A Graphical Approach to Euler’s Method

Abstract

To solve differential equations and study transcendental curves appearing in problems of geometry, celestial mechanics, ballistics and physics, mathematicians have imagined numerous approaches since the 17 th century. Alongside integration by quadratures and the series method, we can notably quote the polygonal method formalised by Euler in 1768. He directly used Leibniz's vision of curves as polygons made up of segments of infinitely tiny tangents. After an historical introduction and the study of an appropriate extract from the work by Euler on integral calculus, this chapter recounts a teaching experiment with 18 year olds, the aim of which was to introduce the notion of differential equations with support from the graphic version of the polygonal method. Through the purely geometric construction of integral curves formed from tiny segments of tangents, the students were able to make useful transfers between algebra and geometry and actively discover the first concepts of infinitesimal calculation.