Isaac Newton: The ‘Principia’ Mathematical Principles of Natural Philosophy 
translated by I. Bernard Cohen.
California, 974 pp., £22, September 1999, 0 520 08817 4
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‘This incomparable author having at length been prevailed upon to appear in public, has in this treatise given a most notable instance of the extent of the powers of the mind.’ This is how the very first review of the Principia began, in summer 1687: from the start, you were forced to admire Newton’s modesty, and his genius. The reviewer, the young astronomer Edmond Halley, knew what he was talking about. Three years earlier, during a visit to Cambridge, he had posed the puzzle which started Newton on the path to his Principia. What is the orbit of a planet under the influence of an attractive force varying inversely as the square of the distance? Halley reminded his readers that the great Kepler had long before identified vital patterns in planets’ motion: their elliptical paths with the Sun at their focus, the mathematical regularities which governed their orbital speeds.

Newton himself was never quite so generous in honouring his predecessors. Others, including Halley and his London colleague, the irascible natural philosopher Robert Hooke, had the idea that orbiting bodies tended to move in straight lines while at the same time being pulled towards some force-centre. Only Newton fully exploited its implications. He defined forces by the changes they produced in the motions of bodies on which they acted, then invented the term ‘centripetal force’ to describe this pull. Using Kepler’s regularities and the most recent data from the Royal Observatory, French expeditions to Africa and the West Indies, and cometary and tidal observations from the Straits of Magellan to the Gulf of Tonkin, he was able to identify the centripetal force acting on planets, moons and the Earth itself with a universally acting gravity to which every particle of matter was susceptible. It was proportional to the bodies’ masses and weakened as the square of their distance from the force-centre. Newton did not, at least publicly, claim to know the cause of gravity, an omission which would generate much heat and some light among his readers. If all matter gravitated, gravity could not have a material cause; if gravity depended directly on God, the proper topic of natural philosophy, then, some alleged, the Principia would turn the world-order into an inexplicable miracle.

Newton could afford to shrug. Using a newfangled geometrical treatment of continuously changing quantities he analysed the motions of the solar system, the path of the Moon and the shape of the Earth. He never doubted that the world showed God’s wise design – he was impressed, for example, by the fact that only if a centripetal force varied exactly as the inverse square of the distance would bodies act as though their mass was concentrated at their centres or generate stable orbits. This wisdom was best evidenced by the success of Newton’s own sums. Even though he grossly mistook the Moon’s mass, he proved what Galileo had sneeringly denied as mere astrology, the lunar influence on the tides, and explained for the first time why there are two tides a day. Even though he could not compute a comet’s orbit precisely from just one transit, he knew what Kepler did not, that comets move under gravitation in orbits round the Sun. For the first time, a work of natural philosophy presented astonishingly, sometimes suspiciously precise numbers to back up its claims about universal forces acting through empty space. Armed with a pendulum to time echoes in a court of Trinity College, Newton ingeniously got his theoretical and experimental estimates of the speed of sound to match. He had a local joiner build a large glass water-tank to test his theory of the fall of bodies in resisting fluids, and was not deflected when the tank smashed, nor when his numbers didn’t quite answer experiment. ‘No closer to the gods can any mortal rise,’ Halley exclaimed. It had been hard for him to get the five hundred copies of Newton’s masterwork to the public, who, at least in his own country, greeted them with adulation. Halley cajoled Newton into releasing his labours and ignoring his critics, offered advice on correction and revision, wrote a fulsome poem for the front of the book, paid for its publication, presented it to the King, and reviewed it in the Royal Society’s journal, which he himself edited. He made a profit of at least £10 out of the sales of the book.

