co-ordinates from the result. The Committee omit this example, without any notice of omission, though they give the passages

between which it lies.

We are obliged frequently to recur to the assertion of the Committee that Newton's example, which we have translated, was description enough of the method of fluxions for any intelligent person. That this, which we shall believe to be the most reckless assertion ever made on a mathematical subject, until some one produces its match, was solemnly put forward by the Committee, is not in our day excuse enough for dwelling upon it. But the sufficient excuse is that writers of note, upon the Newtonian side of the question, still quote the assertion with approbation. In Sir David Brewster's life of Newton, for instance, the whole report of the Committee is printed, and a virtual adhesion given to it. On the other hand, the defenders of Leibnitz, most of whom are not English, prefer to establish his rights independently, and evade an encounter which is rendered repulsive by its dealing more with the comparison of old letters than with mathematical explanations.

Some little question has arisen as to the position in which the Royal Society stands in this matter. According to Leibnitz, Chamberlayne wrote to him to the effect that the Royal Society did not wish the report to pass for a decision of its own. Mr. Weld (Phil. Mag. 1847; Hist. R. S. vol. i. p. 415) found the minute in question (passed May 20, 1714), in which it is stated that" if any person had any material objection against the Commercium, or the Report of the Committee, it might be reconsidered at any time." This Mr. Weld considers as an adoption of the Report of the Committee: in which we cannot join, though we admit that it throws the question open, which as long as Chamberlayne's communication stood unanswered, was settled: and enables us to infer adoption from previous acts. In all probability he informed Leibnitz that the report of the Committee was not to pass for a decision, meaning the stress to lie there, and stating why: and this would be correct, for a question which may be reconsidered at any time is not decided, except in a technical sense. And very likely he added "of the Society:" for it was the full impression of the time that the Society was one with its Committee. There can be no doubt that the hearty adherence given by the Society to the conclusions, the circulation of the Comm. Epist. throughout Europe, the admission of Keill's recensio into the Transactions, the sanction of the reprint ten years after, and the obstinate determination, which lasts down to our own time, not to confess one atom of the error nor right one atom of the wrong, amount to an adoption which could not be more than adequately represented by any quantity of minutes.

It seems the fate of this controversy that whatever the English partisans of the eighteenth century supposed to have happened between the two parties really happened the other way, the places of the parties being changed, and to no effect upon the question. Much stress was laid on Collins transmitting from Newton to Leibnitz an example of the method of tangents: it appears that the example was not sent, that the abridgment sent did not contain it;

but it appears that Collins really forwarded a result from Leibnitz to Newton, which was the only one that passed between them. Not that this gave Newton any information; but neither would Newton's example, if sent, have given any to Leibnitz, after Sluse's publication and Hudde's oral communications.

Again, it was frequently stated that the differential calculus was only the method of fluxions with the notation changed. Now the fact is, that as to every thing elementary that was published with demonstration under the name of fluxions, up to the year 1704 (when Newton himself first published anything under that name) the method of fluxions was nothing but the differential calculus with the notation changed. We know that Newton's letters did not treat of fluxions, nor contain anything from which the writer of a system could draw his materials. No one ventured to print an elementary treatise in England until the seed had grown into a strong plant under the care of Leibnitz, the Bernoullis, &c. When De L'Hopital, in 1696, published at Paris a treatise so systematic, and so much resembling one of modern times, that it might be used even now, he could find nothing English to quote, except a slight treatise of Craig on quadratures, published in 1693. He mentions all that he could of Newton, and even says of the Principia that it was full of the calculus, which is not true; he should have said it was full of the principles on which the calculus is founded, and of application of them in which the reader (whatever might have been the case with the author) is directed by thought without calculus. But the distinction is one which was not then appreciated: in fact it needed the calculus, such as it became, to show it. It must be remembered that when De L'Hopital wrote (for he could not then have seen the first volume of Wallis), there neither was, nor had been, one word of accusation or of national reflection, to create any bias for or against any one. The first thing of this kind took place in 1695, when Wallis, in the preface to the first volume of his collected works, not only claimed the differential calculus as derived from the method of fluxions, but (in ignorance, as he afterwards knew) grounded the claim upon the two celebrated letters of Newton to Oldenburg, of which little notice is taken here, because not even the Committee of the Royal Society venture a mention of them in their report, as any ground of confirmation against Leibnitz.

