DEVELOPMENT OF MATHEMATICAL THOUGHT. 661 



Analytical geometry, Ijy 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- j the habit of considering real and 



.strument for the discovery of ! imaginary quantities as equally 



new truths, whicii nevertheless did ' legitimate led to that principle 



not make stringent proofs super- , which, without analytical geometry, 



fluous." Cauchy's report seems to | could never have been discovered, 



have aroused Poncelet's indignation, j Thus pure geometry was compen- 



Hankel ('Elemente der Projectiv- ! sated for the fact that analysis 



ischeu Geometrie,' 187.5, p. 9) 

 says: "This principle, which was 

 termed by Poncelet the ' Prin- 

 ciple of Continuity,' inasmuch as 



had for a long time absorbed the 

 exclusive interest of mathemati- 

 cians ; indeed it was perhaps an 

 advantage that geometry, for a 



it brings the various concrete i time, had to lie fallow." Kotter 



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 



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." 



