Unit II — Theaetetus: Theodorus’s Radical Constructions & Classification of Powers¶
Incommensurable Square Sides from Three to Seventeen Square Feet & Theaetetus’s Dichotomy of Lengths (Mēkē) and Powers (Dynameis) (147c–148b)
«ὅσαι μὲν γραμμαὶ τὸν ἰσόπλευρον καὶ ἐπίπεδον ἀριθμὸν τετραγωνίζουσι, μῆκος ὡρισάμεθα, ὅσαι δὲ τὸν ἑτερομήκη, δυνάμεις, ὡς μήκει μὲν οὐ συμμέτρους ἐκείναις, τοῖς δ’ ἐπιπέδοις ἃ δύνανται.»
“All lines that form in square an equilateral plane number we defined as length, while all that form an oblong number we defined as powers—incommensurable with the former in length, yet commensurable by the plane areas they have the power to form.” — Theaetetus 148a9–b2
II.1 Theodorus’s Diagram & Lengths vs. Powers (147c–148b)¶
English Translation by Benjamin Jowett (3rd ed., 1892) · Project Gutenberg #1726 (Jowett 3rd ed., 1892)
SOCRATES: Moreover, he might answer shortly and simply, but he makes an enormous circuit. For example, when asked about the clay, he might have said simply, that clay is moistened earth–what sort of clay is not to the point.
THEAETETUS: Yes, Socrates, there is no difficulty as you put the question. You mean, if I am not mistaken, something like what occurred to me and to my friend here, your namesake Socrates, in a recent discussion.
SOCRATES: What was that, Theaetetus?
THEAETETUS: Theodorus was writing out for us something about roots, such as the roots of three or five, showing that they are incommensurable by the unit: he selected other examples up to seventeen–there he stopped. Now as there are innumerable roots, the notion occurred to us of attempting to include them all under one name or class.
SOCRATES: And did you find such a class?
THEAETETUS: I think that we did; but I should like to have your opinion.
SOCRATES: Let me hear.
THEAETETUS: We divided all numbers into two classes: those which are made up of equal factors multiplying into one another, which we compared to square figures and called square or equilateral numbers;–that was one class.
SOCRATES: Very good.
THEAETETUS: The intermediate numbers, such as three and five, and every other number which is made up of unequal factors, either of a greater multiplied by a less, or of a less multiplied by a greater, and when regarded as a figure, is contained in unequal sides;–all these we compared to oblong figures, and called them oblong numbers.
SOCRATES: Capital; and what followed?
THEAETETUS: The lines, or sides, which have for their squares the equilateral plane numbers, were called by us lengths or magnitudes; and the lines which are the roots of (or whose squares are equal to) the oblong numbers, were called powers or roots; the reason of this latter name being, that they are commensurable with the former [i.e., with the so-called lengths or magnitudes] not in linear measurement, but in the value of the superficial content of their squares; and the same about solids.
SOCRATES: Excellent, my boys; I think that you fully justify the praises of Theodorus, and that he will not be found guilty of false witness.