LEONARDO DA VINCI 1452-1519

Leonardo da VINCI Vinci, 1452–Amboise, 1519 Doubling the Cube; an Empirical Solution Pen and brown ink About 1505 Doubling the cube was one of the classical problems of geometry. Given a cube A with an edge length a, the edge length b of cube B needs to be determined so that the volume of B is double the volume of A – which amounts to determining the cubic root of 2. Leonardo found a close solution. Cube A: edge length 4, volume 64, double volume 128. Cube B: edge length very slightly greater than 5, volume slightly greater than 125. Biblioteca Ambrosiana, Milan, Codex Atlanticus, fol. 161R Leonardo da VINCI Vinci, 1452–Amboise, 1519 Doubling the Cube; the Theorems of Hippocrates of Chios and Apollonius of Perga Pen and brown ink About 1505 The Greek mathematician Hippocrates of Chios demons- trated that the equation for doubling the cube amounted to finding two values, b and c, between a and 2a, so that a/b=b/c=c/2a. Great Greek mathematicians such as Apollo- nius, the author of Conics (which Leonardo reused – and contested! – here), proposed various geometrical (rather than canonical) methods for finding these values, using only a straightedge and compass. Biblioteca Ambrosiana, Milan, Codex Atlanticus 110 111 i i, i , b/c = c/2a. Great Greek mathematician such as Apollonius, e auth r of Conics (which Le nardo reuse – a d contested! – here), proposed various geomet ical (rather tha nonical) methods for finding these values, using only a s rai htedge and compass. i li i , il , l i