V 



202 On the ORIGIN and 



has it arifen, that proportions which are in themfelves fo im- 

 portant, and that actually occupied fo confiderable a place in 

 the ancient geometry, have been fo little remarked in the mo- 

 dern ? It cannot indeed be faid, that proportions of this kind 

 were wholly unknown to the moderns before the reftoration of 

 what Euclid had written concerning them ; for befide M. Bos- 

 covich's proportion, of which fo much has been already faid, the 

 theorem which afTerts, that in every fyftem of points there is a 

 centre of gravity, has been fhewn above to be a Porifm ; and we 

 (hall fee hereafter, that many of the theorems in the higher geo- 

 metry belong to the fame clafs of propofitions. We may add, that 

 fome of the elementary propofitions of geometry want only the 

 proper form of enunciation to be perfect: Porifms. It is not there- 

 fore ftriclly true, that none of the propofitions called Porifms have 

 been known to the moderns ; but it is certain, that they have 

 not met, from them, with the attention they met with, from 

 the ancients, and that they have not been diftinguifhed as 

 a feparate clafs of propofitions. The caufe of this difference is 

 undoubtedly to be fought for in a comparifon of the methods 

 employed for the folution of geometrical problems in ancient, 

 and in modern times. 



In the folution of fuch problems, the geometers of antiquity 

 proceeded with the utmoft caution, and were careful to remark 

 every particular cafe, that is to fay, every change in the con- 

 ftruction, which any change in the ftate of the data could pro- 

 duce. The different conditions from which the folutions were 

 derived, were fuppofed to vary one by one, while the others 

 remained the fame ; and all their poffible combinations being 

 thus enumerated, a feparate folution was given, wherever any 

 confiderable change was obferved to have taken place. 



This was fo much the cafe, that the feftio rationis y a geo- 

 metrical problem of no great difficulty, and one of which the 

 folution would be difpatched, according to the methods of the 

 modern geometry, in a fingle page, was made, by Apollo- 



Nius, 



