126 SIMSON. 



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 

 occasion to observe that this is an illogical and imper- 

 fect definition, not coextensive with the thing defined ; 

 the above proposition, however, answers every defini- 

 tion 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 

 geometers, and afterwards to converse with Theodoras 

 at Gyrene, and the Pythagorean School in Italy. But 

 it can hardly be supposed that all the preceding geo- 

 meters had worked their problems and theorems at 

 random; that Thales and Pythagoras with their dis- 

 ciples, a century and a half before Plato, and Hip- 

 pocrates, half a century before his time, had no 

 knowledge of the analytical method, and pursued no 

 systematic plan in their researches, devoted as their 

 age was to geometrical studies. Plato may have im- 

 proved and further systematized the method, as he 

 was no doubt deeply impressed with the paramount 

 importance of geometry, and even inscribed upon the 

 gates of the Lyceum a prohibition against any one 

 entering who was ignorant of it. The same spirit of 

 exaggeration which ascribes to him the analytical 

 method, has also given rise to the notion that he was 

 the discoverer of the Conic Sections ; a notion which 

 is without any truth and without the least probability. 



Of the works written by the Greek geometers some 

 have come down to us ; some of the most valuable, as 

 the 'Elements' and 'Data' of Euclid, and the ' Conies' 

 of Apollonius. Others are lost ; but, happily, Pappus, 

 a mathematician of some merit, who flourished in the 



