Book of Lemmas (Liber Assumptorum) — Philological & Mathematical Notes

1. Arabic-to-Latin Transmission (Kitāb al-Maʾkhūdhāt → Florence 1661)

No Greek manuscript of the Book of Lemmas survives. Its preservation is owed to ninth-century Baghdad geometer Thābit ibn Qurra (836–901 AD), whose Arabic translation (Kitāb al-Maʾkhūdhāt li-Arshimīdis) was commented upon in the tenth century by Abū al-Ḥasan ʿAlī ibn Aḥmad al-Nasawī (“Almochtasso”). In 1661, Abraham Ecchellensis and Giovanni Alfonso Borelli discovered an Arabic codex in the Laurentian Medici Library at Florence and published the Latin editio princeps (ecchellensis-borelli:1661), later re-edited by Heiberg (heiberg:1881, Vol. II, pp. 426–445; 2nd ed. pp. 510–525).

Why does the text say in Props. 4 and 14 “what Archimedes called an arbelos” and “which Archimedes calls a salinon”? As Heath (heath:1897, pp. xxxii, 301) explains, a late Greek or Arabic compiler gathered genuinely Archimedean theorems—attested independently by Pappus (pappus:synagoge:iv, IV.14–18 on the arbelos and IV.36 on neusis trisection)—into a handbook of planar lemmas, adding third-person glosses on Archimedes’ picturesque Syracusan names.

2. GEOMETOR Cross-Corpus Concordance

  • Euclid (`geometor/euclid`) (geometor:euclid): Resolves archimedes:liber-assumptorum cited in Heath’s Euclid on III.31/III.35, IV.10–11, VI.33 (angle trisection), and XIII.9 (golden section).

  • Pappus (`geometor/pappus`) (geometor:pappus, pappus:synagoge:iv): Propositions 1, 4, 5, 6 form the direct foundation of Synagoge IV.14–20 (the infinite Steiner/Pappus circle chain in the arbelos, effortlessly proved today by circle inversion centered at A or B), while Proposition 12 is an early synthetic proof of the polar-orthocenter theorem for cyclic quadrilaterals inscribed in a semicircle!

  • Divine Proportion (`geometor/divine` & `geometor/elements`): Proposition 15 connects the regular pentagon and decagon directly to Euclid XIII. 9 over the golden number field .

  • Rational Trigonometry (`geometor/wildberger`) (geometor:wildberger): Proposition 11 () and the rational twin-circle harmonic mean in Proposition 5 are pure polynomial identities over requiring zero square roots!