De motu corporum in gyrum

De motu corporum in gyrum ("On the motion of bodies in an orbit") is the presumed title of a manuscript by Isaac Newton sent to Edmond Halley in November 1684. It followed a visit by Halley earlier in that year, when Halley had questioned Newton about problems then exercising the minds of Halley and his scientific circle in London, including Sir Christopher Wren and Robert Hooke.

The title of the document is only presumed because the original is now lost. Its contents are inferred from surviving documents, which are two contemporary copies and a draft. Only the draft has the title now used; both copies are without title.

This manuscript (De Motu for short, but not to be confused with several other Newtonian papers carrying titles that start with these words) gave important mathematical derivations relating to the three relations now known as "Kepler's laws" (before Newton's work, these had not been generally regarded as laws). Halley reported the communication from Newton to the Royal Society on 10 December 1684 (Old Style). After further encouragement from Halley, Newton went on to develop and write his book Philosophiæ Naturalis Principia Mathematica (commonly known as the Principia) from a nucleus that can be seen in 'De Motu' – of which nearly all of the content also reappears in the Principia.

One of the surviving copies of De Motu was made by being entered in the Royal Society's register book, and its (Latin) text is available online.

For ease of cross-reference to the contents of De Motu that appeared again in the Principia, there are online sources for the 'Principia' in English translation, as well as in Latin.

De motu corporum in gyrum is short enough to set out here the contents of its different sections. It contains 11 propositions, labelled as 'theorems' and 'problems', some with corollaries. Before reaching this core subject-matter, Newton begins with some preliminaries:

(Newton's later first law of motion is to similar effect, Law 1 in the Principia.)

Then follow two more preliminary points:

Then follows Newton's main subject-matter, labelled as theorems, problems, corollaries and scholia:

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