TELESCOPE. 



A C, AD, and alfo E N and E M, the fines of incidence 

 and refrailion, for which put ii and m ; then bccaiife the 

 triangles E T M, ATP, are fimikr, it will be as E T : 

 TAor TD :: (EM : APorEN ::) EF : FC; and 

 disjointly, T F : E F :: (F C - T D or) T F - C D : 

 F C J and alternately, T F : T F - C D :: E F : F C ; and 

 disjointly, T F : CD :: ( E F : E C ::)»«: m - n. Again, 

 fince (P D : P C :: C E : D T or F C and conjointly) C D 

 :CP:: (EF: FC::)m:n; by compounding this and 

 the foregoing proportion, it will be as T F : C P :: ??/ m : 

 m — n, n. Q. E. D. 



Carol. I — The fegment ACBPA may be confidered 

 as a plano-convex lens ; and when rays fall parallel upon its 

 plane fide, the longitudinal aberration of the extreme ray 

 falling upon A is equal to ^ of its thicknefs P C, as appears 

 by putting 3 and 2 for m and « refpeftively. 



ffi ffj A P'* 



Carol. 2. — Alfo this aberration FT = ^r^=^ x -^rp^ 



m — n,n 2 EC 



AP' 



r^^' 2CF 



A P' 

 For P C = — j:^-^ very nearly, and 



EC 



X CF. 



Corol. 3 Let the refradled ray ATG produced, cut 



the line F G, perpendicular to the axis, in G, and the lateral 



, . T-^ mm APJ mm A P3 



aberration F G = — x — Fr-;v = . -_ x 



2 EC 

 FG:TF :: AP : TP, orCFor 



2CF' 



For 



X CE. 



Carol. 4. — When the femi-diameter of the convexity or 

 the focal diftance is given, the longitudinal aberrations arif- 

 ing from the figure are as the fquares, and the lateral aber- 

 rations as the cubes, of the linear apertures of a plano- 

 convex ler.s. 



Prop. III. 



" When parallel rays Q A, E C [Jig. 5. ) are rejle6led from 

 a fpherical concave A C B, whofe centre is E, and vvhofe 

 aperture, A C B, is but fmall, the longitudinal aberration 

 T F, of the extreme ray A T, from the geometrical focus F, 

 is equal to half the verfed fine C P of the femi-aperture 

 A C very nearly." 



In Jig. 4. imagine E M, the fine of refraftion, to be di- 

 minilhed to nothing, and then to become negative and equal 

 to E N, the fine of incidence, and the refraftion of the ray 

 to be changed to refleftion, as ^njlg. 5 ; and by the former 

 propofition it will be, as T F : C P :: »n m : — m — n, n :: 

 /in: 2 H « :: I : — 2. 



But the particular proof is this : By the laft lemma, the 

 verfed fine C P nearly equals half the verfed fine P D of the 

 arc AD, whofe centre is T, and femi-diameter TA or 

 T E, or half the femi-diameter of the arc A C very nearly. 

 But2TF = 2TE-2EF=:ED -EC = CDex- 

 aftly, or C P nearly. Therefore T F = i C P nearly. 



Corol. I.— We had 2 T F ^ C D exadly, which is the 

 excefs of the fecant E D of the arc A C above its radius 

 E A. For joining A D, the angle D A E in the femi-circle 

 D A E is a right one. 



A P' 



Corol. 2. — The longitudinal aberration T F = -ttv-- 



4 C ii 



A P* 

 For C P = — i=rF nearly. 



2C£. 



A P' 

 Corol. t. — The lateral aberration F G = — 



2CE' 



For 



F G : F T :: A P : P T, or ' C E nearly. 



Corol. 4 — When tlie diameter of the concave or its focal 

 diftance is given, the longitudinal aberrations are as the 

 fquares, and the lateral ones as the cubes of tlic diameters 

 of tlie aj)ertures. 



Pitoi". IV. 



" When parallel rays of any one fort are refradled by a 

 plano-convex objeft-glafs, or when rays of all forts are rc- 

 flefted by a fpherical concave, the diameter of each circle 

 of aberration caufed by the fphericalnefs of the figures, is 

 equal to half the lateral aberration of tlie extreme ray in 

 each, and therefore is given by tlie former propofitions." 



Let a Y T be any refrafted or reflefted ray, cutting tlie axis 

 E C T in T (Jigs. 6 and 7.), and the extreme ray A T G, 

 that comes from the contrary fide of the axis, in Y. Draw 

 Y X perpendicular to the axis ; and fuppofing tlie line 

 A T G immpveable, as the point of incidence u moves from 

 the vertex C, the perpendicular X Y will firll incieafe, be- 

 caufe the angle Cra continually increafcs, and afterwards 

 will decreafe, becaufe the line Tt continually detreafes; and 

 when X Y is the greatell, it is evident tliat all the rays, in- 

 cident upon the lame fide of the axis as itfelf, will pafs 

 through it. To find its greateil quantity, let tlie incident 

 ray qa cut the chord A P B in B, and fuppofing the variable 

 aperture F 13 = v, the variable T X ~ x, and the given lines 

 PA =«, PT =/, TF =i; by Cor.4. Props. II. and 

 III. the aberration F t is to the aberration FT (b) as ■ktj'' 

 or Pp^ (vv) to PA- (<Ja). 



Wherefore F t = — i, and thence TF — Ft^Tt 

 a a 



= — Xna ~vv. Again, P T (/) : P A (n) :: T X 



(.v) : XY = y; alfo^«(^) : 



PT (/) :: XY 



r-f") : X T = — . Hence again, T t, or X t + X T = 



—1 -fK = — X aa ~~ -uv found before ; or — x a + v 

 •u aa 11 



: — X a 4- -u X a 



a a 



Whence x = — -u x a — v, 

 aa 



and therefore x or T X is the greateft poifible when tlie 

 reftangle •uxa — ■u, orP(3x/SBis greateft, that is, when 

 its fides PiS, /S B, are equal, or when v — ^a. Subftitute 

 this value for -v in the laft equation, and it gives the greateft 

 value of .V = ^ l>, or the greateft T X = ^ T F ; and there- 

 fore the greateft X Y = A F G, becaufe T X : X Y :: T F 

 : F G ; and this X Y, turned about the axis P X, defcribes 

 the circle of aberrations through which all the rays falling 

 upon A B will juft pafs. Q. E. D. 



Prop. V. 



" The circle of aberrations caufed by tlie fphericalnefs of 

 the figure of the objeft-glafs of a telefcope, compared with 

 the circle of aberrations caufed by the unequal refrangibility 

 of rays, is altogether inconfiderable." 



For if the objedt-glafs be plano-convex, and the plane fide 

 be turned towards the objeft, and the di.imeter of a iphere, 

 whereof this glafs is a fegment, be called D, and the femi- 

 ■liameter of the aperture of the glafs be called S, and the 



fine 



