Commentary on Propositions I.33–I.48: Areas, Application & Pythagoras — GEOMETOR Commentary & Concordance Notes

GEOMETOR Philosophical & Mathematical Commentary

  1. Locus Theorems (`Friedlein 394–396`) & Textual Lacuna at `I.36`–`I.37` (`Friedlein 403, 5`): Proclus’s recognition of Prop. I.35 as the prototype of locus theorems (τοπικὰ θεωρήματα) links elementary area-shearing to higher Greek locus geometry (Hermotimus, Aristaeus, Apollonius). Note at Friedlein 403, 5 (cited by Heath, Vol. I, p. 332) a famous manuscript lacuna spanning the end of Proclus’s note on I.36 and the opening of I.37, present in all extant Greek codices and noted by Barocius and Friedlein. (Also note Taylor’s 1792 heading misprint PROPOSITION LXVIII for XLVIII, normalized in our heading.)

  2. Application of Areas (`Friedlein 419–420`) & The Golden Section: Proclus’s preservation of Eudemus's report on Prop. I.44—that parabole, hyperbole, and elleipsis belong to the ancient Pythagorean muse—unlocks the unity of Greek geometry: - Exact application (parabole, I.44–I.45) solves $ax = S$ geometrically via the gnomon complement (I.43). - Application with square excess (hyperbole) in II.11 and VI.30 ($x(a + x) = a^2$) constructs division in extreme and mean ratio—the Divine Proportion $\phi$ at the heart of GEOMETOR!

  3. Euclid’s Proof of `I.47` & Pythagorean Triples (`Friedlein 426–429`): Proclus is our sole ancient source crediting Euclid personally with the area-shearing Book I proof of Prop. I.47 (necessary because Book V’s Eudoxian proportion theory has not yet been introduced in Book I!) as well as recording side-by-side the Pythagorean odd generator $(2k+1,\; 2k^2+2k,\; 2k^2+2k+1)$ and the Platonic even generator $(2m,\; m^2-1,\; m^2+1)$.

Friedlein Loci Cited in GEOMETOR (14 Loci in pp. 384–436)

  • Friedlein 393 (text) — Commentary on Prop. I.34: why Euclid introduces the term ‘parallelogrammic areas’ (parallelogramma choria) and how Prop. I.34 constructs the concept. - euclid (proclus:in-euclid:393): Heath Vol. I, p. 325 [folio 0339]

  • Friedlein 394–396 (text) — Commentary on Prop. I.35: birth of Locus Theorems (topika theoremata) in the Elements and distinction between equality of area and congruence of figure. - euclid (proclus:in-euclid:394-6): Heath Vol. I, p. 329 [folio 0343]

  • Friedlein 396–398 (text) — Analysis of the paradoxical wonder of Prop. I.35 (shearing parallelograms of arbitrarily large perimeter having equal area, anticipating isoperimetric theory). - euclid (proclus:in-euclid:396-8): Heath Vol. I, p. 26 [folio 0040]

  • Friedlein 399–400 (text) — Completion of the remaining two positional cases of Prop. I.35 omitted in Euclid’s concise text. - euclid (proclus:in-euclid:399-400): Heath Vol. I, p. 328 [folio 0342]

  • Friedlein 403, 5 (text) — Textual transition between Props. I.36 and I.37 on equal parallelograms and triangles between the same parallels. - euclid (proclus:in-euclid:403.5): Heath Vol. I, p. 332 [folio 0346]

  • Friedlein 407, 4–17 (text) — Commentary on Prop. I.39: partial conversion when two hypotheses yield a conclusion, and exclusion of triangles on opposite sides of the base. - euclid (proclus:in-euclid:407.4-17): Heath Vol. I, p. 337 [folio 0351]

  • Friedlein 414, 15–415, 16 (text) — Commentary on Prop. I.41: Proclus’s geometric exercise extending triangle-parallelogram area relations to trapezia. - euclid (proclus:in-euclid:414.15-415.16): Heath Vol. I, p. 338 [folio 0352]

  • Friedlein 419 (text) — Commentary on Prop. I.44: Eudemus’s testimony that Application of Areas (parabole), Exceeding (hyperbole), and Falling-Short (elleipsis) are ancient discoveries of the Pythagorean muse.

  • Friedlein 423, 18 (text) — Commentary on Prop. I.46: philological distinction between ‘constructing’ (systesasthai) a triangle from parts and ‘describing on’ (anagrapsai apo) a square from a single side. - euclid (proclus:in-euclid:423.18): Heath Vol. I, p. 348 [folio 0362]

  • Friedlein 424–425 (text) — Proofs that squares described on equal straight lines are equal, and conversely that equal squares have equal sides. - euclid (proclus:in-euclid:424-5): Heath Vol. I, p. 348 [folio 0362]

  • Friedlein 426, 6 (text) — Commentary on Prop. I.47: critical assessment of the story that Pythagoras sacrificed an ox, and Proclus’s glowing tribute to Euclid’s Book I proof and Book VI generalization. - euclid (proclus:in-euclid:426.6): Heath Vol. I, p. 350 [folio 0364]

  • Friedlein 426, 6–9 (text) — Proclus’s cautious phrasing (‘if we listen to those who wish to recount ancient history…’) regarding the legend of Pythagoras’s sacrifice on discovering I.47. - euclid (proclus:in-euclid:426.6-9): Heath Vol. I, p. 319 [folio 0333]

  • Friedlein 429, 9–15 (text) — Two ancient methods for generating integer right-angled triangles: the Pythagorean odd-number rule and the Platonic even-number rule. - euclid (proclus:in-euclid:429.9-15): Heath Vol. I, p. 26 [folio 0040]

  • Friedlein 432, 9 (text) — Closing peroration of the Commentary on Book I (Prop. I.48), honoring Syrianus’s oral teaching and inviting successors to expound the remaining Books of Euclid. - euclid (proclus:in-euclid:432.9): Heath Vol. I, p. 32 [folio 0046]