180 On Fluxions. {Serr. 
momentum a be multiplied by 7, an indeterminate quantity, and if 
A Aa 2h POneig 
x be substituted for A, we shall have : = = ts — We next, 
by Maclaurin’s process, Fluxions, art. 708, get the fluxion of an 
oblong, thence that of acube, &c. ‘Thus Newton’s demonstration 
seems superior in brevity, and equal in rigour, to that of any of his 
contemporaries and successors at home or abroad; for it has evi- 
dently no dependance whatever on motion, or on infinitesimals, or 
on vanishing quantities, or even on limits. It is wholly algebraical, 
but may, by a diagram, be rendered geometrical. I think the de- 
monstration in Newton’s second lemma one of the finest produc- 
tions of his unequalled genius. The conception of motion, from 
which Maclaurin demonstrated so very tediously, belongs not to 
Newton’s demonstration, but to his idea of the continuous genera- 
tion of quantity. It seems to be through Maclaurin that some very 
eminent foreign mathematicians see and blame Newton. 
Robins, from what Newton says himself, observes that Newton 
in his Mathematics uses the word momentum in two senses: first, 
for an infinitely small quantity, when he solves; and secondly, 
when he demonstrates, for an indeterminate quantity which is to be 
conceived to vanish: in the first sense, - | = for example ; 
here the quantities really employed are oe not <: but it is 
evident that in the second lemma he uses the word momentum in a 
third sense : for it is there neither a quantity which is to be con- 
ceived to vanish, nor is it ¥ or # till it be multiplied by an indeter- 
minate quantity 2. 
From Newton’s second lemma we obtain the easiest demonstra- 
tion of the binomial theorem for any exponent; because from the 
first fluxion we obtain the second, &c. Now these are the succes- 
sive fluxional coefficients. We have therefore only to multiply 
them by the successive powers of 7, and to divide the terms by 1, 
i x 2, 1 x 2 x 3, respectively. This would not be a legitimate 
demonstration, if the binomial theorem had been previously em- 
ployed to find the fluxions. No one, I think, will say this is de- 
monstrating the binomial theorem by employing the higher mathe- 
matics ; for in my former paper I showed that much of fluxions 
belonged properly to the very elements of geometry and algebra. 
From fig. 3, p. 330, it is easy to demonstrate that any term 
a.n—12"~* 
Fi 
in any proportion than the sum of all the succeeding terms; for if 
na"—* be transferred, with the negative sign, to the other side, 
and if the equation be then divided by 2, the thing is evident. 
Lagrange’s demonstrations are not so easy: it is extremely tedious 
and teasing for a learner to proceed by his method to tangents, 
quadratures, &e.; a proof that his method of investigation and de- 
monstratiog, how refined and convincing soever, is not short and 
i, for example, may be not greater only, but greater 
