CHAPTER XIX

Isaac Newton

1642–1727

I. THE MATHEMATICIAN

HE WAS born in a small farm at Woolsthorpe, in the county of Lincoln, on December 25, 1642 (Old Style)—the year in which Galileo died; cultural, like economic, leadership was passing from the south to the north. At birth he was so small that (his mother later told him) he might have been put into a quart mug, and so weak that no one thought he would live beyond a few days. 1 As his father had died some months earlier, the boy was brought up by his mother and an uncle.

At twelve he was sent to the public school at Grantham. He did not do well there; he was reported as “idle” and “inattentive,” neglecting prescribed studies for subjects that appealed to him, and giving much time to mechanical contrivances like sundials, water wheels, and homemade clocks. After two years at Grantham he was taken from school to help his mother on the farm, but again he skimped his duties to read books and do mathematical problems. Another uncle, recognizing his ability, sent him back to school, and arranged for Newton’s admission to Trinity College, Cambridge (1661), as a subsizar—a student who earned his expenses by various services. He took his degree four years later, and was soon thereafter elected a fellow of the college. He dealt chiefly with mathematics, optics, astronomy, and astrology; in this last study he maintained interest till late in his life.

In 1669 his mathematics teacher, Isaac Barrow, resigned; and on Barrow’s recommendation of him as an “unparalleled genius,” Newton was appointed to succeed him; he held this chair at Trinity for thirty-four years. He was not a successful teacher. “So few went to hear him,” his secretary recalled, “and fewer that understood him, that ofttimes he did in a manner, for want of hearers, read to the walls.” 2 On some occasions he had no auditors at all, and returned sadly to his room. There he built a laboratory—the only one then to be found in Cambridge. He made many experiments, mainly in alchemy, “the transmuting of metals being his chief design”; 3 but also he was interested in the “elixir of life” and the “philosopher’s stone.” 4 He continued his alchemist studies from 1661 to 1692, and even while writing the Principia; 5 he left unpublished manuscripts on alchemy totaling 100,000 words or more, “wholly devoid of value.” 6 Boyle and other members of the Royal Society were feverishly engaged in the same quest for manufacturing gold. Newton’s aim was not clearly commercial; he never showed any eagerness for material gains; probably he was seeking some law or process by which the elements could be interpreted as transmutable variations of one basic substance. We cannot be sure that he was wrong.

Outside his rooms at Cambridge he had a small garden; there he took short walks, soon interrupted by some idea which he hurried to his desk to record. He sat little, but rather walked about his room so much that (said his secretary) “you might have thought him . . . among the Aristotelian sect” of Peripatetics. 7 He ate sparingly, often skipped a meal, forgot that he had missed it, and grudged the time he had to give to eating and sleeping. “He rarely went to dine in the hall; and then, if he had not been [re]minded, would go very carelessly, with shoes down at the heels, stockings untied . . . and his head scarcely combed.” 8 Many stories were told, and many were invented, about his absent-mindedness. On awakening from sleep, we are assured, he might sit on his bed for hours undressed, engrossed in thought. 9 When he had visitors he would sometimes disappear into another room, jot down ideas, and quite forget his company. 10

During those thirty-five years at Cambridge he was a monk of science. He drew up “rules of philosophizing”—i.e., of scientific method and research. He rejected the rules which Descartes in his Discours had set up as a priori principles from which all major truths were to be derived by deduction. When Newton said, “Non fingo hypotheses”—I do not invent hypotheses 11—he meant that he offered no theories as to anything beyond observation of phenomena; so he would hazard no guess as to the nature of gravitation, but would only describe its behavior and formulate its laws. He did not pretend to avoid hypotheses as cues to experiments; on the contrary, his laboratory was devoted to testing a thousand ideas and possibilities, and his record was littered with hypotheses tried and rejected. Nor did he repudiate deduction; he merely insisted that it should start from facts and lead to principles. His method was to conceive possible solutions of a problem, work out their mathematical implications, and test these by computation and experiment. “The whole burden of [natural] philosophy,” he wrote, “seems to consist in this—from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena.” 12 He was a compound of mathematics and imagination, and no one can understand him who does not possess both.

