SH. :lv.2 tee composition OF MIND, 101 



otherwise the two trimglea would uot be similar aud equal, 

 and the conditions of the case would be violated. All the 

 sides being thus equal, each to each, the two triangles must 

 everywhere coincide, and consequently the two basal angles 

 must be equal, both in the triangle which has been turned 

 over and in the one which has kept its original position. 

 Now, each step of this demonstration is a cognition of ttio 

 equality of a pair of relations of length or of direction ; and 

 in each case this cognition is established, not by any anterioi 

 demonstration, but by direct inspection. Or, in other words, 

 when it is said that two lines of equal length, starting from 

 the same point, and running in the same direction, must 

 coincide at their farther extremities, the truth of the state- 

 ment is at once recognized simply because the states of con- 

 sciousness which we call the ideas of the tvjo lines are totally 

 indistinguishable from each other. This immediate perception 

 of the equality — or, in some cases, of the inequality — between 

 two or more relations of position or magnitude is the goal 

 toward which every geometrical demonstration tends. And, 

 still more, it is the mental act implied 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 directly into consciousness, so that their 

 vquality or inequality may be recognized by direct inspection. 

 Manifestly the case is the same in that algebraic 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 rela- 

 tion are equal to each other " is an axiom which twice involves 

 tiie 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 artifices " When a 



