62 CARNOT. 



the imaginaries sometimes disappear amongst the trans- 

 formations which they undergo, and the result is then 

 held to be quite as certain as if it had been arrived at 

 without the help of these algebraic hieroglyphics. It 

 must be confessed that, though thousands on thousands 

 of applications of the calculus justify this confidence, few 

 geometers fail to take credit to themselves for the ab- 

 sence of imaginary quantities in the demonstrations 

 where they have been able to avoid them. 



The " irifinite " first made its irruption into geometry 

 on the day when Archimedes determined the approxi- 

 mate proportion of the diameter to the circumference by 

 assimilating the circle to a polygon " ^vith an infinite 

 number of sides." Bonaventura Cavalieri afterwards 

 went much farther in the same field of research ; various 

 considerations led him to distinguish some " infinitely 

 great quantities " of several orders, from some infinite 

 quantities which were nevertheless infinitely smaller than 

 other quantities. Can we be astonished that, at sight of 

 such results, and notwithstanding his lively predilection 

 for combinations, which had led hira to veritable dis- 

 coveries, the ingenious Italian author should have ex- 

 claimed, in the style of that period, " Here are difip^cidties 

 of which even the arms of Achilles coidd make nothing!" 



The '■'■infinitely small" quantities had, for their part, 

 slipped into geometry even before the " infinitely great" 

 and this not only to facilitate or abridge such and such 

 demonstrations, but as the immediate and necessary re- 

 sult of certain elementary properties of curves. 



Let us examine, in effect, the properties of the most 

 simple of all — the circumference of a circle ; and by that 

 we will not understand the rugged clumsy curve which 

 we should succeed in drawing by the aid of our com- 



