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MODERN MATHEMATICS ALGEBRA 25 

and like one part the speculative and not the experimental part 
of modern physical science, may be considered to begin with 
Rene* Descartes (1596-1650). Of course, as \ve should expect, 
Descartes had many and worthy predecessors. Perhaps the 
greatest of them was the French mathematician Fran9ois Viete 
(1540-1603), better known by his Latinized name of " Vieta." 
But it is simpler and shorter to confine our attention to Descartes. 

Descartes always plumed himself on the independence of his 
ideas, the breach he made with the old ideas of the Aristotelians, 
and the great clearness and simplicity with which he described 
his ideas. But we must not uiider-estimate the part that 
"ideas in the air" play; and, further, we know now that 
Descartes' breach with the old order of things was not as great 
as he thought. 

Descartes, when describing the effect which his youthful 
studies had upon him when he came to reflect upon them, said : 

" I was especially delighted with the mathematics, on account 
of the certitude and evidence of their reasonings : but I had 
not as yet a precise knowledge of their true use ; and, thinking 
that they but contributed to the advancement of the mechanical 
arts, I was astonished that foundations so strong and solid 
should have had no loftier superstructure reared on them." 

And again : 

"Among the branches of philosophy, I had, at an earlier 
period, given some attention to logic, and, among those of the 
mathematics, to geometrical analysis and algebra three arts 
or sciences which ought, as I conceived, to contribute something 
to my design. But, on examination, I found that, as for logic, 
its syllogisms and the majority of its other precepts are of avail 
rather in the communication of what we already know, or even 
in speaking without judgment of things of which we are ignorant, 
than in the investigation of the unknown : and although this 
science contains indeed a number of correct and very excellent 
precepts, there are, nevertheless, so many others, and these 
either injurious or superfluous, mingled with the former, that it 
is almost quite as difficult to effect a severance of the true from 
the false as it is to extract a Diana or a Minerva from a rough 
block of marble. Then as to the analysis of the ancients and 
the algebra of the moderns ; besides that they embrace only 
matters highly abstract, and, to appearance, of no use, the