he had found by theory for the motion of the moon's apogee, is omitted in the new scholium.

It is interesting to find among the papers on the Lunar Theory a good many containing Newton's calculations relating to the inequalities which are described in the above scholium. These papers are unfortunately very imperfect, and they have greatly suffered from fire and damp, but there is enough remaining to give a general idea of Newton's mode of proceeding. The most interesting of these papers relate to the motion of the moon's apogee. Two lemmas are first established which give the motion of the apogee in an elliptic orbit of very small eccentricity due to given small disturbing forces acting, (1) in the direction of the radius vector, and (2) in the direction perpendicular to it.

These lemmas are carefully written out, as if in preparation for the press, and they were probably at first intended to form part of the Principia.

Next follows the application of the lemmas to the particular case of the Moon, in which the supposition that the disturbances are represented by changes in the elements of a purely elliptic orbit of small eccentricity would lead to practical inconvenience, and consequently Newton is led to modify that supposition. In the Principia he shows that if the moon's orbit be supposed to have no independent eccentricity, its form will be approximately an oval with the earth in the centre, the smaller axis being in the line of syzygies and the larger in that of quadratures, the ratio of these axes being nearly that of 69 to 70. Now when the proper eccentricity of the orbit is taken into account, supposing that eccentricity to be small, Newton assumes that the form of the orbit in which the moon really moves will be related to the form of the oval orbit before mentioned, nearly as an elliptic orbit of small eccentricity with the earth in its focus is related to a circular orbit about the earth in the centre. He then attempts to deduce the horary motion of the moon's apogee for any given position of the apogee with respect to the sun, and his conclusion is that if C denote the cosine of double the angle of elongation of the sun from the moon's apogee, then the mean hourly motion of the

inoon's apogee when in that position is to the mean hourly motion of the moon as

1+10 238.

The investigation on this point is not entirely satisfactory, and from the alterations made in the MS. Newton evidently felt doubts about the correctness of the coefficient 11 which occurs in this formula.

From this, however, he deduces quite correctly that the mean annual motion of the apogee resulting would amount to 38° 51′ 51′′, whereas the annual motion given in the Astronomical Tables is 40° 41'.

The result stated in the scholium to the 1st Edition appears to have been found by a more complete and probably a much more complicated investigation than that contained in the extant MSS.

The papers also contain a long list of propositions in the Lunar Theory which were evidently intended to be inserted in a second edition, upon which Newton appears to have been engaged in 1694. This list, together with the two lemmas on the motion of the apogee mentioned above, will be found in the Appendix.

Halley inserted in the Philosophical Transactions of 1721 a Table of Refractions by Newton, without giving any idea of the method of its formation.

Kramp, in his Analyse des Réfractions, published in 1799, investigates by a new and powerful analytical method the law of atmospheric refraction for rays in the neighbourhood of the horizon.

On comparing his theoretical results with Newton's Table, he finds a remarkably close agreement, which is enough to show that the Table was also the result of theory, and therefore that Newton must have had some method of his own of solving the difficult problem of horizontal refraction.

Nothing was known of this method, however, until the publication of the correspondence between Newton and Flamsteed by Mr Baily in 1835. In a letter to Flamsteed, dated December 20th, 1694', Newton tries to explain the foundation of

1 Baily's Flamsteed p. 145.

his theory of refraction by giving a theorem from which it is clear that Newton then understood how to form the differential equation to the path of a ray of light through our atmosphere. It is true that, for the sake of greater simplicity in this communication to Flamsteed, Newton restricts the enunciation of his theorem to the particular case where the density decreases uniformly as the height increases, but it is obvious from the form of the enunciation of Newton's theorem that the method is general, provided that the differential of the density which is appropriate to any given law of diminution be employed in finding the corresponding differential of the refraction. In an interesting article in the Journal des Savants for 1836, M. Biot directs particular attention to this subject, and tries to reproduce the method which Newton may be supposed to have employed in order to calculate his table of refractions. closes his article in the following terms:

M. Biot

"Il est donc prouvé, par ce qui précède, que Newton a formé l'équation différentielle exacte de la réfraction pour les atmosphères de composition uniforme; qu'il l'a appliquée exactement au cas où les densités des couches sont proportionelles aux pressions, ce qui rend leur température constante; et qu'enfin, pour ce cas, il a obtenu les vraies valeurs des réfractions à toute distance du zénith, sans avoir eu besoin d'employer les intégrations analytiques qu'il a dû très-vraisemblablement ignorer. Il est donc le créateur de cette théorie importante de l'astronomie physique, qui serait probablement aujourd'hui plus perfectionée, si l'on avait connu plus tôt ses premiers efforts."

