268 THE PRINCIPLES OF SCIENCE. [chap. 



and we thus account for the extraordinary precision with 

 which intervals of time in astronomy are compared to- 

 gether. 



The fourth mode of measurement, in which we equate 

 submultiples of two magnitudes, is comparatively seldom 

 employed, because it does not conduce to accuracy. In 

 the photometer, perhaps, we may be said to use it ; we 

 compare the intensity of two sources of light, by placing 

 them both at such distances from a given surface, that the 

 light falling on the surface is tolerable to the eye, and 

 equally intense from each source. Since the intensity of 

 light varies inversely as the square of the distance, the 

 relative intensities of the luminous bodies are propor- 

 tional to the squares of their distances. The equal in- 

 tensity of two rays of similarly coloured light may be 

 most accurately ascertained in the mode suggested by 

 Arago, namely, by causing the rays to pass in opposite 

 directions through two nearly flat lenses pressed together. 

 There is an exact equation between the intensities of the 

 beams when Newton's rings disappear, the ring created 

 by one ray being exactly tlie complement of that created 

 by the other. 



The Method of Repetition. 



The ratio of two quantities can be determined with 

 unlimited accuracy, if we can multiply both the object 

 of measurement and the standard unit without error, and 

 then observe what multiple of the one coincides or nearly 

 coincides with some multiple of the other. Although per- 

 fect coincidence can never be really attained, the error 

 thus arising may be indefinitely reduced. For if the 

 equation py = qx be uncertain to the amount e, so 



that p// = qx ± e, then we have p = q'-± -, and 



as we are supposed to be able to make y as great as we 

 like without increasing the error e, it follows that we 

 can make e -^ y as small as we like, and thus approxi- 

 mate within an inconsiderable quantity to the required 

 ratio X -— y. 



This method of repetition is naturally employed when- 

 ever quantities can be repeated, or repeat themselves 



