Unit I — Meno: Definition of Figure, Doubling the Square & Geometric Hypothesis — Critical Commentary & Notes

Exact Symbolic Reconstruction, Greek Mathematical Philology & Canon Reception · Meno 75d–76a, 81e–85d, 86e–87b

Meno 82b-85b: Doubling the Square by the Diagonal (Q = 4 + 4 = 8) [ B C ]: (1)*x + (0)*y + (-2) = 0 [ D C ]: (0)*x + (1)*y + (-2) = 0 [ M S ]: (1)*x + (0)*y + (-3) = 0 [ P V ]: (0)*x + (1)*y + (-3) = 0 [ A U ]: (0)*x + (1)*y + (0) = 0 [ A D ]: (1)*x + (0)*y + (0) = 0 [ E J ]: (1)*x + (0)*y + (-4) = 0 [ G L ]: (0)*x + (1)*y + (-4) = 0 [ B D ]: (-2)*x + (-2)*y + (4) = 0 [ B H ]: (-2)*x + (2)*y + (4) = 0 [ H K ]: (-2)*x + (-2)*y + (12) = 0 [ K D ]: (2)*x + (-2)*y + (4) = 0 M = (3, 0) E = (4, 0) P = (0, 3) G = (0, 4) H = (4, 2) K = (2, 4) F = (4, 4) N = (3, 3) A = (0, 0) U = (1, 0) B = (2, 0) D = (0, 2) C = (2, 2) M E P G H K F N A U B D C
Socrates' Geometric Demonstration in Meno 82b–85b: the 2×2 given square < A B C D > (Q = 4), the 4×4 fourfold square < A E F G > (Q = 16), the 3×3 nine-foot trial < A M N P > (Q = 9), and the inscribed diagonal square < B H K D > (Q = 8).

I. Exact Geometric & Symbolic Reconstruction

In geometor.plato.constructions.construct_meno_diagonal_doubling (see Demonstrations), Socrates’ sand diagram (82b–85b) is modeled with 100% exact rational coordinates over \(\mathbb{Q}\) without introducing irrational square roots until linear magnitude is requested—directly vindicating Rational Trigonometry’s primacy of quadrance:

  • Given Two-Foot Square ABCD (82c–d): Anchored at A \(= (0,0)\) and B \(= (2,0)\), every side has quadrance \(Q(A,B) = 4\), enclosing area \(4\) square feet.

  • First Refutation — Doubling the Side (82e–83c): Extending A B to E \(= (4,0)\) produces the fourfold square AEFG of quadrance \(Q(A,E) = 16\) (composed of four \(4\)-foot squares meeting at interior center C \(= (2,2)\)).

  • Second Refutation — The Three-Foot Trial (83d–e): Searching strictly between \(2\) and \(4\), the attendant proposes the arithmetic midpoint M \(= (3,0)\) (sesquialter \(3:2\)), which generates square AMNP of quadrance \(Q(A,M) = 9 \neq 8\), bringing on the purifying aporia (84a–c).

  • Constructive Resolution via the Four Diagonals (84d–85b): Drawing the four diagonals BD, BH, HK, and KD across the four sub-squares cuts each \(4\)-foot square into two equal isosceles right triangles of area \(2\). The inscribed tilted square BHKD therefore has exact area and side quadrance:

    \[Q(B, D) = Q(A, B) + Q(A, D) = 4 + 4 = 8 = 2 \, Q(A, B).\]

Socrates’ Elliptic Application Hypothesis (86e–87b): As Sir Thomas Heath demonstrated (History of Greek Mathematics, Vol. I, pp. 298–303), Socrates’ famous geometric obscurity at 87a—inscribing a given area \(C\) as a triangle in a circle of diameter \(AB = d\)—reduces to choosing a point \(X\) on diameter \(AB\) and erecting ordinate \(XY\) to the circle so that rectangle \(AX \cdot XY = C\). Since in the semicircle \(XY^2 = AX \cdot XB\), squaring gives \(AX \cdot XB = (C / AX)^2\), meaning rectangle \(AX \cdot XY\) is applied (parateinein) along \(AB\) while falling short (ἐλλείπειν) on \(XB\) by a rectangle \(XB \cdot XY\) similar to the applied rectangle \(AX \cdot XY\) (\(AX : XY = XY : XB\))—an exact fourth-century instance of hyperbolic/elliptic application of areas (Elements VI.28).

II. Literal Greek Mathematical Philology (Contra Paraphrasim)

  • 76a7 — στερεοῦ πέρας σχῆμα εἶναι: Jowett translates ‘figure is the limit of solid’, faithfully preserving Burnet's technical pairing of stereon and peras. Compare Elements XI, Def. 2 (στερεοῦ δὲ πέρας ἐπιφάνεια, ‘the extremity of a solid is a surface’) and I, Def. 14 (‘a figure is that which is contained by any boundary or boundaries’).

  • 85b4 — διάμετρον ὄνομα, καλοῦσιν δὲ ταύτην οἱ σοφισταί: Here σοφισταί denotes technical mathematical experts/professors rather than eristic rhetoricians; diametros means the diagonal of the square (preserved throughout Euclid and Aristotle).

  • 87a3–6 — παρὰ τὴν δοθεῖσαν αὐτοῦ γραμμὴν παρατείναντα ἐλλείπειν τοιούτῳ χωρίῳ οἷον ἂν αὐτὸ τὸ παρατεταμένον ᾖ: Jowett renders loosely ‘whether, if the figure be applied to the given line of the circle, it is deficient by another figure similar to that which is applied’. Literally, παρατείνειν παρὰ γραμμήν is the technical Pythagorean-Academic idiom for parabolic/elliptic application of an area along a given straight line (παραβολή / ἔλλειψις τῶν χωρίων), later standardized by Euclid in Book VI and by Apollonius in naming the parabola and ellipse.

III. Reception in Aristotle, Euclid, Proclus & Hoenen

  • Aristotle (Anal. Post. I.1, 71a29; plato:meno): Resolves Meno’s dilemma by distinguishing universal pre-existent knowledge (that every triangle has 2R) from simultaneous recognition (ἅμα ἐπαγόμενος) that the drawn figure falls under the universal.

  • Proclus (In Euclidem, Friedlein 45.15–46.3; plato:meno:81e-85d): Cites Socrates’ questioning of the boy as proof that mathematical logoi reside substantially within the soul rather than being stamped from outside by sense-impressions.

  • Hoenen (De Noetica Geometriae, Cap. I & VII; plato:meno:82b-85b): Highlights 82b–85b as the locus classicus for intellectual vision in the singular phantasm: the boy counts neither decimal approximations of \(2\sqrt{2} \approx 2.828\) nor syllogistic major premises, but beholds universal necessity (\(4 \times \frac{1}{2} \times 4 = 8\)) directly in the bisected grid.

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 I live in src/geometor/plato/ and the companion walkthrough Demonstrations.