The note of alarm thus sounded, our countrymen began to write upon fluxions. Some writings are so advanced that they do not define their terms: from these therefore we cannot tell whether means the velocity with which ≈ changes, or an infinitely small increment of x. Such (atleast so we suppose from the enlarged second edition of 1718) was the little tract of Craig, to which De L'Hopital refers, as we have seen: and such were Dr. Cheyne's tract on fluents (1703) and De Moivre's answer to it (1704). Newton himself, in the Principia, was not a fluxionist, but a differentialist. Though imagining quantity

F*"C'est encore une justice dûë au sçavant M. Newton, et que M. Leibnis luy a renduë luy-même: Qu'il avoit aussi trouvé quelque chose de semblable au calcul différentiel, comme il paroit par l'excellent Livre intitulé...... Principia..... lequel est presque tout de ce calcul."—Preface.

generated by motion or flux (in the celebrated Lemma in which he gives a brief description) he calculates, not by velocities but by moments, or "momentaneous increments and decrements," which are infinitely small quantities, for "moments, so soon as they become finite magnitudes, cease to be moments." Of Wallis we shall presently speak. De Moivre (Phil. Trans. 1695, No. 216) represents fluxions as momentaneous increments or decrements. And the only elementary writers, Harris* and Hayes,† are strictly writers on the differential calculus, as opposed to fluxions, in every thing but using instead of dz. Harris says, "By the Doctrine of Fluxions we are to understand the Arithmetick of the Infinitely small Increments or Decrements. These he says Newton properly calls fluxions; and he proceeds to show that his own ideas are not very clear, by asserting that ""Tis_much more natural to conceive the Infinitely small Increments or Decrements of the variable and Flowing Quantities, under the notion of Fluxions (that is, according to him, of infinitely small increments or decrements) than under that of Moments or Infinitely small Differences, as Leibnitz chose rather to take them." And then he proceeds to speak of velocities: in fact he jumbles De L'Hopital, whom he did understand, with Wallis, whom he did not. Hayes, a much clearer writer, begins thus: "Magnitude is divisible in infinitum the infinitely little Increment or Decrement is called the Fluxion of that Magnitude Now those infinitely little Parts being extended, are again infinitely Divisible; and these infinitely little Parts of an infinitely little Part of a given Quantity, are by Geometers called Infinitesima Infinitesimarum or Fluxions of Fluxions." And again (p. 5), ..suppose half the infinitely little increment of X to be *, and half the Fluxion or infinitely little Increment of Z to be

1

2

And thus it appears that all explanation that was tendered in print, up to the year 1704, whether by Newton himself, or by any of his followers (except only Wallis as presently mentioned), was Leibnitian in principle. But when Newton, in 1704, published the treatise on the Quadrature of Curves which he had written before Leibnitz communicated the differential calculus to him, he starts with nothing but the notion of quantity increasing or diminishing with velocity, and this velocity or celerity is the fluxion. And in the Introduction, written at the time of publication, he says, “I do not consider mathematical quantities as consisting of the smallest possible parts (partes quam minima) but as described by continuous

.....

*The first elementary work on fluxions in England is a tract of twenty-two pages in "A New short treatise of Algebra Together with a Specimen of the Nature and Algorithm of Fluxions." By John Harris, M.A. London, 1702, octavo (small). "A Treatise of Fluxions; or, an Introduction to Mathematical Philosophy. Containing A full Explication of that Method by which the Most Celebrated Geometers of the present Age have made such vast Advances in Mechanical Philosophy. A Work very Useful for those that would know how to apply Mathematicks to Nature. By Charles Hayes, Gent." London, folio, 1704. This work, which has had very little notice (Hayes, born 1678, died 1760, wrote many works, but never set his name to any but this), is a very full treatise, nearly three times as large as that of De L'Hopital, having 315 closely printed folio pages on fluxions, besides an introduction on conic sections.

motion. This is the first public declaration of the meaning of a fluxion that was made by the author of the word, in his own name.

It may appear strange that we defer till now to mention a very fluxional view of fluxions which appeared as early as 1693. But we wish to give prominence to what is really Newton's first publication on the subject, though it has received but little notice until lately. The second volume of Wallis's works, containing the Algebra, in which the matter spoken of occurs, was published in 1693, the first in 1695, but false title-pages* make them appear as of 1699. Again, those who look at the preface to the first volume see that Wallis excuses himself from mentioning the differential calculus, because it was nothing but the fluxions which Newton, he says, had communicated to Leibnitz in the celebrated Oldenburg letters, and which he (Wallis) had described, from those letters, nearly word for word, in his Algebra. No one of later times would thereupon refer to this Algebra for information; since they would know that nothing upon fluxions could be given word for word, but only letter for letter. For all that is said upon fluxions, in those celebrated epistles, is, as is well known, in two anagrams, one of which is

