Philosophiae Naturalis Principia Mathematica

's own copy of his Principia, with hand written corrections for the second edition.
Newton's own copy of his Principia, with hand written corrections for the second edition.

The Philosophiae Naturalis Principia Mathematica (Latin: "mathematical principles of natural philosophy", often Principia or Principia Mathematica for short) is a three-volume work by Isaac Newton published on July 5, 1687. It contains the statement of Newton's laws of motion forming the foundation of classical mechanics as well as his law of universal gravitation. He derives Kepler's laws for the motion of the planets (which were first obtained empirically).

For an in-depth account of Newton's authoring of this work, see the writing of Principia Mathematica.

In formulating his physical theories, Newton had developed a field of mathematics known as calculus. However, the language of calculus was largely left out of the Principia. Instead, Newton recast the majority of his proofs as geometric arguments.

It is in the Principia that Newton expressed his famous Hypotheses non fingo ("I feign no hypotheses", that is, "I do not assert that any hypotheses are true"). Here is the translated passage containing this famous remark:

I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction.

The Philosophiae Naturalis Principia Mathematica is composed of three volumes.

  1. Of The Motion Of Bodies
  2. Of The Motion Of Bodies (contd.)
  3. The System Of The World

Location of copies

Missing image
A page from the Principia

Many national rare book collections contain original copies of Newton's Principia Mathematica. Notable examples are

  • The Wren Library in Trinity College Cambridge has Newton's own copy of the first edition, with hand written notes for the second edition.
  • The Whipple Museum of the History of Science ( in Cambridge has a first edition copy which used to belong to Robert Hooke.
  • Fisher Library in the University of Sydney has a first edition copy, annotated by a mathematician of uncertain identity and corresponding notes from Newton himself.
  • The Pepys Library in Magdalene College Cambridge, has Samuel Pepys' copy of the third edition.
  • The Martin Bodmer Library ( keeps a copy of the original edition that was owned by Leibniz. In it, we can see hand-written notes by Leibniz, in particular concerning the controversy of who invented calculus (although he published it later, Newton claimed that he invented it earlier). As an interesting side-note, the copy shows clear signs that Leibniz was an avid smoker.

A facsimile edition was published in 1972 by Alexandre Koyré and I. Bernard Cohen, Cambridge University Press, 1972, ISBN 0674664752.

Two more editions were published during Newton's lifetime:

Second Edition

Richard Bentley, master of Trinity College, influenced Roger Cotes, Plumian professor of astronomy at Trinity, to undertake the editorship of the second edition. Newton did not intend to start any re-write of the Principia until 17091. Under the weight of Cotes' efforts, but impeded by priority disputes between Newton and Leibniz2, and by troubles at the Mint3, Cotes was able to announce publication to Newton 30 June 17134. Bentley sent Newton only six presentation copies; Cotes was unpaid; Newton omitted any acknowledgement to Cotes.

Among those who gave Newton corrections for the Second Edition were:

However, Newton omitted acknowledgements to some because of the priority disputes. John Flamsteed, the Astronomer Royal, suffered this especially.

Third Edition

The third edition was published 25 March 1726, under the stewardship of Henry Pemberton, M.D., a man of the greatest skill in these matters ...; Pemberton later said that this recognition was worth more to him than the two hundred guinea award from Newton.5


Newton's geometric proofs for the motion of bodies are not easy, if not near impossible, to follow for a modern day mathematician as it assumes the reader has a very high knowledge about the geometry of cones, spheres and ellipses - a hot topic in Newton's time. Nearly 300 hundred years later Richard Feynman, a famous Nobel Prize winning physicist, failed in his attempt to follow Newton's proofs during a sabbatical year, and instead spent a significant amount of time producing his own geometric proofs, as was his custom when mastering a subject ("What I cannot create, I do not understand" -- Feynman).

Feynman had given a lecture on the motion of bodies, but the notes and pictures were lost for a time and consequently not included in The Feynman Lectures on Physics series. The lecture notes were later found, but unfortunately without the photographs of his illustrative chalkboard drawings. An attempt was made to publish the notes; ironically however one of the editors, David L. Goodstein, found that he could not completely understand Feynman's derivations and so rewrote them, following Feynman closely, but not exactly. The book was published as Feynman's Lost Lecture: The Motion of Planets Around the Sun.

Some of the other difficulties in following Newton's work are that many of the concepts were not named. For example he clearly grasped the concept of angular momentum, but it was not a concept that he ever named or used mathematically directly. The word 'motion' in his works clearly means what we now call momentum, yet the fundamental formulas such as F=m·a are never mentioned in his works.

This is all mentioned to point out how difficult it is to deal with the motion of bodies in pure geometry. Modern day physicists have a much more powerful and elegant toolkit to deal with such problems.

See also

Whitehead and Russell's Principia Mathematica (1910-1913)


External link


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