TELESCOPE. 



fc-inaiii io long as one jubilance only remained to be the me- 

 tliam of refraction. The ingenious Huygens, however, 

 fuppofing that the diminution of the fpherical aberration 

 would contribute greatly to the improvement of the tele- 

 fcope, in-llituted lome experiments and calculations, which 

 greatly promoted the fcience of Dioptrics. He found, that 

 Sie lengthening of the radius of convexity of an objeft-glafs 

 Ihortened the verled fine of the curvature, or lefl'cned the 

 thicknefs of the glafs, on which, with equal apertures, the 

 fpherical aberration feemcd to depend ; and alfo that, in a 

 limple eye-glafs, the aberration from the figure was greateft 

 in a double convex lens, when the curves of the two faces 

 were from the fame radius ; and alfo that it increafed as the 

 radius (hortened. The ratio l : l being found to have the 

 greateft aberration, and I : 2 to have lefs, an inveftigation 

 was inftituted, from which it was at length proved, that the 

 aberration in a double convex lens is the fmalleft poffible, 

 when the radii of convexity are to each other as i : 6 ; the 

 face I being turned to the radiant or objedt to be viewed. 

 From thefe experiments originated the famous Huygenian 

 telefcope of 123 feet focal diilance, and a table of apertures 

 correfponding to the refpedlive focal lengths of the objeft 

 and eye glafles, that would exhibit an image equally well 

 defined : which calculations were the bafis of all the long or 

 (Kr/'a/ telefcopes that were in repute for a whole century ; but 

 which are now fuperfeded by the (hort achromatic refractors. 



The fame ingenious author of dioptrics difcovered, that the 

 aberration arifing from the curved figure of a lens might be 

 Itill further diminiftied, by fubflituting two lenfes in the eye- 

 piece of a telefcope inilead of one ; which difcovery was the 

 foundation of all the improved eye-pieces that have been 

 fince adopted, under different arrangements of intermediate 

 diftance, and with different degrees of curvature. But 

 before we can explain how the indiftinftnefs arifing from 

 both the fpherical and prifmatic aberrations of mixed rays, 

 may be in a great meafure counteracted, (on which im- 

 portant confideration, the excellence of modern improved 

 telefcopes depends,) it is neceflary to examine this fubjeft 

 further, and to (hew how the circle of aberration of mixed 

 rays arifing from their unequal refrangibihty, and alfo the 

 lateral and longitudinal aberrations arifing from the fpherical 

 figure of refracting and reflecting furfaces, may be mathe- 

 matically determined. In doing this, we (hall avail ourfelves 

 of Dr. Smith's propofitions, which are at the fame time per- 

 fpicuous and conclufive. 



Prop. I. 



Aberrations. — " Let the common fine of incidence be to 

 the fine of refraCtion of the leajl refrangible rays, as I 

 to R, and to the fine of refraCtion of the mojl refrangible 

 rays, as I to S ; and the diameter of the leaft circulai- fpace, 

 into which heterogeneal parallel rays can be coUeCted by a 

 fpherical furface, or by a plano-convex lens, will be to the 

 diameter of its aperture in the conftant ratio of S — R to 

 S -t- R- 2 I." 



For let an heterogeneal ray P A ( Plate XXVI./^. i.) 

 fall upon a fpherical furface A C B, and let it be feparated 

 by refraCtion into the rays A F, A/, cutting the axis E C, 

 drawn parallel to FA, in F and /. Take the arc C B 

 equal to C A, and let another heterogeneal ray P B, coming 

 parallel to P A, be refraCted into the lines B F, B/, cutting 

 the two former rays in R and S. Join R S, and produce it 

 till it meets the incident rays produced in I and K, and the 

 perpendiculars EA, E B, to the refraCting furface at the 

 points A, B, in H and L. And when A B, the breadth of the 

 •iperture or of the pencil, is but moderate, and confequently 

 the refractions at A B but fmall, the angles of incidence and 

 10 



refraCtion HA I, HA R, HA S, or the arcs th.it mea- 

 fure them, or their perpendicular fubteules H I, H R, H S, 

 will be to each other very ne;u-ly in the fame given ratios as 

 thofe of the fines I, R, S, of thofe angles. And disjointly, 

 the dilTerences of thofe fubtenfes will be proportionable to 

 the di(fcrences of thofe fines ; that is, the fine R S : R I :: 

 S — R : R — I, and doubling the confequents, R S : 

 2RIorIK-RS :: S-R : 2R- 2I; and con- 

 jointly, R S : I K, or A B :: 6 - R : S + R - 2 I. 

