58 GEEEK GEOMETRY. 



the Ancient Geometry than its beauty and clearness deserve ; 

 and if he could justly make this complaint a century and 

 a half ago, when the old method had but recently, and only 

 in part, fallen into neglect and disuse, how much more are 

 such regrets natural in our day, when the very name of the 

 Ancient Analysis has almost ceased to be known, and the 

 beauties of the Greek Geometry are entirely veiled from the 

 mathematician's eyes ! It becomes, for this reason, necessary 

 that the life of Simson, the great restorer of that geometry, 

 should be prefaced by some remarks upon the nature of the 

 science, in order that, in giving an account of his works, we 

 may say his discoveries, it may not appear that we are record- 

 ing the services of a great man to some science different from 

 the mathematical. 



The analysis of the Greek geometers was a method of 

 investigation of peculiar elegance, and of no inconsiderable 

 power. It consisted in supposing the thing as already done, 

 the problem solved, or the truth of the theorem established ; 

 and from thence it reasoned until something was found, some 

 point reached, by pursuing steps each one of which led to the 

 next, and by only assuming things which were already 

 known, having been ascertained by former discoveries. The 

 thing thus found, the point reached, was the discovery of 

 something which could by known methods be performed, or of 

 something which, if not self-evident, was already by former 

 discovery proved to be true ; and in the one case a construc- 

 tion was thus found by which the problem was solved, in the 

 other a proof was obtained that the theorem was true, because 

 in both cases the ultimate point had been reached by strictly 

 legitimate reasoning, from the assumption that the problem 

 had been solved, or the assumption that the theorem was true. 

 Thus, if it were required from a given point in a straight line 

 given by position, to draw a straight line which should be cut 

 by a given circle in segments, whose rectangle was equal to 

 that of the segments of the diameter perpendicular to the 

 given line the thing is supposed to be done; and the 

 equality of the rectangle gives a proportion between the 



