Bibliography & Critical Editions

Critical Greek & Latin Editions, Public-Domain Translations, Ancient Commentaries & Canon Sources

Authoritative editions, ancient commentaries, and companion mathematical treatises (17 entries) cited across Introductio Arithmetica.

Nicomachus of Gerasa (ed. Richard Hoche) — Nicomachi Geraseni Pythagorei Introductionis Arithmeticae Libri II (1866)

  • Citation Key: hoche:1866 | Locus: Leipzig: B. G. Teubner, 1866 (pp. i–xi, 1–147; CTS urn:cts:greekLit:tlg0358.tlg001.1st1K-grc1)

  • Significance: Standard critical Teubner Greek edition collated from Munich (Cizensis, Monacensis G), Göttingen, Hamburg, and Paris manuscripts, preserving ancient α-pebble figurate diagrams and the synoptic table of Ten Means.

Martin Luther D’Ooge (trans.), with F. E. Robbins & L. C. Karpinski — Nicomachus of Gerasa: Introduction to Arithmetic (1926)

  • Citation Key: dooge:1926 | Locus: New York: Macmillan / University of Michigan Studies, Humanistic Series Vol. XVI, 1926 (pp. 181–286)

  • Significance: First English translation of Nicomachus (Public Domain: published 1926, >95 years in US), accompanied by Studies in Greek Arithmetic (pp. 3–177) and critical mathematical footnotes.

Anicius Manlius Severinus Boethius (ed. Gottfried Friedlein) — De Institutione Arithmetica Libri Duo, De Institutione Musica Libri Quinque (c. 505 CE / ed. 1867)

  • Citation Key: boethius:friedlein:1867 | Locus: Leipzig: B. G. Teubner, 1867 (pp. 1–173)

  • Significance: Boethius’s authoritative Latin adaptation of Nicomachus, coining the term quadrivium (I.1, p. 7 Friedlein) and transmitting Nicomachean ratio species, equality unfolding, figurate numbers, and the Ten Means to the Latin West.

Iamblichus of Chalcis (ed. Ermenegildo Pistelli) — In Nicomachi Arithmeticam Introductionem Liber (c. 300 CE / ed. 1894)

  • Citation Key: iamblichus:pistelli:1894 | Locus: Leipzig: B. G. Teubner, 1894

  • Significance: Fourth book of Iamblichus’s On Pythagoreanism, preserving crucial historical testimonies on Archytas, Eudoxus, Hippasus, Myonides, Euphranor, and Eratosthenes’ equality rule.

John Philoponus (ed. Richard Hoche) — Ioannis Philoponi in Nicomachi Arithmeticam Introductionem Libri Duo (6th cent. CE / ed. 1864–1867)

  • Citation Key: philoponus:hoche:1864 | Locus: Wesel / Berlin: Bagel & Calvary, 1864–1867

  • Significance: Extensive Alexandrian Neoplatonic lecture commentary on Books I and II of Nicomachus edited by Richard Hoche immediately prior to his 1866 Teubner edition.

Theon of Smyrna (ed. Eduard Hiller) — Expositio Rerum Mathematicarum ad Legendum Platonem Utilium (2nd cent. CE / ed. 1878)

  • Citation Key: theon:hiller:1878 | Locus: Leipzig: B. G. Teubner, 1878 (pp. 107–111 on Adrastos & Equality)

  • Significance: Parallel Middle Platonic mathematical handbook preserving Adrastos of Aphrodisias’s proof of the three-term equality unfolding and side-and-diagonal numbers.

Archytas of Tarentum — Harmonics, Fragment B1 & B2 (DK 47 B1–B2) (c. 390 BCE)

  • Citation Key: archytas:dk47b1 | Locus: apud Nicomachum I.3.4 (Hoche pp. 6–7) & Porphyry In Ptol. Harm.

  • Significance: Authentic Doric fragment quoted by Nicomachus declaring astronomy, geometry, arithmetic, and music to be ‘sister sciences’ (ἀδελφεαὶ τὰ μαθήματα), plus definition of the three classical means.

