Philosophiæ Naturalis Principia Mathematica

• 1. Introduction

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  Philosophiæ Naturalis Principia Mathematica arrived in the summer of 1687, and the world it entered would not survive it intact. Isaac Newton's book - three volumes of dense Latin prose, authorized by Samuel Pepys on the 5th of July 1686 - set out to do something that had never been attempted at this scale: to show that the same force bending a comet around the Sun also pulled the tides in and out of every harbor on Earth. That one invisible principle could hold all of it together.

  A work described by one French mathematician as "the greatest production of the human mind" and by another as marking "the epoch of a great revolution in physics" raises an obvious question: how did it come to exist at all? The answer involves a debtor too proud to pay, a publisher who lost his salary, a rivalry stretching across decades, and one man who could not sit still long enough to eat his meals. What the Principia achieved is famous. How it was made is something else entirely.

• 2. The Spur of 1684

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  In January 1684, Edmond Halley sat in a room with Christopher Wren and Robert Hooke, and the conversation turned to planetary motion. Hooke claimed he had already derived the inverse-square law of gravitation and all the laws of planetary motion. Wren was unconvinced. Hooke never produced the proof. Halley, who could derive the law only for the restricted circular case, resolved to ask someone who might actually know.

  Halley's visit to Newton in Cambridge probably took place in August 1684. When Halley described the problem discussed with Hooke and Wren, Newton told him that he had already worked it out some time earlier - but could not find his papers. The result arrived in November 1684: a nine-page manuscript titled De motu corporum in gyrum, meaning "Of the motion of bodies in an orbit". It derived what are now called Kepler's three laws from an inverse square law of force, and extended those results to conic sections.

  Halley was so struck by the mathematical and physical originality of that short document that he returned to Newton in November 1684 to urge him to let the Royal Society have more. What followed was a period of work that Newton's secretary and copyist of the period, Humphrey Newton, later described in vivid terms: Isaac Newton sometimes forgot his food, forgot his sleep, and sometimes rushed back from a walk in his garden without even waiting to sit before beginning to write down a new idea. Newton's chemical notebooks, which normally contained dated entries, show no entries at all from May 1684 to April 1686. He had effectively set aside every other pursuit for well over a year and a half.

• 3. Inside the Three Books

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  Book 1, subtitled De motu corporum - "On the motion of bodies" - opens not with physics but with mathematics. Its early pages lay out a geometric form of infinitesimal calculus built from what Newton called "the method of first and last ratios", dealing with "vanishingly small" shapes. From those foundations, the book builds up relationships between centripetal forces and what is now recognized as Kepler's second law of planetary motion, then moves through conic sections and eccentric orbits.

  Propositions 57 through 69 take the first steps in what would later become famous as the three-body problem: the problem of determining the movements of three massive bodies subject to their mutually perturbing gravitational attractions. Propositions 70 through 84 establish what is now called the Shell theorem - Newton's proof that a massive spherically symmetrical body attracts other bodies outside itself as if all its mass were concentrated at its center. That result alone is what allows the inverse-square law to be applied to the real solar system with precision.

  Book 2 largely concerns motion through resisting mediums. A later analysis found that of its 53 propositions, almost all are correct, with only two or three open to question. It pioneers the study of fluid mechanics, estimates the speed of sound at around 1088 feet per second, and derives Boyle's law - though incorrectly, because Newton treated an ideal gas as an elastic fluid rather than what it actually is. The book concludes by attacking Descartes' theory of vortices, which held that planets were carried through space by whirling fluid; Newton argued this was flatly incompatible with the astronomical evidence.

  Book 3, subtitled De mundi systemate - "On the system of the world" - applies the previous two books to the actual Solar System. It establishes the inverse-square law of universal gravitation by starting with the satellites of Jupiter and working outward step by step. It accounts for the tides, the precession of the equinoxes, the oblateness of the Earth, and the highly eccentric, near-parabolic orbits of comets. Much of the comet data came from John Flamsteed and Edmond Halley.

• 4. Newton's Own Voice

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  Newton built the Principia on his own definitions of mass, momentum, and force - definitions still recognizable in textbooks today. His definition of mass stated that "the quantity of matter is that which arises conjointly from its density and magnitude". From mass he derived momentum, and from momentum the concept of force through change in motion over time. Curiously for modern readers, his exposition appears dimensionally incomplete, since he does not introduce the dimension of time in rates of change.

  His treatment of space and time was equally deliberate and equally strange. Newton insisted on distinguishing "absolute" time and space from the relative and apparent versions that ordinary people experience. He wrote that "in philosophical discussions, we ought to step back from our senses, and consider things themselves, distinct from what are only perceptible measures of them".

  In Book 3's second and third editions Newton added a section titled "Rules of Reasoning in Philosophy", which laid out four principles for how to handle unknown phenomena and reach explanations. The fourth rule, which appeared only in the 1726 edition, states that propositions inferred by induction from phenomena should be treated as accurately or very nearly true until other phenomena either refine them or introduce exceptions. These rules have been recognized as a foundation for the modern approach to scientific inquiry.

