Nicomachus of Gerasa

← Encyclopedia

Nicomachus of Gerasa was a Greek mathematician of the Neo-Pythagorean revival, active around the year 100 in the Roman city of Gerasa, in what is now Jordan. Little is known of his life beyond the writings that carry his name. He belongs to the wider resurgence of Pythagorean thought in the early Roman Empire — the movement that read the cosmos as built on number, and treated mathematics not as a tool but as the doorway to the divine.

The city itself supplies the first fact. Gerasa was one of the cities of the Decapolis, a prosperous Greek-speaking town of the Roman East with colonnaded streets and a temple to Zeus, planted on the eastern edge of the empire among Semitic and Nabataean neighbors. A mathematician working there around the turn of the second century stood at a crossroads where Greek learning, the older arithmetic of the Near East, and the administrative Latin of Rome met in a single marketplace. That Nicomachus wrote in Greek, thought as a Pythagorean, and was read first by Greeks and then by Latins is the shape of his whole afterlife in miniature. He is dated only by inference — he cites Thrasyllus, the court astrologer of Tiberius, and is in turn quoted by Apuleius in the middle of the second century — which fixes his floruit to roughly 100, no more precisely.

The Pythagorean revival

The tradition Nicomachus inherited was already six centuries old and largely reconstructed. Pythagoras of Samos had left no writings; the school he founded at Croton in the sixth century BCE survived as a cloud of doctrine, ascetic rule, and number-lore transmitted through Philolaus, Archytas, and the long Hellenistic afterlife of the name. By the first century BCE that afterlife had hardened into a movement — Neopythagoreanism — that fused the recovered number-doctrine with the metaphysics of Plato, read backwards as if Plato had been a Pythagorean all along. Figures such as Moderatus of Gades and Numenius of Apamea in this revival treated arithmetic and the dialogues as a single inheritance, and it is to this current, more than to any verifiable sixth-century original, that Nicomachus belongs. What he preserved was therefore not Pythagoras unmediated but Pythagoreanism in its imperial-age form: the systematized arithmology in which the first numbers carry fixed qualitative meaning, and the conviction that the structure of number is the structure of the world.

He is the most influential transmitter of that conviction, not its boldest thinker. His genius was pedagogical. Where the older Pythagorean material was scattered, oracular, and guarded, Nicomachus laid it out as a textbook — clear, ordered, teachable, with worked examples and a definite sequence — and in doing so made it portable across a thousand years.

The Introduction to Arithmetic

His surviving works are two. The Introduction to Arithmetic (Arithmetike eisagoge) sets out the properties of whole numbers — even and odd, prime and composite, the figured numbers, the means and ratios — and frames them throughout as the foundation beneath geometry, music, and astronomy.

The book is built as an ascent. It opens by dividing quantity into the discrete and the continuous, and assigns to each its science: arithmetic governs the discrete-in-itself, music the discrete-in-relation, geometry the continuous-at-rest, astronomy the continuous-in-motion. These are the four mathematical sciences, and Nicomachus insists on their order — arithmetic first, because the others presuppose it. Numbers do not need points or magnitudes to exist; points and magnitudes need number to be counted. Arithmetic is therefore prior, the others derivative, and the student who would climb to the contemplation of the changeless must begin where the changeless is purest.

From there the text descends into the kinds of number. It classifies the even and the odd, then subdivides each — the even-times-even, the even-times-odd, the odd-times-even; the prime, the composite, and the numbers prime to one another. It treats the perfect numbers, those equal to the sum of their proper divisors, and gives the rule for generating them; it names the deficient and the superabundant by the same measure. It develops the figured numbers — the triangular, square, pentagonal, and the solid figures built from them — showing how number takes shape, so that arithmetic already contains geometry in seed. The second book turns to ratio and proportion, cataloguing the ten kinds of proportional relation and the three classical means — arithmetic, geometric, and harmonic — that the Pythagoreans took to govern both music and the soul. Along the way the text preserves the first surviving Greek multiplication table and states the elegant theorem, still attached to his name, that the sum of the first run of cubes equals the square of a triangular number.

