An unpublished memoir by Lagrange on the successive expansion of curves (Q2077237)

The article is an annotated transcription of an unedited memoire of Lagrange conserved in the library of the Institute de France. The author outlines the work of the commission that decided which manuscripts that Lagrange left unpublished should still be printed. He sketches the historical context of the creation of the manuscript starting with Hyugens' work about the cycloid and discusses then some results of Johann Bernoulli and Leonard Euler, which paved the way for Lagrange's argument. These concern Bernoulli's generation of a epicycloid as well as of a logarithmic spiral and an algorithm for the approximation of \(\pi\) and Euler's generalisation of some results of Bernoulli. Finally, the author explains the proof of Lagrange and that of Puiseux from 1844 using Fourier series.