponents are both pofitives, or, that while y increafes, decreases, when the exponent of x is negative; it can by no means be concluded, that the fluent of the given expreffion, contains either a maximum, or minimum: unlefs fome other condition be annexed to make it fo, which is not mentioned. He gives another example: - That the fluent of xn ym że may be a maximum, or minimum, and that of xp yay to be equal to a given quantity. These two quantities joined together, with the indeterminate co-efficient b, gives xn yn x+ bxt yy for the fum; and proceeding according to the rule, finds byq=mx for the required equation. But as this equation is of the fame nature as the former, it cannot be concluded, that the propofed quantity contains a maximum, or minimum, which, we imagine, the author fhould have fhewn: for it is by no means fufficient to give K 3 |