DEVELOPMENT OF MATHEMATICAL THOUGHT. 661 



Analytical geometry, by substituting an algebraical ex- 

 pression for a geometrical figure say a curve, could 

 apply to it all the artifices of abstract analysis. By 

 varying the co-ordinates you can proceed along the whole 

 extent of the curve and examine its behaviour as it 

 vanishes into infinity, or discover its singular points at 

 which there occurs a break of continuity : you can vary 

 its constants or parameters, and gradually proceed from 

 one curve to another belonging to the same family, as is 

 done in grouping together all curves of the second order, 

 or as was done in the calculus of variation, invented 

 by Euler and Lagrange you can vary the form of the 

 equation, proceeding from one class of curve to another. 

 Now clearly all this operating on equations and sym- 

 bolic expressions was originally abstracted from geom- 

 etry, including the mechanical conception of motion ; in 

 particular the ideas which underlie the method of 

 fluxions were suggested by the motion of a point in 

 space. The conception of continuous motion in space 



the principle as a valuable in- 

 strument for the discovery of 

 new truths, which nevertheless did 

 not make stringent proofs super- 

 fluous." Cauchy's report seems to 

 have aroused Poncelet's indignation. 

 Hankel ('Elemente der Projectiv- 

 ischen Geometric,' 1875, p. 9) 

 says : " This principle, which was 

 termed by Poncelet the ' Prin- 

 ciple of Continuity,' inasmuch as 

 it brings the various concrete 

 cases into connection, could not 

 be geometrically proved, because 

 the imaginary could not be 

 represented. It was rather a 

 present which pure geometry re- 

 ceived from analysis, where im- 

 aginary quantities behave in all 

 calculations like real ones. Only 



the habit of considering real and 

 imaginary quantities as equally 

 legitimate led to that principle 

 which, without analytical geometry, 

 could never have been discovered. 

 Thus pure geometry was compen- 

 sated for the fact that analysis 

 had for a long time absorbed the 

 exclusive interest of mathemati- 

 cians ; indeed it was perhaps an 

 advantage that geometry, for a 

 time, had to lie fallow." Kbtter 

 continues : " Von Staudt was the 

 first who succeeded in subjecting 

 the imaginary elements to the 

 fundamental theorem of projective 

 geometry, thus returning to analyt- 

 ical geometry the present which, 

 in the hands of geometricians, had 

 led to the most beautiful results." 



