This book is the result of fifteen years of studies on art, in collaboration with mathematicians (in particular Jean-Paul Guichard and the IREM of Poitiers). Until the Renaissance, the composition of painting and architecture was based on a specific geometry, called with the eyes. Its adventure began before writing, and books report nothing about. This practice avoids calculation for fear of scaring numbers, thanks to a grid where the figures are constructed. Thales, like many Greek philosophers, went to Egypt to learn it - he will give his name to the axioms. Then, all the properties are demonstrated with visual evidences, qualified as monstrations, in particular the four manifestations of the golden ratio in the triangle 3-4-5. This book offers a rich pedagogical material, as well as a solid basis for reflection on art and history. Rublev, Botticelli and Dürer are the great masters of this art. The color images for the screen, and the B/W for print, are accessible on the Internet.