  In two articles ([Brisson-Ofman1, 2]), we have analyzed the so-called
'mathematical passage' of Plato's Theaetetus, the first dialogue of a trilogy
including the Sophist and the Statesman. In the present article, we study an
important point in more detail, the 'definition' of 'powers'
('$\delta\upsilon\nu\acute\alpha\mu\epsilon\iota\varsigma$'). While in
[Brisson-Ofman2], it was shown that the different steps to get the definition
are mathematically and philosophically incorrect, it is explained why the
definition itself is problematic. However, it is the first example, at least in
the trilogy, of a definition by division. This point is generally ignored by
modern commentators though, as we will try to show, it gives rise, in a
mathematical context, to at least three fundamental questions: the meaning(s)
of 'logos', the connection between 'elements and compound' and, of course the
question of the 'power(s)'. One of the main consequences of our works on
Theaetetus' 'mathematical passage', including the present one, is to challenge
the so-called 'main standard interpretation'. In particular, following
[Ofman2014], we question the claim that Plato praises and glorifies both the
mathematician Theodorus and the young Theaetetus. According to our analysis,
such a claim, considered as self-evident, entails many errors. Conversely, our
analysis of Theaetetus' mathematical mistakes highlights the main cause of some
generally overlooked failures in the dialogue: the forgetting of the 'logos',
first in the 'mathematical part', then in the following discussion, and finally
the failure of the four successive tries of its definition at the end of the
dialogue. Namely, as we will show, the passage is closely connected with the
problems studied at the end of the dialogue, but also to the two other parts of
the trilogy through the method of 'definition by division'. Finally, if our
conclusions are different from the usual ones, it is probably because the
passage is analyzed, maybe for the first time, simultaneously from the
philosophical, historical and mathematical points of view. It had been
considered usually either as an excursus by historians of philosophy (for
instance [Burnyeat1978]), or as an isolated text separated from the rest of the
dialogue by historians of mathematics (for instance [Knorr1975]), or lastly as
a pretext to discuss some astute developments in modern mathematics by
mathematicians (for instance [Kahane1985]).[Brisson-Ofman1]: Luc
Brisson-Salomon Ofman, `Theodorus' lesson in Plato's Theaetetus(147d3-d6)
Revisited-A New Perspective', to appear[Brisson-Ofman2]: Luc Brisson-Salomon
Ofman, `The Philosophical Interpretation of Plato'sTheaetetus and the Final
Part of the Mathematical Lesson (147d7-148b)', to appear[Burnyeat 1978]: Myles
Burnyeat, `The Philosophical Sense of Theaetetus' Mathematics',Isis, 69, 1978,
489-514[Kahane1985]: Jean-Pierre Kahane, `la th{\'e}orie de Th{\'e}odore des
corps quadratiques r{\'e}els',L'enseignement math{\'e}matique, 31, 1985, p.
85-92[Knorr1975]: Wilbur Knorr, The evolution of the Euclidean elements,
Reidel, 1975[Ofman2014]: Salomon Ofman, `Comprendre les math{\'e}matiques pour
comprendre Platon-Th{\'e}{\'e}t{\`e}te (147d-148b)', Lato Sensu, I, 2014, p.
70-80