Judging from Newton's account of the time which he employed in making these calculations, there must have been a considerable mass of papers devoted to them which have not been preserved. Fortunately, however, among the Portsmouth papers we find a detailed calculation of the refraction corresponding to the altitudes 0o, 3o, 12° and 30°. In order to make this calculation the path of a ray of light through the atmosphere is divided into a number of parts subtending given small angles at the centre of the earth. Hence are found by the fluxional method quantities which are proportional to the refractions suffered by the ray in passing over the successive

portions of the path, and from these the actual refractions in passing over these portions are derived by making the total. horizontal refraction equal to the amount given by observation. It should be remarked that the above calculation requires an approximate knowledge of the path of the ray, whereas this path is at first unknown, and cannot be accurately determined without a knowledge of the refraction itself. Newton solves the difficulty by an indirect method, making repeated approximations to the form of the path, and thus at length succeeding in satisfying all the required conditions.

The papers show that the well-known approximate formula for refraction commonly known as Bradley's was really due to Newton. This formula is only applicable when the object is not very near to the horizon, but the method of calculation employed by Newton is equally valid whatever be the apparent zenith distance.

It is well known that in the Principia Newton determines the form of the solid of least resistance, thus affording the first example of a class of problems which we now solve by means of the Calculus of Variations. He there gives what is equivalent to the differential equation to the curve by the revolution of which the above-named solid is generated, without explaining the method by which he has obtained it. Now among the Newton papers we have found the draft of a letter to a correspondent at Oxford, no doubt Professor David Gregory, in which Newton gives a clear explanation of his method, which is very simple and ingenious. The draft has no date, but from internal evidence it was probably written about 1694. A small part of the letter has perished but it is very easy to restore the missing portion. The letter will be found in the Appendix at the end of this preface. It may be remarked that a similar method is immediately applicable to the problem of finding the line of quickest descent.

A great many of the Newton papers relate to the dispute with Leibnitz about the discovery of Fluxions or the Differential Calculus. They show that Newton's feelings were greatly excited on this subject, and that he considered that

N.

b

Leibnitz had shown towards him in reference to it great unfairness and want of candour. Newton always maintained that Leibnitz was the aggressor in this dispute, and that he had, by his language in the Leipsic Acts, covertly accused him of plagiarism, whereas he might have known from the correspondence that formerly took place between them, that Newton's method was in his possession long before he himself became acquainted with the Differential Calculus.

On the other hand Leibnitz, without avowing himself the author of the article in the Leipsic Acts, denied that it really bore the meaning attributed to it by Newton, and maintained that Newton had either been deceived by a false friend into imagining that he had been accused of plagiarism, or else that he was not sorry to find a pretext for attributing to himself the invention of the new Calculus, contrary to the avowal he had made in the Scholium in the 1st Edition of the Principia.

From a paper by Leibnitz, which has been published by Dr Gerhardt, it appears that the article in the Leipsic Acts, of which Newton complained, was really written by Leibnitz, and it also seems probable that the ambiguity of its language was not unintentional. We cannot wonder, then, that Newton, firmly believing that Leibnitz had charged him with plagiarism, should have experienced a strong feeling of resentment, and should have been induced to retort the charge upon his accuser1. It was not unnatural that this embittered feeling should still survive even after the death of Leibnitz.

It is clear from these Portsmouth papers that Newton believed that Leibnitz, during his second visit to England in October 1676, had obtained access to his MS. entitled De Analysi per Equationes numero terminorum infinitas, which was in the hands of Collins, and that he had thus been materially assisted in discovering the Differential Calculus. This tract of Newton's is printed in full in the Commercium Epistolicum, and is there used merely in order to prove Newton's priority to Leibnitz. It is nowhere asserted or even implied in the Commercium that this tract of Newton had ever

1 In connection with this Newton makes the following quotation from Ovid: "Nec lex est justior illâ, etc." (Artis Amatoriæ, 1. 656.)