Johann Kepler had proved by an elaborate series of measurements that each planet revolves in an elliptical orbit round the sun, whose centre occupies one of the foci of the orbit, that the radius vector of each planet drawn from the sun describes equal areas in equal times, and that the squares of the periodic times of the planets are in the same proportion as the cubes of their mean distances from the sun. The fact that heavy bodies have always a tendency to fall to the earth, no matter at what height they are placed above the earth’s surface, seems to have led Newton to conjecture that it was possible that the same tendency to fall to the earth was the cause by which the moon was retained in its orbit round the earth. Newton, by calculating from Kepler’s laws, and supposing the orbits of the planets to be circles round the sun in the centre, had already proved that the force of the sun acting upon the different planets must vary as the inverse square of the distances of the planets from the sun. He therefore was led to inquire whether, if the earth’s attraction extended to the moon, the force at that distance would be of the exact magnitude necessary to retain the moon in its orbit. He found that the moon by her motion in her orbit was deflected from the tangent in every minute of time through a space of thirteen feet. But by observing the distance through which a body would fall in one second of time at the earth’s surface, and by calculating from that on the supposition of the force diminishing in the ratio of the inverse square of the distance, he found that the earth’s attraction at the distance of the moon would draw a body through 15 ft. in 1 min. Newton regarded the discrepancy between the results as a proof of the inaccuracy of his conjecture, and “laid aside at that time any further thoughts of this matter.” But in 1679 a controversy between Hooke and Newton, about the form of the path of a body falling from a height, taking the motion of the earth round its axis into consideration, led Newton again to revert to his former conjectures on the moon. The measure of the earth, which had hitherto been accepted by geographers and navigators, was based on the very rough estimate that the length of a degree of latitude of the earth’s surface measured along a meridian was 60 m. More accurate estimates had been made by R. Norwood and W. Snell, and more recently by P. Picard. At a meeting of the Royal Society on the 11th of January 1672, Oldenburg the secretary read a letter from Paris describing the method followed by Picard in measuring a degree, and specifically stating the precise length that he calculated it to be. It is probable that Newton had become acquainted with this measurement of Picard’s, and that he was therefore led to make use of it when his thoughts were redirected to the subject. This estimate of the earth’s magnitude, giving 69·1 m. to 1°, made the two results, the discrepancy between which Newton had regarded as a disproof of his conjecture, to agree so exactly that he now regarded his conjecture as fully established.
In January 1684 Sir Christopher Wren, Halley and Hooke were led to discuss the law of gravity, and, although probably they all agreed in the truth of the law of the inverse square, yet this truth was not looked upon as established. It appears that Hooke professed to have a solution of the problem of the path of a body moving round a centre of force attracting as the inverse square of the distance; but Halley, finding, after a delay of some months, that Hooke “had not been so good as his word” in showing his solution to Wren, started in the month of August 1684 for Cambridge to consult Newton on the subject. Without mentioning the speculations which had been made, he asked Newton what would be the curve described by a planet round the sun on the assumption that the sun’s force diminished as the square of the distance. Newton replied promptly, “an ellipse,” and on being questioned by Halley as to the reason for his answer he replied, “Why, I have calculated it.” He could not, however, put his hand upon his calculation, but he promised to send it to Halley. After the latter had left Cambridge, Newton set to work to reproduce the calculation. After making a mistake and producing a different result he corrected his work and obtained his former result.
In the following November Newton redeemed his promise to Halley by sending him, by the hand of Mr Paget, one of the fellows of his own college, and at that time mathematical master of Christ’s Hospital, a copy of his demonstration; and very soon afterwards Halley paid another visit to Cambridge to confer with Newton about the problem; and on his return to London on the 10th of December 1684, he informed the Royal Society “that he had lately seen Mr Newton at Cambridge, who had showed him a curious treatise De Motu,” which at Halley’s desire he promised to send to the Society to be entered upon their register. “Mr Halley was desired to put Mr Newton in mind of his promise for the securing this invention to himself, till such time as he could be at leisure to publish it,” and Paget was desired to join with Halley in urging Newton to do so. By the middle of February Newton had sent his paper to Aston, one of the secretaries of the Society, and in a letter to Aston dated the 23rd of February 1685, we find Newton thanking him for “having entered on the register his notions about motion.” This treatise De Motu was the germ of the Principia, and was obviously meant to be a short account of what that work was intended to embrace. It occupies twenty-four octavo pages, and consists of four theorems and seven problems, some of which are identical with some of the most important propositions of the second and third sections of the first book of the Principia.
The years 1685 and 1686 will ever be memorable in the history of science. It was in them that Newton composed almost the whole of his great work. During this period Newton had a very extensive correspondence with John Flamsteed, who was then the astronomer-royal. Many of the letters are lost, but it is clear from one of Newton’s, dated the 19th of September 1685, that he had received many useful communications from Flamsteed, and especially regarding Saturn, “whose orbit, as defined by Kepler,” Newton “found too little for the sesquialterate proportions.” In the other letters written in 1685 and 1686 he applies to Flamsteed for information respecting the orbits of the satellites of Jupiter and Saturn, respecting the rise and fall of the spring and neap tides at the solstices and the equinoxes, respecting the flattening of Jupiter at the poles (which, if certain, he says, would conduce much to the stating the reasons of the precession of the equinoxes), and respecting the difference between the observed places of Saturn and those computed from Kepler’s tables about the time of his conjunction with Jupiter. On this last point the information supplied by Flamsteed was peculiarly gratifying to Newton; and it is obvious from the language of this part of his letter that he had still doubts of the universal application of the sesquialteral proportion. “Your information,” he says, “about the errors of Kepler’s tables for Jupiter and Saturn has eased me of several scruples. I was apt to suspect there might be some cause or other unknown to me which might disturb the sesquialteral proportions, for the influences of the planets one upon another seemed not great enough, though I imagined Jupiter’s influence greater than your numbers determine it. It would add to my satisfaction if you would be pleased to let me know the long diameters of the orbits of Jupiter and Saturn, assigned by yourself and Mr Halley in your new tables, that I may see how the sesquialteral proportion fills the heavens, together with another small proportion which must be allowed for.”
Upon Newton’s return from Lincolnshire in the beginning of April 1685, he seems to have devoted himself to the preparation of his work. In the spring he had determined the attractions of masses, and thus completed the law of universal gravitation. In the summer he had finished the second book of the Principia, the first book being the treatise De Motu, which he had enlarged and completed. Excepting in the correspondence with Flamsteed we hear nothing more of the preparation of the Principia until the 21st of April 1686, when Halley read to the Royal Society his Discourse concerning Gravity and its Properties, in which he states “that his worthy countryman Mr Isaac Newton has an incomparable treatise of motion almost ready for the press,” and that the law of the inverse square “is the principle on which Mr Newton has made out all the