Few others got a look behind the scenes of this amazing work. Its impenetrability, and that of its author, became clichés. One Cambridge student recalled that ‘we gazed on him, never enough satisfied, as on somewhat divine’; another, less cowed, reportedly pointed him out in the street: ‘There goes the man that writ a book that neither he nor anyone else understands.’ John Locke, then a political refugee in the Netherlands, simply couldn’t master the mathematics, though this didn’t stop him reviewing the book in the Dutch press. He was glad to be reassured by a local expert that he could take the sums and proofs on trust, then turned himself into a loyal, and credulous, Newtonian. This was a work whose reputation travelled, from Leipzig to Lisbon, from St Petersburg to Paris. Dubious anecdotes about apples and rhetorical flourishes about God’s handiwork proliferated in its wake. One witty Frenchman, Newton’s first biographer, compared his subject’s genius with the Nile – none knew its source, and it only showed itself in full flood. Another, gawping at the Principia’s contents, wondered whether its author ‘eats and drinks and sleeps. Is he like other men?’

His apostles made sure that Newton’s compatriots, not all of whom read Latin and few of whom could follow the proofs, were properly in awe of his mathematical analysis of the motion of bodies on Earth and in Heaven. Halley, for one, pulled no punches. Forget about Moses and the Ten Commandments, the inventors of writing or the founders of cities; even the gift of wine, to which Halley was notoriously partial, was as nothing. These were but the ‘few comforts of a wretched life’. But thanks to Newton, ‘we are now admitted to the banquets of the gods.’ Catchpenny handbooks and gaudy charts, offering fast-food versions of Newton’s divine dinner, poured from the London presses into the coffee houses. Self-made experts announced that the Principia had tamed comets and tides, planets and pendulums by celestial law and geometrical order. However surprising it may seem, even the English title, ‘The Mathematical Principles of Natural Philosophy’, was, according to Newton, designed to help the book sell. Long before scientific popularisation became the common salvation of hard-pressed publishers, Newtonianism spawned a strenuous commercial attempt to vulgarise the exacting physics of an esoteric cosmology. In 1729, with the immortal Newton two years dead, a London mathematics teacher and draughtsman, Andrew Motte, even managed to bring out an English translation of the whole of his greatest work. There has never, until now, been another.

It has understandably taken the Harvard historian of science I. Bernard Cohen and his collaborators much longer to produce this new translation than Newton took to compose the original, or Motte his version. In 1956, when Cohen had already established his reputation as a historian of Newtonian natural philosophy, he started working with the great émigré scholar Alexandre Koyré on a new presentation of the variations between the three Latin editions of the Principia published in Newton’s lifetime. Koyré died in 1964 and the variorum edition eventually appeared from Harvard in 1971 with the invaluable aid of Anne Whitman, a fine classicist. In a 400-page introduction Cohen foresaw a re-edition of Motte’s translation and a commentary on Newton’s work, including an English rendition of those sections currently of most interest to scholars. These scholars, however, urged Cohen and Whitman to make a completely new translation of the whole book, taking as their text the third Latin edition of 1726. This was all but ready in 1984, when Whitman died. Meanwhile, a definitive biography by Richard Westfall, publication of Newton’s correspondence and, decisively, a Cambridge edition of his mathematical papers were all completed.

The new translation makes use of this scholarship. It opens with another introduction by Cohen, occupying more than a third of the book, which offers a summary of the Principia’s composition, reflections on its translation, an analytical summary of its contents and some worked examples drawn from its principal propositions. The edition copes ably with puzzles of Newtonian commentary. It is now possible to follow in detail just how Newton proved that Keplerian paths for planets were evidence of a centripetal force of gravitation in the Sun, then generalised this force to allow for planets’ pull on each other. The translator’s choice between the terms ‘to gravitate’ or ‘to be heavy’ matters here, since the former rather masks the achievement of Newton’s demonstration that gravity exists and the latter reads awkwardly (as in ‘the Moon is heavy towards the Earth’). Cohen’s team offer both. As in any new version of a sacred text, familiar phrases regrettably vanish. In his discussion of the methods of calculus, Newton is traditionally understood as saying that ‘the hypothesis of indivisibles is rather harsh [durior].’ The new translation prefers the modish term ‘problematical’. Here on the page is Newton’s use of calculus to define the path a body must follow under a centripetal force whatever its variation with distance. The edition painstakingly traces the tortuous route through which he sought to reconcile his tidal data with his lunar theory, and the reasons he suppressed an entire section on the resistance of fluids to bodies moving through them. It remains to be seen what effect these labours will have on public and scientific notions of Newton’s achievement.

