28 A POPULAR ACCOUNT 



and its cause, have been already indi- 

 cated in (11.) Let LL',/'. 31, be the 

 section of a plano-convex object-glass 

 made by a plane passing through its 

 axis, and let parallel rays of pure ho- 

 mogeneous light be supposed to fall on 



Fig. 31. 



L 

 A' 



Square of the radius of the 

 spherical surface of the 

 lens, (radius being 1200 

 inches) 1440000 



Square of the sine (2) of re- 

 fraction 4 



Their product 



5760000 



the plane side, perpendicular to the sur- 

 face. If the surface of the lens be con- 

 ceived to be divided into a number of 

 concentrical rings, as described in (11), 

 the foci of each ring will be more distant 

 from the lens, the nearer the ring is to 

 the edge of the lens. Let/ be the focus 

 of the marginal ring, and F that of the 

 central rays. The foci of all the inter- 

 mediate rings will lie between F and /. 

 The rays diverging from all the foci 

 between / and F are collected in a 

 circle having the line A A' for its dia- 

 meter, and this is evidently the smallest 

 space within which all these rays are 

 collected. The diameter of this circle, 

 therefore, measures the lateral aberra- 

 tion which parallel rays would sustain 

 from the sphericity of the lens ; and 

 Newton calculated this, in order to 

 compare the imperfection of telescopes, 

 arising from this cause, with that im- 

 perfection which arises from the un- 

 equal refrangibility of light. 



By geometrical reasoning, the details 

 of which we cannot properly introduce 

 here, it is proved that the square of the 

 radius of the spherical surface of the 

 lens, multiplied by the square of the 

 sine of refraction, has to the square of 

 half the breadth of the lens L L/ mul- 

 tiplied by the square of the sine of the 

 angle of incidence the same proportion 

 as half that breadth bears to the aberra- 

 tion A A'. 



Newton then proceeds to show that if 

 the object-glass were a plano-convex 

 lens, having its plane side turned to- 

 wards the object, having the radius of 

 its convex surface 100 feetor 1200 inches, 

 and the diameter of the lens four inches, 

 the diameter of the smallest circle into 

 ^vhich equally refrangible rays would be 

 collected, would be about goooo^h of an 

 inch. The calculation is as follows, the 

 proportion of the sine of incidence to 

 that of refraction being supposed to be 

 3 to 2. 



Square of half the breadth of the 

 lens, (the breadth being 4) . . 4 



Square of the sine (3) of the angle 

 of incidence 9 



Their product .... 36 



The proportion of these products is 

 that of 160, 000 to 1 ; and such is the 

 proportion of half the breadth of the 

 lens (i. e. two inches) to the aberration, 

 which is, therefore, the 160,000th part 

 of two inches, or the 80,000 th part of 

 an inch. 



The diameter of the lateral aberration 

 arising from unequal refrangibility of 

 light, would, in the case of the lens just 

 described, be the fifty-fifth part of four 

 inches, or four fifty-fiths of an inch. 

 The lateral aberration produced by the 

 spherical form of the lens has, there- 

 fore, to that produced by the unequal 

 refrangibilily of light, so small a pro- 

 portion as 1 to 5800.* 



It follows, therefore, that the im- 

 perfection of telescopes, which arises 

 from the spherical form of lenses, bears 

 an exceedingly small proportion to that 

 which is caused by the unequal refran- 

 gibility of light. But even the small er- 

 ror arising from the spherical form may 

 be almost removed, as Newton suggests, 

 by a compound object-glass, formed by 

 two glass lenses with water between them. 

 So that thus all the labours of Descartes, 

 and others who devoted themselves to 

 the formation of spheroidal lenses were 

 fruitless, since even had they succeeded 

 in producing lenses absolutely free from 

 spherical aberration, the effect would not 

 have been perceptible. 



(37.) Reasoning thus, Newton did not 

 hesitate to pronounce the improvement 

 of refracting telescopes desperate, a con- 

 clusion which forms a striking exception 

 to the almost superhuman sagacity 

 which characterised all the philosophical 

 researches of this extraordinary man. 

 What renders this error the more won- 

 derful, is that the property of light, 



* This proportion is calculated with reference to 

 the green or mean rays. If, however, it be taken 

 with reference to those rays which produce the 

 strongest effect in vi*ion, it will only be as 1 to 1200, 



