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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 investiga- 

 tion 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 extra- 

 ordinary 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 facul- 

 ties more severely, because it was less mechanical than 

 the operations of the analyst. That it afforded evi- 

 dence of a higher character, more rigorous in its na- 

 ture than that on which algebraic reasoning rests, 



