470 SIMSON. 



demonstrated by supposing the thing true, and thus 

 reasoning till we find that the angle in a semicircle is 

 a right angle, a known truth. Lastly, suppose we 

 change the hypothesis, and leave out the position of 

 the point as given, and inquire after the point in the 

 given straight line from which a line being drawn 

 through a point to be found in the circle, the seg- 

 ments will contain a rectangle equal to the rect- 

 angle under the perpendicular segments we find that 

 one point answers this condition, but also that the 

 problem becomes indeterminate ; for every line drawn 

 through that point to every point in the given straight 

 line has segments, whose rectangle is equal to that 

 under the segments of the perpendicular. The enun- 

 ciation of this truth, of this possibility of finding such 

 a point in the circle, is a Porism. The Greek geo- 

 meters of the more modern school, or lower age, defined 

 a Porism to be a proposition differing from a local 

 theorem by a defect or defalcation in the hypothesis ; and 

 accordingly we find that this porism is derived from the 

 local theorem formerly given, by leaving out part of 

 the hypothesis. But we shall afterwards have occa- 

 sion to observe that this is an illogical and imperfect 

 definition, not coextensive with the thing defined ; the 

 above proposition, however, answers every definition of 

 a Porism. 



The demonstration of the theorem or of the construc- 

 tion obtained by investigation in this manner of pro- 

 ceeding, is called synthesis, or composition, in opposi- 

 tion to the analysis, or the process of investigation ; 

 and it is frequently said that Plato imported the whole 

 system in the visits which he made, like Thales of 

 Miletus and Pythagoras, to study under the Egyptian 



