Mathematical & Philosophical Authorities

Prosopography of Pythagorean, Academic, Neoplatonic, Scholastic & Modern Authorities

Chronological and philological register of 27 authorities cited by Nicomachus or central to the transmission and mathematical analysis of the Introductio Arithmetica.

Pythagoras of Samos (c. 570–495 BCE)

• Greek / Latin: Πυθαγόρας ὁ Σάμιος | Role: Founder of Pythagorean philosophy & mathematical science

• Nicomachus Loci: I.1 (pp. 1–2), II.21–29 (pp. 119–147)

Honored by Nicomachus (I.1.1) as the first to define philosophy as the ‘love of wisdom’ (φιλοσοφία), restricting wisdom to the exact science of eternal, immaterial realities accessible through the four mathemata.

Philolaus of Croton (c. 470–385 BCE)

• Greek / Latin: Φιλόλαος ὁ Κροτωνιάτης | Role: Pythagorean philosopher & harmonic theorist

• Nicomachus Loci: II.18, II.26.2 (p. 135), II.29 (pp. 144–147)

Formulated the metaphysics of Limit and Unlimited; called the cube ‘geometric harmony’ (γεωμετρικὴ ἁρμονία) because its 6 faces, 8 vertices, and 12 edges embody the harmonic mean (II.26.2, II.29).

Archytas of Tarentum (c. 428–347 BCE)

• Greek / Latin: Ἀρχύτας ὁ Ταραντῖνος | Role: Pythagorean mathematician, harmonic theorist & statesman

• Nicomachus Loci: I.3.4 (pp. 6–7), II.22–25

Quoted verbatim by Nicomachus in Book I, Ch. 3.4 (‘For these sciences seem to be sisters’: arithmetic, geometry, sphaerics, and music); defined the three classical means (arithmetic, geometric, harmonic, originally ‘subcontrary’) and proved no rational mean exists in a superparticular ratio.

Androcydes the Pythagorean (4th cent. BCE)

• Greek / Latin: Ἀνδροκύδης ὁ Πυθαγορικός | Role: Author On Pythagorean Symbols

• Nicomachus Loci: I.3.3 (p. 6)

Cited by Nicomachus (I.3.3) alongside Eubulides, Aristoxenus, Hippobotus, and Neanthes as confirming the Pythagorean fourfold curriculum of quantity and magnitude.

Plato of Athens (c. 428–348 BCE)

• Greek / Latin: Πλάτων | Role: Founder of the Athenian Academy

• Nicomachus Loci: I.2–3 (pp. 2–9), I.5, II.18, II.24, II.29

Repeatedly invoked by Nicomachus: Timaeus 27d (Being vs. flux, I.2), Republic VII 527d–533d (‘eye of the soul’ & ‘steps and bridges’ of the mathemata, I.3.5–7), Epinomis 991d–992a (‘bond of all these’, I.3.5), Timaeus 35a–36b (Same & Other and World-Soul scale, II.18 & II.29), and the Nuptial Number (Republic VIII 546b–c, II.24).

Eudoxus of Cnidus (c. 390–337 BCE)

• Greek / Latin: Εὔδοξος ὁ Κνίδιος | Role: Geometer, astronomer & theorist of proportion

• Nicomachus Loci: II.22.1 (p. 122), II.28 (pp. 139–144)

Credited in ancient mathematical history (Iamblichus on Nicomachus II.28) with adding the 4th, 5th, and 6th subcontrary means to the three archaic Pythagorean means.

Aristotle of Stagira (384–322 BCE)

• Greek / Latin: Ἀριστοτέλης | Role: Founder of the Lyceum & logician

• Nicomachus Loci: I.2–5 (pp. 4–12), II.18

Distinguished discrete and continuous quantity in Categories 6 (4b20) and formulated the principle of ontological priority by non-reciprocal elimination (συναναιρεῖ μὲν, οὐ συναναιρεῖται δέ) deployed by Nicomachus in I.4–5 to rank the Quadrivium.

Euclid of Alexandria (fl. c. 300 BCE)

• Greek / Latin: Εὐκλείδης | Role: Author of the Elements

• Nicomachus Loci: I.11–16 (cf. Elem. VII & IX.36), II.21–27 (cf. Elem. V & VI.13)

Demonstrated the dyadic sufficiency theorem for even perfect numbers in Elements IX.36 and foundational number theory across Books VII–IX using line-segment diagrams, contrasted with Nicomachus’s inductive α-pebble and tabular style.

