Among the brilliant achievements of Greek culture, ancient Greek Mathematics holds a prominent position. The mathematical discoveries of the first three centuries, from Thales to Euclid, are recorded in a magnificent way in the Elements. However, they are presented in their final form, without revealing their original origin and the paths of their creation. Therefore, the classical period of Greek Mathematics up to Euclid, despite being studied by significant mathematicians and/or historians of Mathematics, continues to maintain a mysterious and enigmatic character, displaying open problems and lively scientific debates at every step.

With this work of ours, the product of long study, research, and teaching, we attempt to identify the ancient paths that lead to the Propositions of the Elements, and to follow them, in the manner of ancient analysis, towards the initial discoveries, in order to achieve a more valid reconstruction of ancient Greek Mathematics of the classical period. For this purpose, we study Euclid's Elements, the ancient sources - especially Proclus' commentaries - and take into account important contemporary and current related studies. However, at critical steps, it proved necessary to introduce new questions or arguments or to differentiate old ones.

Given our mathematical background, we have tried to maintain a delicate and fragile balance between mathematical and historical ethics in writing this book, serving both the unity and timelessness of Mathematics, as well as the distinctiveness of its History, avoiding fallacious associations, but also discovering admirable convergences.

The great mathematical discoveries of this ancient period, especially the greatest discovery of asymmetry, had a decisive influence on ancient Philosophy, particularly Pythagorean, Eleatic, and Platonic. For this reason, detecting and identifying these influences allows us to recover, through Philosophy, ancient mathematical knowledge that has not been preserved in mathematical sources, sometimes with impressive results.

In Volume I, the theory of rational numbers (Book 7 of the Elements) is presented, as well as its origin from Pythagorean music, the geometry of Thales and the Pythagoreans (Books 1 and 2 of the Elements), Pythagorean parabola of sections and Geometric Algebra, Pythagorean antiphonetic asymmetry, lateral and diametrical numbers, and Pythagorean philosophy.

Our work is addressed to all those interested in ancient Greek Mathematics, especially students, mathematicians, mathematics teachers, and historians of Mathematics. It is self-contained and can be read independently.