De motu corporum in gyrum

De motu corporum in gyrum (from Latin: "On the motion of bodies in an orbit"; abbreviated De Motu) is the presumed title of a manuscript by Isaac Newton sent to Edmond Halley in November 1684. The manuscript was prompted by a visit from Halley earlier that year when he had questioned Newton about problems then occupying the minds of Halley and his scientific circle in London, including Sir Christopher Wren and Robert Hooke. This manuscript gave important mathematical derivations relating to the three relations now known as "Kepler's laws of planetary motion" (before Newton's work, these had not been generally regarded as scientific 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: 3 Definitions: 1: 'Centripetal force' (Newton originated this term, and its first occurrence is in this document) impels or attracts a body to some point regarded as a center. (This reappears in Definition 5 of the Principia.) 2: 'Inherent force' of a body is defined in a way that prepares for the idea of inertia and of Newton's first law (in the absence of external force, a body continues in its state of motion either at rest or in uniform motion along a straight line).