Isaac Newton

Newton, Isaac (1642–1727)

Newton is best known for having invented the calculus and formulated the theory of universal gravity – the latter in his Principia, the single most important work in the transformation of natural philosophy into modern physical science. Yet he also made major discoveries in optics, and put no less effort into alchemy and theology than into mathematics and physics.

Throughout his career, Newton maintained a sharp distinction between conjectural hypotheses and experimentally established results. This distinction was central to his claim that the method by which conclusions about forces were inferred from phenomena in the Principia made it ’possible to argue more securely concerning the physical species, physical causes, and physical proportions of these forces’. The law of universal gravity that he argued for in this way nevertheless provoked strong opposition, especially from such leading figures on the Continent as Huygens and Leibniz: they protested that Newton was invoking an occult power of action-at-a-distance insofar as he was offering no contact mechanism by means of which forces of gravity could act. This opposition led him to a tighter, more emphatic presentation of his methodology in the second edition of the Principia, published twenty-six years after the first. The opposition to the theory of gravity faded during the fifty to seventy-five years after his death as it fulfilled its promise on such issues as the non-spherical shape of the earth, the precession of the equinoxes, comet trajectories (including the return of ’Halley’s Comet’ in 1758), the vagaries of lunar motion and other deviations from Keplerian motion. During this period the point mass mechanics of the Principia was extended to rigid bodies and fluids by such figures as Euler, forming what we know as ’Newtonian’ mechanics.

1 Life

Isaac Newton entered Trinity College Cambridge in 1661, and began investigations of mathematics in 1664. These investigations culminated two years later in the binomial theorem and the fundamentals of the calculus. During the so-called annus mirabilis of 1666, while the university was closed because of the plague, and in the years immediately following, he extended his mathematical work; he also conducted optical experiments and worked on several basic problems in mechanics, including impact and circular motion. He became Lucasian Professor of Mathematics at Cambridge in 1669.

Although some of Newton’s mathematical manuscripts were in circulation, yielding him some renown, his only notable publications before the Principia were a series of communications in the Philosophical Transactions of the Royal Society from 1672 to 1676 on his experiments on light and colours and the reflecting telescope. The debate which this work provoked led Newton to begin articulating what he called his ’experimental philosophy’, which focused on establishing propositions by means of experiment.

In an exchange of letters in late 1679, Robert Hooke, himself an eminent scientist, asked Newton to use his mathematical methods to determine the trajectory of a body under a combination of inertial motion and an inverse-square force directed towards a central point – that is, the force Newton later named ’centripetal’. But the intense effort that culminated in the publication of the Principia (1687) did not begin until 1684, after a visit from young Edmond Halley, who later became Astronomer Royal.

Newton spent most of the years after 1689 in London. He was elected to represent Cambridge University in Parliament in 1689 and again in 1701, the year in which he resigned his professorship. He became Warden of the Mint in 1696, and Master of it in 1699. In 1703 he became President of the Royal Society, a post he held until he died. He was knighted in 1705.

During his London years Newton engaged in an acrimonious dispute with Leibniz over who had priority for inventing the calculus. One element fuelling this dispute was Newton’s failure to publish his work, save for a three-page summary of a handful of results in Book II of the Principia. His first formal publications on the calculus appeared in 1704, when two earlier manuscripts were included as supplements in the first edition of the Opticks. (A Latin edition of the Opticks appeared two years later.)

Newton gave some thought to a restructured edition of the Principia in the early 1690s. But the second edition was not published until 1713, after four years of effort under the constructively critical eye of its editor, Roger Cotes. A third edition followed in 1726. These editions sharpened the contrast between his approach and that of Leibniz and the Cartesians. The second English edition of the Opticks (1717/18) included Queries that summarized his conjectures on atomism. These Queries end with a concise statement of his method for establishing scientific knowledge on the basis of experiment and induction; so too does his final riposte in the priority dispute with Leibniz, his anonymous ’An Account of the Book Entitled Commercium epistolicum’.

2 Experimental philosophy in the light and colours debate

Newton’s 1672 paper on light and colours reported only a small fraction of the optical experiments he had conducted. The debate it initiated concerned what the reported experiments had established. According to Newton, these experiments had conclusively shown that the oblong shape of the image cast by sunlight that has passed through a round hole and has then been refracted by a prism is caused by sunlight’s consisting of rays that are refracted in different degrees by the prism. (The correspondence between these different refrangibilities and different colours led Newton to invent the first reflecting telescope, which eliminates the problems of chromatic aberration that had marred the refractive telescopes of the era.)

