5 
80 CARNOT. 
pose at the same time that — 10, the first term of the 
second ratio, surpasses + 10, the second term of the 
same ratio; — 10 cannot be, at the same time, both infe- 
rior and superior to + 10.* 
Such is, in substance, one of the principal arguments 
on which our member grounds his view, that the notion 
of absolute or comparative magnitude should not be ap- 
plied to negative quantities any more than to imaginary 
ones ; that we cannot examine whether they are greater 
or less than zero; that they must be considered “as 
creations of our reason, as mere algebraical forms.” 
When the genius of Descartes ‘had shown that the posi- 
tions of all possible curves, their forms, and the whole of 
their properties, might be exactly included in analytical 
equations, the question of negative quantities presented 
itself under an entirely new light. The illustrious philos- 
* — 10 is neither inferior nor superior to + 10; it is equal to it; 
though not algebraically =; but in taking, as our author does, the 
sense of mathematical formulz, — 10 is just as good and as strong in 
its way as + 10 in its other way. Indeed — and + are merely sym- 
bols of action one way or the other; notwithstanding the ordinary 
translation of minus, — being “less,” whereas it simply means nega- 
tive, the opposite of positive. And though it is most habitual to our 
ideas to consider every thing in a positive light, the negative value is 
just as real; a correct appreciation of it only requiring the knowledge 
of where the zero of the peculiar subjects treated of is placed, which 
should always be one of the data in a mathematical question; thus 
10 feet below the level of the sea are just as efficient as 10 feet above, 
and 10 degrees below any level in the thermometer are a perfect match 
for 10 degrees above. In fact, — 10 may be less than + 10 in our 
usual manner of viewing positive things; yet mathematically and 
truly it is not less, nor greater, but just as great. Perhaps calling 
a — quantity less than nothing, has occasioned a confusion of terms; 
for it is merely a quantity on the other side of zero, which is only a 
symbol of equilibrium, or of no power one way or the other. The 
place and value of zero depend on the class of subjects treated of, and 
are previously known from experience.— Translator. 
. 
