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 “infinite” 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 “with 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 him to veritable dis- 
coveries, the ingenious Italian author should have ex- 
claimed, in the style of that period, “ Here are difficulties 
of which even the arms of Achilles could make nothing!” 
The “ ¢nfinitely 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- 
