64 GREEK GEOMETRY. 



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 geo- 

 metrical methods. The application of algebra to geometry, it 

 would be ridiculous to suppose that either he or his celebrated 

 pupil Matthew Stewart disliked or undervalued. That appli- 

 cation forms the most valuable service which modern analysis 

 has rendered to science. But they did object, and most reason- 

 ably and consistently, to the introduction of algebraic reason- 

 ing 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 constructions 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 carry- 

 ing geometrical reasoning into the fields peculiarly appro- 

 priate 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 sti-aight lino drawn from 

 one focus this is to be taken only as an homage to the 

 undervalued potency of the Greek analysis, or at most, as a 

 feat of geometrical force, and by no means as an indication of 

 any wish to substitute so imperfect, however beautiful, an 

 instmment, for the more powerful, though more ordinary one 

 of the calculus which " alone can work great marvels." At 

 the same time, and with all the necessary confession of the 



