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 diffi- 

 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 

 " irrational" 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 



