THE COMPOSITION OF MIND 



tends. And, still more, it is the mental act im- 

 plied in every step of every such demonstration. 

 All the devices familiar to the reader of Euclid 

 — the bisecting of lines and angles, the drawing 

 of parallels and the circumscribing of circles for 

 argumentative purposes — are simply devices 

 for bringing a given pair of space relations di- 

 rectly into consciousness, so that their equality 

 or inequality may be recognized by direct in- 

 spection. 



Manifestly the case is the same in that alge- 

 braic reasoning which our astronomer will often 

 find it desirable to employ in the course of his 

 computation of the moon's distance. The axiom 

 that " relations which are equal to the same 

 relation are equal to each other " is an axiom 

 which twice involves the immediate recognition 

 of the equality of two given relations. And, if 

 any proof were needed that the whole science 

 of algebra is based upon this axiom, it may be 

 found in one of the most common algebraic arti- 

 fices. " When a simpHfication may be thereby 

 achieved, it is usual to throw any two forms of 

 an equation into a proportion — a procedure in 

 which the equality of the relations is specifically 

 asserted." To cite Mr. Spencer's simple illus- 

 tration, — if we take any equation, 2 xy =y, 

 and, dividing it byjy, obtain a second equation, 

 2 X =jy, the legitimacy of our proceeding is at once 

 rendered apparent when the two equations are 

 147 



