THE DIFFERENTIAL CALCULUS. 63 



passes or best geometrical drawing-pens ; but really the 

 circumference of a circle endowed with an ideal per- 

 fection, really a curve without thickness and without 

 roughness of any sort. Let us, in imagination, draw a 

 tangent to this curve. At the point where the tangent 

 and the curve touch one another, they will form an angle, 

 which has been called the " angle of contact." This 

 angle, since the first origin of mathematical science, has 

 been the object of the most serious reflections of geom- 

 eters. Since two thousand years ago it has been rigor- 

 ously demonstrated that no straight line, drawn from the 

 apex of the angle of contact, can be included between its 

 two sides, and that it cannot pass between the curve and 

 the tangent. Well, I ask, what else is that angle into 

 which an infinitely fine straight line cannot be introduced 

 or insinuated, but an infinitely small quantity. 



The infinitely small angle of contact, into which no 

 straight line can be introduced, may nevertheless include 

 between its two sides millions of circumferences of cir- 

 cles, all greater than the first. This truth is established 

 by reasoning of an incontestable and uncontested force. 

 Here, then, we have, in the very heart of elementary 

 geometry, an infinitely small quantity, and, what is still 

 more incomprehensible, susceptible of being divided as 

 much as we please ! The human intellect was humili- 

 ated and lost in face of such results ; but, at any rate, 

 these were results, and it submitted. 



The infinitely small quantities which Leibnitz intro- 

 duced into his differential calculus excited more scruples. 

 This great geometer distinguished several orders of them, 

 those of the second order might be neglected in relation 

 with the infinitely small of the first; these infinitely 

 small of the first order in their turn disappeared before 