Hooke, interpreting Newton as claiming that the experiments established a corpuscular theory of light, insisted that a wave theory could account for the results just as well. Newton responded that the hypothesis that light is a body was put forward only as a conjecture suggested by the experiments, and not as part of what he claimed to have been established by them. He granted that Hooke’s wave hypothesis could explain the conclusion the experiments had established; but this conclusion spoke of light only abstractly as ’rays’ propagating in straight lines from luminous bodies, with no commitment to any specific ’mechanical’ hypothesis.

His Dutch contemporary, Christiaan Huygens argued that Newton had failed to show the nature and difference of colours because he had offered no ’hypothesis by motion’ to explain them. Newton responded that he ’never intended to shew, wherein consists the nature and difference of colours, but only to shew that de facto they are original and immutable qualities of the rays which exhibit them’ (1958: 144).

Newton’s contemporaries had trouble understanding his attempt to construe light rays abstractly in a way that would allow experiments to decide claims about them – this, independently of any mechanical account of light. In his replies, Newton outlined how, according to his experimental philosophy, diligently establishing properties of things by experiment takes precedence over framing hypotheses to explain them. Yet he also made clear that the propositions he regarded as conclusively established by experiment were nevertheless subject to correction based on detailed criticism of the experimental reasoning that had established them or on further experimental results challenging them (see Optics §§1–2).

3 Space, time and the laws of motion

Two aspects of the Principia provoked philosophical controversy in the decades following its publication: first, the appeal to absolute space and motion, and second, the insistence on establishing properties of gravity, especially its universality, without appeal to any mechanical hypothesis that could begin to explain how gravity is produced.

The Principia opens with two short sections, ’Definitions’ and ’Axioms, or Laws of Motion’, that have drawn philosophical fire ever since. The distinctions between absolute and relative time, space and motion are drawn in the first of these, following his introduction of the concept of mass and definitions pertaining to quantity of motion and force. For Newton, the distinction between absolute (or true) and relative (or apparent) motion is primary, and the parallel distinctions concerning space and time serve mostly to support this one (see Mechanics, classical; Space §§2–3). Newton was acutely aware of the empirical difficulties raised by such distinctions:

It is certainly very difficult to find out the true motions of individual bodies and actually to differentiate them from the apparent motions, because the parts of that immovable space in which the bodies truly move make no impression on the senses. Nevertheless, the case is not utterly hopeless. For it is possible to draw evidence partly from apparent motions, which are differences between the true motions, and partly from the forces that are the causes and effects of the true motions …. But in what follows, a fuller explanation will be given of how to determine true motions from their causes, effects, and apparent differences, and, conversely, how to determine from motions, whether true or apparent, their causes and effects. For this was the purpose for which I composed the following treatise.

(Newton 1687: Scholium to Definitions)

The reference here is to the laws of motion and their corollaries, which immediately follow this last remark, as well as to the ninety-eight demonstrated Propositions of Book I and the fifty-three of Book II.

Like Descartes, Newton appealed to forces to distinguish true from apparent motions. And, again like Descartes, the true motion of greatest importance to him in the sequel is curvilinear motion, most notably the true motion of the planets that would distinguish between their equivalent relative motions in the Copernican and Tychonic systems. Unlike Descartes, however, Newton refused to offer hypotheses concerning the forces in question (see Descartes, R. §11). He required that the forces be inferred from phenomena with the help of the mathematical principles of Books I and II, many of which licence inferences from observed motions to measures of force. Inconsistencies among the inferred quantities of force or the motions subsequently inferred from them would indicate a failure to be dealing with true motions. But this is an empirical question, to be decided by carrying out the investigation of motions under forces to its fullest extent, insisting on no less than complete agreement between observed and calculated motions. Thus, the successes, and also the limitations, of the appeal to absolute space, time and motion were, for Newton, empirical issues that the long-term development of an exact science of motion would decide, and not something he thought was open to a priori resolution.

4 Inferences from phenomena and rules of natural philosophy

The Propositions of Books I and II are powerful resources for establishing conclusions about forces from phenomena of motion. For example, according to Propositions 1 and 2, Kepler’s area rule holds if and only if the force acting on the moving body is centripetal. A corollary adds that the areal velocity is increasing when the force is off-centre in the direction of motion and decreasing when it is in the opposite direction. The variation of the areal velocity is thus a measure of the direction of the force. Similar systematic dependencies are involved in the inferences from Kepler’s 3:2 power rule and the absence of discernible orbital precession to the inverse-square variation of celestial centripetal forces (see Kepler, J.).

Rules of reasoning, which in the second and third editions are singled out at the beginning of Book III under the title Regulae philosophandi, strengthen the inferences that can be drawn from phenomena by licensing inductive generalizations (see Scientific method §2). The first two rules, for example, underlie the inference that the force holding the moon in orbit is terrestrial gravity – this, on the basis of the inverse-square relation between the centripetal acceleration of the moon and the acceleration of gravity at the earth’s surface. The third rule, appearing for the first time in the second edition, supports the inference that all bodies gravitate towards each planet with weights proportional to their masses – this, on the basis of pendulum experiments and the common acceleration of Jupiter and its satellites toward the sun.

