Why Newton Rewrote the Principia Three Times

Three Editions, Forty Years, One Obsession

Isaac Newton published the first edition of the Philosophiae Naturalis Principia Mathematica in 1687. He published a second edition in 1713 and a third in 1726, a year before his death. Each edition was substantially revised. Newton added new material, corrected errors, sharpened arguments, responded to critics, and removed passages he regretted. The differences between the three editions reveal how Newton’s thinking evolved over four decades and how the scientific and philosophical context around the Principia changed.

Most people who refer to “the Principia” are referring to the third edition, which is the version most commonly reprinted and translated. But the first and second editions are different books in important respects, and understanding what changed between them illuminates both Newton’s character and the development of physics.

The First Edition (1687): The Revolution

The first edition was produced under pressure. Edmund Halley, who financed the publication out of his own pocket, pushed Newton to complete the manuscript. Newton wrote much of it in an extraordinary burst of sustained effort between 1684 and 1686. The result was a masterpiece, but it bore the marks of haste.

The first edition contained the three laws of motion, the law of universal gravitation, the derivation of Kepler’s laws from inverse-square attraction, the theory of tides, the shape of the Earth, and the orbits of comets. It also contained Book II on fluid mechanics and the refutation of Descartes’s vortex theory. The mathematical arguments were presented in geometric form, using proportions and constructions rather than the algebraic calculus that Newton had developed privately.

The first edition also contained passages that Newton would later regret. A section in Book III discussed the properties of a “subtle spirit” pervading all bodies, which Newton suggested might explain gravity, magnetism, and other phenomena. This passage, vague and speculative, was criticized by contemporaries and removed from later editions.

The print run was approximately 300 to 400 copies. The book sold slowly at first but gradually became recognized as the most important work of natural philosophy ever written. By the early 1690s, copies were scarce and expensive.

The Second Edition (1713): The Counterattack

The second edition was prepared under the editorship of Roger Cotes, a brilliant young mathematician at Trinity College, Cambridge. Newton and Cotes exchanged hundreds of letters over two years, discussing revisions, corrections, and additions. The collaboration was productive but tense; Newton was defensive about criticism, and Cotes was not afraid to point out errors.

The major changes in the second edition were:

• Corrected lunar theory: Newton’s treatment of the Moon’s orbit in the first edition contained significant errors. He had struggled with the three-body problem (Earth, Moon, Sun) and had been unable to derive the Moon’s motion with the accuracy he wanted. The second edition included improved calculations, though the problem was not fully solved.
• Revised cometary theory: Newton updated his analysis of cometary orbits, incorporating new observational data from Halley and others.
• Improved density calculations: Newton revised his estimates of the densities of planets, incorporating better data.
• Response to Leibniz: By 1713, the priority dispute over the invention of calculus was raging. The second edition contained subtle changes designed to assert Newton’s priority and diminish Leibniz’s claims. Some propositions that had been stated without proof in the first edition were now proved using methods that emphasized Newton’s own mathematical techniques.
• The General Scholium: The most famous addition to the second edition was the General Scholium, a philosophical essay appended to Book III. In it, Newton addressed the nature of gravity, the existence of God, and the limits of scientific explanation. The General Scholium contains Newton’s famous declaration: “Hypotheses non fingo” (I do not feign hypotheses). Newton was responding to critics (particularly Leibniz) who objected that his theory of gravity did not explain how gravity works, only what it does. Newton’s reply was that it was enough to have described gravity’s effects mathematically; the cause of gravity was a separate question that he would not speculate about without evidence.

The General Scholium also contains Newton’s argument for the existence of God as the creator and sustainer of the solar system. Newton argued that the orderly arrangement of the planets (all orbiting in the same direction, in nearly the same plane) could not have arisen by chance and must be the work of an intelligent being. This theological passage has been debated by philosophers ever since.

Roger Cotes wrote a preface to the second edition that attacked the vortex theory of Descartes and the philosophical objections of Leibniz with considerable force. Newton approved the preface, though its combative tone may have exceeded what he would have written himself.

The Third Edition (1726): The Final Word

The third edition was prepared by Henry Pemberton, a physician and mathematician who had attracted Newton’s attention by publishing a paper defending the Principia against Continental critics. Newton was now in his eighties, and the revision process was slower and less thorough than the Cotes collaboration had been.

The major changes in the third edition were:

• Further refinement of lunar theory: Newton continued to improve his calculations of the Moon’s orbit, incorporating data from the Astronomer Royal, John Flamsteed (with whom Newton had a bitter and public feud).
• Updated cometary data: New observations of comets, including the predicted return of Halley’s Comet, were incorporated.
• Revised General Scholium: Minor changes to the theological and philosophical passages.
• Removal of some speculative material: Newton was increasingly cautious about speculation in his old age and trimmed passages that went beyond what the data supported.
• Addition of a preface by Pemberton, less polemical than Cotes’s preface to the second edition.

The third edition was published in March 1726, about a year before Newton’s death on March 31, 1727. It was the version that Newton considered definitive, and it is the basis for most subsequent reprints and translations, including Andrew Motte’s influential 1729 English translation.

What the Revisions Reveal

The differences between the three editions tell us several things about Newton:

He was never satisfied. Newton revised the Principia continuously over forty years, always seeking greater precision, better data, and sharper arguments. The book was not a finished product delivered to the world in 1687. It was a living document that Newton refined as long as he was able.

He was responsive to criticism. Despite his reputation for defensiveness, Newton incorporated suggestions from Cotes, Pemberton, Halley, and others. The second edition is significantly better than the first in many technical respects, largely because Cotes identified and corrected errors that Newton had missed.

He was combative. The revisions to the second and third editions were partly motivated by the desire to answer critics and assert priority. The calculus dispute with Leibniz, the philosophical objections of the Cartesians, and personal feuds with Hooke and Flamsteed all left traces in the text.

He struggled with the Moon. The lunar theory was the one area where Newton never achieved the accuracy he wanted. The three-body problem (which he himself had identified as analytically intractable) defeated his best efforts. He came close, but the precise prediction of the Moon’s motion would require the perturbation methods developed by Clairaut, d’Alembert, and Euler in the following century.

The Principia You Can Hold

Kronecker Wallis’s edition of the Principia presents the work as three individually bound books within a single cover, echoing Newton’s own division of the text into three parts. The design invites the reader to engage with each book separately: the abstract mathematics of Book I, the fluid mechanics of Book II, and the cosmic synthesis of Book III. It is a way of reading the Principia that honors the structure Newton chose and that reveals, book by book, the architecture of the argument that changed the world three times over.