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 
