Jp^ Carnot on the Tfjcory ef 



an auxiliary line NQ is drawn parallel to, and at an arbitrary 

 diftancc from, the ordinate MPy to which it {NQ) may con- 

 tinually approach, till the two lines coincide; the line NO, 

 or NQ — MP will then be an infinitely fniall quantity (fee 

 art. 19). Now as NO is the difference of the two values^ 

 MP and NQ, fucceflively attributed to the ordinate, it has 

 been agreed to diftinguifli it in difcourfe by the diminutive 

 word, the differential, of the variable line MP, and to repre- 

 fent it in calculation by the fame variable line, with the cha- 

 ra<fter d prefixed *. Jhus, putting y for the ordinate MPf 

 dy will fignify the Differential of MP. 



But to fuppofe, as we have done, that NQ continually ap- 

 proaches MP, is alfo to fuppofe that AQ continually ap- 

 })roaches to AP -, for the firft of thefe fuppolitions neceflarily 

 implies the fecond. Putting, therefore, x for the abfcifle APy 

 the little line PQ or MO will be the differential of x, and 

 We fliall have MO=^.v in the fame time that NO~dj>. 



If we farther fuppofe NQ =y, and AQ = x\ we Ihall have 

 j/ - ^'4 dy, and x^ ~ x-\-dx\ fo that the differentials dy and dx^ 

 are nothing clfe than the increments of their correfpondent 

 Variables y and .r, or the quantities by which they are in^ 

 creafed when they become y and x^ f- 



50. Now, let there be attributed to the ordinate a new 

 value RSy fuch that PQ and Q^ may differ infinitely littles 



■ * Here one i.s almoft tempted to aflc,Whetb.er the ingenious author con* 

 fiders the Bhtifli marhematidans as mere Differentia' s? For they have 

 never agreed to ufe the notation he mentions ; but, inftead o( dx, dx dy^ 

 &ic. write, with Newton, the immortal inventpr of fluxions, a-, xj &c. 

 The d's only ferve to embarrafs the combixiations, which fhould be ex- 

 prcKcdwith the utmoil clearnefs. W. D. . 



-{■ The author, after partly explaining the doftrine of prime and ulti- 

 mate ratios, fcems to decline applying that doctrine^ when he calls dx and 

 dy mere inerements. It may, however, be obicrved that, had he con- 

 fidercd them in tlieir extreme ratios, he would have en£\rely deviated from 

 the" infinitefimal or differential theory, which is properly his fubjeft," into 

 the fluxionary. But, as already hinted, his mixed Way of elucidating his 

 Tdo6lrine, may be of great ufe, if his readers take care not to confound the 

 ^differentials (or increments) of quantities confidered as formed by an ap- 

 ^ofition of parts, with the fluxions, which are accurately in the prime 

 ratio of the nafcent, or the ultimate ratio of the evanefcent, increments (or 

 ■dGcremeiirs^, uf qcaniitics, ccnlldci-ed as gencraccd-by m.otiori. — W. D. 



fronv 