Nevertheless we proceed. His fame has two foci—calculus and gravitation. He began his work on the calculus in 1665 by finding the tangent and radius of curvature at any point on a curve. He called his method not calculus but “fluxions,” and gave for this term an explanation upon which we cannot improve:

Lines are described, and thereby generated, not by the apposition of parts, but by the continued motion of points; superficies [planes] by the motion of lines; solids by the motion of superficies; angles by the rotation of the sides; portions of time by continual flux; and so in other quantities . . . Therefore, considering that quantities, which increase in equal times, and by increasing are generated, became greater or less according to the greater or less velocity with which they increase or are generated, I sought a method of determining quantities from the velocities of the motions or increments with which they are generated; and calling these velocities of the motions or increments Fluxions, and the generated quantities Fluents, I fell by degrees upon the Method of Fluxions . . . in the years 1665 and 1666. 13

Newton described his method in a letter to Barrow in 1669, and referred to it in a letter to John Collins in 1672. He probably used the method in reaching some of the results in his Principia (1687), but his exposition there (probably for the convenience of his readers) followed accepted geometrical formulas. He contributed a statement of his fluxions procedure—but not over his own name—to Wallis’ Algebra in 1693. Not till 1704, in an appendix to his Opticks, did he publish the account just quoted. It was characteristic of Newton to delay publication of his theories; perhaps he wished first to resolve the difficulties suggested by them. So he waited till 1676 to announce his binomial theorem,* though he had probably formulated it in 1665.

These deferments embroiled the mathematicians of Europe in a disgraceful controversy that for a generation disrupted the international of science. For between Newton’s communication of his “fluxions” to his friends in 1669 and the publishing of the new method in 1704, Leibniz developed a rival system at Mainz and Paris. In 1671 he sent to the Académie des Sciences a paper containing the germ of the differential calculus. 14 On a visit to London, January to March, 1673, Leibniz met Oldenburg; he had already corresponded with him and Boyle; Newton’s friends later believed—historians now doubt 15—that Leibniz on this trip received some suggestion of Newton’s fluxions. In June, 1676, at the request of Oldenburg and Collins, Newton wrote a letter for transmission to Leibniz, explaining his method of analysis. In August Leibniz replied to Oldenburg, including some examples of his own work in calculus; and in June, 1677, in a further letter to Oldenburg, he described his form of differential calculus, and his system of notation, which differed from Newton’s. In the Acta Eruditorum for October, 1684, he again expounded the differential calculus, and in 1686 he published his system of integral calculus. In the first edition of the Principia (1687) Newton apparently accepted Leibniz’s independent discovery of calculus:

In letters which went between me and that most excellent geometer, G. W. Leibniz, ten years ago, when I signified that I was in the knowledge of a method of determining maxima and minima, of drawing tangents, and the like, . . . that most distinguished man wrote back that he had also fallen upon a method of the same kind, and communicated his method, which hardly differed from mine except . . . in his forms of words and symbols. 16

This gentlemanly acknowledgment should have contracepted controversy. But in 1699 a Swiss mathematician, in a communication to the Royal Society, suggested that Leibniz had borrowed his calculus from Newton. In 1705 Leibniz, in an anonymous review of Newton’s Opticks, implied that Newton’s fluxions were an adaptation of the Leibnizian calculus. In 1712 the Royal Society appointed a committee to examine the documents involved. Before the year was out the Society published a report, Commercium Epistolicum, asserting the priority of Newton, but leaving open the question of Leibniz’ originality. In a letter of April 9, 1716, to an Italian priest in London, Leibniz protested that Newton’s scholium had settled the question. Leibniz died November 14, 1716. Soon afterward Newton denied that the scholium “allowed him [Leibniz] the invention of the calculus differentialis independently of my own.” In the third edition of the Principia (1726) the scholium was omitted. 17 The dispute was hardly worthy of philosophers, since either claimant might have bowed to Fermat’s priority.

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