Paulian’s Commentaire on L’Hospital’s Analyse des infiniment petits: On How to Teach Differential Calculus in the 1760s

In 1696, Guillaume de L’Hospital (1661–1704) published the first treatise on differential calculus, Analyse des infiniment petits. This work presented the foundations of differential calculus in a structured manner. L’Hospital's work played a crucial role in disseminating and teaching the new Leibnizian calculus. However, it was generally considered difficult for beginners, primarily due to the presentation of the concept of infinitely small quantity. Over the course of nearly a century, several commentaries were published in France as supplements to the Analyse. In 1768, Aimé-Henri Paulian (1722–1802) wrote a commentary that was appended to the third edition of the Analyse. The publication of Paulian's commentary raises several questions regarding the teaching of differential calculus to beginners. Why did Paulian consider it necessary to write his commentary? How did he approach the concept of infinitely small quantity? What strategies should be employed to enhance understanding of the Analyse? This contribution aims to examine Paulian’s work to gain deeper insights into the teaching of differential calculus to beginners in 1760s France.