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 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 $\scriptstyle{\frac{0}{0}=nx^{n-1}}$,|undefined 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 that time approach nearer the one to the other than by any given difference, become ultimately equal."

In this, as well as in the lemmas following this, there are obscurities and difficulties. Newton appears to teach that a variable quantity and its limit will ultimately coincide and be equal. But it is now generally agreed that in the clearest statements which have been made of the theory of limits, the variable does not actually reach its limit, though the variable may approach it as near as we please.

The full title of Newton's Principia is Philosophiæ Naturalis Principia Mathematica. It was printed in 1687 under the direction, and at the expense, of Dr. Edmund Halley. A second edition was brought out in 1713 with many alterations and improvements, and accompanied by a preface from