corporis particulas æquales est reciproce ut quadratum distantiæ locorum a particulis’ (Principia, bk. iii. prop. viii. cor. 2). The force between the earth and the moon is the resultant of the infinite number of forces between the particles of these bodies. Newton was the first to show that the force of attraction between two spheres is the same as it would be if we supposed each sphere condensed to a point at its centre (ib. bk. iii. prop. viii.). Up to this time it had only been possible for him to suppose as Hooke had stated, that the theorems he had discovered as to motion were approximately true for celestial bodies, inasmuch as the distance between any two such bodies is so great, compared with their dimensions, that they may be treated as points.
But now these propositions were no longer merely approximate, save for the slight correction introduced into the simple theory by the fact that the bodies of the solar system are not accurately spherical. The explanation of the system of the universe on mechanical principles lay open to Newton, and in about a year from this time it was published to the world.
In the opinion of Professor Adams (bicentenary address of Dr. Glaisher) it was the inability to solve, previous to this date, the question of the mutual attraction of two spheres which led Newton to withhold so long his treatise on ‘Motion,’ and his proof that gravity extends to the moon. As soon as he mastered this problem he returned to the calculations respecting gravitation and the moon laid by in 1665, and of course he now used Picard's value for his length of a degree of latitude (Pemberton, A View of Sir Isaac Newton's Philosophy, Preface). The theorem which he had just found gave him the power of applying his analysis to the actual universe, and the problem became one of absorbing interest.
The ‘Principia’ was to consist of three books. The treatise ‘De Motu,’ enlarged in the autumn of 1685, forms the first book; the second book, ‘being short,’ was finished in the summer of 1685, it was written out for press next year (Newton to Halley, 20 June 1686, Rigaud, Essay on the First Publication of the Principia, App. p. 29). The work of preparing his great discovery for publication thus proceeded with amazing speed. To quote again from Dr. Glaisher, ‘the “Principia” was the result of a single continuous effort. Halley's first visit to Cambridge took place in August 1684, and by May 1686 the whole of the work was finished, with the exception of the few propositions relating to the Theory of Comets. It was therefore practically completed within 21 months of the day when Newton's attention was recalled to the subject of central forces by Halley. We know also, from a manuscript in Newton's handwriting in the Portsmouth collection, that, with the exception of the eleven propositions sent to Halley in 1684, the whole was completed within seventeen or eighteen months. The total interval from Halley's first visit to the publication of the book is less than three years.’ The first book of the ‘Principia’ was exhibited at the Royal Society on 28 April 1686 (Birch, Hist. of Roy. Soc. iv. 479): ‘Dr. Vincent presented to the society a manuscript treatise entitled “Philosophiæ Naturalis Principia Mathematica,” and dedicated to the society by Mr. Isaac Newton, wherein he gives a mathematical demonstration of the Copernican hypothesis, and makes out all the phenomena of the celestial motions by the only supposition of a gravitation to the centre of the sun decreasing as the squares of the distances reciprocally. It was ordered that a letter of thanks be written to Mr. Newton, that the printing of his book be referred to the consideration of the council, and that in the meantime the book be put into the hands of Mr. Halley to make a report thereof to the council.’ And on 19 May 1686 it was ordered (ib. iv. 484) that ‘Mr. Newton's “Philosophiæ Naturalis Principia Mathematica” be printed forthwith in quarto in a fair letter; and that a letter be written to him forthwith to signify the Society's resolution, and to desire his opinion as to the print, volume, cuts, &c.’ Halley, who was secretary, wrote on 22 May to Newton that the society ‘resolved to print it at their own charge in a large quarto of a fair letter. … I am intrusted to look after the printing of it, and will take care that it shall be performed as well as possible.’
The minute of 19 May required the ratification of the council, and on 2 June it was ordered ‘that Mr. Newton's book be printed, and that Mr. Halley undertake the business of looking after it and printing it at his own charge, which he engaged to do’ (ib. iv. 486). At the time the society were in difficulties for want of funds (Rigaud, Essay, p. 34), and it appears that the council must have declined to undertake the risk of publication, and have left it to the generosity of Halley to provide for the cost.
But Halley had other difficulties to surmount. In his official letter to Newton of 22 May he felt bound to refer to the conduct of Hooke, who, when the manuscript was presented to the society, claimed to have first discovered the law of inverse squares, and to have communicated it to Newton in the cor-
