# LEIBNIZ ON INFINITESIMALS

## [LEIBNIZ, GOTTFRIED WILHELM].

# Observatio, quod rationes sive proportiones non habeant locum circa quantitates nihilo minores, et de vero sensu methodi infinitesimalis.

Leipzig, Grosse & Gleditsch, 1712. 4to. In: "Acta Eruditorum Anno MDCCXII". The entire volume offered in contemporary full vellum. Hand written title on spine. A yellow label pasted on to top of spine. A small stamp to title-page and free front end-paper. Library label to pasted down front free end-paper. As usual with various browning to leaves and plates. Pp. 167-69. [Entire volume: (2), 555, (35) pp. + five engraved plates.].

*First printing of one of Leibniz's latest publications in which he proposed an interpretation of infinitesimals by a comparison of bodies of different extensions. *

The paper is a response to to a problem raised by French philosopher and mathematician Antoine Arnauld, who wrote an important philosophical work known as "The Logic of Port-Royal" from 1662 and "Geometry", 1667. In the book he includes an example of symbolic rules that he considers to be against our basic intuitions on magnitudes and proportions. His reasoning goes as follows "Suppose we have two numbers, a larger and a smaller one. The proportion of the larger to the smaller one should evidently be larger than the proportion of the smaller to the larger one. But if we use 1 as the larger number and - 1 as the smaller one this would lead to (1/-1) > (-1/1) which is against the rules of algebra". (Heeffer, The Methodological Relevance of the History of Mathematics for Mathematics Education, 1992).

Leibniz saw this as a genuine mathematical problem but argued that the division should be performed as a symbolic calculation.

"Following Leibniz, the infinite appeared in two forms as the (i) Contiuous infinite and (ii) the discrete infinite. The status of the differentials is closely related to the status of the infinite. [...] As a consequence, there is no clear and consistence distinction between continua of different kind related to (i) geometry and to (ii) mechanics. [...] Leinbiz did neither consequently argue mathematically or arithmetically nor consequently geometrically, phenomenologically and mechanically. [But] The correlation between mathematics and physics is as impressive as possible. (Suisky, Euler as physicist, 2009, p. 89-90).

The volume also contains:

Bernoulli, Johann. Angulorum arcuumque sectio indefinita per formulam universalem expressa. Pp. 274-277; 329-30.

And many other papers by influential contemporary mathematicians, philosophers and historians.

Order-nr.: 44073