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Overview of Prima calculi magnitudinum elementa

Thanks to the progressive edition of Leibniz’ papers, and in particular of his correspondence, we have now access to the circumstances surrounding the writing of the Prima calculi magnitudinum elementa. The occasion leading to this text was the arrival in Helmsdted of the theologian Johann Andreas Schmidt in 1695. Schmidt was asked to take in charge the teaching of mathematics and, since he was not a specialist of this topic, he turned to Leibniz for help (Schmidt to Leibniz, 31 august/10 September 1697; A I, 14, 467). Leibniz worked on this project in the following year and sent to Schmidt at the end of 1698 the beginning of a treatise on Mathesis Universalis (A I, 16, 295; 341 and 393; see [Leibniz 2018, p. 113-120] for a French translation and a presentation of this draft, which was edited by Gerhardt in GM VII, 53-76). Schmidt was very pleased with the result and urged Leibniz to achieve his project (A I, 16, 393 and 607-608). It is in this precise context that Leibniz announces at the beginning of 1699 a continuation of his drafts containing the proofs for the foundation of the calculus: “I will give a work where I supplement in some way what I began to write at some point as an Introduction to Mathesis Universalis; and I have already redacted a few things pursuing the project; in particular, the proofs of the foundations of the calculus, which seem to me of primary necessity in order to give to science its complete solidity” (To Schmidt, 3(13) march 1699; A I, 16, 633).

Leibniz sent some drafts on this topic to his secretary R. C. Wagner (who Schmidt introduced to him) in April 1699 and told him that Schmidt was willing to make them transcribed by a famulus, as he already did for the first manuscript on Mathesis Universalis (A III, 8, 96). The very same day, Wagner went to Schmidt with these “pages about Mathesis universalis and in them the proofs of the calculus of magnitudes” (A I, 16, 730-731). Thanks to a subsequent letter, we can identify these documents as being the Prima calculi magnitudinum elementa (GM VII, 77-82). Indeed, in a letter dated 14/24 April 1699 (A III, 8, 103), Wagner mentions to Leibniz that he should delete a redundant expression in the second page of the draft (“In plagula 2da sub initium (ni) (24) delenda erunt verba si subtractio vel signum – evanescat”) – a sentence which is clearly in the manuscript of the Prima calculi under the said number and that Leibniz forgot to erase after correcting the passage.

This remark holds for the copy of the manuscript (LH XXXV, 4, 13, fol. 1-4, the original draft by Leibniz being in fol. 5 and 6), which was presumably done by Schmidt’s famulus and on which Leibniz indicated several (sometimes substantial) corrections. These corrections induced some inconsistencies in the numbering of sections (and of subsequent internal references to these sections: we indicate the corrected numbers in red in brackets). We also dispose of several other drafts, which clearly belong to the same set of preparatory versions (in particular the Calculus magnitudinum, LH XXXV, 4, 13, fol. 11-12 and the Caput primum. De aequalitate, fol. 9-10). They are also very close to what can be found in the manuscript Mathesis generalis (LH XXXV 1, 9, fol. 9-14) – itself close to “numerus integer…” (LH XXXV 1, 9, fol. 7). The version copied by Schmidt’s famulus with Leibniz’ corrections is what Gerhardt transcribed in his Mathematische Schriften. Awaiting a critical edition of this set of manuscripts, we propose here a translation of Gerhardt’s version.

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