BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 277 



Descartes had attempted the solution of a historic geometrical 

 problem propounded by Pappus. From a point P perpendiculars 

 are dropped on m given straight lines and also on n other given 

 lines. The product of the m perpendiculars is in a constant 

 ratio to the product of the n; it is required to determine the locus 

 of P. Pappus had stated without proof that for m = n = 2 the 

 locus is a conic section, Descartes showed this algebraically, 

 Newton afterwards conquering the difficulty by unaided geometry. 



Descartes distinguished geometrical curves for which x and y 

 may be regarded as changing at commensurable rates, or as we 

 should say, curves for which the slope is an algebraic function 

 of the coordinates, from curves which do not satisfy this condition. 

 These he called "mechanical," and did not discuss further. For 

 the accepted definition of a tangent as a line between which and 

 the curve no other line can be drawn, he introduced the modern 

 notion of limiting position of a secant. In connection with this 

 he considered a circle meeting the given curve in two consecutive 

 points, a perpendicular to the radius of the circle being a common 

 tangent to the circle and the given curve. The circle was not 

 however that of curvature, but had its centre on an axis of sym- 

 metry of the given curve. He recognized the possibility of ex- 

 tending his methods to space of three dimensions, but did not work 

 out the details. His geometry contained also a discussion of the 

 algebra then known, and gave currency to certain important inno- 

 vations, in particular the systematic use of a, b, and c, for known, 

 x, y, and z, for unknown quantities ; the introduction of exponents ; 

 the collection of all terms of an equation in one member ; the free 

 use of negative quantities ; the use of undetermined coefficients in 

 solving equations ; and his rule of signs for studying the number 

 of positive or negative roots of equations. He even fancied that 

 he had found a method for solving an equation of any degree. 



It is important to distinguish just what Descartes contributed 

 to mathematics in his analytic geometry. Neither the com- 

 bination of algebra with geometry nor the use of coordinates was 

 new. From the time of Euclid quadratic equations had been 

 solved geometrically, while latitude and longitude involving a 



