It is a quite long story and not a "one-shot" discovery, as suggested by the "apple tale".
There are two widely held opinions concerning the development of Newton's scientific ideas: that he found the law of universal "gravitation" in the 1660s and then refrained from publishing it for twenty years, and that he found this law by "deducing"
it from Kepler's "laws" (or, possibly, from only one of Kepler's laws). The analysis presented in this chapter will show that according to any reasonable definition of universal "gravitation", Newton did not find this law until some time after November
1684, and before 1686, and then published it forthwith. [page 222]
In Newton's memorandum [written in 1718 in the draft of a letter to Des Maizeaux] and in his early manuscript of the 1660s, there is as yet no explicit suggestion that there is a solar "force" exerted on the planets that is identical to the terrestrial force exerted on the moon (which he alleges that he supposed to be ordinary terrestrial gravity). The "endeavours to recede", in which he believed in the 1660s, are very different in concept from the simple centripetal forces continually deflecting planets and moons from their inertial (rectilinear) paths in which he came to believe some time later, in 1679-1680 or thereafter. Accordingly, there is no legitimate ground for saying that Newton had "known''
of the law (or a law) of universal gravitation in 1665 and had "delayed" announcing it for twenty years. In fact, far from believing in a force that is "universal", Newton did not then even have an awareness of the possibility of a planetary action on the sun, a
lunar action on the earth, much less an action of one planet on another.
In the memorandum quoted above, Newton says that he 'began to think of gravity extending' to the moon. Presumably, he would have guessed - by analogy - that if the planetary "endeavours" to recede from the sun (in his mind they were not at that time "forces")
vary inversely as the square of the distance, then it must be the same for the moon. Hence, if gravity extends to the moon, and varies inversely as the square of the distance, the intensity of gravity at the moon's orbit should be 1/r*r times the intensity here on
earth, r being the earth's radius. [page 232]
In an untitled MS document of the mid- or late 1660s (U.L.C. MS Add. 3958, sect. 5, fol. 87), Newton computes the 'endeavour of receding from the center' ('conatus a centro' or 'conatus recedendi a centro') by determining how far out along a tangent a body would move in some given time if it had that same 'endeavour' in a linear direction along the tangent and there were no impediment. In short, Newton is measuring the 'endeavour of receding'
(not yet "centrifugal force") by the acceleration, and the acceleration by the distance through which a body would move, according to Galileo's rule for uniform acceleration, freely along a straight line in a given time, 'the time of one revolution'. Newton
then computes how far a body would descend if its 'endeavour of approaching toward the centre in virtue of its gravity' ('conatus accedendi ad centrum virtute gravitatis') were equal in magnitude to its 'endeavour of receding from the center' at the equator, as a result of the earth's daily rotation. [page 238]
In the manuscript in which the foregoing calculations are described, Newton does derive an inverse-square law for the planets, by combining Kepler's third law for 'the primary planets' with their 'endeavours of receding' from the sun. This occurs in some
brief or summary paragraphs that are added onto the rather detailed discussion and exposition of the moon's 'endeavour of receding' and the 'endeavour of receding' at the earth's surface. Newton neither says expressly nor in any way implies in this document
either that the earth's gravity may extend as far out as the moon's orbit or that the moon's 'endeavour of receding' is in accord with an inverse-square law of the distance. The only application that he makes of his calculations concerning the moon's orbital 'endeavour
to recede' is an attempt to account for the fact that the moon always 'turns the same face toward the earth'. [page 240]
Newton's interest in astronomy was a long-standing one, easily traceable to at least his student days at Cambridge in the 1660s. [page 241]
Newton's attention was forcibly drawn to astronomical problems three years later, in 1679, when Robert Hooke (recently appointed secretary of the Royal Society) wrote to Newton, expressing the hope that Newton would renew his former 'philosophicall' exchanges with the Society. To start things off, Hooke invited Newton to comment on a 'hypothesis or opinion of mine . . . of compounding the celestiall motions of the planetts [out] of a direct
motion by the tangent & an attractive motion towards the centrall body'. In his reply, Newton declined to discuss Hooke's 'hypothesis'. Instead, he advanced a 'fansy' of his own: the effects of 'the Earth's diurnal motion' on the path of freely falling bodies (Newton
to Hooke, 28 November 1679). Newton had erred, however, as Hooke was quick to discover.
Newton had proposed a spiral path, usually accompanying the publication of Newton's letter to Hooke (28 November 1679) [...]. Hooke corrected Newton; he showed that the curve would not be (as Newton thought) 'a kind of spirall', which after a few revolutions would bring the falling object to 'the Centre of the Earth'. The curve would be 'rather a kind [of] Elleptueid'. [page 242]
Once Newton had acknowledged Hooke's correction, Hooke was emboldened to write to Newton of 'my supposition' concerning the attractive force that keeps the planets in their orbits; it was '... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall and Consequently that the Velocity will be in a sub-duplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance' (Hooke to Newton, 6 January 1679/80). Newton did not comment directly on this statement. His eventual opinion may be readily gathered from the fact that he proved that the velocity of a body (such as a planet or planetary satellite) moving in an elliptical orbit under the action of an inverse-square force is not 'as Kepler [and as Hooke] Supposes Reciprocall to the Distance', but is rather reciprocally as (or in inverse proportion to) the perpendicular distance from the center of force to the tangent to the planetary orbit. [page 243]
In various unpublished documents, Newton admitted that in 1679-1680 Hooke had provided the occasion for his study of planetary dynamics, although he would not admit that Hooke had
made any substantive contribution to his thinking. [page 248]
it is a plain fact that he [Newton] did undertake an analysis of planetary motion according to the Hookean mode of conceiving a centrally directed force to be acting on a planet that had a component of linear inertial motion, and he was able to do so even
though he still had not fully given up his adherence to the concept of aethereal vortices and though he apparently did not really believe in centripetal forces acting at a distance. Whatever his beliefs, the Newtonian style enabled him to explore the properties of this kind of force and eventually to discover [...] that universal gravity is useful and even necessary, and that it 'really exists' (as he later declared in the concluding general scholium of the Principia in 1713) and acts according to the laws he had set forth. [page 254]
