Decoding Book II of the Elements

Abstract

The paper is a commentary to the Polish translation of the Elements Book II, included in this volume.We focus on relations between figures represented and not represented on diagrams and identify rules which enable
Euclid to bridge these two kinds of objects. Also, we argue that the main mathematical problem addressed in Book II is constructing a leg of a rightangled triangle, given its hypotenuse and the other leg. In proposition II.14,
Euclid solves it through the construction called the geometric mean. We trace the problem in Book III and beyond the Elements: in Heron’s Metrica, Descartes’ La Géométrie, and modern foundations of mathematics. We show
that Descartes, by novel interpretation of the Pythagorean theorem, provides a modern solution to this problem.