[ 335 ] 



% Dr. Dickson's Tranflatlon of Carnot on the. 

 Infinitefiimal Calculus. 



[Continued from Page 240.] 

 The fundamental Principles of the htfinitefimal Analyfis* 



33. J. HEOREM I. — -If in any imperfecl equation what* 

 ever, there be fuhjlituted for any one of its component quanti- 

 ties, another quantity differing infinitely little from it, or 

 ivbofe ratio to the firjl hath unity for its limit, or ultimate- 

 value; then, I fay, that the equation refulting from this trans- 

 formation cannot be a falfe equation, that is, it will becomt 

 abfolutely exacl, or, at leaf, will remain what I call an 

 imperfect equation. 



For fince, by the hypothecs, there has been fubftituted for 

 one quantity another of the fame ultimate value, and whofe 

 ratio to the former hath unity for its limit, it is evident that 

 fuch fubftitution could not change either the ultimate values 

 of the fides of the propofed equation, or their ultimate ratio. 

 Now, by the hypothefis, this ultimate ratio was (1) unity 

 before the fubftitution ; therefore it will ftill remain unity; 

 and confequently the equation will preferve the character of 

 what I call an imperfect one, if it do not become rigoroufly 

 exact, a. e.d. 



$$. Theorem II. — An equation which contains only of* 

 figned quantities, cannot he an imperfecl equation. 



For, by the definition of imperfecl; equations (article 31)^ 

 their fides are unequal, but differ infinitely little from each 

 Olher, their ratio approaching, as nearly as we pleale, to the 

 ratio of equality ; therefore there enters into fuch an equa- 

 tion fome quantity which makes no part of the fyftem of the 

 quantities propofed. But by the hypothefis the propofed 



I confefs that I mail fufpend my judgment on this point, notwithftanding 

 the inclination which I firft had to believe thefe two acids to be of th* 

 fame nature. For this reafon, before any thing can be determined, we 

 tnuft wait until experiments made on a larger fcale throw more light un 

 the fubjeft. 



X x % equation,. 



