The History of a Lemma from Archimedes to Newton

Abstract

In this work, we identify some traces of the history of Archimedes’ straight line, a lemma introduced by Archimedes within the solution of prop. II 4 of On the Sphere and Cylinder, in which it is asked to cut a sphere with a plane in order to obtain two spherical segments such that their ratio is equal to a given ratio. In the course of solving this proposition with the method of analysis, assuming given what is required, as specified in this method, Archimedes traces the path along which one goes in search of the final point, from which what is required comes, expressed in a lemma which in history is referred to as Archimedes’ straight line or the square problem. The Syracusan mathematician promised to later give a solution to the lemma, but he did not keep his promise. The lemma was taken up again by Eutocius of Ascalon towards the end of the Hellenistic period, it had considerable resonance among some Arab mathematicians, who traced a specific line of research around it, finally finding attention in the Latin tradition up to Newton within the first part of the history of the path of the analytic method.