TELESCOPE. 



angletwe, therefore, cqu'iangukr, and confequently /m:/<7r. 

 Hence we derive the foUowing analogies for determining the 

 refraded ray MF; ^.,•=. M E : M C :: M R : M N ; 



that is, a : r :: J : — = MN. 



a 



" Again, from the triangles M G C and M O N, we have 



rt 

 MG : MC :: MO : MN; that is, i : r ::/:-- = 



M O :: M F - G F (or M G) : M F ; that is, in fymbols 

 bmds — aans — ands it , mdbb 



<.db 



nda 



— .•. sb =■ at ; and io a : b :: s 

 a 



/, or — = MO. 

 a 



And in the triangles B M R, BQ,r, we have BM : Bf 

 (= BE) :: MR : Q-r; that k, d : d ^- a :: s - 



^i±-ii = Q,-. But C^ : C^ (:: CE : CG) : 



m : n :: C<:-CE : C^-CG :: Q^ .: S^ : 



ds + as nas + nds 

 d m d 



= Sg. 



" Laftly, the fimilar triangles E MO, F S^, give MO 

 S^ :: MF : SF or GF; therefore, M - S^ 



am d a 



= M F, the focal diftance required. 



" As the right angles at E and G arc both fubtended by 

 the fame hypolheniife, Or right line M C, it is evident that 

 this line is the diameter of a femicircle, M E G C, paffing 

 through them, as in Jig. 3 ; and if the curve AMD be a 

 circle, then C will be its centre ; and when the point M is 

 extremely near to the vertex A, there will be M E = M G 

 = M C, or a = i = >-. In this cafe, the theorem becomes 



— ; ' • = A F — f; and the point F, or focus 



md — nd — iir 



of refrafted rays, is then in the axis B C produced." 



From this original theorem for finding the finiple refraftion 

 of a pencil of diverging rays out of a rare into a denfe medium, 

 may be derived other theorems for finding the fimple re- 

 fraAion out of a denfe into a rare medium, and for the 

 refraftion of lenfes of any of the common fliapes, either at 

 the firft or fecond furface. We will fubjoin a fmall tabic 

 of fuch of thofe theorems as apply to glafies of the ordinary 

 conftrudlion. 



Theorems for one fimple Refraftion. 



Hitherto we have confidered the refraftion of a ray at 

 only one furface of a lens ; but as every lens has two fur- 

 faces, or radii, r and R, it is neceflary to carry our invefti- 

 gation farther, ancf fee what theorems can be obtained for 

 finding the foci of glaffcs of the different fhapes, when double 

 refraftion takes place, which is the cafe in all inftances of 

 complete traiifmiflion. By way of diftinftion, we will con- 

 fider r as the radius of the firft furface, or that which re- 

 ceivei the rays from the radiant ; and R as the fecond fur- 

 fice, or that which is fuppofed to be turned from the 

 radiant, in all our fubfequent theorems. We muft now 



confider a ray, as M N, in ^g. 4, coming out of a denfe 

 medium Y, after proceeding in a direftion towards F, into 

 the rare medium X ; but meeting with a Ipherical furface 

 N D, on quitting the denfe medium, is relrafted into the 

 direftion N_/, to interfeft the axis Dy", in the focal pointy. 

 When two Ipherical refrafting furfaceB are near to each 

 other, as A M, N D, in Jig. 5, they conflitute a lens 

 A M N D, of which the radius of the curve A M is r, 

 when the radiant is on that fide, but that of N D is denomi- 

 nated R ; and the hne B A D F, paffing at right angles 

 through the middle of the Icbs, is called the axis. Now to 



find 



