  This article is the first part of a study of the so-called 'mathematical
part' of Plato's Theaetetus (147d-148b). The subject of this 'mathematical
part' is the irrationality, one of the most important topics in early Greek
mathematics. As of huge interest for mathematicians, historians of mathematics
as well as of philosophy, there had been an avalanche of studies about it. In
our work, we revisit this question, for we think something is missing: a global
analysis of Plato's text, from these three points of view simultaneously:
history, mathematics and philosophy. It is what we have undertook through a new
translation, a new interpretation of the mathematical lesson about irrational
magnitudes and a novel interpretation of the whole passage from these three
points of view. Our guideline is considering Plato's writings seriously, not as
some playful work. This simple rule is indeed surprisingly constraining, but it
allows us to get a rare direct glance inside pre-Euclidean mathematics, in
contradiction with the 'Main Standard Interpretation' prevailing in history of
mathematics as well as in history of philosophy. This study had been divided in
two parts for editorial reasons. In the present article, we propose an analysis
of the first part of this 'mathematical part', Theodorus' lesson. In the second
article (Brisson-Ofman (to appear)), we present the sequel of the lesson and a
philosophical interpretation of the 'mathematical part' within the framework of
the entire dialogue. Both articles form a whole. They are both aimed to an
audience without any particular mathematical background, and require only
elementary mathematical knowledge, essentially of high school-level. Some more
delicate points are nevertheless developed in the Appendices.
