THE INFINITESIMAL CALCULUS. 61 
ties from the pleasure of “measuring himself with” the 
grand problems whose solution requires years of con- 
tinuous and persevering effort, Carnot chose those dif_i- 
cult but circumscribed questions which may be taken up, 
abandoned, and taken up again, by fits and starts ; which 
an elevated mind capable of coping with difficult sub- 
jects, develops and fathoms without paper or pencil, 
either during a walk, in the midst of the excitements of 
a crowd, the gayeties of a banquet, or the vigils of labo- 
rious nights; in a word, he directed his meditations to- 
wards the “ metaphysics of the calculus.” In the present 
day such researches would be, I fear, but little relished ; 
nevertheless, if we recur to the times when mathematical 
studies gradually led to the consideration of quantities of 
such different natures, we shall be amply aware of the 
apprehension with which they inspired exact philosophers, 
and must acknowledge that, on many points, it is rather 
habit than true science which has rendered us more con- 
fident. 
Amongst the quantities to which I have alluded, the 
“¢rrational” presented themselves first. The ancients 
scrupulously avoided using them; the ‘moderns would 
also have wished to avoid the use of them; “ but they” 
(the quantities) “ gained the day by their numbers,” says 
the ingenious author of the “ Geometry of Infinites.” 
To the quantities which were not numerically assign- 
able, succeeded the impossible quantities, the “ imaginary 
quantities,” regular symbols of which it would be vain to 
attempt to give, not only the exact values, but even mere 
approximations. ‘These imaginaries are nevertheless used 
in combination by addition and subtraction; they are 
multiplied and divided, the one by the other in the same 
manner as real quantities; at the end of the calculation 
