GEOMETRY OF POSITION. 79 
talent than Voltaire, or Rousseau, or Bonaparte, and by 
the assent of the generality of the public, that Carnot 
has combated with the keen weapons of logic. 
Certainly nothing is more simple than the notion of a 
negative quantity, when it is attached to a positive quan- 
tity greater than itself; but a detached negative quantity, 
a detached quantity looked upon as isolated, must it be 
really considered less than zero, and @ fortiort, inferior 
to a positive quantity? Carnot, agreeing on this point 
with D’Alembert, who, most amongst the great mathe- 
maticians of the last century, occupied himself with the 
philosophy of science, maintains that negative isolated 
quantities figure in operations admitted by everybody, 
and in which, nevertheless, it would be impossible to 
suppose them beneath zero. Notwithstanding the dry- 
ness of such details, I will quote one of these operations. 
No one denies that 
+ 10 is to — 10 as — 10 is to + 10. 
In order that four numbers should form a proportion, 
it is necessary, and, in fact, it suffices that, if the four 
numbers are fittingly ranged in order, the product of the 
extremes should be equal to that of the means. We 
must. not be startled at this, Gentlemen ; the principle I 
call in here, is no other than that of the famous rule of 
three of the teachers of writing and arithmetic; it is the 
principle of the calculation which is executed some hun- 
dreds of thousands of times daily in the shops of the 
metropolis. Now, in the proportion which I have just 
cited, the product of the extremes is + 100, as it is also 
of the means; therefore 
+ 10:—10::— 10: + 10. 
Nevertheless, if ++ 10, the first term of the proportion, 
surpasses the second term — 10, it is impossible to sup- 
