388 Arithmetical Relations between Chemical Equivalents. 



There next arises the question "What signification is to be 

 attached to a quantity presenting itself under these circumstances 

 with a negative sign ? I need not remark that the theory of 

 negative signs has been a stumbling block in the way c " 



most exalted intelligence ; such as those which occurred between 

 Leibnitz and Bernouilli, and between Euler and D'Alembert. 



But the ideas which prevail at the present day are sufficiently 

 clear to throw a light on this subject, and I cannot do better 

 than quote verbatim the following passage which I translate 

 from Carnot. 



"The true sense which is to be attached to this expression, 

 (that of a negative quantity), is that this absolute quantity does 

 not belong to the system on which the reasonings have been 

 established ; but to another which stands with it in a certain re- 

 lation ; such that in order to render applicable to it the formuJas 

 found for this first system, it is necessary to change from + to 

 the sign which precedes it. . , 



"But from the necessity of placing for example -y 'P *°® 

 place of +y it does not follow that the quantity represented hjy 

 has become negative ; but only that, as has just been proved, it is 

 the difference between two other quantities a, z, of which that 

 which was the greater in the system on which the reasoning was 

 established and the formulas found, has become the least in tne 

 system to which it is desired to apply these formulas. For tne 

 quantity represented by y being constantly, by hypothesis, tne 

 diflerence between the two (quantities, a, 2, will be now a jj 

 now z—a, according as z is less or greater than a; '^'^* ^"^ 

 cases it will be the greater of these two quantities less the lesse j 

 and consequently always positive, and the expression -y y 

 never be anvthing more than a simple algebraical expressio , 

 without signification in itself but having the property ^^.^^ A 

 being substituted in the formulas found, in place of +2/, 1* j^ 

 render thera applicable to cases not previously foreseen, or w 

 at least were not included in those on which the reasoning waa 

 primarily established."* . -. ,3 



The above is precisely the case in the question of fq^^^^^^^': 

 before us. The above quantity, a, may be taken as the common 

 difference in a series of equivalents, and z any term in that sen^ 

 y the following term. Now the sign of y depends upon i 

 relative greatness of the numbers a and z, but in the words J^^ 

 quoted, ''from the necessity of placing —y in the place of+y^ „ 

 not follow that the quantiiu represented bu ?/ has become «^5'f f^^* .- 



The criticism to which I am now referring has probably been 

 founded upon the very general assumption that a ^^S^^}J^^^^ 

 tity is less than nothing. This assumption is so plausible 

 * Camot, G6omatrie de Position, Paris An xi (1803). Dissert Pr^L ^^^ 



