The notes and criticisms that I have made in these six chapters on the manuscripts of Leibniz may give the impression that I am an anti-Leibnizian. This is quite wrong. My prime object was to show, to the best of my power, that the charges of plagiarism brought against Leibniz by partisans of Newton, and indeed by Newton himself in the Recensio published in the Philosophical Transactions, were unfounded. I considered that the charges in the Recensio were perhaps the hardest to be answered, since they were not only direct charges, backed with circumstantial evidence, but they were also set forth very cleverly. Also I thought that the method of defense adopted by Gerhardt and other partisans of Leibniz did as much harm to him as the strongest attack of avowed opponents, such as Sloman. The weak case made out by Gerhardt is deplorable. Never surely did any man have such a glorious opportunity as Gerhardt, in the whole history of scientific controversies; surely there never was an advocate who left himself so open to the attacks of the opponents. Gerhardt starts with the theory that every single word of Leibniz represents gospel truth; and that it is almost blasphemy to doubt it; in consequence he is soon in difficulties when it comes to reconciling the varying statements of the sequences of events that are made by Leibniz at different times. But, once the idea is accepted that Leibniz, while perfectly reliable on the general run of events, is unreliable when it comes to un important details, and then all difficulty disappears. I therefore set out with the determination to break down, if I could, the credibility of Leibniz as a witness in his own defense, when it came to unimportant details; then to show that he had opportunities for obtaining everything necessary to the development of the Calculus, that he could not be expected to supply for himself by original work, without having need to know anything of the work of Newton; then to show that these sources of information were set out in a form far more suitable to the requirements of Leibniz than the work of Newton; finally, to clinch the matter, that the analogy of Leibniz’s work was so close to these sources, that it was idle to suppose that he made use of any other sources. In other words, (i) the Analysis per aequationes was unnecessary to Leibniz, (ii) Newton’s method of evading fractions and roots by means of infinite series was clever, but futile for the needs of Leibniz when developing an operational calculus.
The unreliability of Leibniz with regard to details may be in some measure due to his apparently bad memory (which is suggested by his habit of committing everything to writing), and to passage of time. But in a far greater degree it must be ascribed to the circumstances and characteristics of Leibniz. We know that he designed to compile an encyclopedia of all science, and for this he considered not at all the nationality or the personality of the discoverer or the author: all he was interested in were the facts or principles discovered.
That he was unreliable with regard to details is proved by the facts I have adduced:
i. the confusion between Mouton and Mercator in the account of the assertion that he had been anticipated (see above, Chapter III, p. 36, and Note 73, p. 37) ;
ii. the varied assortment of figures that he gives to illustrate how he found the Characteristic Triangle (see above, Chapter III, pp. 15 and 39, and compare them with the figures, given in the accounts quoted by Gerhardt in his essay “Leibniz and Pascal,” on pp. 211, 217, and 221) ;
iii. the circumstantial detail of the context of the Archimedean measurement of the surface of the sphere being absent from the author he quotes;
iv. the several different accounts of the order in which he obtained his different books for study, and even the persons from whom he obtained them;
v. the error with regard to the time of the presentation of the copy of the Horologium (see above, Chapter III, p. 36, where, in the Historia, it is stated that he received it before he left for England on his first visit) ;
vi. the confusion as to the date at which he obtained his Barrow (see above, Chapter II, p. 20, where, in the Bernoulli postscript, he states that he found the greater part of his theorems anticipated in “Barrow, when his Lectures appeared”) ; and many other things, all unimportant details singly; but, when taken in combination, they show distinctly that we must only take Leibniz’s word as accurately describing the general course of events.
Another characteristic of Leibniz seems to have been insistent at all times; he burned to distinguish himself as a discoverer of new things. I have suggested that there may have been an ulterior motive to this desire, namely, to get himself taken into the select circle of mathematicians who corresponded with one another. Thus, when he studied an author, and came across some new idea, he would break off his reading to follow that idea to the limit and exhaust all its possibilities, committing his results to writing, whether they were important or not; there is some evidence, too, that while doing this, he would refer to other authors who had discussed the point under consideration, before returning to his reading.
My motive in trying to show that he got everything from Barrow, except his methods, was to remove any charge of plagiarism; for, I consider that even if he had merely rewritten Barrow in terms of Descartes, adding his own notation for the sake of convenience, he would still have done a great thing, and would no more have been guilty of plagiarism from either Descartes or Barrow than Stephenson was from Watt, or Parsons from either of these. Leibniz’s Calculus was his own, and would have been his own even on the supposition above. Lastly, it was not only more complete than that of Newton, in that it was an operational calculus, though it did perhaps miss the idea of rate; but also from an intellectual standpoint it was greater, in that it was developed, after its first principles were found out, as a practical theory, while Newton’s was developed as a mere instrument for his own purposes.
