Joseph Ehrenfried Hofmann Demonstrates that Gottfried Wilhelm Leibniz Did Not Plagiarize Isaac Newton
I have previously written, and will continue to write, about critiques by Gottfried Wilhelm Leibniz (1646-1716) of Isaac Newton’s (1643-1727) theory of gravity. The previous posts can be found here:
Another confrontation between Leibniz and Newton took place over the priority of the infinitesimal calculus. In this case, circles around Newton accused Leibniz of plagiarism; these accusations were formalized in 1711 by the Royal Society, whose conclusions were written by Newton himself.
But were these accusations valid? The German historian of mathematics, Joseph Ehrenfried Hofmann (1900-1973), addressed this issue head-on in a book published in 1949, entitled Die Entwicklungsgeschichte der Leibnizschen Mathematik während des Aufenthalts in Paris (1672-1676) [The history of the development of Leibniz’s mathematics during his stay in Paris (1672-1676)]. A significantly revised version was published in English in 1974 under the title Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity1. This latter version took advantage of an accessibility to documents in libraries and archives across Europe that did not exist during World War Two, when much of the original version was being written.
This book is anything but light reading, and is certainly not for the faint-hearted. One of the most fascinating parts is Chapter 23, entitled “Chronological Index”. The initial paragraph reads as follows:
This index consists of five parts. The main section (1) gives a chronological register of all letters or original publications in contemporary periodicals that have been adduced in the text or in the footnotes; plain figures denote pages of text, pairs refer to the notes, thus 7:23 signifies note 23 of chapter 7; bracketed figures indicate an indirect or implied reference. The second portion (2) lists hitherto unpublished Hanover manuscripts according to Rivaud’s Catalogue Critique II; here dates printed in italics are those actually found on the documents, other dates had to be inferred and so are to be taken as provisional. Locations for other manuscripts are briefly given in section (3). We add (4) a calendar of relevant meetings of the Royal Society of London and the Académie Royale des Sciences of Paris, followed by a chronological checklist (5) of publications in the periodical literature of the time. [p.307]
This chapter covers a full 36 pages, of which 30 are the first section, i.e., the “chronological register of all letters or original publications in contemporary periodicals that have been adduced in the text or in the footnotes”. This register begins with correspondence in 1638 between René Descartes (1596-1650), Marin Mersenne (1588-1648), Pierre de Fermat (1601-1655) and Gilles Personne de Roberval (1602-1675), all the way to correspondence in 1716 between Leibniz, Newton, Sebastian Kortholt (1675-1750) and Antonio Schinella Conti (1677-1749).
The book focuses on the five years that Leibniz spent in Paris working on mathematics, and moving from a near-neophyte under the guidance of Christiaan Huygens (1629-1695) to the development of the infinitesimal calculus with the introduction of the notation dx/dy to denote a derivative and of the long-s (∫ ) as integration symbol. But to make these five years comprehensible, the technical developments made by various mathematicians, mostly in Italy, France and Britain, are presented, as are the correspondence between these mathematicians, along with their private notes.
With respect to the relationship between Leibniz and the British mathematicians, Hofmann also relates in detail the visits to London made by Leibniz, along with the correspondence between Leibniz and Henry Oldenburg (1619-1677), the permanent secretary of the Royal Society, and John Collins (1625-1683), effectively Oldenburg’s mathematical adviser. As Collins was not at all in the same league as the famed British mathematicians James Gregory (1638-1675), John Wallis (1616-1703) or Newton, the fact that Leibniz was corresponding through the intermediaries of Oldenburg and Collins did not in any way facilitate the communications process. Hofmann writes:
Collins had started as an accountant and arithmetical practitioner and though his understanding of the higher mathematics was never very deep he had come to be ardently interested in all new developments in it, especially at the hands of British mathematicians. Through his close contact with London booksellers he made vigorous efforts to get his friends’ works quickly and adequately published. His contact, in person and by letter, with the most distinguished British scientists may have flattered his ambition no less than the position of trust he occupied as adviser to Oldenburg who had never himself done any serious mathematics and in his foreign mathematical correspondence with Sluse, Leibniz, Tschirnhaus and others had of necessity to rely to a large extent on Collins’ prior drafts. Through an exaggerated national pride, unfortunately, Collins was incapable of an unprejudiced assessment of the achievement and character of foreign scholars; he was heavily biased against Frenchmen altogether and Descartes and the Cartesians in particular, and endeavoured in every way to extol the performance of his countrymen. The narrowness of his scientific horizon, in spite of his sincere wish to advance the truth and the new trends of thought, many times let him down badly. Indeed that more than a generation later there broke out a squabble over priority between Leibniz and Newton must in part be ascribed to Collins’ intervention. [p.30]
Chapter 13 is entitled “The Invention of the Calculus”. Therein, Hofmann presents the evolution in Leibniz’s thought and—most importantly—notation. It becomes clear over time to Leibniz that he is developing general techniques for the uniform presentation of already proven specific theorems. After working through the details of Leibniz’s notes, Hofmann concludes:
These, then, are Leibniz’ famous notes whose composition led him, struggling to attain the simplest and most obvious way of presentation, to invent the Calculus. At first there can be no question on his part of consciously creating something new; it was simply a matter of suitably and formally abbreviating various integral transforms which he had formulated in the reduction of certain inverse-tangent problems, and in particular of eliminating the opaque and long-winded verbal descriptions which barred the way to a general viewpoint. Once this first, crucial step towards the ‘algebraization’ of infinitesimal problems had been taken, a new vision disclosed itself to a man experienced in identifying general, characteristic elements in a medley of similar things. [p.194, my emphasis]
So for Hofmann, the invention of the calculus, as understood by Leibniz, consists in the “ ‘algebraization’ of infinitesimal problems”. Hofmann continues:
Much was still wanting which only a later age contributed, as for instance, any distinction between the definite and the indefinite integral. And much needed to be done, such as superseding Sluse’s rule, unsuitable as a formalism, however useful as a calculating device. But the creator of this new and ingenious tool was a man blessed with an abundant store of viable ideas and the will to give them shape and form. He had a clear sense of much of what was still lacking in his calculus, but he knew that its defects could be cured, and that the way into new country was now open. [pp.194-195, my emphasis]
In Chapter 20, “The Second Visit to London”, Hofmann relates that Collins allowed Leibniz access to notes written by Newton and Gregory themselves, some of which Leibniz excerpted, i.e., partially copied. Hofmann writes:
On the other hand through Collins Leibniz gained sight of certain papers by Newton and Gregory. An accurate understanding of his excerpts from these is of crucial significance in coming to a correct appreciation of the whole situation. [p.278]
And here is the crucial point that Hofmann makes: Leibniz did not excerpt sections pertaining to infinitesimals.
