Overview of Qvelques remarques sur les Memoires de Trevoux de l’an 1701 (1702)
The Mémoires pour l’histoire des sciences et des beaux-arts, better known as the Journal de Trévoux or Mémoires de Trévoux, is a journal founded by the Jesuits in 1701 in Trévoux. This bimonthly publication was composed of articles on various literary, scientific and religious topics.
As soon as the first issues of the Journal appeared, Leibniz wished to be kept informed of articles published in it and he obtained them through his friend François Pinsson. His interest was aroused by certain articles and he did not hesitate to send his point of view to the Journal on many occasions.
The present text is an extract from a manuscript of notes and remarks that Leibniz wrote about articles published in the issues of September-October and November-December 1701, in which he also contributed [Leibniz 1701a], [Leibniz 1701b].
The article entitled “Mémoire de Mr Leibnitz touchant son sentiment sur le Calcul différentiel” [Leibniz 1701b] is an answer to an article from the months of May-June 1701 [Gouye 1701]. In it, Father Gouye criticized the Analyse des infinis in which, according to the Preface to the book by L’Hospital [L’Hospital 1696], “things got taken further” [a porté les choses plus loin], “not only embracing the infinite, but the infinity of the infinite, or an infinity of infinities” [n’embrassant pas seulement l’infini, mais l’infini de l’infini ou une infinité d’infinis]. Nevertheless, despite “his admirable fecundity”, Gouye judged that the new analysis lacked “in its demonstrations the evidence that one expects from them” [dans ses démonstrations cette évidence que l’on attend] and that it was therefore preferable to favour other safer methods than to “to embark on the new roads of the infinity of the infinite” [s’engager dans les nouvelles routes de l’infini de l’infini] “where one can easily get lost, without noticing” [où l’on peut aisément s’égarer, sans qu’on s’en apperçoive] [Gouye 1701, 430]. Leibniz’s well-known answer [Leibniz 1701b] consists in explaining that mathematics does not depend on metaphysical considerations, and that in the practice of calculus it is not necessary to consider rigorous infinities (but quantities as small or large as desired to be less than a given one). By doing “so that one only differs from the manner of Archimedes only in the “expressions”.
Leibniz compares the consideration of the infinity of infinities with what happens when one considers the ratio between a ball and the diameter of the Earth or the ratio between the Earth and the distance of the fixed stars – or when one considers that the ratio between the ball and the distance from the fixed stars is infinitely infinitely small. Its development, however, based on a comparison with finite quantities, caused confusion amongst the members of the Académie des science, who understood, to their great surprise, that Leibniz was claiming that a differential is a fixed and determined finite. Leibniz’ text is followed by a comment which is almost a request for a reply
“Some geometers, who have examined with great care the Analyse des infiniment petits by Mr. Le Marquis de l’Hôpital, & who declare following his method, say that it is necessary to take infinity à la rigueur, & not as Mr. Leibnitz explains here”.
The text we publish, Qvelqves remarques sur les Memoires de Trevoux de l’an 1701, has never been published. In the marginalia, Leibniz wrote that he had lost the piece of paper and thus the remarks written on it have not been communicated. Leibniz would finally respond to this last comment in March 1702 [Leibniz 1702] and we find in our short extract some ideas which are developed at greater length in the March response. Differentials are not rigorous infinites; they lack reality as much as the imaginary roots of algebra. The calculus is well founded because “the thing can be reduced to the incomparable”.