Eratosthenes of Cyrene (c. 276–194 BCE)

• Greek / Latin: Ἐρατοσθένης ὁ Κυρηναῖος | Role: Alexandrian polymath, geographer & number theorist

• Nicomachus Loci: I.13.2–9 (pp. 29–35), I.23

Inventor of the Sieve (κόσκινον, preserved solely in its odd-only form by Nicomachus I.13) AND documented in Iamblichus/Pappus for deriving all ratios from Equality via the three-term rule expounded by Nicomachus in I.23.

Myonides & Euphranor (Hellenistic / Neo-Pythagorean)

• Greek / Latin: Μυωνίδης καὶ Εὐφράνωρ | Role: Neo-Pythagorean theorists of proportion

• Nicomachus Loci: II.22.1, II.28.6–11 (pp. 141–144)

Credited by Iamblichus (In Nicomachi Arith. Introd.) with introducing the four later Neo-Pythagorean means (Means 7–10) catalogued by Nicomachus in Book II, Ch. 28.

Nicomachus of Gerasa (c. 60–120 CE)

• Greek / Latin: Νικόμαχος ὁ Γερασηνός | Role: Neo-Pythagorean philosopher, arithmetician & harmonic theorist

• Nicomachus Loci: Books I–II passim (Hoche pp. 1–147)

Born at Gerasa in the Syrian Decapolis; author of the Introductio Arithmetica (Books I–II), the Enchiridion Harmonices, and the lost Theologumena Arithmeticae. His treatise served via Boethius as the universal textbook of arithmetic for a thousand years.

Theon of Smyrna (fl. c. 115–140 CE)

• Greek / Latin: Θέων ὁ Σμυρναῖος | Role: Middle Platonist mathematician

• Nicomachus Loci: cf. I.23, II.8–12, II.19–20

Contemporary of Nicomachus and author of Mathematics Useful for Understanding Plato, preserving parallel accounts of side-and-diagonal numbers, figurate gnomons, and the derivation of ratios from equality.

Apuleius of Madauros (c. 124–170 CE)

• Greek / Latin: Ἀπουλήιος | Role: Latin Platonist & first translator of Nicomachus

• Nicomachus Loci: Latin transmission

Composed the earliest (now lost) Latin translation of Nicomachus’s Introductio Arithmetica within a single generation of its composition, as attested by Cassiodorus and Isidore of Seville.

Iamblichus of Chalcis (c. 245–325 CE)

• Greek / Latin: Ἰάμβλιχος ὁ Χαλκιδεύς | Role: Neoplatonic philosopher & commentator on Nicomachus

• Nicomachus Loci: Commentary on Books I–II

Elevated Nicomachus’s Introductio to Book IV of his ten-book Collection of Pythagorean Doctrines (In Nicomachi Arithmeticam Introductionem, ed. Pistelli 1894), recording the history of the Ten Means.

Pappus of Alexandria (fl. c. 320 CE)

• Greek / Latin: Πάππος ὁ Ἀλεξανδρεύς | Role: Alexandrian geometer (Mathematical Collection)

• Nicomachus Loci: cf. II.21–28

In Synagoge Book III (Props. 12–27), constructs all three classical means in a single semicircle and systematically analyzes the Ten Nicomachean Means (geometor/pappus).

Proclus Lycius Diadochus (412–485 CE)

• Greek / Latin: Πρόκλος ὁ Διάδοχος | Role: Head of the Athenian Academy; reported in tradition as Nicomachus reborn

• Nicomachus Loci: cf. I.1–5 (Proclus Friedlein 35–37)

Marinus’s Vita Procli (§28) records that Proclus believed his own soul belonged to the Pythagorean chain of Nicomachus of Gerasa; his Euclid Prologue (geometor/proclus, Friedlein 35–37) adopts Nicomachus’s exact division of the Quadrivium.

John Philoponus (c. 490–570 CE)

• Greek / Latin: Ἰωάννης ὁ Φιλόπονος | Role: Alexandrian commentator on Aristotle & Nicomachus

• Nicomachus Loci: Alexandrian commentary on Books I–II

Authored the largest surviving Greek scholia/commentary on both books of Nicomachus (ed. Richard Hoche, 1864–1867), derived from the Alexandrian lectures of Ammonius Hermeiou.

Anicius Manlius Severinus Boethius (c. 480–524 CE)

• Greek / Latin: Βοήθιος | Role: Roman statesman, philosopher & Latin adapter of Nicomachus

• Nicomachus Loci: De Inst. Arith. I–II (paralleling Nicomachus I–II)

Adapted Nicomachus into De Institutione Arithmetica (ed. Friedlein 1867) and De Institutione Musica, coining the enduring Latin word quadrivium (‘fourfold path’, De Inst. Arith. I.1) directly rendering Nicomachus I.3–4.

