Mathematical Principles of Natural Philosophy

1687 Light and sky theory

Share

A book of 1687 in which three laws of motion and the law of universal gravitation are derived from planetary motion and tested on the satellites of Jupiter, the Moon, comets and the tides. Its author, the Englishman Isaac Newton, wrote it in the language of the geometry of limits, not of his own calculus of fluxions.

The data came from Kepler, Flamsteed and Halley; the printing was ordered by the Royal Society and paid for by Halley. One and the same force describes falling bodies on Earth and the revolution of the planets: hence the idea of laws of nature that can be expressed mathematically.

Why this date

exact for the year of publication: Book I was presented to the Royal Society at the end of April 1686, the order to print was given within a month, Books II and III appeared the following year (Royal Society blog); the draft De motu was sent to Halley in November 1684 (SEP)

What it grew from

• Spyglass turned to the sky

  influenced · Confidence: high

  The argument of Book III of the Principia opens with telescopic data: Phenomenon I (the satellites of Jupiter, measured with a micrometer in a telescope) extends the inverse-square law beyond the planets; Kepler's laws themselves, however, were derived from Tycho's pre-telescopic observations.

  Dispute: For a strong dependence: the satellites of Jupiter (Phenomenon I) and Saturn (Phenomenon II) cannot be observed without a telescope, and it is with them that Newton begins the proof of the universality of gravitation. Against it: Kepler's three laws, the Moon test and the inverse-square law for the planets rest on Tycho Brahe's pre-telescopic data and on the ancient estimate of the Moon's distance; without the telescope the book would have appeared in a narrower form, but it would have appeared. The quote about the micrometer and the 123-foot telescope comes from the third edition (1726), where Pound's data replaced the Cassini and Flamsteed measurements of the first edition of 1687.

  Link sources

  • Newton I. Philosophiae Naturalis Principia Mathematica, 3-. 1726, III, I (1726)

    The diameter of Jupiter was often taken from the micrometer in the 123 feet telescope, and reduced according to the mean distance of Jupiter from the sun or the earth, always produced an image less than 40'', at no time less than 38", more often 39".

  • Britannica, «Isaac Newton: The Principia» (2024) Verified 24 Sep 2026

    Because the satellites of Jupiter also obey Kepler's third law, an inverse square centripetal force must also attract them to the center of their orbits.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• The laws of planetary motion

  prerequisite for · Confidence: high

  Newton took Kepler's laws as empirically established and derived the law of gravitation from them: the 3/2 power rule combined with Huygens's formula for uniform circular motion yields a force falling off as the square of the distance, while the ellipse with the Sun at a focus and the area rule fix its direction and form.

  Dispute: Newton himself played down the size of the debt: in the sections where he treats all three of Kepler's rules mathematically he never once names Kepler (Christie 2018), and in his letter to Halley of 1686 he reduces Kepler's contribution to a guess that the orbit was elliptical. Moreover, according to Smith (SEP), by 1684 all three rules were held to be no more than good approximations and were disputed as a method of calculation, and it was after the Principia that they became laws; that is, Newton did not so much take over ready-made laws as confer their status on them retrospectively.

  Link sources

  • «Математические начала натуральной философии», III, 4 (1687) Verified 20 Sep 2026

    This proportion, which was found by Kepler, is accepted by everyone.

  • Smith G. E. «Newton's Philosophiae Naturalis Principia Mathematica», Stanford Encyclopedia of Philosophy (2007) Verified 20 Sep 2026

    The inverse-square part of this question came from combining the mathematical theory of uniform circular motion, which Huygens had published in his Horologium Oscillatorium of 1673, with Kepler's 3/2 power rule: the force in a string retaining a body in a uniform circular orbit varies directly as the radius of the circle and inversely as the square of the period; but the squares of the periods of the planets vary as the cubes of their mean distances; and hence, at least to a first approximation, the forces retaining the planets in their orbits vary inversely with the square of the radii of their nearly circular orbits.

  • Smith G. E. (2007) Verified 20 Sep 2026

    So, all three of Kepler's rules that came to be called 'laws' after the Principia were known to be nothing more than holding to high approximation when Newton started on the project in 1684.

  • Christie T. «400 Years of The Third Law: An overlooked and neglected revolution in astronomy», The Renaissance Mathematicus (2018) Verified 20 Sep 2026

    By the time Newton wrote his great work the empirical truth of Kepler's third law had been accepted and Newton uses this to establish the empirical truth of the law of gravity.

  • Christie T. (2018) Verified 20 Sep 2026

    Newton shows that his law of gravity and Kepler's third law are equivalent, although in this whole section where he deals mathematically with Kepler's three laws of planetary motion he never once mentions Kepler by name.

