Unit I — Meno: Definition of Figure, Doubling the Square & Geometric Hypothesis — Structural Synopsis

Argumentative Architecture & Interactive Diagram Guide · 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).

Architectural Thesis

Unit I gathers the three foundational mathematical episodes of Plato's Meno—the earliest surviving Greek text to stage an explicit geometric diagram step by step:

  1. Dialectical Definition of Figure (75d–76a): Before defining virtue, Socrates offers Meno a model geometric definition: roundness (strongylotēs) is merely a figure, whereas universal figure (schēma) is the limit of solid (στερεοῦ πέρας, 76a).

  2. Anamnesis & Doubling the Square by the Diagonal (81e–85d): Answering Meno’s eristic paradox of inquiry, Socrates summons an untutored household attendant and constructs the \(2 \times 2\) square ABCD (\(Q = 4\)). After exposing two false arithmetic intuitions—doubling the side to AE (\(4 \times 4 = 16\), AEFG) and taking the sesquialter side AM (\(3 \times 3 = 9\), AMNP)—Socrates draws the four diagonals BD, BH, HK, and KD, cutting each \(4\)-foot square in half (\(2 + 2 + 2 + 2 = 8\)) to enclose the exact double square BHKD.

  3. Inquiry from Hypothesis & Application of Area (86e–87b): Pressed to ask whether virtue is teachable prior to knowing its essence, Socrates adopts the geometer’s method of hypothesis (ἐξ ὑποθέσεως, 86e): reducing the inscription of a triangular area inside a given circle to an elliptic application of area along the diameter.

Stephanus Conspectus

Section

Stephanus Span

Mathematical & Philosophical Argument

Tracks

I.1 Definition of Figure

75d–76a

Genus vs. species; surface, solid (stereon), and limit (peras).

EN · EL

I.2 Diagonal Doubling

81e–85d

Given square (\(Q=4\)), refutation of \(Q=16\) & \(Q=9\), aporia, and diagonal proof (\(Q=8\)).

EN · EL

I.3 Geometric Hypothesis

86e–87b

Analytic reduction of circular inscription to elliptic application of areas (Elements VI.28).

EN · EL