 From this given ratio of R S to A B, in which they increafe 

 or decreafe together, it appears that all the intermediate 

 rays which fall upon A B, will pafs through R S. And 

 when parallel rays fall perpendicularly upon the plane fide of 

 a plano-convex lens, they are refracted only at their emer 

 gence from its convex furface ; and fo' the aberrations are 

 the fame in both cafes. Q. E. D. 



Carol. I. — Hence the diameter R S, of the circle of prif- 

 matic aberrations that contains all the incident rays, is a 

 55th part of the diameter A B of the aperture of a plano- 

 convex glafs, whatever be its focal diftance. For fuppofing 

 with Newton the prifmatic ipeCtrum divided into feven 

 colours, and AR and A S to be the outermoft red and 

 violet rays, their fines of incidence and refraCtions I, R, S, 

 are to each other as- 50, 77, 78. Whence S — R is to 

 S -t- R — 2 I, as I to 55. 



Carol. 2. — The diameter of the leaft circle that can receive 

 the rays of any fingle colour, or of feveral contiguous colours, 

 is alfo determinable from the proportions of their finei. 

 Thus all the orange and yellow is contained in a circle, 

 whofe breadth is the 260th part of the breadth of the aper- 

 ture of the plano-convex glafs ; the fines of the outermoft 

 orange A R, and yellow A S, being to the common fine of 

 incidence, as 77^ and 77-i- to 50. 



Carol. 3. — In diftercnt furfaces, or plano-convex glaftes, 

 the angles of prifmatic aberration R A S are as the breadths 

 of the apertures A, B, diredtly, and as the focal diftances 

 C, F, inverfely ; becaufe any angle, as R A S, is as its fubtenfe 

 R S direCtly, and as its radius A R or C F inverfely. 



Lemma. — The verfed fines A B, AC, of very fmall 

 arcs B D, C D, {Jigs. 2. and 3.) of unequal circles B D G, 

 C D H, that have the fame right fine A D, are reciprocally 

 proportionable to their diameters B G, C H, very nearly ; 

 that is, A B : AC :: CH : BG. 



For fince the reCtangles under BAG and C A H are 

 each equal to the fquare of A D, and confequently to each 

 other, their fides are reciprocally proportionable ; that is, 

 A B is to A C as A H to A G, or as C H to B G very 

 nearly, when the verfed fines are incompai'ably lefs than the 

 diameters themfelves. Q. E. D. 



Prop. II. 



" When homogeneal parallel rays N A, EC, {Jig. 4. ) 

 fall upon a fpheric:! furface A C, whofe centre is E, 

 the longitudinal aberration F T, of any refraCted ray A T 

 from F, the focus of the pencil, is to the verfed fine of the 

 arc A C, intercepted between the point of incidence and 

 the axis E C F, in the given ratio of the fquare of the fine of 

 refraCtion, to the reCtangle under the fine of incidence, and 

 the difference of the fines very nearly ; and the aberration is 

 the fame when the rays fall perpendicularly upon the plar»e 

 fide of a plano-convex lens." 



For when the refraCtion is made in the paffage of a ray 

 N A from a denfer to a rarer medium, then the interfeCtion 

 T, of the refraCted ray A T, with the axis E C F, lies be- 

 tween the refracting furface and its focus F. With the 

 centre T and femi-diameter T A, having defcribed the arc 

 A D, cutting the axis in D, draw the fine A P of the arcs 



AC, 



