between square numbers (perfect squares) and the rest, which they called oblong; given a
geometric representation of these numbers as quadrilaterals they called the sides of squares
equal to them “lengths” and “powers” respectively. Plato’s representation of this accomplish-
ment has been taken to foreshadow Theaite ̄tos’ subsequent mathematical achievements.
In his history of geometry, P mentions Theaite ̄tos together with L
T and A T immediately after praising Plato’s accomplishments, and
says that the three of them increased the number of theorems and brought them into a more
scientific arrangement (In Eucl. p. 66.14–18 Fr.). Later he tells us that E completed or
perfected many of Theaite ̄tos’ results. According to the Souda Theta-93, Theaite ̄tos was the
first to describe (graphein) the five regular solids, information which the first scholion on Book
13 of the Elements presents more precisely by saying that the cube, pyramid, and dodeca-
hedron are Pythagorean, but the octahedron and icosahedron belong to Theaite ̄tos. The
scholion credits Euclid with extending the elemental ordering to this subject, making it
plausible to suppose that Book 13 is based on Theaite ̄tos’ work. Book 13 depends on Book
10, which develops an elaborate classification of irrational straight lines (lines which are
incommensurable with a given straight line and such that a square with one of them as side
is incommensurable with the square with the given straight line as side), focusing on lines
called medials, binomials, and apotomes. P (In Eucl. X 1.1; cf. 2.17) tells us that
Theaite ̄tos not only made the distinction ascribed to him in Plato’s Theaite ̄tos, but also made
the distinction between the three fundamental lines of Book 10. Although Pappos’ account
of Theaite ̄tos’ characterization of these lines differs from the characterization found in the
Elements, there is again good reason to suppose that Book 10 is based on Theaite ̄tos’ work.
Histories of Greek mathematics now commonly ascribe to Theaite ̄tos another achievement,
the first theory of proportion applicable to both commensurable and incommensurable
magnitudes. The theory which Euclid develops in Book 5 is thought to be dependent on the
work of E, but a passage in A (To p. 8.3 [158b24–35]; cf. A
A ad loc.) suggests the existence of an earlier theory in which (to state matters
anachronistically) the ratio of two magnitudes x and y was expressed in terms of the applica-
tion of the Euclidean algorithm of alternating subtraction to them (Elem. 7.2; cf. 10.1–3)
and one said that x:y :: z:w if and only if application of the algorithm to each pair produces
the same result. No ancient text associates such a theory with Theaite ̄tos; but even if he did
not develop it, his accomplishments more than justify Theodo ̄ros’ high praise of him in
Plato’s dialogue.
O. Becker, “Eine voreudoxische Proportionenlehre und ihre Spuren bei Aristoteles und Euklid,” Quellen
und Studien zur Geschichte der Mathematik, Astronomie, und Physik B.2 (1933) 311–333; B.L. van der
Waerden, Science Awakening (trans. Arnold Dresden), (1963) 165–179; W.C. Waterhouse, “The dis-
covery of the regular solids,” AHES 9 (1972–3) 212–221; M.F. Burnyeat, “The philosophical sense
of Theaetetus’ mathematics,” Isis 69 (1978) 489–513; DSB 13.301–307, I. Bulmer-Thomas;
Lasserre (1987) 3; Mueller (1997) 277–285.
Ian Mueller
Theano ̄, pseudo (200 BCE – 100 CE?)
P’ wife (or daughter, according to some traditions) and daughter of Brontinus
of Kroto ̄n, the popular idealized wife and mother. To her were ascribed various writings (On
virtue, Exhortations to women, On Pythagoras) and apophthegms. I S (1.10.13)
transmits under her name a spurious Dorian fragment On Piety (peri eusebeias). Equally
THEANO ̄, PSEUDO