62 HISTORY of the SOCIETY. 



ticians of the firft name had been engaged, and the utmoft re- 

 fources of the integral calculus had been employed. Exit though 

 many excellent folutions had been given, there was none of 

 them at once direct in its method and fimple in its principles. 

 Mr STEWART was fo happy as to attain both thefe objects. 

 He founds his folution on a general property of curves, which, 

 though very fimple, had perhaps never been obferved ; and 

 by a moft ingenious application of that property, he mows 

 how the approximation may be continued to any degree 

 of accuracy, in a feries of refults which converge with prodigi- 

 ous rapidity. Whoever examines this folution will be aftonifh- 

 ed to find a problem brought down to the level of elementary 

 Geometry, which had hitherto feemed to require the finding of 

 fluents and the reverfion of feries ; he will acknowledge the rea- 

 fonablenefs of whatever confidence Mr STEWART may be here- 

 after found to place in thofe fimple methods of inveftigation, 

 which he could conduct with fo much ingenuity and fuccefs ; and 

 will be convinced, that the folution of a problem, though the 

 moft elementary, may be the leaf! obvious, and, though the eafieft 

 to be underftood, may be the moft difficult to be difcovered. 



THIS folution appeared in the fecond volume of the Eflays 

 of the Philofophical Society of Edinburgh, for the year 1756. 

 In the firft volume of the fame Collection, there are fome 

 other propofitions of Mr STEWART'S, which are an extenfion of 

 a curious theorem in the fourth book of PAPPUS. They have 

 a relation to the fubjecl of porifms, and one of them forms the 

 pift of Dr SIMSON'S Refloration. They are befides very beau- 

 tiful propofitions, and are demonftrated with all the elegance 

 and fimplicity of the ancient analyfis. 



IT has been already mentioned, that Mr STEWART had form- 

 ed the plan of introducing into the higher parts of mixed Ma- 

 thematics the ftrict and fimple form of ancient demonftration. 

 The profecution of this plan produced the Tratts Phyftcal and 

 Mathematical^ which were publiflied in 1761. In the firft of 



thefe, 



