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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:
"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
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.)
been seen by Leibnitz. There can now be no doubt, however, that Newton was right in thinking that Leibnitz had been shown this MS., since a copy of part of it, in Leibnitz's hand, has been found among the papers of Leibnitz preserved in the Royal Library at Hanover'. It is, of course, possible that at the time when this copy was taken Leibnitz was already acquainted in some degree with the Differential Calculus, but it is difficult to acquit him of a want of candour in never avowing in the course of the long controversy respecting the discovery of Fluxions, that he had not only seen this tract of Newton's, but had actually taken a copy of part of it. He must have seen, also, at the same time, that the MS. was an old one, and although it does not contain the pointed letters which Newton sometimes but by no means invariably employed to denote Fluxions, Leibnitz could hardly fail to see, if he was acquainted with the Differential Calculus, that the principle of Newton's method was the same as that of his own. It is repeatedly stated by Newton that what he claims is the first invention of the method, and that he does not dispute about the particular signs and symbols in which the method may be expressed. Again, he often states that although, in the sense which he employs, the method can have but one inventor, yet the method may be improved, and the improvements belong to those who make them.
In some of these papers relating to the dispute with Leibnitz, Newton gives us some interesting information respecting the times when several of his discoveries were made. Thus in a passage, which has been quoted by Brewster2, he states that he wrote the Principia in seventeen or eighteen months, beginning in the end of December 1684, and sending it to the Royal Society in May 1686, excepting that about ten or twelve of the propositions were composed before, viz. the 1st and 11th in December 1679, the 6th, 7th, 8th, 9th, 10th, 12th, 13th and 17th, Lib. I, and the 1st, 2nd, 3rd and 4th, Lib. II, in June and July 1684. The following extract will give an idea of Newton's prodigious mental activity at an earlier period of his life.
1 See Gerhardt, Mathem. Schriften Leibnitzens, 1. p. 7.
2 Brewster's Life, Vol. 1. p. 471.
"In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions, and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions. And the same year I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere, from Kepler's Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their orbs I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666', for in those days I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since. What Mr Hugens has published since about centrifugal forces I suppose he had before me. At length in the winter between the years 1676 and 1677o I found the Proposition that by a centrifugal force reciprocally as the square of the distance a Planet must revolve in an Ellipsis about the center of the force placed in the lower umbilicus of the Ellipsis and with a radius drawn to that center describe areas proportional to the times. And in the winter between the years 1683 and 16843 this Proposition with the Demonstration was entered in the Register book of the R. Society. And this is the first instance upon record of any Proposition in the higher Geometry found out by the method in dispute. In the year 1689 Mr Leibnitz, endeavouring to rival me, published a Demonstration of the same Proposition upon another supposition, but his Demonstration proved erroneous for want of skill in the method."
The above extract has been given here on account of its intrinsic interest, although in writing it so many years after
1 In 1666 Newton was in the 24th year of his age.