Thomas Aquinas (1225–1274)

• Greek / Latin: Θωμᾶς Ἀκινάτης | Role: Scholastic philosopher & theologian

• Nicomachus Loci: Reception via Boethius I.1–2

Received Nicomachean discrete-vs.-continuous quantity, relative ratio, and the hierarchy of the Quadrivium through Boethius in his Commentary on Boethius’s De Trinitate (geometor/aquinas).

Leonardo Pisano (Fibonacci) (c. 1170–1250)

• Greek / Latin: Λεονάρδος ὁ Πισανός | Role: Author of Liber Abaci & Liber Quadratorum

• Nicomachus Loci: cf. II.28.10–11 (p. 143–144: γ, ε, η = 3, 5, 8)

Developed additive recurrences whose 3-term proportion (z − x):(z − m) = m:x <-> x + m = z was already isolated as the Tenth Mean (3, 5, 8) in Nicomachus II.28.10–11 (geometor/fibonacci).

Luca Pacioli (c. 1447–1517)

• Greek / Latin: Λουκᾶς Πατσιόλι | Role: Franciscan mathematician (Summa & De Divina Proportione)

• Nicomachus Loci: Reception of I.17–23 & II.21–29

Built Renaissance proportion nomenclature, figurate progressions, and the contrast of rational superparticulars vs. the irrational Divine Proportion directly upon Boethius–Nicomachus (geometor/pacioli).

Johannes Kepler (1571–1630)

• Greek / Latin: Ἰωάννης Κέπλερ | Role: Astronomer & cosmic harmonic geometer

• Nicomachus Loci: cf. I.3–5 & II.29

Unified the Quadrivium’s geometry, music, and astronomy in Harmonices Mundi (1619), critiquing purely arithmetic Neo-Pythagorean number symbolism while fulfilling Nicomachus’s harmonic cosmos through constructible polygons (geometor/kepler).

Richard Gottfried Hoche (1834–1906)

• Greek / Latin: Ῥιχάρδος Χόχε | Role: Classical philologist & editor of the 1866 Teubner Greek text

• Nicomachus Loci: Critical Edition, Leipzig: Teubner, 1866 (pp. i–xi, 1–147)

Edited both John Philoponus’s Commentary on Nicomachus (1864–1867) and the standard critical Teubner edition Nicomachi Geraseni Pythagorei Introductionis Arithmeticae Libri II (Leipzig, 1866), establishing the canonical text and α-pebble diagrams.

Gottfried Friedlein (1828–1875)

• Greek / Latin: Γοτφρῖδος Φρίντλαϊν | Role: Editor of Boethius (1867) and Proclus (1873)

• Nicomachus Loci: Boethius Teubner Edition, 1867

Published the critical Teubner edition of Boethius’s De Institutione Arithmetica Libri Duo and De Institutione Musica Libri Quinque (Leipzig, 1867) immediately following Hoche’s 1866 Greek Nicomachus.

Martin Luther D’Ooge (1839–1915)

• Greek / Latin: Μαρτῖνος Λούθηρος Ντ’Οόγκε | Role: Professor of Greek at the University of Michigan & first English translator of Nicomachus

• Nicomachus Loci: English Translation & Notes, 1926 (pp. 181–286)

Completed the authoritative first English translation of Nicomachus’s Introduction to Arithmetic (published posthumously in 1926 by Macmillan / University of Michigan Studies, Humanistic Series Vol. XVI, with studies by F. E. Robbins and L. C. Karpinski).

Frank Egleston Robbins & Louis Charles Karpinski (1884–1963 / 1878–1956)

• Greek / Latin: ** | Role: Co-authors of Studies in Greek Arithmetic (1926)

• Nicomachus Loci: Michigan Edition Part I, pp. 3–177 (1926)

Completed and annotated D’Ooge’s 1926 Michigan edition with comprehensive philological, mathematical, and historical studies on Greek arithmetic, sources of Nicomachus, and medieval reception.

Sir Thomas Little Heath (1861–1940)

• Greek / Latin: ** | Role: Historian of Greek mathematics

• Nicomachus Loci: Hist. Greek Math. I, pp. 97–115

Provided standard modern mathematical analyses of Nicomachus’s Introductio Arithmetica in A History of Greek Mathematics (1921, Vol. I, pp. 97–115) and The Thirteen Books of Euclid’s Elements (geometor/euclid).