# Doctoral studies > PhD dissertations UAB

**Mónica Blanco Abellán***Hermenèutica del càlcul diferencial a l'Europa del segle XVIII: de l'Analyse des infiniment petits de L'Hôpital (1696) al Traité élémentaire de calcul différentiel et de calcul intégral de Lacroix (1802)***Supervisor/s: **Ferran Cedó Giné (tutor) **Date of defense: **28-10-2004

The aim of the PhD thesis “Hermeneutics of the differential calculus in 18th century Europe: from the Analyse des infiniment petits by L’Hôpital (1696) to the Traité élémentaire de calcul différentiel et de calcul intégral by Lacroix (1802)” is to analyse the mathematical development of differential calculus through a number of works which were written in 18th century in France, Germany, Italy and Great Britain. Placing the study in the historical context of the development of calculus and the institutional context of mathematics during this period, it has been compared how, from the analysed works, these countries expound the elements of calculus. The works which have been chosen to that purpose are:

- Analyse démontrée (1708) by Charles René Reyneau.

- Cours de mathématiques à l’usage du corps de l’artillerie (1799-1800) by Étienne Bézout.

- Leçons sur le calcul des fonctions (1800) by Joseph Louis Lagrange.

- Traité élémentaire de calcul différentiel et de calcul intégral (1802) by Sylvestre François Lacroix.

- Elementa analyseos (1713-1715) by Christian Wolff.

- Anfangsgründe der Analysis des Unendlichen (1760) by Abraham Gotthelf Kästner.

- Anfangsgründe der Analysis des Unendlichen (1770) by Georg Friedrich Tempelhoff.

- Anfangsgründe der mathematischen Analysis und höhern Geometrie (1786) by Wenceslau J. G. Karsten.

- Instituzioni Analitiche (1748) by Maria Gaetana Agnesi.

- Principj di analisi sublime (1759) by Giuseppe Luigi Lagrange.

- Institutiones Analyticae (1765-67) by Vincenzo Riccati and Girolamo Saladini and Compendio d’analisi (1775) by Girolamo Saladini.

- An Institution of Fluxions (1706) by Humphry Ditton.

- A Treatise of Fluxions (1742) by Colin Maclaurin.

- The Doctrine and Application of Fluxions (1750) by Thomas Simpson.

Leonhard Euler’s Institutiones calculi differentialis (1755) is analysed as well, due to the author’s influence on the development of calculus and his internationality.

This PhD thesis begins with a comparative analysis of the Analyse des infiniment petits (1696) by L’Hôpital, the first systematical treatise on differential calculus, and the Lectiones de calculo differentialium by Johann Bernoulli, based on the lessons he offered L’Hôpital between 1691 and 1692. This analysis suggests the main lines which have been taken into account when analysing and comparing the chosen works:

1. the way they present the foundations of calculus and the theoretical corpus they include;

2. whether the language they use is geometric or algebraic;

3. the criteria for choosing coordinates in connection with the way they treat algebraic and transcendental curves;

4. the problems and applications they work with, and their layout;

5. methodological aspects, such as structure, didactical skills and notation (applying multivariate statistical techniques to quantify the structure of the different texts by means of the frequencies of some words).

There is a revision of the factors which fostered their reading, such as the public they were addressed to. The influence of L’Hôpital’s Analyse on these works is also assessed.

The comparison of the works on differential calculus, taking Leibniz and Euler as references, provides the grounds to conclude that three groups are detected: 1) the works of L’Hôpital, Reyneau, Wolff, Agnesi, Saladini and Bézout; 2) the Italian work of Lagrange and those of Kästner and Tempelhoff; 3) the works of Karsten and Lacroix.

On the other hand, from the study of the British works it can be seen that the development of fluxional calculus was by no means homogeneous. Likewise, as far as military context is concerned, the role given to foundations and applications by the corresponding authors differed. Finally, the chronological development of the problems and applications which are included in the analysed works is outlined, as well as that of the treated curves, in connection with the choice of coordinates.