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
Mr. Cotes. It was sold out in a few months, but a pirated edition published in Amsterdam supplied the demand.34 The third and last edition which appeared in England during Newton's lifetime was published in 1726 by Henry Pemberton. The Principia consists of three books, of which the first two, constituting the great bulk of the work, treat of the mathematical principles of natural philosophy, namely, the laws and conditions of motions and forces. In the third book is drawn up the constitution of the universe as deduced from the foregoing principles. The great principle underlying this memorable work is that of universal gravitation. The first book
was completed on April 28, 1686. After the remarkably short period of three months, the second book was finished. The third book is the result of the next nine or ten months' labours. It is only a sketch of a much more extended elaboration of the subject which he had planned, but which was never brought to completion.