(123) 



S I M S N. 



THE wonderful progress that has been made in the 

 pure mathematics since the application of algebra to 

 geometry, begun by Vieta in the sixteenth, completed 

 by Des Cartes in the seventeenth century, and espe- 

 cially the still more marvellous extension of analytical 

 science by Newton and his followers, since the inven- 

 tion of the Calculus, has, for the last hundred years 

 and more, cast into the shade the methods of investi- 

 gation which preceded those now in such general use, 

 and so well adapted to afford facilities unknown while 

 mathematicians only possessed a less perfect instrument 

 of investigation. It is nevertheless to be observed that 

 the older method possessed qualities of extraordinary 

 value. It enabled us to investigate some kinds of 

 propositions to which algebraic reasoning is little 

 applicable; it always had an elegance peculiarly its 

 own ; it exhibited at each step the course which the 

 reasoning followed, instead of concealing that course 

 till the result came out ; it exercised the faculties more 

 severely, because it was less mechanical than the 

 operations of the analyst. That it afforded evidence 

 of a higher character, more rigorous in its nature than 

 that on which algebraic reasoning rests, cannot with 

 any correctness be affirmed ; both are equally strict ; 

 indeed if each be mathematical in its nature, and con- 

 sist of a series of identical propositions arising one out 

 of another, neither can be less perfect than the other, 

 for of certainty there can be no degrees. Nevertheless 

 it must be a matter of regret and here the great 

 master and author of modem mathematics has joined 



