Newton to Euler

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

"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

necessary to introduce into geometry infinitely small quantities." This mode of differentiating does not remove all the difficulties connected with the subject. When $0$ becomes nothing, then we get the ratio ${\displaystyle \frac{0}{0}}=n{x}^{n-1}$, which needs further elucidation. Indeed, the method of Newton, as delivered by himself, is encumbered with difficulties and objections. Among the ablest admirers of Newton, there have been obstinate disputes respecting his explanation of his method of "prime and

ultimate ratios."

The so-called "method of limits" is frequently attributed

to Newton, but the pure method of limits was never adopted by him as his method of constructing the calculus. All he did was to establish in his Principia certain principles which

are applicable to that method, but which he used for a different purpose. The first lemma of the first book has been made the foundation of the method of limits:–-

"Quantities and the ratios of quantities, which in any finite time converge continually to equality, and before the end of