Scientific classics are often left unread. Physicists do not treat the Principia as analysts treat The Interpretation of Dreams, or even as biologists deal with The Origin of Species. Newtonians worked hard to make close reading of Newton’s own words dispensable. For a century (mainly Francophone) mathematicians used his materials creatively to build a new science of motion. Many principles uttered in Newton’s name, such as the definition of force as the product of mass and acceleration, or the demonstration of the long-term stability of the entire solar system, were absent from the Principia, present only in the works of his great successors. As the introduction to this new translation points out, it was only with the triumphant work of Pierre Simon Laplace, more than a century later, that the tasks set by the ‘Principia of legend’ were finally accomplished. Though published in other European languages, within Britain the founding text of the nation’s mathematical physics was digested in study guides and chopped up into standardised examination questions, its errors tidied up and its genesis deemed devoid of scientific interest. More than almost any other text of the scientific canon, successive editions were thoroughly overhauled by its author, hypotheses added or removed, entire sections replaced, long commentaries developed on salient puzzles in mechanics or metaphysics. Newton used references in the Principia as a way of rewarding, or penalising, his colleagues. Disciples helped make the data fit with the precision Newton needed and often silently altered passages critics had challenged.

The index and notes on variants included in this new translation help us see just how these changes worked. Many of them, and Newton’s memoranda about the Principia’s composition, were partisan moves in the controversies which wracked Baroque mechanics. Sometimes, Newton wanted to stress the extraordinarily short time he took to compose the work – roughly between autumn 1684, after Halley’s visit to Cambridge, and spring 1686, when the first section of the book was handed to the printer. He blamed the text’s errors on this haste, and on the innumeracy of his amanuensis. But he also wanted to emphasis, especially against his principal enemy, Leibniz, that he had already known the true mathematical principles of motion three decades earlier, as a young Cambridge scholar. On these occasions the elderly sage would regale his confidants with stories about the fall of an apple and the behaviour of the Moon, reminiscing about a time when he was ‘in the prime of my age for invention’. To bolster his claim to be the inventor of the calculus, he would declare, against any evidence we now possess, that the published propositions of the Principia differed markedly from the mathematical form in which he first discovered them. There thus developed the badly posed question of a great delay in the work’s appearance, between a sudden insight in the annus mirabilis of 1666 and the work’s eventual appearance in 1687. Cohen correctly reminds us that there was no youthful comparison between the gravitational pull on the fruit and the satellite, since Newton then supposed the Moon was tending to recede from its orbit’s centre. The fabled revelation of the 1660s is distinctly less impressive than the genuine speed with which Newton put his work together in the 1680s. He only developed some of the most important resources for his mature mechanics, such as the crucial notion of centripetal force, three or four years before it went to press. ‘Every translation is a continuous interpretation,’ Cohen acknowledges. Editors and translators must bring out the Principia’s remarkable quality as a mutable project rather than a finished tract.

There is a dirty little secret in previous Newton scholarship for which this new and handsome edition atones. In 1934, the same press issued a revised version of Motte’s translation, a revision whose many and sometimes scandalous errors have not prevented it remaining by far the most commonly read version of Newton’s work. To explain the need for his own efforts, Cohen selects some of the juiciest disasters of the 1934 text. This has Newton writing of the heavens ‘crumbling’, when he was really referring to their optical effects (‘refractio’); implying falsely that the acceleration of falling bodies depends on their mass, when he was in fact discussing the proportionality of mass and weight; and uttering platitudes about the ‘accuracy of language’ when he was instead, as a pious if heretical natural philosopher, discussing proper respect for the Scriptures (‘sacris literis’).

