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 majestically in the Elements. However, they are presented in their final form, without reflecting their original source 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, presenting open problems and lively scientific disputes at every step.

With this work, the product of many years of study, research, and teaching, we seek to identify the ancient paths leading to the Propositions of the Elements and to follow them, in the manner of ancient analysis, towards the original discoveries, in order to achieve a more valid reconstruction of ancient Greek mathematics from the classical period. To this end, we study the Elements of Euclid, the ancient sources - particularly the comments of Proclus - and consider the significant modern 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 tried during the writing of the book to maintain the delicate and fragile balance between mathematical and historical ethics, serving both the unity and timelessness of mathematics and the diversity of their history, avoiding unfounded associations while discovering remarkable convergences.

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

In Volume I, the theory of ratios of numbers (Book 7 of the Elements), its origin from Pythagorean music, the geometry of Thales and the Pythagoreans (Books 1 and 2 of the Elements), the Pythagorean parable of sections, and geometric algebra, the Pythagorean anti-contradictory asymmetry, side and diagonal numbers, and Pythagorean philosophy are presented.

Our work is aimed at all those interested in ancient Greek mathematics, particularly students, mathematicians, mathematics teachers, and historians of mathematics. It is self-sufficient and can be read independently.