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tities is here renounced in a manner which would lead one to suppose that Newton had never held it himself. Thus it appears that Newton's doctrine was different in different periods. Though, in the above reasoning, the Charybdis of infinitesimals is safely avoided, the dangers of a Scylla stare us in the face. We are required to believe that a point may be considered a triangle, or that a triangle can be inscribed in a point; nay, that three dissimilar triangles become similar and equal when they have reached their ultimate form in one and the same point.

In the introduction to the Quadrature of Curves the fluxion of $\scriptstyle {x^{n}}$ is determined as follows:—

"In the same time that x, by flowing, becomes $\scriptstyle {x+0}$, the power $\scriptstyle {x^{n}}$ becomes $\scriptstyle {(x+0)^{n}}$, i.e. by the method of infinite series

$\scriptstyle {x^{n}+n0x^{n-1}+{\frac {n^{2}-n}{2}}0^{2}x^{n-2}+}$etc.,

and the increments

0 and $\scriptstyle {n0x^{n-1}+{\frac {n^{2}-n}{2}}0^{2}x^{n-2}+}$etc.,

are to one another as

1 to $\scriptstyle {nx^{n-1}+{\frac {n^{2}-n}{2}}0x^{n-2}+}$etc.

"Let now the increments vanish, and their last proportion will be 1 to $\scriptstyle {nx^{n-1}}$: hence the fluxion of the quantity x is to the fluxion of the quantity $\scriptstyle {x^{n}}$ as $\scriptstyle {1:nx^{n-1}}$.

"The fluxion of lines, straight or curved, in all cases whatever, as also the fluxions of superficies, angles, and other quantities, can be obtained in the same manner by the method of prime and ultimate ratios. But to establish in this way the analysis of infinite quantities, and to investigate prime and ultimate ratios of finite quantities, nascent or evanescent, is in harmony with the geometry of the ancients; and I have endeavoured to show that, in the method of fluxions, it is not