It is little to the credit of English science that the fundamental optical theorems of Cotes and Smith should have passed almost into oblivion, until rediscovered in a somewhat 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 exchanges 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 forgotten, 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 surfaces are plane or spherical. With an application to Telescopes and Microscopes.”