  The General Scholium, added to the second edition of 1713 and amended in 1726, contains Newton's famous phrase hypotheses non fingo - "I frame no hypotheses". He wrote it in direct response to critics who accused him of introducing "occult agencies" into science by positing a gravitational force that acted across empty space without any visible medium. Newton's reply was that the phenomena themselves implied the attraction; the phenomena did not yet indicate its cause; and it was both unnecessary and improper to speculate about things not implied by the phenomena.

• 5. Halley Pays the Bill

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  The text of Book 1 was presented to the Royal Society at the close of April 1686, and Hooke immediately made priority claims over the inverse-square law. Newton, who hated disputes, threatened to suppress Book 3 entirely. Halley, showing what the record describes as considerable diplomatic skill, persuaded Newton to withdraw the threat. Samuel Pepys, as president, gave his formal imprimatur on the 30th of June 1686.

  Then the money ran out. The Royal Society had recently spent its book budget on a work called De Historia piscium, leaving nothing for the Principia. Edmund Halley paid for the publication himself. He was also acting at the time as publisher of the Philosophical Transactions of the Royal Society. When the book appeared in summer 1687, Halley learned that the Society could no longer pay him the promised annual salary of fifty pounds. His compensation was leftover copies of De Historia piscium - the very book that had consumed the budget in the first place.

  The first edition has been estimated at around 500 copies, with a survey published in 1953 locating 189 surviving copies and a 2020 survey finding nearly 200 more. A 2020 study identified 387 copies of the first edition, significantly more than the long-held estimate of roughly 250, suggesting a wider early readership than previously assumed. In 2016, one first edition sold for 3.7 million dollars. The second edition of 1713 was printed in 750 copies, and the third edition of 1726 in 1,250 copies.

• 6. Controversy with Hooke

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  Robert Hooke had been arguing for a principle of gravitational attraction since the 1660s. In Micrographia of 1665, in a 1666 Royal Society lecture, and again in 1674, he postulated mutual attractions between the Sun and planets that increased with proximity to the attracting body. But Hooke's own words in 1674 were frank about what he lacked: "Now what these several degrees are I have not yet experimentally verified." He had no mathematical demonstration and no evidence.

  In November 1679, Hooke began a correspondence with Newton that eventually, on the 6th of January 1680, included Hooke's suggestion that gravitational attraction "always is in a duplicate proportion to the Distance from the Center Reciprocall" - that is, an inverse-square law. Newton later acknowledged that this correspondence had reawakened his dormant interest in astronomical matters. But he was pointed in saying that Hooke had given him no new idea: "yet am I not beholden to him for any light into that business but only for the diversion he gave me from my other studies to think on these things."

  In 1686, when Book 1 was presented to the Royal Society, Hooke pressed his claim again. He agreed, according to Halley's contemporary report, that the mathematical demonstrations of the curves generated by an inverse-square law were wholly Newton's. Newton denied Hooke credit as the originator of the idea, pointing to prior work by others before Hooke and arguing that mathematical demonstration - not conjecture - was what gave the law its scientific standing.

  Decades later, in 1759, the mathematician Alexis Clairaut reviewed Hooke's published work and offered what became the settled judgment: "The example of Hooke serves to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated."

• 7. Revisions and Translations

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  Newton had been urged toward a second edition since the early 1690s, partly because copies of the 1687 printing had become rare and expensive within a few years of publication. Richard Bentley, master of Trinity College, eventually persuaded Newton to proceed, and appointed Roger Cotes, Plumian professor of astronomy at Trinity, to handle the editing. The correspondence of 1709-1713 shows Cotes managing a large and important set of revisions while reporting to two masters: Bentley and Newton. Cotes was able to announce publication on the 30th of June 1713. Newton sent Bentley only six presentation copies. Cotes received no payment. Newton omitted any acknowledgement to him.

  Newton revised again in the autumn of 1723, aged eighty, following a serious illness. The third edition appeared on the 25th of March 1726 under the stewardship of Henry Pemberton. In 1739-1742, two French priests, Pères Thomas LeSeur and François Jacquier, produced an extensively annotated version of the 1726 text sometimes called the Jesuit edition, reprinted more than once in Scotland during the nineteenth century.

  Émilie du Châtelet made the only complete French translation of all three books and their prefaces, adding a commentary section and an analytical section that applied calculus to Newton's most controversial theories. Her translation remains the standard French translation to this day.

  Four full English translations have appeared, all based on the 1726 edition. The first, by Andrew Motte in 1729, was described by Newton scholar I. Bernard Cohen in 1968 as still of great value for conveying the sense of Newton's words in their own time. A fourth complete English translation, published in 2021 by Cambridge University Press, came from Charles Leedham-Green, professor emeritus of mathematics at Queen Mary University of London, who worked on it for twenty years. In 1977, the spacecraft Voyager 1 and 2 left Earth carrying a picture of a page from the Principia as part of the Golden Record - a collection of messages from humanity addressed to extraterrestrials.