GEOMETRY OF POSITION. 79 



talent than Voltaire, or Eousseau, 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 a fortiori, 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 



_j_ 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 -f- 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- 