Plato — Timaeus (c. 360 BCE)

  • Citation Key: plato:timaeus:27d-36d | Locus: 27d–28a, 31b–32c, 35a–36d (`geometor/plato`)

  • Significance: Quoted verbatim by Nicomachus at I.2.3 (27d: eternal Being vs. continual Becoming) and expounded across II.18–29 (Same & Other, two cubic means, and harmonic division of the World-Soul).

Plato — Republic (c. 375 BCE)

  • Citation Key: plato:rep:vii-525a-534e | Locus: VII, 525a–534e & VIII, 546b–c (`geometor/plato`)

  • Significance: Quoted by Nicomachus at I.3.7 (527d–e: purifying the ‘eye of the soul’ blinded by other pursuits) and II.24 (Nuptial Number).

Plato (or Philip of Opus) — Epinomis (c. 347 BCE)

  • Citation Key: plato:epinomis:991d-992a | Locus: 991d–992a

  • Significance: Quoted by Nicomachus at I.3.5 on the ‘single natural bond’ (δεσμὸς πεφυκώς) uniting all mathematical sciences and rendering the learner blessed.

Aristotle — Categories & Posterior Analytics (c. 335 BCE)

  • Citation Key: aristotle:cat:4b20 | Locus: Cat. 6, 4b20–5b10; Cat. 12, 14a29; Anal. Post. I.27, 87a31 (`geometor/aristotle`)

  • Significance: Foundational distinction between discrete and continuous quantity and ontological priority by non-reciprocal elimination (συναναιρεῖ μὲν, οὐ συναναιρεῖται δέ) underlying Nicomachus I.2–5.

Euclid — Elements (Στοιχεῖα), Books V, VI, VII–IX (c. 300 BCE)

  • Citation Key: euclid:elements:vii-ix | Locus: VI.13, VII.1–2, VII.Def.22, IX.20, IX.35–36 (`geometor/euclid`, `geometor/elements`)

  • Significance: Euclidean deductive counterparts to Nicomachus: mutual subtraction/anthyphairesis (VII.1–2), infinitude of primes (IX.20), geometric series sum & even perfect numbers (IX.35–36), and geometric mean construction (VI.13).

Pappus of Alexandria — Synagoge (Mathematical Collection), Book III (c. 320 CE)

  • Citation Key: pappus:synagoge:iii | Locus: Book III, Props. 12–27 (`geometor/pappus`)

  • Significance: Constructive geometric synthesis of the three classical means in a single semicircle and systematic geometric derivation of all ten Nicomachean means.

Proclus Diadochus — Commentary on the First Book of Euclid’s Elements (c. 470 CE)

  • Citation Key: proclus:in-euclid:35-37 | Locus: Prologue I, Ch. XII, Friedlein pp. 35–37 (`geometor/proclus`)

  • Significance: Reproduces Nicomachus’s exact Pythagorean fourfold schema of discrete (per se / ad aliud) and continuous (stationary / moving) quantity.

Thomas Aquinas — Super Boetium De Trinitate (c. 1257–1259)

  • Citation Key: aquinas:in-boethium-trinitate | Locus: qq. 5–6 (`geometor/aquinas`)

  • Significance: Scholastic epistemology of the Quadrivium, intelligible matter, and subalternation of music to arithmetic and astronomy to geometry.

Luca Pacioli — Summa de Arithmetica (1494) & De Divina Proportione (1509) (1509)

  • Citation Key: pacioli:divina-proportione:1509 | Locus: Venice, 1494 / 1509 (`geometor/pacioli`)

  • Significance: Renaissance culmination of Boethian–Nicomachean proportion terminology alongside Euclid’s extreme and mean ratio.

Sir Thomas Little Heath — A History of Greek Mathematics, Vol. I: From Thales to Euclid (1921)

  • Citation Key: heath:hgm:1921 | Locus: Oxford: Clarendon Press, 1921 (pp. 97–115)

  • Significance: Authoritative modern algebraic analysis of Nicomachus’s Introductio Arithmetica, gnomon recurrences, sum of cubes, and Ten Means.