What distinguishes the Introduction is not the mathematics, much of which descends from Euclid and the earlier tradition, but its frame. Euclid had proved; Nicomachus expounds. Rigor is not his aim, and by the standard of the Elements the book is loose — it asserts where Euclid demonstrates, and generalizes from examples. But it was never meant as a treatise in the Euclidean sense. It is a philosophy of number wearing the dress of a handbook, and its looseness is the price of its reach: it could be read by anyone, and it was.

The Manual of Harmonics

The Manual of Harmonics (Encheiridion harmonikes) applies the same arithmetic of ratio to the intervals of music. Here the doctrine of the Introduction is put to work on sound. The consonances — the octave, the fifth, the fourth — are shown to rest on the simplest whole-number ratios, 2:1, 3:2, 4:3, and the difference between fifth and fourth on the ratio 9:8, the whole tone. Nicomachus recounts the old story of Pythagoras at the smithy, hearing the hammers ring at consonant pitches and discovering that the harmonies of the audible world are governed by number — a tale that is acoustically impossible but doctrinally exact, since its point is that what the ear hears as beauty the mind grasps as ratio. The arithmetic of the Introduction and the harmonics of the Manual are thus a single teaching seen from two sides: the same proportions that order the abstract decad order the strings of the lyre and, by extension, the motions of the heavens. This is the genuine harmony-of-ratio motif that runs from Pythagoras through Nicomachus into the medieval doctrine of the music of the spheres — and the same motif that the Greek imagination personified as the goddess Harmonia, born of strife and love, married into the founding house of Thebes; the Pythagorean-Platonic harmonics that make number, sound, and being a single structure are its technical core.

The lost Theology of Arithmetic

A third work, a Theology of Arithmetic assigning divine significance to each of the first ten numbers, is lost; what is known of it comes from later writers who quoted and expanded it, so the line between Nicomachus’s own teaching and his successors’ is hard to draw. The reconstruction is genuinely vexed. A surviving compilation transmitted under the title Theologoumena arithmeticae — handed down in some manuscripts under the name of Iamblichus, though more probably a later cento drawing on Nicomachus, on Anatolius of Laodicea, and on others — preserves much of the material but mingles it past disentangling. From these secondary witnesses the doctrine can be sketched but not securely attributed. Each of the numbers from one to ten carried a cluster of names and powers: the monad as source and seed of all number, neither odd nor even but the principle of both; the dyad as the first division, the origin of otherness and matter; the triad as the first number proper, beginning, middle, and end; the tetrad as the root of the decad, since one, two, three, and four together make ten — the tetractys, the figure the Pythagoreans held sacred; and the decad itself as the perfect number, the completion of the series within which all later number merely repeats. To name each number was, in this teaching, to name a stage in the unfolding of the cosmos.

The doctrine: number before the world

What Nicomachus held, in the Pythagorean manner, was that number is prior to the physical world rather than abstracted from it — that the things counted exist because the numbers do, and that to study arithmetic is to study the pattern by which the cosmos was ordered. This is the load-bearing claim, and it is worth holding it apart from the modern habit of treating number as a convenience the mind invents to keep track of things. For Nicomachus the order runs the other way. Number is not extracted from a heap of apples; the apples are countable because number already structures what they are. The numbers exist before and above the sensible world, in the planning intelligence by which the cosmos was laid out, and the visible order — the seasons, the intervals of music, the distances of the planets — is the trace of that arithmetic written into matter. To learn arithmetic, then, is not to acquire a skill but to retrace the thought of the maker. Counting and contemplation become, at the limit, the same act.

This places his work on the threshold of the metaphysics that the later Neoplatonists would build. The doctrine that all multiplicity flows from a single source, and that reality descends in ordered ranks from that source, has its arithmetical rehearsal in the generation of all number from the monad; the One of the Neoplatonists and the monad of the arithmeticians stand a degree apart as principles, yet the road from the second to the first runs straight through this material. Nicomachus himself stops short of the full systematic emanation; he is a mathematician with a theology of number, not a metaphysician of the One. But the emanation scheme that organizes the later tradition found in his arithmetic a ready ladder.