The fourth rule authorizes the practice of treating propositions that are supported properly by reasoning from phenomena as ’either exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions’. It was added in the third edition to justify treating universal gravity as an established scientific fact in the face of complaints that it was unintelligible without an explanation of how it results from mechanical action by contact. This rule, and the related discussion of hypotheses at the end of the General Scholium added in the second edition, distinguish Newton’s experimental philosophy most sharply from the mechanical philosophy of his critics.

5 Gravity as a universal force of interaction

The systematic dependencies by means of which Keplerian phenomena become measures of celestial forces are one-body idealizations. Universal gravity entails interactions among bodies, producing perturbations that require corrections to the Keplerian phenomena. The Principia includes a successful treatment of two-body interactions and some promising, though limited, results on three-body effects in lunar motion. But the full significance of the inferences that could be drawn from universal gravity did not become clear until Clairaut’s analysis of the precession of the lunar orbit in 1749, and Laplace’s determination in 1785 of the roughly 880-year ‘Great Inequality’ in the motions of Jupiter and Saturn.

Such successful treatments of perturbations do more than provide corrections to Keplerian phenomena. They also show that Newtonian measurements of inverse-square centripetal forces continue to hold to high approximation in the presence of perturbations. Interactions with other bodies account for the precessions of all the planets except Mercury. Even in the case of Mercury, the famous 43 seconds-of-arc-per-century residual in its precession yields -2.000000157 as the measure of the exponent, instead of the exact -2 measured for the other planets. That such a small discrepancy came to be a problem at all testifies to the extraordinary level to which Newton’s theory of gravity demonstrates an ideal of empirical success.

6 Mathematics

Newton engaged in extensive mathematical research throughout much of the period from 1664 until he left for London in 1696, and even after that he produced new results on some problems. Besides the many lines along which he developed and applied the calculus, he made substantial discoveries in algebra and in pure as well as analytic geometry. The mathematics of the Principia is itself a new form of synthetic geometry, incorporating limits. (Contrary to the still-persisting myth, there is no evidence that Newton first derived his results on celestial orbits within the symbolic calculus and then recast them in geometric form.)

Newton’s invention of the calculus grew out of his attempts to solve several distinct problems, often employing novel extension of the ideas and methods of others. For example, his initial algorithms for derivatives of algebraic curves combined Cartesian techniques with the idea of an indefinitely small, vanishing increment. He exploited a method of indivisibles that John Wallis had used in obtaining integrals of algebraic curves; but he reconceptualized the method to represent an integral that grows as the curve extends incrementally, and then joined this with the binomial series to yield integrals via infinite series of a much wider range of curves. Geometrical representations of these results revealed the inverse relation between integration and differentiation. Then, adapting his Lucasian predecessor Isaac Barrow’s idea to treat curves as arising from the motion of a point, Newton recast the results on derivatives, replacing indefinitely small increments with his ’fluxions’. The first full tract on fluxions, ‘To Resolve Problems by Motion’, is dated 1666, but the first manuscript to circulate was De analysi per aequationes infinitas of 1669.

Mathematicians in England used Newtonian methods and notation into the nineteenth century. But the Leibnizian tradition had gained so much momentum by the time Newton’s works appeared that the calculus, as we know it, stems far more from that tradition. Ironically, it was such figures as the Bernoullis and Euler (belonging to the Leibnizian tradition) who recast the Principia into the language of the calculus.

7 Studies in alchemy and theology

Newton’s unpublished manuscripts contain voluminous studies on alchemy, theology, prophecy and Biblical chronology. His alchemical work led to a number of elaborate chemical experiments carried out from the mid-1670s until 1693. His notes from these efforts display his great skill as an experimenter, but they appear to include nothing that would have altered the course of chemistry had they become public at the time (see Alchemy).

He first became preoccupied with theology in the early 1670s, probably in response to the requirement that he accept ordination to retain his Trinity fellowship. (He was granted a dispensation in 1675.) By 1673 he had rejected the doctrine of the Trinity and concluded that Christianity had become a false religion through a corruption of the Scriptures in the fourth and fifth centuries. He returned to these studies in subsequent decades, especially in the last years of his life. During his lifetime, however, he conveyed his radical views to only a few. They became widely known only when Observations upon the Prophecies was published in 1733.

Recent investigations of the alchemical and theological writings suggest that Newton’s hope in natural philosophy was to look through nature to see God, that it was to be part of a larger investigation that would give meaning to life. His engagement with these larger issues may have helped him to free himself from the narrower restraints of the mechanical philosophy.