Assuming, then, that Leibniz did not remember, or did not really care, what his text-books were, so long as he was not accused of using somebody else’s methods, I will try and reconstruct the progress of his reading and his discoveries. His text-books were,
i. Lanzius and Clavius in algebra, and Leotaud for geometry, in his early youth; he also looked through, more or less.without understanding them, Descartes and Cavalieri’s Geometria Indivisibilibus.
ii. On his return from London he brought back with him Barrow, some portions of which he had glanced at in London and on his journey; he obtained Pascal, St. Vincent, and Cavalieri’s Exercitationes Sex, perhaps a little later than the others; besides these, Wallis and Mercator specially.
He read portions of the Barrow afresh, and obtained the Characteristic Triangle, and found his general theorem from this; meanwhile he is also studying Descartes, and we have the materials for the manuscript of August, 1673. Probably he has had a look through Pascal during this time. He remembers the similarity between the complicated diagrams of Barrow and some of those of Pascal, and starts studying the Traitté des Sinus, in which he finds the second variant of the differential triangle that appears in the manuscript of October, 1674. Previous to this, however, his attention has been arrested by Barrow’s proof of the inverse nature of the operations of finding a tangent and an area, and the analogy between this and sums and differences strikes him. He has also considered the examples on the differential triangle given by Barrow; one of them suggests the method of Mercator to him, he has already got an idea from Wallis of the summation of the several powers of the variable; he applies this to Barrow’s expression, equivalent to
d(tan-1x)/dx = 1/(1 + x2),
in modern notation, performs the division as Mercator had done, and obtains the series for the inverse-tangent by a summation according to Wallis, i. e., practically an integration. This answers the charge made by Newton that somehow or other he got this series from him or James Gregory. In the same way, he thought that he could obtain other series, but later found that it was beyond his power. We find in this manuscript of October, 1674, an attempt to get something out of an analogous series, the logarithmic series, showing that it is very probable that he has been studying Mercator during the interval between August, 1673, and October, 1674. And in the Historia he definitely states that he came upon the Arithmetical Tetragonism in 1674; so that I think that I have offered a reasonable suggestion as to the course his studies took so far. Also in the meanwhile he has been doing much work on series, and has invented his Harmonic Triangle. I now suppose that he completes his study of Pascal, is led by a remark in it to study the Exercitationes Sex of Cavalieri (he has already got some acquaintance with the Geometria Indivisibilibus, read as a youth), he does not find much in that to his liking, except the notion of moments. He breaks off his reading and proceeds to work out an application of Descartes’s algebra to this new idea of moments, the result being the manuscripts of October and November, 1675, here he is led on to the introduction of the symbols for summation and differentiation, though as yet applied to series, and sums of powers. The consideration of the Quadratrix, leads him to make a further study of Barrow; and he is led to x/d, by a consideration of Barrow’s propositions on the inverse nature of the operations of integration and differentiation. This, combined with the analogy to the inverse nature of summations and differences, leads him to search for a reason why x/d should represent a difference such as he has considered to be denoted by dx. This at a later date necessitates the discussion of what the result of operating with d on a product or a quotient will be. Meanwhile the study of Barrow brings him to that proposition which gives the polar differential triangle; in it he perceives at once the method of “transmutation of figures.” I now suppose that he appreciates Barrow more fully and begins to apply Cartesian geometry to Barrow’s theorems; in a manuscript dated November, 1675, he attacked the problem of tangents, and in connection with it considered the method of Descartes. In the next manuscript that we have, dated June, 1676, he practically obtained the differentiation of the sine and the inverse sine; his figure, if he had given one, would have been the same as that of Barrow for the differentiation of the tangent. In July, 1676, he attacked the inverse-tangent problem, still considering the work of Descartes. I think, however, that his work on Barrow has taken effect, for from now on he includes the differential factor dx under the integral sign. This is the last manuscript before he went to London for the second time.
Thus, I take it that all Leibniz’s work is the result of his own original methods on ideas that have been suggested chiefly by two books, those of Barrow and Descartes; at least, everything could have been suggested by these two books alone, except the notion of “moment,” which came from Cavalieri. Thus it was unnecessary for him to have known anything about the work of Newton before he went to London for the second time. What he saw there may have had the effect of corroborating his own work; it could have had little other effect. The final polishing of his method I put down to a study of the Differential Triangle method of Barrow, which Leibniz perceived to be powerful, but found distasteful on account of the geometrical nature of the work.