  • Smith G. E. (2007) Verified 20 Sep 2026

    Newton had real grounds for claiming in a letter to Halley in June 1686 a 'right' to the ellipse, remarking that 'Kepler knew the Orb to be not circular but oval, and guest it to be Elliptical.'

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• The Elements: definition, axiom, proof

  influenced · Confidence: high

  The Mathematical Principles of Natural Philosophy is built in Euclidean form: first a section of definitions, then the axioms, or laws of motion, and after them propositions derived from these and labelled, as in Euclid, either theorems or problems.

  Dispute: The title of the book alludes not to Euclid but to Descartes. According to the Stanford Encyclopedia, Newton changed the title in 1686 to Philosophiae Naturalis Principia Mathematica in allusion to Descartes' Principia Philosophiae, that is as a challenge to Cartesian physics. What is Euclidean here is the architecture of the book, definitions, axioms, theorems and problems, not its name; the common claim that the title too points back to Euclid's Elements is not supported by the sources.

  Link sources

  • Grabiner J. «How Euclid once ruled the world», Plus Magazine, 9 2016 (2016) Verified 20 Sep 2026

    In science, Isaac Newton's famous work Principia Mathematica clearly demonstrates Euclid's influence.

  • Grabiner J. (2016) Verified 20 Sep 2026

    Newton called his famous laws of motion 'axioms' and deduced his law of gravitation in the form of two mathematical theorems.

  • Smith G. «Newton's Philosophiae Naturalis Principia Mathematica», Stanford Encyclopedia of Philosophy Verified 20 Sep 2026

    In keeping with the Euclidean tradition, the propositions mathematically derived from the laws of motion are labeled either as theorems or as problems.

  • Smith G. Verified 20 Sep 2026

    The Principia opens with a section called 'Definitions' that includes Newton's discussion of absolute space, time, and motion.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• Logarithms and tables

  influenced · Confidence: high

  Napier's logarithms let Kepler reduce decades of astronomical observations into the Rudolphine Tables: Kepler calculated eight-figure logarithms and published them with the tables, and answered critics by proving how logarithms work from Euclid's Elements. These reduced observations gave Newton the empirical foundation for the theory of gravitation set out in the Principia of 1687.

  Link sources

  • O'Connor J. J., Robertson E. F. «John Napier», MacTutor History of Mathematics Verified 27 Sep 2026

    It was through the use of logarithms that Kepler was able to reduce his observations and make his breakthrough which then in turn underpinned Newton's theory of gravitation.

  • O'Connor J. J., Robertson E. F. «Johannes Kepler», MacTutor History of Mathematics Verified 27 Sep 2026

    Kepler calculated tables of eight-figure logarithms, which were published with the Rudolphine Tables (Ulm, 1628).

  • O'Connor J. J., Robertson E. F. «Johannes Kepler», MacTutor History of Mathematics Verified 27 Sep 2026

    Kepler's answer to the second objection was to publish a proof of how logarithms worked, based on an impeccably respectable source: Euclid's Elements Book 5.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

What grew out of it

• The method of least squares

  influenced · Confidence: low

  In the preface to Theoria motus Gauss names Newton's demonstration of motion in conic sections as what generalized the problem of orbit determination, and it is such an orbit that he fitted to observations; no independent source for the connection was found, the testimony comes from Gauss himself.

  Dispute: The Newtonian formulation can be traced only in Gauss. Legendre's book of 1805 is titled as a work on the orbits of comets, but the method itself is set out in the appendix Sur la méthode des moindres quarrés and is shown there on a numerical example drawn from the French meridian measurements, that is, on the problem of the figure of the earth rather than on a cometary orbit, so this descent does not hold through Legendre.

  Link sources

  • Carl Friedrich Gauss, Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, Charles Henry Davis, Boston 1857 (1857) Verified 25 Sep 2026

    On the other hand, Newton, starting from the principle of universal gravitation, demonstrated à priori that all bodies controlled by the attractive force of the sun must move in conic sections, of which the planets present one form to us, namely, ellipses, while the remaining forms, parabolas and hyperbolas, must be regarded as being equally possible, provided there may be bodies encountering the force of the sun with the requisite velocity;

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• The lunar tables of Tobias Mayer

  influenced · Confidence: medium

  Mayer himself names the foundation of his tables: the famous theory of the great Newton, reduced by Euler to general analytic equations. Gravitation supplied the form of the equations, observation supplied the coefficients.

  Dispute: The confidence is medium for two reasons. First, the claim about Newton comes from Mayer himself rather than from a historian: Higgitt, speaking of his tables, names Euler and the treatment of observations and does not name Newton at that point. Second, the Royal Museums Greenwich state plainly that the theory of the Principia was not enough for longitude, and that what made the tables work was empirical fitting rather than derivation from theory. The connection exists, but it supplies the form of the equations, not the result.

