A unique book that turns our notions about Euclid’s Elements and non-Euclidean geometry. Proclus’ hypothesis leads to the new view on the mathematics history, starting from Euclid. According to this hypothesis, Euclid’s main goal, while writing the Elements, was to create a complete geometric theory of "Platonic solids,” which are associated in the ancient Greek science with the Universe Harmony. Euclid’s Elements is a source for the Classical Mathematics and the Mathematics of Harmony based on the “golden ratio” and “Platonic solids.” The Mathematics of Harmony, as a new interdisciplinary direction of modern science, is a reflection of the “harmonic ideas” by Pythagoras and Plato in modern science and mathematics. New classes of hyperbolic and spherical Fibonacci functions, based on the “golden proportion” and its generalization – the “metallic proportions,” underlie the original solution of Hilbert’s Fourth Problem for hyperbolic and spherical geometry. The challenge searching for new hyperbolic and spherical worlds of Nature follows from this solution. The "golden" hyperbolic geometry with the base 1.618 ("Bodnar geometry") underlies botanical phenomenon of phyllotaxis.