THE NEW GEOMETRY.
The seventeenth century, which had advanced with such spirit and success in combating prejudice, detecting error, and establishing truth, was destined to conclude with the most splendid series of philosophical discoveries yet recorded in the history of letters. It was about to witness, in succession, the invention of Fluxions, the dis covery of the Composition of Light, and of the principle of Universal Gravitation,— all three within a period of little more than twenty years, and all three the work of the same individual. It is to the first of these that our attention at present is to be par titularly directed.
The notion of infinite Quantity had as we have already seen, been for some time introduced into Geometry, and having become a subject of reasoning and calculation, had, in many instances, after facilitating the process of both, led to conclusions from which, as if by magic, the idea of infinity had entirely disappeared, and left the geome ter or the algebraist in possession of valuable propositions, in which were involved no magnitudes but such as could be readily exhibited. The discovery of such results had increased both the interest and extent of mathematical investigation.
It was in this state of the sciences, that Newton began his mathematical studies, and, after a very short interval, his mathematical discoveries. 1 The book, next to the elements, which was put into his hands, was Wallis's Arithmetic of litfinites, a work well fitted for suggesting new views in geometry, and calling into activity the powers of mathematical invention. Wallis had effected the quadrature of all those curves in which the value of one of the co-ordinates can be expressed in terms of the other, without involving either fractional or negative exponents. Beyond this point neither his researches, nor those of any other geometer, had yet reached, and from this point the discoveries of Newton began. The Savilian Professor had himself been extremely desirous to advance into the new region, where, among other great objects, the quadrature of the circle must necessarily be contained, and he made a very noble effort to pass the barrier by which the undiscovered country appeared to be defend ' ed. He saw plainly, that if the equations of the curves which he had squared were ranged in a regular series, from the simpler to the more complex, their areas would constitute another corresponding series, the terms of which were all known. He farther
remarked, that, in the first of these series, the equation to the circle itself might be in troduced, and would occupy the middle place between the first and second terms of the series, or between an equation to a straight line and an equation to the common parabola. He concluded, therefore, that if, in the second series, he could interpolate • a term in the middle, between its first and second terms, this term must necessarily be no other than the area of the circle. But when he proceeded to pursue this very re fined and philosophical idea, he was not so fortunate ; and his attempt toward the re quisite interpolation, though it did not entirely fail, and made known a curious pro perty of the area of the circle, did not lead to an indefinite quadrature of that Newton was much more judicious and successful in his attempt. Proceeding on the same general principle with Wallis, as he himself tells us, the simple view which he took of the areas already computed, and of the terms of which each consisted, enabled him to discover the law which was common to them all, and under which the expres sion for the area of the circle, as well as of innumerable other curves, must needs be comprehended. In the case of the circle, as in all those where a fractional exponent appeared, the area was exhibited in the form of an infinite series.
The problem • of the quadrature of the circle, and of so many other curves, being thus resolved, Newton immediately remarked, that the law of these series was, with a small alteration, the law for the series of terms which expresses the root of any binomial quan tity whatsoever. Thus he was put in possession of another valuable discovery, the Binomial Theorem, and at the same time perceived that this last was in reality, in the order of things, placed before the other, and afforded a much easier access to such quadratures than the method of interpolation, which, though the first road, appeared now neither to be the easiest nor the most direct.