IV. 



GREEK GEOMETRY. ANCIENT ANALYSIS. PORISMS. 



THE wonderful progress that lias 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 especially the still more 

 marvellous extension of analytical science by Newton and 

 his followers, since the invention of the Calculus, has, for 

 the last hundred years and more, cast into the shade the 

 methods of investigation 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 opera- 

 tions 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 mathe- 

 matical in its nature, and consist of a series of identical pro- 

 positions 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 modern mathematics has joined 

 in expressing it that so much less attention is now paid to 



