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In two articles ([Brisson-Ofman1, 2]), we have analyzed the so-called'mathematical passage' of Plato's Theaetetus, the first dialogue of a trilogyincluding the Sophist and the Statesman. In the present article, we study animportant 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 definitionare mathematically and philosophically incorrect, it is explained why thedefinition itself is problematic. However, it is the first example, at least inthe trilogy, of a definition by division. This point is generally ignored bymodern commentators though, as we will try to show, it gives rise, in amathematical context, to at least three fundamental questions: the meaning(s)of 'logos', the connection between 'elements and compound' and, of course thequestion of the 'power(s)'. One of the main consequences of our works onTheaetetus' 'mathematical passage', including the present one, is to challengethe so-called 'main standard interpretation'. In particular, following[Ofman2014], we question the claim that Plato praises and glorifies both themathematician Theodorus and the young Theaetetus. According to our analysis,such a claim, considered as self-evident, entails many errors. Conversely, ouranalysis of Theaetetus' mathematical mistakes highlights the main cause of somegenerally overlooked failures in the dialogue: the forgetting of the 'logos',first in the 'mathematical part', then in the following discussion, and finallythe failure of the four successive tries of its definition at the end of thedialogue. Namely, as we will show, the passage is closely connected with theproblems studied at the end of the dialogue, but also to the two other parts ofthe trilogy through the method of 'definition by division'. Finally, if ourconclusions are different from the usual ones, it is probably because thepassage is analyzed, maybe for the first time, simultaneously from thephilosophical, historical and mathematical points of view. It had beenconsidered usually either as an excursus by historians of philosophy (forinstance [Burnyeat1978]), or as an isolated text separated from the rest of thedialogue by historians of mathematics (for instance [Knorr1975]), or lastly asa pretext to discuss some astute developments in modern mathematics bymathematicians (for instance [Kahane1985]).[Brisson-Ofman1]: LucBrisson-Salomon Ofman, `Theodorus' lesson in Plato's Theaetetus(147d3-d6)Revisited-A New Perspective', to appear[Brisson-Ofman2]: Luc Brisson-SalomonOfman, `The Philosophical Interpretation of Plato'sTheaetetus and the FinalPart of the Mathematical Lesson (147d7-148b)', to appear[Burnyeat 1978]: MylesBurnyeat, `The Philosophical Sense of Theaetetus' Mathematics',Isis, 69, 1978,489-514[Kahane1985]: Jean-Pierre Kahane, `la th{\'e}orie de Th{\'e}odore descorps 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 pourcomprendre Platon-Th{\'e}{\'e}t{\`e}te (147d-148b)', Lato Sensu, I, 2014, p.70-80

Powers and division in the 'mathematical part' of Plato's Theaetetus