  Link sources

  • «Tobias Mayer», MacTutor History of Mathematics Archive, University of St Andrews Verified 20 Sep 2026, quotation assembled from fragments

    I have constructed theses tables ... with respect to the inequalities of motions, from that famous theory of the great Newton ... which that eminent mathematician Eulerus first elegantly reduced to general analytic equations.

  • Rebekah Higgitt, «Longitude», Encyclopedia of the History of Science, vol. 4, issue 1 (April 2021) (2021) Verified 20 Sep 2026

    One went to the widow of Tobias Mayer, whose application of Leonhard Euler's mathematical techniques to lunar observations had allowed him to produce greatly improved lunar tables.

  • «Longitude found: Nevil Maskelyne and the lunar method», Royal Museums Greenwich (2024) Verified 20 Sep 2026

    Isaac Newton's theories within his book Principia failed to describe the Moon's motions well enough to find longitude accurately.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.

Links under review

• Differential and integral calculus

  influenced · Confidence: medium

  The link between Newton's early calculus and the Principia is disputed: Whiteside found no drafts supporting Newton's 1715 claim that the propositions were found by analysis; the sources confirm only the presence of "moments" and series.

  The Principia took over the notions of limit and of "moments" from Newton's early work on the calculus, but the apparatus of the calculus itself is not used in the book: it is written in the language of the geometry of limit ratios, not in the algorithm of fluxions.

  Dispute: Newton himself in 1715 (the anonymous Account in the dispute with Leibniz) claimed to have found the propositions of the Principia by the new analysis and only then rewritten them geometrically. Whiteside, editor of Newton's mathematical papers, found no draft supporting this; Guicciardini (1989, 1999) and the Stanford Encyclopedia describe the mathematics of the Principia as a geometry of ultimate ratios without the algorithm of fluxions. On the other hand, Section I of Book I (the lemmas on first and last ratios) and Lemma II of Book II on moments are calculus in geometric dress, and in that sense the book would have been different without Newton's work of 1665-1671. A further difficulty: the calculus is counted from Leibniz's publication of 1684, which Newton did not use when writing the Principia, so the connection makes sense only through Newton's own unpublished fluxions.

  Link sources

  • Guicciardini N. The Development of Newtonian Calculus in Britain, 1700-1800. Cambridge University Press, p. 1 (1989) Verified 5 Oct 2026

    In the Principia there was no proper algorithm of fluxions, even though series and some algebraical manipulations with 'moments' occurred, especially in the second book.

  • Smith G. E., «Newton's Philosophiae Naturalis Principia Mathematica», Stanford Encyclopedia of Philosophy (2007) Verified 20 Sep 2026

    But the geometric mathematics used in the Principia offered no way of representing second derivatives.

  • Newton I. «An Account of the Book entituled Commercium Epistolicum», Philosophical Transactions 29 (1715), The Renaissance Mathematicus, «Newton Lied!» (1715) Verified 20 Sep 2026

    By the help of the new Analysis Mr. Newton found out most of the Prepositions in his Principia Philosophiae: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of the Heavens might be founded upon good Geometry.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

Node sources

• Smith G. E., «Isaac Newton», Stanford Encyclopedia of Philosophy (2007) Verified 5 Oct 2026

  The roughly three hundred copies of the Principia came off the press in the summer of 1687, thrusting the 44 year old Newton into the forefront of natural philosophy and forever ending his life of comparative isolation.

• Smith G. E., «Newton's Philosophiae Naturalis Principia Mathematica», Stanford Encyclopedia of Philosophy (2007) Verified 5 Oct 2026

  The answer in the nine page tract 'De Motu Corporum in Gyrum' that Newton sent to Halley in November 1684 is, an ellipse, provided the velocity is not too high (and if it is, then instead a parabola or a hyperbola, depending on the velocity).

• Britannica, «Isaac Newton: The Principia» (2024) Verified 24 Sep 2026

  Nearly five years later, in August 1684, Newton was visited by the British astronomer Edmond Halley, who was also troubled by the problem of orbital dynamics.

• Royal Society, «Principia» (2014) Verified 5 Oct 2026

  Newton presented Book 1 of the text to the Society at the end of April 1686 and the order to print it was given within a month.

• Guicciardini N. The Development of Newtonian Calculus in Britain, 1700-1800. Cambridge University Press, p. 1 (1989) Verified 5 Oct 2026

  In the Principia there was no proper algorithm of fluxions, even though series and some algebraical manipulations with 'moments' occurred, especially in the second book.

"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

Open this node on the map · Wikidata: Q205921