Transmission: Iamblichus, Boethius, and the quadrivium

The Introduction was the channel through which much of that inheritance traveled. Within two centuries it had become a school text, and the Neoplatonists treated it as a classic to be expounded. Iamblichus of Chalcis, the fourth-century theurgist, wrote a commentary on it — his In Nicomachi arithmeticam introductionem, which forms part of his larger nine-book collection of Pythagorean doctrine — folding Nicomachus’s arithmetic into a program for the recovery of Pythagoreanism as a way of life. The wider Neoplatonism line carried the same reverence: Proclus, the great fifth-century systematizer of the Athenian school, was said by his biographer to have believed he was Nicomachus reborn, and Proclus’s own number-theology stands in the line the Introduction fixed. In the Greek East the book remained a standard primer down through the Byzantine centuries.

The decisive turn came in the Latin West. In the early sixth century Boethius — Roman senator, philosopher, and the last great Latin transmitter of Greek learning before the long medieval night — rendered the Introduction into Latin as the De institutione arithmetica. It was a free adaptation rather than a literal translation: Boethius compressed, reorganized, and Christianized his source, but the substance is Nicomachus’s, and the De institutione arithmetica opens with the same insistence that arithmetic is the foundation of the mathematical sciences. It was in this work that Boethius coined the word that would name them — quadrivium, the four roads by which the mind travels to wisdom. Through Boethius the Introduction became the standard arithmetic of the medieval schools, one of the four mathematical arts of the quadrivium, and the place where generations first met the idea that counting and contemplation might be the same act. For a thousand years a student who learned arithmetic in the Latin West learned Nicomachus, at one remove, without knowing the name.

Scholarship and texts

The two surviving works are well served by modern editions, and the Greek Introduction survives intact. The standard Greek text remains Richard Hoche’s Teubner edition, Nicomachi Geraseni Pythagorei Introductionis arithmeticae libri II (Leipzig, 1866), long in the public domain. The single English translation is the posthumous volume of Martin Luther D’Ooge, Nicomachus of Gerasa: Introduction to Arithmetic (New York, 1926), issued in the University of Michigan’s Humanistic Series, whose accompanying studies by Frank Egleston Robbins and Louis Charles Karpinski remain the fullest account in English of Nicomachus’s place between Euclid and Boethius. For the Manual of Harmonics the standard modern rendering is Flora R. Levin’s translation and commentary, The Manual of Harmonics of Nicomachus the Pythagorean (Phanes Press, 1994), and the harmonic text is set among its ancient peers in Andrew Barker’s Greek Musical Writings, volume two (Cambridge, 1989). The lost Theology of Arithmetic and the pseudonymous Theologoumena arithmeticae that preserves its matter are anatomized in Joel Kalvesmaki’s open-access study, The Theology of Arithmetic: Number Symbolism in Platonism and Early Christianity (Center for Hellenic Studies, 2013), which traces the number-theology from Nicomachus forward into the early Christian writers who took it up. The Greek of Iamblichus’s commentary survives in Hermenegildus Pistelli’s Teubner edition of 1894, the principal witness to how late antiquity read the Introduction line by line.

The long afterlife

His later reputation outran the record. The Pythagorean number-mysticism that runs through Renaissance and modern esoteric writing on the meaning of numbers draws, often at several removes, on the tradition he helped fix in writing. When Marsilio Ficino and the circle of Renaissance hermetism revived the ancient theology of number in fifteenth-century Florence, the arithmetic they inherited through Boethius was Nicomachus’s, and the conviction that the cosmos is a written number descended through the same line. The later popular numerology that assigns meanings to numbers and to the numerical value of names invokes the Pythagorean inheritance, but operates as a divinatory technique cut loose from the arithmological metaphysics that gave the original its seriousness — it shares the method and not the cosmology. Through all of it the Introduction persisted: copied, commented, translated, abridged, the handbook that taught the West to count and to believe that counting reached the divine — though how much of it he would have recognized as his own is another matter.

→ In the library: Westcott — Numbers: Their Occult Power and Mystic Virtues (1911)

→ Related: Neoplatonism · The One · Emanation · Pythagoras · Pythagoreanism · Neopythagoreanism · Iamblichus · Boethius · Proclus · Plato · Numerology · Pythagorean Arithmology · Pythagorean Platonic Harmonics · Marsilio Ficino · Renaissance Hermetism