Unit II — Theaetetus: Theodorus’s Radical Constructions & Classification of Powers — Critical Commentary & Notes¶
Exact Symbolic Reconstruction, Greek Mathematical Philology & Canon Reception · Theaetetus 147c–148b
I. Exact Geometric & Symbolic Reconstruction¶
In geometor.plato.constructions.construct_theodorus_spiral and classify_theaetetus_powers (see Demonstrations), Theodorus's lesson (147d) is modeled by iterating pure Euclidean perpendicular erection (I.11) and unit compass transfer (Post. 3):
Starting from O \(= (0,0)\) and rational unit P1 \(= (1,0)\) (\(Q(O, P_1) = 1\)), each orthogonal unit leg P1 P2, …, P16 P17 satisfies unit quadrance \(Q(P_n, P_{n+1}) = 1\) and exact perpendicularity \(\overrightarrow{OP_n} \perp \overrightarrow{P_n P_{n+1}}\).
By Pythagoras’s Theorem (Elements I.47), the spoke quadrances step through every integer from \(1\) to \(17\):
\[Q(O, P_{n+1}) = Q(O, P_n) + Q(P_n, P_{n+1}) = n + 1 \quad (1 \le n \le 16).\]Why Did Theodorus Stop at 17 Square Feet? (μέχρι τῆς ἑπτακαιδεκαπόδος, ἐν δὲ ταύτῃ πως ἐνέσχετο, 147d6): Modern scholarship (Heath 1921, Wilbur Knorr 1975) highlights two converging reasons visible directly in our exact construction: 1. Geometric Self-Overlap: Summing the interior angles \(\sum_{k=1}^{16} \arctan(1/\sqrt{k}) \approx 351.15^\circ < 360^\circ\), whereas adding the \(18\)-foot triangle (\(\arctan(1/\sqrt{17}) \approx 13.63^\circ\)) pushes the cumulative angle to \(364.78^\circ > 360^\circ\), causing the next spoke to cross the initial baseline O P1! 2. Pebble/Parity Arithmetic: In Pythagorean odd-even congruence modulo \(8\), every odd square satisfies \((2k+1)^2 = 4k(k+1) + 1 \equiv 1 \pmod{8}\). For \(N = 3, 5, 7, 11, 13, 15\), assuming \(p^2 = N q^2\) immediately fails modulo \(4\) or \(8\), whereas \(N = 17 \equiv 1 \pmod{8}\) is the first prime congruent to \(1 \pmod{8}\), requiring an infinite-descent peeling of powers of \(2\) before contradiction emerges!
II. Literal Greek Mathematical Philology (Contra Paraphrasim)¶
147d3–5 — περὶ δυνάμεών τι ἡμῖν Θεόδωρος ὅδε ἔγραφε, τῆς τε τρίποδος πέρι καὶ πεντέποδος: Jowett translates ‘Theodorus was writing out for us something about roots, showing that the sides of squares containing three or five square feet…’. Crucially, ἔγραφε means was drawing geometric diagrams, and δύναμις (dynamis) does not mean an algebraic number ‘root’, but a line segment named from the square area it has the power (*δύναται*) to form. Likewise τρίπους and πεντέπους denote squares of three and five square feet (literally ‘the three-foot [power]’, where linear vs. areal ‘foot’ is determined by feminine δύναμις).
147e5–148a4 — τὸν ἴσον ἰσάκις… προμήκη δὲ ἀριθμόν: Burnet’s exact multiplicative syntax ἴσος ἰσάκις γιγνόμενος (‘becoming equal-times-equal’, \(k \times k\)) vs. πλείων ἐλαττονάκις ἢ ἐλάττων πλειονάκις (‘greater times less or less times greater’) is preserved verbatim in Euclid Elements VII, Defs. 16–19. Note that Plato uses promēkēs (oblong) broadly for any non-square integer, whereas later Neopythagoreans restricted heteromēkēs to \(k(k+1)\).
148b1–2 — ὡς μήκει μὲν οὐ συμμέτρους ἐκείναις, τοῖς δ’ ἐπιπέδοις ἃ δύνανται: Literally ‘incommensurable with those in length, but [commensurable] by the plane figures which they have the power to form’—the verbatim ancestor of Euclid Elements X, Defs. 1–2 (εὐθεῖαι δυνάμει σύμμετροί εἰσιν, ὅταν τὰ ἀπ’ αὐτῶν τετράγωνα τῷ αὐτῷ χωρίῳ μετρῆται, ‘straight lines are commensurable in square/power when the squares on them are measured by the same area’).
III. Reception in Euclid Books VII, X & Sister Repositories¶
All five sister-corpus citations (plato:theaetetus, 147.d, 147D, 147.d.sqq, 147-e-148-b in geometor/euclid and geometor/elements) converge on Elements X.9 (attributed explicitly to Theaetetus by an ancient scholium going back to Pappus): Squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which do not have the ratio of square to square have sides incommensurable in length.
IV. Executable Symbolic Verification¶
In accordance with GEOMETOR’s strict content/code separation for historical editions, executable geometor.model SymPy constructions and Rational Trigonometry verification routines for Unit II live in src/geometor/plato/ and the companion walkthrough Demonstrations.