Fundamental Propositions in Optics. 469 



in t ; and supposing the glasses were removed, let an unre- 

 fracted ray PMN cut the line G t in M. Then the quantity 

 of refracted rays which fall upon the line N is to the quan- 

 tity of unrefracted rays which would fall upon it as the angle 

 C P t to the angle CPM, that is as the apparent magnitude 

 of the line N seen from P, to the true. And therefore, by 

 turning the figure round the axis P, the quantity of refracted 

 rays which fill the pupil is to the quantity of unrefracted rays 

 which would fill it (as the apparent magnitude of any surface 

 at seen from P, to the true ; or as the apparent magnitude 

 of any surface at P seen from to the true ; and consequently) 

 as the apparent magnitude of the least surface, or physical 

 point P, to the true ; that is as the picture of the point P 

 formed upon the retina by those refracted rays, to its picture 

 formed by the unrefracted rays. These pictures of the point 

 P are therefore equally bright and cause the appearance of P 

 to be equally bright in both cases. Now let the pupil be 

 larger than the greatest area inlightened at by the pencil 

 that flows from P ; and supposing a smaller pupil equal to this 

 area, we have shown that the pictures of P made upon the retina 

 by refracted and unrefracted rays would be equally bright ; 

 and consequently each of them would be less bright than 

 when the larger pupil is filled with unrefracted rays, in the 

 same proportion as the smaller pupil, or area inlightened by 

 the refracted rays, is less than the larger pupil, inlightened by 

 unrefracted rays." 



" § 267. Corol. 6. It is evident that an object seen through 

 glasses may appear as bright as to the naked eye ; but never 

 brighter, even though all the incident light be transmitted 

 through the glasses." 



Smith's splendid work was evidently unknown to Helm- 

 holtz, when, after establishing the law of apparent brightness, 

 he remarks * : — 



" Diese Folgerung ist schon von Lagrange gezogen wor- 

 den. Leider hat er den zweiten Fall, der gerade bei starken 

 Yergrosserungen der gewohnliche ist, nicht besprochen, n'am- 

 lich den, wo das in die Pupille eindringende Strahlenbiindel 

 diese nicht ganz ausfiillt. Das mag nicht wenig zu der 

 Vergessenheit beigetragen haben, in welche seine wichtige 

 Abhandlung gefallen ist." It is indeed astonishing that a 

 theorem of such obvious importance should need to be dis- 

 covered three times f. 



* Pogg. Ann., Jubelband, 1874, p. 566. 



t In his biography of Lagrange (Suppl. Enc. Brit. ; Young's Works, 

 vol. ii. p. 575) Young, with, some want of appreciation, thus refers to 

 the memoir * On a General Law of Optics, 1803 ' : — " A demonstration of 



Phil. Mag. S. 5. Vol. 21. No. 133. June 1886. 2 L 



