632 THE PRINCIPLES OF SCIENCE. [CHAP. 



the means ef dividing a circle by purely geometric means 

 into 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30 parts, but he 

 was totally unacquainted with the theory of the roots of 

 unity exactly corresponding to this division of the circle. 



During the middle ages, on the contrary, algebra ad- 

 vanced beyond geometry, and modes of solving equations 

 were gradually discovered by those who had no notion that 

 at every step they were implicitly solving geometric prob- 

 lems. It is true that Regiomontanus, Tartaglia, Bombelli, 

 and possibly other early algebraists, solved isolated geo- 

 metrical 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 repre- 

 sented 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 a curve or 

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

 peculiarity in the curve indicates some peculiarity in the 

 equation. It is impossible to describe in any adequate 

 manner the importance of this discovery. The advantage 

 was two-fold : algebra aided geometry, and geometry gave 

 reciprocal aid to algebra. Curves such as the well-known 

 sections of the cone were found to correspond to quadratic 

 equations ; and it was impossible to manipulate the equa- 

 tions without discovering properties of those all-important 

 curves. The way was thus opened for the algebraic 

 treatment of motions and forces, without which Newton's 

 Principia could never have been worked out. Newton 

 indeed was possessed by a strong infatuation in favour of 

 the ancient geometrical methods ; but it is well known 

 that he employed symbolic methods to discover his theo- 

 rems, and he now and then, by some accidental use of 

 algebraic expression, confessed its greater power and 

 generality. 



Geometry, on the other hand, gave great assistance to 

 algebra, by affording concrete representations of relations 

 which would otherwise be too abstract for easy compre- 

 hension. 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 represents some algebraic 



