200 THE PRINCIPLES OF SCIENCE. 



geometric problems. It is true that Regiomontanus, Tar- 

 taglia, Bombelli, and possibly other early algebraists, solved 

 isolated geometrical problems by the aid of algebra, but 

 particular numbers were always used, and no consciousness 

 of a general method was displayed. Vieta in some degree 

 anticipated the final discovery, and occasionally represented 

 the roots of an equation geometrically, but it was reserved 

 for Descartes to show, in the most general manner, that 

 every equation may be represented by some curve or 

 figure in space, and that every bend, point, cusp, or other 

 peculiarity in the curve indicates some peculiarity in the 

 values of the algebraic symbols. It is impossible to describe 

 in any adequate manner the importance of this discovery. 

 The advantage was twofold : algebra aided geometry, and 

 geometry gave reciprocal aid to algebra. Curves such as 

 the long described sections of the cone were found to 

 correspond to quadratic equations of no great difficulty ; 

 and it was impossible to manipulate the symbolic equa- 

 tions without discovering properties of those all important 

 curves. The way was thus opened for the algebraic treat- 

 ment of motions and forces, without which Newton's 

 ' Principia ' could never have been worked out. Newton 

 indeed was possessed by a strange and, to some extent, 

 unfortunate infatuation in favour of the ancient geome- 

 trical methods ; but it is weh 1 known that he employed 

 symbolic methods to discover his profound truths, and he 

 every now and then, by some accidental use of algebraic 

 expressions, confessed its greater powers and generality. 



Geometry, on the other hand, gave the greatest assist- 

 ance to algebra, by affording concrete representations of 

 relations which would otherwise be too abstract for easy 

 comprehension. A curve of no great complexity may 

 give the whole history of the variations of value of a 

 troublesome mathematical expression. As soon as we 

 know, too, that every regular geometrical curve repre- 



