A r I* i . \ i , 



NOTE A 



Historical sketch.* Analytic Geometry, in the form in which it is 

 now known, wan invented by Kent Descartes (1506-1600) and flirt pub- 

 lished by In... in 1'..::. ... the third section of a treatise on universal 

 science entitled - Disoours de la tntfthod poor bien oonduire sa raisoo si 

 chercher la refit* dans la science*." He made the invention while 

 attempting to solve a certain problem, proposed by Pappus, the most 

 important case of which is: to find the locus of a point such that the 

 product of the perpendiculars drawn from it upon m given straight lines 

 shall bear a constant ratio to the product of the perpendiculars drawn 



it upon n other given straight lines. By pure geometry this prob- 

 lem hsd already been solved for the special oases when = 1 and n = 1 

 or 2. Pappus had also asserted, but without pro- nen m = >. 



then the loons of this point is a conk. In his effort to prove this fact 

 Descartes introduced his system of coordinates and found the equation 

 of the locus to be of the second degree, thus proving -a conic. 



Analytic geometry does not consist merely (as Is sometimes loosely 

 said) in the application of algebra to geometry: that had been done by 



tnedes and many others, and had become the usual method of pro- 

 cedure in the works of mathematicians of the sixteenth century. Hut in 

 sll these earlier applications a special set of axes were required for each 



lu.il curve. The great advance made by Descartes was that he 

 saw that a point >uM be completely determined if its distances, say * 

 and * from two fixed lines, drawn at right angles to each other, in the 

 plane, were given : and that though an equation /(x, y ) =0 is indet. 

 nate and can be satisfied by an infinite number of values of x and y, yet 

 these values of x and y determine the coordinates of a number of points 



i form a curve of which the equation f(r, y) = expresses some 

 geometric property, t.*., a property true for every point of the curve. 

 Moreover, he saw that this method enables one to refer all the curves 

 that may be under investigation to the tarn* set of axes; and that in 



Taken chiefly from Ball's History of 

 881 



