The Life of Sir Isaac Newton

by David Brewster

In the year 1707, Mr. Whiston published the algebraical lectures which Newton had, during nine years, delivered at Cambridge, under the title of Arithmetica Universalis, sive de Compositione et Resolutione Arithmetica Liber. We are not accurately informed how Mr. Whiston obtained possession of this work; but it is stated by one of the editors of the English edition, that “Mr. Whiston thinking it a pity that so noble and useful a work should be doomed to a college confinement, obtained leave to make it public.” It was soon afterward translated into English by Mr. Ralphson; and a second edition of it, with improvements by the author, was published at London in 1712, by Dr. Machin, secretary to the Royal Society. With the view of stimulating mathematicians to write annotations on this admirable work, the celebrated S’Gravesande published a tract, entitled, Specimen Commentarii in Arithmeticam Universalem; and Maclaurin’s Algebra seems to have been drawn up in consequence of this appeal.

Among the mathematical works of Newton we must not omit to enumerate a small tract entitled, Methodus Differentialis, which was published with his consent in 1711. It consists of six propositions, which contain a method of drawing a parabolic curve through any given number of points, and which are useful for constructing tables by the interpolation of series, and for solving problems depending on the quadrature of curves.

Another mathematical treatise of Newton’s was published for the first time in 1779, in Dr. Horsley’s edition of his works.[60] It is entitled, Artis Analyticæ Specimina, vel Geometria Analytica. In editing this work, which occupies about 130 quarto pages, Dr. Horsley used three manuscripts, one of which was in the handwriting of the author; another, written in an unknown hand, was given by Mr. William Jones to the Honourable Charles Cavendish; and a third, copied from this by Mr. James Wilson, the editor of Robins’s works, was given to Dr. Horsley by Mr. John Nourse, bookseller to the king. Dr. Horsley has divided it into twelve chapters, which treat of infinite series; of the reduction of affected equations; of the specious resolution of equations; of the doctrine of fluxions; of maxima and minima; of drawing tangents to curves; of the radius of curvature; of the quadrature of curves; of the area of curves which are comparable with the conic sections; of the construction of mechanical problems, and on finding the lengths of curves.

In enumerating the mathematical works of our author, we must not overlook his solutions of the celebrated problems proposed by Bernouilli and Leibnitz. On the Kalends of January, 1697, John Bernouilli addressed a letter to the most distinguished mathematicians in Europe,[61] challenging them to solve the two following problems:

1. To determine the curve line connecting two given points which are at different distances from the horizon, and not in the same vertical line, along which a body passing by its own gravity, and beginning to move at the upper point, shall descend to the lower point in the shortest time possible.

2. To find a curve line of this property that the two segments of a right line drawn from a given point through the curve, being raised to any given power, and taken together, may make every where the same sum.

On the day after he received these problems, Newton addressed to Mr. Charles Montague, the President of the Royal Society, a solution of them both. He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it. He solved also the second problem, and he showed that by the same method other curves might be found which shall cut off three or more segments having the like properties. Leibnitz, who was struck with the beauty of the problem, requested Bernouilli, who had allowed six months for its solution, to extend the period to twelve months. This delay was readily granted, solutions were obtained from Newton, Leibnitz, and the Marquis de L’Hopital; and although that of Newton was anonymous, yet Bernouilli recognised in it his powerful mind, “tanquam,” says he, “ex ungue leonem,” as the lion is known by his claw.

The last mathematical effort of our author was made with his usual success, in solving a problem which Leibnitz proposed in 1716, in a letter to the Abbé Conti, “for the purpose, as he expressed it, of feeling the pulse of the English analysts.” The object of this problem was to determine the curve which should cut at right angles an infinity of curves of a given nature, but expressible by the same equation. Newton received this problem about five o’clock in the afternoon, as he was returning from the Mint; and though the problem was extremely difficult, and he himself much fatigued with business, yet he finished the solution of it before he went to bed.

Such is a brief account of the mathematical writings of Sir Isaac Newton, not one of which were voluntarily communicated to the world by himself. The publication of his Universal Arithmetic is said to have been a breach of confidence on the part of Whiston; and, however this may be, it was an unfinished work, never designed for the public. The publication of his Quadrature of Curves, and of his Enumeration of Curve Lines, was rendered necessary, in consequence of plagiarisms from the manuscripts of them which he had lent to his friends, and the rest of his analytical writings did not appear till after his death. It is not easy to penetrate into the motives by which this great man was on these occasions actuated. If his object was to keep possession of his discoveries till he had brought them to a higher degree of perfection, we may approve of the propriety, though we cannot admire the prudence of such a step. If he wished to retain to himself his own methods, in order that he alone might have the advantage of them in prosecuting his physical inquiries, we cannot reconcile so selfish a measure with that openness and generosity of character which marked the whole of his life. If he withheld his labours from the world in order to avoid the disputes and contentions to which they might give rise, he adopted the very worst method of securing his tranquillity. That this was the leading motive under which he acted, there is little reason to doubt. The early delay in the publication of his method of fluxions, after the breaking out of the plague at Cambridge, was probably owing to his not having completed the algorithm of that calculus; but no apology can be made for the imprudence of withholding it any longer from the public. Had he published this noble discovery even previous to 1673, when his great rival had not even entered upon those studies which led him to the same method, he would have secured to himself the undivided honour of the invention, and Leibnitz could have aspired to no other fame but that of an improver of the doctrine of fluxions. But he unfortunately acted otherwise. He announced to his friends that he possessed a method of great generality and power; he communicated to them a general account of its principles and applications; and the information which was thus conveyed directed the attention of mathematicians to subjects to which they might not have otherwise applied their powers. In this way the discoveries which he had previously made were made subsequently by others; and Leibnitz, in place of appearing in the theatre of science as the disciple and the follower of Newton, stood forth with all the dignity of a rival; and, by the early publication of his discoveries had nearly placed himself on the throne which Newton was destined to ascend.

It would be inconsistent with the popular nature of a work like this, to enter into a detailed history of the dispute between Newton and Leibnitz respecting the invention of fluxions. A brief and general account of it, however, is indispensable.