Leibniz’ excerpts are confined exclusively to series-expansions and the general remarks accompanying them: the sections relating to infinitesimals in the De Analysi remain completely disregarded—for the obvious reason that they offered nothing new to Leibniz. [p.278-279, my emphasis]
The final textual chapter, Chapter 21, is the Conclusion. Therein Hofmann writes:
The persistent study of the available literature not only furnishes numerous factual details, but beyond these, shapes that instinctively right attitude towards infinitesimal problems without which a unifying synthesis would have been unthinkable. Leibniz’ preoccupation with questions of algebra increases his algorithmic skill and opens up the way to the Calculus: the decisive feature is his introduction of suitable symbols to be formed on the model of algebraic ones. He now rethinks the basis of his ars inveniendi from a metaphysical, logical and mathematical viewpoint. In the treatment of one particular inverse tangent question the notation ‘omn.’ for (infinite) summation which had hitherto been used is replaced by the ∫ sign, and after a few further groping attempts the symbols ∫ y.dx and dy/dx make their appearance; with their introduction the inverse character of the two operations they symbolize is recognized and the related formal apparatus is gradually constructed. Simple differential equations are changed into relations between integrals and then by a step-by-step integration their solutions are expanded into power series. By means of the newly introduced symbols the whole domain of infinitesimal problems so far explored is dealt with simply and elegantly. [p.296, my emphasis]
The point made in Chapter 20, “The Second Visit to Leibniz”, is reiterated in the conclusion:
The accusation of plagiarism against Leibniz relates not at all to the doctrine of series (where the priority of the English discovery has never been questioned), but solely and entirely to the infinitesimal methods which paved the way to the calculus. That Leibniz here, wholly uninfluenced by others, gained his crucial insights unaided, is beyond all doubt. That he made no excerpts of any of the infinitesimal portions of the papers he inspected only confirms what we already know—that he did not expect new insights into calculus methods from others working in this field. [p.306, my emphasis]
Here is Hofmann’s concluding paragraph:
Leibniz’ aim in making these and many other extracts was quite different. His method so far had been to examine the results of others for their basic insights, to penetrate and assimilate their very essence, and to extract from them the general viewpoints that were methodologically of value to him. Whoever studies his sharply outlined summaries of the content of what he has read, and ponders them will realize that his concern is to bring out its underlying concepts, formal structure and method of presentation. Turned into the mathematical: that his intention is not to reproduce the full text of the original in its chronological sequence of composition, but to delineate in its full purity the characteristic idea which is there described. In this sense Leibniz, enthusiastically and unreservedly gathering and absorbing all the knowledge in any way accessible to him and then forming it in a grand new synthesis into a unified whole, is the first true historian of science. From him has been borrowed the method used in this present study: namely, of reporting all essential details as faithfully as possible and connecting the links which clarify the origin and growth of central ideas, their structure, their effectiveness and their final aim—a laborious task since it is not sufficient to confine one’s treatment of a topic to sketching its principal developments in broad outline, but also a rewarding one for anybody who is prepared to devote adequate attention to the smaller details and to elaborate their meaning and significance in the total pattern. [pp.306-307, my emphasis]
For Hoffman, Leibniz’s interactions with other mathematicians, including the British ones, were not to steal their ideas, but to help clarify his own.
For me, it is interesting that the formal accusation of plagiarism was only made in 1711, long after Oldenburg and Collins had died. By that time, Newton had become a powerful political figure, being both Master of the Mint and President of the Royal Society. In the meantime, Leibniz had published in 1710 the Théodicée [Theodicy], his main philosophical work, which rejects the empiricist, tabula rasa, ideas of John Locke (1632-1704), one of the founders of liberalism. I do not think that the attack against Leibniz can be separated from the political context of the time.
The discussion of the development of infinite series, in which the British mathematicians did play a leading part, will be dealt with in a separate post.
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Joseph E. Hofmann. Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity. Cambridge University Press, 1974. Revised translation of Die Entwicklungsgeschichte der Leibnizschen Mathematik während des Aufenthalts in Paris (1672-1676). Munich: R. Oldenbourg Verlag, 1949.
These are fascinating questions. Thanks for looking into them.