SIMSON. 137 



and above all, by the absolute rigour of its demonstra- 

 tions. He never undervalued modern analysis ; it is 

 a great mistake to represent him as either disliking 

 its process, or insensible to its vast importance for the 

 solution of questions which the Greek analysis is 

 wholly incapable of reaching. But he considered it 

 as only to be used in its proper sphere ; and that 

 sphere he held to exclude whatever of geometrical 

 investigation can be, with convenience and elegance, 

 carried on by purely geometrical methods. The appli- 

 cation of algebra to geometry, it would be ridiculous 

 to suppose that either he or his celebrated pupil 

 Stewart disliked or undervalued. That application 

 forms the most valuable service which modern analysis 

 has rendered to science. But they did object, and 

 most reasonably and consistently, to the introduction 

 of algebraic reasoning wherever the investigation 

 could, though less easily, yet far more satisfactorily, 

 be performed geometrically. They saw, too, that in 

 many instances the algebraic solution leads to con- 

 structions of the most complex, clumsy, unmanageable 

 kind, and therefore must be, in all these instances, 

 reckoned more difficult, and even more prolix than 

 the geometrical, from the former being confined to the 

 expression of all the relations of space and position, 

 by magnitudes, by quantity and number, (even after 

 the arithmetic of sines had been introduced,) while the 

 latter could avail itself of circles and angles directly. 

 They would have equally objected to carrying geome- 

 trical reasoning into the fields peculiarly appropriate 

 to modern analysis ; and if one of them, Stewart, did 

 endeavour to investigate by the ancient geometry 

 physical problems supposed to be placed beyond its 

 reach as the sun's distance, in which he failed, and 

 Kepler's problem, in which he marvellously succeeded, 

 that of dividing the elliptical area in a given ratio by 

 a straight line drawn from one focus this is to be 

 taken only as an homage to the undervalued potency 



