sect, i.] DISSERTATION SECOND. 9 



The elementary truths of that science were connected by 

 Euclid into one great chain, beginning from the axioms, 

 and extending to the properties of the five regular solids ; 

 the whole digested into such admirable order, and explain- 

 ed with such clearness and precision, that no similar work 

 of superiour excellence has appeared, even in the present 

 advanced state of mathematical science. 



Archimedes had assailed the more difficult problems of 

 geometry, and by means of the method of Exhaustions, 

 had demonstrated many curious and important theorems, 

 with regard to the lengths and areas of curves, and the 

 contents of solids. The same great geometer had given a 

 beginning to physico-mathematical science, by investigat- 

 ing several propositions, and resolving several problems in 

 Mechanicks and Hydrostaticks. 



Apollonius had treated of the Conick Sections, — the 

 Curves which, after the circle, are the most simple and 

 important in geometry ; and, by his elaborate and profound 

 researches, had laid the foundation of discoveries which 

 were to illustrate very distant ages. 



Another great invention, the Geometrical Analysis, as- 

 cribed very generally to the Platonick school, but most 

 successfully cultivated by the geometer just named, is one 

 of the most ingenious and beautiful contrivances in the 

 mathematicks. It is a method of discovering truth by 

 reasoning concerning tilings unknown, or propositions 

 merely supposed, as if the one were given, or the other 

 were really true. A quantity that is unknown, is only to 

 be found from the relations which it bears to quantities that 

 are known. By reasoning on these relations, we come at 

 last to some one so simple; that the thing sought is thereby 

 determined. By this analytical process, therefore, the 

 thing required is discovered, and we are at the same time 

 put in possession of an instrument by which new truths 



