8 • Optica! Ah.riJiivufnm Figure. 



hx convex IciilM, t'lofe rays which pafs. »t a diftance from tho axis are converge*! to « 

 point nearer to the lens than its geometrical focus. The ililbncc between the point at 

 whicli the external ray ol" a pencil incident on a lens intcneds its axis, and the geometri- 

 cal focus, is called the linear aberration of that lens. 



Ilugenius demonllrate';, that \vhci\ a plano-convex lens is cxpofed to parallel rays, with 

 its plane lule towards them, this aberration will amount to four times and a half the thick- 

 nefs of the glafs. Dy the thlcknefs of a convex lens, is meant its greateft thickncfs in the 

 middle, after fubtrading its tliicknefs, if it has any, at the outer edge; and by the thick- 

 ncfs of a concave lens, is meant its thicknefs at the external cdgii, after deducing its thick- 

 ncfs in the middle. 



On turning the eonvcx fide of the lens towards the light, t!ie linear aberration will only 

 exceed the thickncfs of the lens by one fixth part. 



When both fides of a lens are convex, and the proportion of their convexities is as one to 

 fix; if the moll convex fide be expofcd to paruilel rays, the aberration will exceed the thick- 

 ncfs of the lens one fourteenth, which is the fmalleft. poflible aberr.uion of any convex lens. 

 " 1.' it is required to increafe the aberration, this may be done by grinding one fide of 

 the Itns convex, and the other fide concave, to a longer radius. Such a lens, with its con- 

 cave fide turned towards parallel rays, wdl have more aberration than any plano-convex 

 or d.ouble convex lens of the fame focal dillance. 



Ilugcnius proceeds to fliew, that th.e fame aberration is produced by concave lenfes as 

 bv fimilar convex ones. When a plano-convex lens is expofed to parallel rays with its plane 

 fide outward, the external ray of the pencil being produced backward, after refraftion, will 

 interfect the a^is of the lens nearer to it than its focus by four times and a half the thick- 

 ncfs of the lens. But if its concave fide be expofed to the parallel rays, the aberration 

 will only exceed the thicknefs of the lens one fourteenth part. A double concave, whofe 

 ladii are as one to fix, with the moft concave fide turned outward, difperfes the rays with 

 the lead aberration ; and a concave menifcus, with its convex fide outward, produces 

 more aberration than any plano-concave or double concave lens of an equal focal diftaiice. 



Thefe are fufficient data for corre£ling the aberration from the fpherical figure, in cafes 

 where both a convex and concave lens are required in the conftruiftion of the compound 

 object-glafs. 



Fig. 3. PI. I. Let AB reprefent a convex lens receiving a pencil of rays from the objeiTt: S, 

 and converging rays incident near the axis at ST to the point F, and external rays as SB 

 to (he point D, fo that DF reprefents the- gteateft linear aberration in this cafe. 



/\g.iin, let GH (Fig. 4.) reprefent a concave lens receiving the parallel rays SH, RK, 

 ■which it refracls in the lines HX and-KV. This ray KV being produced backward, will 

 interfecl the axis of the lens nearly at the point N, which is called the virtual focus of the 

 conc.ive -, and the external ray HX produced backward, will intcrfedl tlie axis in fonie point 

 P nearer to the lens than its focus PN, being the linear aberration. 



It may here be obferved, that the convex is in that pofition which produces the lead 

 aberration, and the concave in the pofition which produces moft aberration. Hence, to 

 render the aberrations DF (Fig. 3.) and PN (Fig. 4.) equal, the focal diftanee of the con- 

 vex mull be much fhoricr than that of the concave ; and if the diftanccs of the points F 

 4 and 



