The laws of planetary motion

1609/1619 Light and sky theory

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Three empirical rules of planetary motion: a planet moves on an ellipse with the Sun at one focus; the radius vector from the Sun to the planet sweeps out equal areas in equal times; the squares of the orbital periods are as the cubes of the mean distances.

The first two laws come from a reworking of Tycho Brahe's observations of Mars: Johannes Kepler gave up the circle and uniform motion and printed them in Astronomia nova (1609). The decisive argument was a discrepancy of eight arc minutes, which at Tycho's accuracy could no longer be blamed on observational error.

The third law, found in 1618 and printed in the Harmonice mundi (1619), joined the planets into a single system rather than a set of independent orbits. The Rudolphine Tables (1627) are built on these rules; from them Newton obtained the law of universal gravitation.

Why this date

exact: the first two laws in Astronomia nova (Heidelberg, 1609), and this first publication is the axis value. Kepler found the third law on 8 March 1618, rejected it because of a calculation error, found it again on 15 May 1618 and printed it in Harmonices mundi libri V (Linz, 1619); the book was finished on 27 May 1618 and took about a year to print, hence the discrepancy between 1618 and 1619 in different sources (Britannica gives 1618 for the third law). The observational basis is pre-telescopic: the measurements of Tycho Brahe, made before his death in 1601.

What grew out of it

• Mathematical Principles of Natural Philosophy

  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 method of least squares

  influenced · Confidence: medium

  Both founding publications of the method are works on determining orbits from redundant and discordant observations, and in the preface to Theoria motus Gauss derives his problem directly from Kepler's extraction of orbital elements from the observations of Tycho Brahe.

  Dispute: The descent from the problem of orbit determination is known from Gauss's own preface, that is, from the testimony of a participant rather than from an independent examination by a historian. The priority of the method is disputed as well. In Legendre's favour stands print: he published the method first, in 1805, and wrote to Gauss 'je n'ai jamais appelé principium nostrum un principe qu'un autre avait publié avant moi'. In Gauss's favour stand his own statement in Theoria motus of 1809 ('my principle, which I have made use of since 1795'), the diary entry of 17 June 1798, and Stigler's examination, which leans towards Gauss having had the method well before Legendre while stating outright that this cannot be regarded as conclusive.

  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) Verified 20 Sep 2026

    After the laws of planetary motion were discovered, the genius of Kepler was not without resources for deriving from observations the elements of motion of individual planets.

  • A. Abdulle, G. Wanner, 200 years of least squares method, Elemente der Mathematik 57, 45-60 (2002) Verified 20 Sep 2026

    In 1805 appeared the work Nouvelles méthodes pour la determination des orbites des comètes by A.-M. Legendre, containing in an appendix an extremely beautiful presentation of the méthode des moindres quarrés.

  • Stephen M. Stigler, Gauss and the Invention of Least Squares, Annals of Statistics 9 (3), 465-474 (1981) Verified 20 Sep 2026

    It is argued (though not conclusively) that Gauss probably possessed the method well before Legendre, but that he was unsuccessful in communicating it to his contemporaries.

  "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

• MacTutor History of Mathematics, «Johannes Kepler» Verified 20 Sep 2026

  Kepler concluded that the orbit of Mars was an ellipse with the Sun in one of its foci (a result which when extended to all the planets is now called "Kepler's First Law"), and that a line joining the planet to the Sun swept out equal areas in equal times as the planet described its orbit ("Kepler's Second Law"), that is the area is used as a measure of time.

• MacTutor History of Mathematics, «Johannes Kepler» Verified 20 Sep 2026

  The Harmony of the World also contains what is now known as 'Kepler's Third Law', that for any two planets the ratio of the squares of their periods will be the same as the ratio of the cubes of the mean radii of their orbits.

• Gingerich O. «Kepler, Johannes», Dictionary of Scientific Biography (via Encyclopedia.com) (1973) Verified 20 Sep 2026

  Kepler had thus arrived at what we now call his first law, that the planetary orbits are ellipses with the sun at one focus.

• Gingerich O. (1973) Verified 20 Sep 2026

  Thus the radius vector swept out equal areas in equal times.

• Gingerich O. (1973) Verified 20 Sep 2026

  In the course of this investigation, Kepler hit upon the relation now called his third or harmonic law: The ratio that exists between the periodic times of any two planets is precisely the ratio of the 3/2 power of the mean distances.

• Gingerich O. (1973) Verified 20 Sep 2026

  Because these 8' could not be ignored, they along have led to a total reformation of astronomy.

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

  He had originally discovered it two months earlier on 8 March but due to a calculation error rejected it. On 15 May he found it again and this time recognised that it was correct.

• Britannica, «Kepler's laws of planetary motion» (2024) Verified 24 Sep 2026

  his first two laws in the year 1609 and a third law nearly a decade later, in 1618

"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.

Further reading

Not evidence: these carry no quotation and support nothing we wrote. They are books and scans to read next.

• Kepler J. «Astronomia nova aitiologetos, seu physica coelestis, tradita commentariis de motibus stellae Martis», Heidelberg (1609)

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