Chap. IV. ANTIENT METAPHYSICS. 403 



-for demonftration. And, accordingly, all Euclid's demonftrations re- 

 fult from definitions and axioms, which, therefore, are very properly 

 prefixed to his work. 



And thus it appears that all demonftration, or fcience properly fo 

 called, is derived from the knowledge of the nature and elTence of the 

 fubjedls of the demonftration, and from felf-evident propofitions. 



Demonftration is of two kinds ; it is either what is called direSi — or it 

 is what Ariftotle calls apagogical, that is, ex abfurdo, as our mathemati- 

 cians exprefs it. The former is, when we demonftrate from the nature of 

 theihing as comprehended in its definition. The other is, when we 

 demonftrate the impofTibility of the contrary, by ihovving the abfurd 

 confequences that would follow upon the fuppofition of the contrary. 

 This method of demonftration muft appear, at firft fight, to be a 

 round-about way of coming to the truth, and, therefore, it is very 

 properly oppofed to the dired: method. It is, however, as convincing, 

 and proves equally that the thing is ; but it does not fliow us ivhy 

 it is,' which the other does, being deduced from the nature of the 

 thing; and, therefore, the other is very properly judged by Ariftotle 

 to be preferable, and more fcientifical. 



Not only is the analytic method ufed in whole fyftems of fcience, fuch 

 -as the philofophy of the human mind, but in the in veftigation of particular 

 propofitions, whether theorems or problems; and the method is the fame 

 in both, namely, to begin where the teaching ends, that is, to fuppofe 

 the theorem to be already demonftratcd, or the problem performed, and 

 then confider what the confequences will be ; and, if we find that thefc 

 confequences do lead us, by one or more fteps, to fome axiomatical 

 truth or propofition before demonftratcd, with which the theorem or 

 problem IS neceflarily conneded, there is an end of the analyfis, and 

 there the teaching begins ; for, by defcending from this principle, tb 



E e c 2 which 



