Unit VI — Parmenides, Philebus & Laws VII: Straightness, Limit & Incommensurability — Critical Commentary & Notes

Exact Symbolic Reconstruction, Greek Mathematical Philology & Canon Reception · Parmenides 137d–138a, 145a–b; Philebus 16c–26d, 51b–52c; Laws VII 817e–820d

Parmenides 137e, Philebus 51c & Laws VII 819d: Compass, Straight Ray & 1-sqrt(2)-sqrt(3) Triad ( O C ): (x - (0))^2 + (y - (0))^2 - (1) = 0 [ C F ]: (1)*x + (0)*y + (-1) = 0 [ B H ]: (0)*x + (1)*y + (-1) = 0 [ O C ]: (0)*x + (1)*y + (0) = 0 [ O B ]: (1)*x + (0)*y + (0) = 0 [ O D ]: (-1)*x + (1)*y + (0) = 0 [ D K ]: (-1)*x + (-1)*y + (2) = 0 [ O E ]: (-1 - sqrt(2)/2)*x + (1 - sqrt(2)/2)*y + (0) = 0 C = (1, 0) A = (-1, 0) B = (0, 1) D = (1, 1) E = (1 - sqrt(2)/2, sqrt(2)/2 + 1) O = (0, 0) C A B D E O
Compass (Tornos), Optical Straight Ray (Kanon) & Three-Dimensional Incommensurability: circle (O, C) of equal radii Q = 1 (Parm. 137e, Phil. 51c), optical diameter [A, O, C], plane diagonal O D (Q = 2), and unfolded cube diagonal O E (Q = 3, Laws VII 819d–820c).

I. Exact Geometric & Symbolic Reconstruction

In geometor.plato.constructions.construct_compass_and_peras (see Demonstrations), the three foundational motifs of Parmenides, Philebus, and Laws VII are unified in a single compass-and-straightedge plate (foundations_peras_compass.svg):

  1. Equidistant Circle & Optical Straight Line (*Parmenides* 137e, 145b):

    • Circle (O, C) centered at O \(= (0,0)\) exhibits equal radial quadrances \(Q(O, A) = Q(O, C) = Q(O, B) = 1\) (‘that whose extremes are everywhere equally distant from the middle’, 137e1–3, verbatim ancestor of Euclid Elements I, Def. 15).

    • Collinear diameter A O C exhibits Plato’s optical straightness (epiprosthēsis, 137e3–4): looking from A toward C, midpoint O blocks (ἐπίπροσθεν) both extremes—bridging Euclidean geometry and Euclidean Optics!

  2. Compass (*Tornos*) & Straightedge (*Kanōn*) Pure Beauty (*Philebus* 51c):

    • Erecting perpendicular radius O B by I.11 and completing unit square OCDB yields plane diagonal O D of exact quadrance \(Q(O, D) = 1 + 1 = 2\).

  3. Three-Dimensional Incommensurability Triad (*Laws* VII 819d–820c):

    • Erecting a unit third-dimensional edge perpendicular to O D at D yields the unfolded cube space diagonal O E of exact quadrance:

      \[Q(O, C) = 1 \;(\text{length } 1), \quad Q(O, D) = 2 \;(\text{plane } \sqrt{2}), \quad Q(O, E) = 3 \;(\text{solid } \sqrt{3}).\]

    • Pairwise ratios \(1 : \sqrt{2}\), \(1 : \sqrt{3}\), and \(\sqrt{2} : \sqrt{3}\) (\(Q_D / Q_E = 2/3\), non-square!) prove that unit length, square diagonal, and cube diagonal are mutually incommensurable across all three pairs—illustrating the Athenian Stranger’s shock (819e–820a) that educated Greeks falsely suppose length, breadth, and depth to be universally commensurable!

II. Literal Greek Mathematical Philology (Contra Paraphrasim)

  • Parmenides 137e3–4 — οὗ ἂν τὸ μέσον ἀμφοῖν τοῖν ἐσχάτοιν ἐπίπροσθεν ᾖ: Jowett translates ‘that of which the middle intercepts the view of the extremes’. As Heath (Euclid Vol. I, pp. 165–166; plato:parm:137E) and Proclus (Friedlein 109.21; plato:parmenides:137e) observe, Euclid deliberately replaced Plato’s optical ἐπίπροσθεν (which appeals to visual rays) with the purely intrinsic geometric definition ἥτις ἐξ ἴσου τοῖς ἐφ’ ἑαυτῆς σημείοις κεῖται (‘a line which lies evenly with the points on itself’, I, Def. 4), while retaining the optical principle as Postulate 1 of Euclid’s Optics!

  • Philebus 25a7–b1 — ὁπόσα… ἀριθμὸν πρὸς ἀριθμὸν ἢ μέτρον πρὸς μέτρον: Notice Plato’s deliberate disjunction: number to number (ἀριθμὸς πρὸς ἀριθμόν, commensurable rational ratios of discrete arithmetic) or measure to measure (μέτρον πρὸς μέτρον, continuous geometric ratios encompassing incommensurable magnitudes via Eudoxus's Book V theory of proportion!).

  • Philebus 51c3–4 — εὐθύ τι λέγω… καὶ περιφερές, καὶ ἀπὸ τούτων δὴ τά τε τοῖς τόρνοις γιγνόμενα ἐπίπεδα καὶ στερεὰ καὶ τὰ τοῖς κανόσι καὶ γωνίαις: Plato names the exact toolkit of constructive geometry: τόρνος (lathe/compass generating circles, cylinders, and spheres), κανών (straightedge), and γωνία (carpenter’s/geometer’s right-angle square).

III. Peripheral Sister-Corpus Optical & Axiom Loci Catalogued Here

Three peripheral sister-corpus citations in geometor/euclid and geometor/elements complement Unit VI’s philosophy of measure and optical geometry:

  • plato:laws:777D (Laws VI 777d): Plato quotes as an old proverb ‘if equals be added to unequals, the wholes become unequal’—cited by Heath on Euclid Common Notion 3–4 as pre-Euclidean testimony to quantitative axioms.

  • plato:rep:X.602c (Republic X 602c–603a): A straight stick partially immersed in water appears bent to sense-perception, and concave/convex shading deceives sight; numbering, measuring, and weighing (τὸ μετρεῖν καὶ ἀριθμεῖν καὶ ἱστάναι) rescue the rational part of the soul from optical illusion.

  • plato:sophist:235.d (Sophist 235d–236c): Distinguishes icastic art (εἰκαστική), which preserves true invariant geometric proportions (τὰς οἰκείας συμμετρίας), from phantastic perspective (φανταστική), which alters upper vs. lower proportions in colossal statues to compensate for visual foreshortening (scenographia).

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