Introduction -- L'Hôpital's Preface -- In Which We Give the Rules of this Calculus -- Use of the Differential Calculus for Finding the Tangents of All Kinds of Curved Lines -- Use of the Differential Calculus for Finding the Greatest and the Least Ordinates, to Which are Reduced Questions De maximis & minimis -- Use of the Differential Calculus for Finding Inflection Points and Cusps -- Use of the Differential Calculus for Finding Caustics by Reflection -- Use of the Differential Calculus for Finding Caustics by Refraction -- Use of the  Differential Calculus for Finding the Points of Curved Lines that Touch an Infinity of Lines Given in Position, Whether Straight or Curved -- The Solution of Several Problems that Depend Upon the Previous Methods -- A New Method for Using the Differential Calculus with Geometric Curves, from Which we Deduce the Method of Messrs, Descartes, and Hudde -- Appendices.

This monograph is an annotated translation of what is considered to be the world’s first calculus textbook, originally published in French in 1696. That anonymously published textbook on differential calculus was based on lectures given to the Marquis de l’Hôpital in 1691-2 by the great Swiss mathematician, Johann Bernoulli. In the 1920s, a copy of Bernoulli’s lecture notes was discovered in a library in Basel, which presented the opportunity to compare Bernoulli’s notes, in Latin, to l’Hôpital’s text in French. The similarities are remarkable, but there is also much in l’Hôpital’s book that is original and innovative. This book offers the first English translation of Bernoulli's notes, along with the first faithful English translation of l’Hôpital’s text, complete with annotations and commentary. Additionally, a significant portion of the correspondence between l’Hôpital and Bernoulli has been included, also for the fi rst time in English translation. This translation will provide students and researchers with direct access to Bernoulli’s ideas and l’Hôpital’s innovations. Both enthusiasts and scholars of the history of science and the history of mathematics will fi nd food for thought in the texts and notes of the Marquis de l’Hôpital and his teacher, Johann Bernoulli.