6 accdæ 13 eff7i319n4o4qrr4s9 t 12 v x, the information given being that whoever can form a certain sentence properly out of six as, two cs, a d, &c., will see as much as one sentence can show about Newton's mode of proceeding. No one but Raphsont imagined that any human being derived any information from this; and probably therefore few would be induced by Wallis's preface to consult the work. They would not know (and we shall see that Wallis himself could hardly have anything to make him remember) that Wallis had been in communication with Newton, had obtained not only the key of the anagrams but their meaning, and had added a brief account of fluxions, with an extract from what Newton afterwards published in the treatise of 1704, besides other matter expressly obtained from Newton in explanation of the second anagram. The reader cannot detect the new information, except in that additional part which explains the second anagram: all that can be said of the rest is, that to a reader who compares chapters 91 and 95 there are a couple of sentences which would perhaps puzzle a person who did not know that a new

*The Comm. Epist. says that two volumes appeared in 1695; probably the second volume got a new title-page in that year. The third volume was published in 1699, and then the first volume certainly got a title-page of that date. This vile practice of altering title-pages will be put down by the scorn of all honest men, so soon as its tendencies are seen. A person who reads Wallis's collected works under the date of 1699 easily convicts the author, as honest a man as ever lived, of the grossest unfairness, upon his own testimony.

+ The sentence was Data Aequatione quotcunque, fluentes quantitates involvente, fluxiones invenire, et vice versa, given any equation involving fluent quantities, to find the fluxions, and vice versa. Many writers have called this a cipher, which it is not: a cipher gives, in some way, the order of the letters as well as substitutes for the letters themselves. Raphson declared that Leibnitz had first deciphered the anagram, and then detected the meaning of the word fluxion, after which he forged a resemblance. But Raphson was the unscrupulous man of the time, if any one could deserve that name. Newton stated distinctly that Leibnitz sent him the details of a Method which was his own in all respects except language. Raphson says (Hist. of Fluxions, p. 1) that Leib. nitz "writ in answer that he had found out a Method not unlike it, as Sir Isaac himself has hinted, page 253, Princip....." The impudence of this paraphrase is one of the minor gems of the controversy: and we could rub it brighter if we had room.

The re

source of information was referred to in these sentences. viewer of Wallis in the Acta Eruditorum, in complaining of the suppression of the differential calculus, hit the real reason, namely, Wallis's ignorance of a good deal of what had been done abroad: and Wallis, who wrote to Leibnitz the day after he saw this review, acknowledges that he knew nothing of what Leibnitz had written, except two slight and old papers, and had never heard the name of the differential* calculus until the preface was in the press, when a friend mentioned with indignation that Newton's fluxions were current in Belgium under that name. Then, and probably without consulting what he had written, Wallis added the sentence we have mentioned to his preface. In the third volume, Wallis printed all his correspondence with Leibnitz, and all the correspondence with others on the subject which he could collect, and mentions fluxions and the differential calculus as two distinct things in the preface. What we have here to do with, however, is the nature of the publication of fluxions which was made in 1693.

We now come to the independent proofs of the separate invention of Leibnitz, as contained in his recently published papers. Preliminarily, however, to these, we may notice one which was published in 1671, and which shows the way in which the current of his ideas was setting. Dr. Hales, in his Analysis Fluxionum (Lond. 1800, 4to.), says that Leibnitz had given no obscure germs of his differential method in his Theoria Notionum Abstractarum, dedicated to the French Academy in 1671: and Dr. Hutton (Math. Dict. Art. Fluxions) refers to this theory of abstract notions. Both are wrong in the name; for the paper which Leibnitz dedicated to the Academy in that year is Theoria Motus Abstracti (Op. Leibn. vol. ii. part ii. p. 35). This paper is certainly a witness to character; throughout it there occurs a frequent approximation to the idea of infinitely small quantities having ratio to each other, but not to finite quantities. One extract (translated) will serve as a specimen : "A point is not that which has no parts, nor of which part is not considered; but which has no extension, or whose parts are indistant, whose magnitude is inconsiderable, inassignable, less than any which has ratio (except an infinitely small one) to a sensible quantity, less than can be given: and this is the foundation of Cavalieri's method, by which its truth is evidently demonstrated, namely, to suppose certain rudiments, so to speak, or beginnings of lines and figures, less than any assignable." So that, in 1671, it was working in Leibnitz's mind that in the doctrine of infinitely small quantities lay the true foundation of that approach to the differential calculus which Cavalieri presented.

Dr. Gerhardt, the editor of the correspondence already referred to, found among the papers of Leibnitz preserved in the Royal Library at Hanover various original draughts, containing problems in

* Nevertheless, Leibnitz and the differential method are mentioned in the second volume, that is, in the account of fluxions on which we are writing; but (as discovered by Professor Rigaud) Wallis's copy preserved in the Savilian library has manuscript additions which note and explain this forgetfulness. It appears that the whole communication is Newton's, and is inserted in Newton's words: an author can hardly remember another person's writing, to which he gives admission, as if it were his own.