3815.) On Fluxions. 179 
great. Newton's expression, therefore, “* The ultimate ratio of 
the increments is the ratio of the fluxions,” is incorrect, and seems 
to have misled the Bishop of Cloyne. If a man is not a soldier, he 
may be the last of the men in atrain, but, in that train, he cannot 
be the last of the soldiers. Newton, therefore, must be understood 
rationally, not literally; the literal interpretation, indeed, is im- 
possible. In Milton too, the literal interpretation of ‘* The fairest 
of her daughters, Eve,” is also impossible. Such incorrectness of 
expression is frequently found in Robins. I do not remember that 
Maclaurin has correeted it till article 505, in the second volume of 
Fluxions, Maseres has rectified it more directly in p. 2] of the 
preface to the fifth volume of the Logarithmic Writers. Euler has 
fallen into the same mistake in his Definition of the Differential 
Calculus, in p. § of the preface. 
I am inclined to think that, in p. 468, Harvey’s idea of deve» 
loping generated quantities is better than mine of generating them. 
It was to avoid the idea of motion that, in the demonstration, which 
I think is new, 1 employed bisection like the ancients. I might 
have avoided the idea of motion in the solution too; for 1 might 
have solved as Lacroix does in the beginning of his Calcul in 8vo. 
As the fluxional calculus was derived from the celebrated problem 
of the tangents, I think that the easiest and shortest demonstration 
is to be obtained from the same source. I consider such a demon- 
stration as au extension of Descartes’ application of algebra to 
geometry. I think that no rigorous demonstration of the fluxional 
problem purely algebraical can be so short as that in pp. 330 and 
331; it oceupies no more than twelve entire lines, as it properly 
begins at line 33, p. 330, and ends at line 8, p. 331; for, in order 
to prove that the limits of a variable quantity are equal, J might 
have referred to Robins, vol. ii. p. 56, art. 120; or to Lacroix 
Calcul, vol.i. p. 18. D’Alembert observes, that all the differential 
caleulus may be referred to the problem of the tangents. 
Without the aid of a diagram, the application to tangents, 
quadratures, cubatures, rectifications, and complanations, is much 
more difficult and tedious to a learner. This is evident from La- 
grange. 
Motion conceived may be rigorously mathematical ; not so, 
motion executed. Now in fluxions it is motion conceived only that 
comes under consideration. 
With regard to Newton’s second lemma, as a square is simpler 
than an oblong, if we subtract the square of A — a from that of 
A + a, there will remain 4 A a, of which the half is 2 Aa; and 
then as the momentum is evidently greater than the decrement, and 
smaller than the increment, when the rate of change thus varies, 
we may prove by reduction to absurdity that the momentum. of 
AA can be neither more vor less than 2 Aa; for it may be de- 
monstrated to differ less from 2A a + aa, the increment, than by 
apy assigned quantity how small soever: and, in wa, if the 
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