124 SIMSON. 



in expressing it that so much less attention is now 

 paid to 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 Sim son, 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 recording 

 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 incon- 

 siderable 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 per- 

 formed, or of something which, if not self-evident, was 

 already by former discovery proved to be true ; and in 

 the one case a construction was thus found by which 

 the problem was solved, in the other a proof was ob- 

 tained that the theorem was true, because in both cases 

 the ultimate point had been reached by strictly legiti- 

 mate 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 