Non-existent characters, such as ‘Professor Gresham’, turn up in the 1934 revision. So, more importantly, does a lot of post-Newtonian physics. Newton, for example, typically referred to a ‘force of inertia’, by which bodies are preserved in a state of rest or uniform rectilinear motion. The term, though not the concept, is Kepler’s. But to modern eyes this is a strange kind of force, since Newton himself dictated that forces are agents which change, rather than preserve, the motion of bodies. Surely he had abandoned the scholastic conceit that all motions, even uniform ones, need a force to keep them going. The revisors of the 1934 edition saved the modern Newton from Newton the ancient, and simply omitted the word ‘force’ whenever he referred to inertia. The new translation adopts a different tactic. It preserves the phrase ‘force of inertia’, but Cohen then adds that ‘Newton (if only on an unconscious or psychological level) has not fully abandoned the ancient notion.’ Speculations about the inner states of Newton’s mind are rarely worthwhile. But much worse is any cavalier attempt to impose post-Newtonian physics on the Principia. Here as elsewhere, this new version often renders accessible the terms in which Newton incompletely but decisively revolutionised natural philosophy.

Right from the start the Principia’s reputation has been such that physicists have often sought warrant for their views in its pages. Though Newton denied that light travelled in waves, wave theorists made him one of their own. When Victorian physicists made the conservation of energy the premise of their entire science, they went to the Principia for a formulation of this principle by translating his term ‘action’ as their watchword, ‘horsepower’. Cohen and his colleagues point out that one proposition does indeed evaluate the product of force and displacement by changes in the squares of a body’s velocity, our contemporary measure of kinetic energy, but this does not warrant treating Newton as the founder of energetics. Reading the Principia well surely needs considerable technical expertise in mathematical physics. Here Cohen has enlisted colleagues such as the particle physicist Michael Nauenberg, who remarks on Newton’s use of graphical methods, while a jet engineer and philosopher of science, George Smith, gives us his views on Newton’s theories of the Moon’s motion and the mutual attraction of Jupiter and Saturn, and evaluates the Principia’s treatment of motion through resisting media.

The highest technical expertise is no guarantee of competent interpretation. Five years ago, the late Subrahmanyan Chandrasekhar, one of the 20th century’s greatest mathematical cosmologists, published what he called Newton’s ‘Principia’ for the Common Reader. Cohen understandably criticises the work, as have other historians of science. It sneers at these historians’ studies, relies solely on scientists’ remarks on Newton’s work and makes no use of the immense scholarship in the recent edition of Newton’s mathematical papers. Chandrasekhar, for example, mistook what Newton meant by ‘change of motion’. Elsewhere, he praised him for ‘surpassing the level of scientific understanding of his time by two hundred years’ in a mathematical proposition about the integration of areas within an ellipse. Thanks to Kepler, Newton could use the area swept by the line joining a planet to the Sun as a measure of the time the planet spent in each part of its orbit. In this proposition, he argued that such areas could not be evaluated exactly. But his argument, unbeknownst to Chandrasekhar, is in fact defective.

It is a common habit either to read back into Newton’s work achievements which considerably postdate his lifetime, or else, in desperation, to tease apart what has survived from what has not, attributing the latter to some psychological defect in the great man’s make-up. Thus Stephen Hawking added to his Brief History of Time a short appendix on what he calls Newton’s ‘deviousness and vitriol’. Hawking claims, oddly, that Kepler discovered his planetary laws by observation alone, that with these Keplerian principles Newton could have predicted the expansion of the Universe, and that, since it is apparently logically incompatible with classical mechanics, Newton’s theory of an absolute space intrinsically at rest was simply irrational. The new translation of the Principia’s famous scholium lets us see just how Newton rationally established that true motion can be understood only with reference to absolutely motionless space. Just because modern physics finds fault with this reasoning, it does not follow that Newton here lapsed into unreason.

Cohen’s introduction tries to strike a balance between modern physics and Baroque natural philosophy. ‘It is a fact that the Principia can be read as a work in physics, as has been done successfully for three centuries, without taking cognisance of what to modern eyes will seem to be extrascientific considerations.’ So it can. This was exactly Chandrasekhar’s aim, for example. But the Principia can also be read as an astonishing product of the natural philosophy of late 17th-century Europe, successfully accomplished then, recoverable now not least thanks to this new translation. The unfamiliarity of that enterprise, its mixture of theology and philosophy, experiment and mathematics, has made some turn Newton into a mystic wizard. It makes others salvage Newton the modern physicist from his unfortunate indulgences in the irrational. If, as the Principia’s first reviewer already supposed, this work is a ‘notable instance of the powers of the mind’, we need a much better account of how such powers work.

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