Fundamental Propositions in Optics. 467 



different form by Lagrange, Kirchhoff, and von Helmholtz. 

 Even now the general law governing apparent brightness 

 seems to be very little understood, although it has acquired 

 additional importance in connection with the theory of ex- 

 changes and the second law of Thermodynamics. In seeking 

 the most natural basis for the law of magnifying, usually 

 attributed to Lagrange, I was struck with the utility of Smith's 

 phrase "apparent distance," which has never been quite for- 

 gotten, and was thus induced to read his ch. v. book ii.*, 

 founded upon Cotes's "noble and beautiful theorem." I 

 think that it may be of service to present a re-statement, as 

 nearly as may be in his own words, of the more important of 

 the laws deduced by Smith, accompanied by some remarks 

 upon the subject regarded from a more modern point of view. 



The general problem is thus stated : — 



" To determine the apparent distance, magnitude, situation, 

 degree of distinctness and brightness, the greatest angle of 

 vision and visible area, of an object seen by rays successively 

 reflected from any number of plane or spherical surfaces, or 

 successively refracted through any number of lenses of any 

 sort, or through any number of different mediums whose sur- 

 faces are plane or spherical. With an application to Telescopes 

 and Microscopes." 



It is divided into three propositions, of which the first is: — 



a Having the focal distances and apertures of any number 

 of lenses of any sort, placed at any given distances from one 

 another and from the eye and object, it is required to find the 

 apparent distance, magnitude, situation, degree of distinctness 

 and brightness of the object seen through all the lenses ; to- 

 gether with the greatest angle of vision and visible area of the 

 object, and the particular aperture which limits them both." 



Apparent distance means the distance at which the object 

 would have to be placed so as to appear by direct vision of 

 the same apparent magnitude as through the lenses. 



Let PL (fig. 1) be an object viewed by the eye at through 



Fig. 1. 



1 L I 



7T 



II! I 



A B P 



any number of lenses placed at A, B, C, . . . . whose focal 



* Smith's ' Oompleat System of Opticks/ Cambridge, 1738. French 

 translations were published by P. Perzenas, Avignon, 1767, and by Duval 

 Leroy, Brest, 1767. 



