Nicomachus of Gerasa

Life and Historical Context

Nicomachus of Gerasa was a Greco-Roman mathematician and Neopythagorean philosopher active in the late 1st and early 2nd centuries CE. Almost nothing certain is known about his life beyond what can be inferred from his own writings and later testimonies. He is associated with Gerasa in the Roman province of Arabia (modern Jerash in Jordan), then a Hellenized city within the broader cultural orbit of the eastern Mediterranean.

Chronologically, Nicomachus belongs to the period sometimes called Middle Platonism, when Platonist, Pythagorean, Aristotelian, and Stoic ideas circulated in complex combinations. He is usually classed as a Neopythagorean, one of several thinkers who revived and reinterpreted Pythagorean doctrines, placing special emphasis on number, harmony, and the moral-spiritual significance of mathematical study.

Later authors such as Iamblichus and Boethius cite Nicomachus as an authority on arithmetic and music. These testimonies, together with the style of his Greek and internal references, support a dating roughly between 60 and 120 CE, though precise dates remain uncertain.

Works and Doctrines

Nicomachus is known primarily for two surviving works and for several lost texts known only by title or fragment.

Introduction to Arithmetic (Arithmētikē eisagōgē)

Nicomachus’ most influential text is the two-book Introduction to Arithmetic, intended as a systematic yet accessible exposition of arithmetic for a broad learned audience. Unlike Euclid’s strictly deductive Elements, Nicomachus writes in a more didactic and discursive style, mixing definitions and classifications with examples, etymologies, and Pythagorean speculation.

Central topics include:

• Classification of numbers: Nicomachus distinguishes odd and even, prime and composite, and discusses perfect, deficient, and abundant numbers. Perfect numbers (e.g., 6, 28) are those equal to the sum of their proper divisors, a category he invests with symbolic and metaphysical significance.
• Figurate numbers: He treats triangular, square, pentagonal, and other polygonal numbers, continuing the Pythagorean interest in visual representations of numerical structures.
• Proportion and ratio: While less geometrically rigorous than Euclid, Nicomachus elaborates basic notions of arithmetic proportion and their application to musical intervals.

An important feature of the Introduction is its fusion of technical arithmetic with number mysticism. Numbers are not only tools for calculation; they are treated as ontologically primary and morally charged. For example, Nicomachus discusses the “characters” of numbers (such as the stability associated with the number 4 or the completeness of 10), reflecting an inherited Pythagorean numerology.

From a modern mathematical standpoint, the work contains little original theory and sometimes presents results without proof or with arguments that fall short of Euclidean standards. Nonetheless, it served as a standard textbook of arithmetic for centuries. The Latin philosopher Boethius drew heavily on Nicomachus in his own De institutione arithmetica, through which Nicomachean arithmetic entered the medieval quadrivium.

Manual of Harmonics (Encheiridion Harmonikon)

Nicomachus’ second extant work, the Manual of Harmonics, is a concise treatise on Pythagorean music theory, examining the mathematical ratios underlying musical intervals and scales. It follows the Pythagorean discovery that consonant intervals can be expressed as simple numerical ratios:

• Octave: 2:1
• Fifth: 3:2
• Fourth: 4:3

Nicomachus presents these ratios as evidence that music is fundamentally arithmetical and that the structure of sound reflects a deeper cosmic harmony. He connects the science of harmonics to the broader Pythagorean notion of the “music of the spheres”, the idea that the motions of the heavenly bodies form a rational, if inaudible, harmony expressible in numerical terms.

The treatise contributed to the formation of the quadrivial discipline of music in late antiquity and the Middle Ages, where music was understood not primarily as performance but as a mathematical science of proportion. As with his arithmetic, Nicomachus prioritizes doctrinal exposition and symbolic interpretation over exhaustive mathematical rigor.

Lost and Attributed Works

Ancient sources mention other works by Nicomachus, now lost, including:

• A larger treatise on arithmetic, sometimes called the Theologoumena arithmētikēs (Theology of Arithmetic), which may or may not be identical with a later compilation transmitted under that title.
• Possible writings on geometry and astronomy, consistent with the Pythagorean ideal of a mathematically ordered cosmos.

The authorship and extent of these works are subject to scholarly debate. Some late antique texts with pronounced number symbolism have been variously attributed to Nicomachus or to his school, but direct evidence is scarce.

Philosophical Significance and Legacy

Nicomachus’ enduring importance lies less in original discoveries and more in his role as a systematizer and transmitter of Pythagorean-Platonic arithmetic and musical theory.

Philosophically, his central conviction is that number underlies being. For Nicomachus, arithmetic is the “first” of the mathematical sciences, prior even to geometry, because it deals with intelligible, abstract entities independent of spatial extension. This priority aligns with Neopythagorean and Platonist hierarchies of knowledge, in which the more abstract and immaterial a subject, the closer it stands to ultimate reality.

Several thematic strands are especially notable:

• Metaphysics of number: Nicomachus treats numbers as having real, quasi-divine status, not merely as human constructs. The properties of specific numbers (e.g., the perfection of 10 as the tetractys) are taken to reveal structural truths about the cosmos.
• Ethical and spiritual dimensions: Study of arithmetic is portrayed as morally ennobling and purifying, training the soul to recognize order, proportion, and measure. This links mathematics to the traditional Greek virtues of moderation and harmony.
• Epistemology of abstraction: By encouraging contemplation of immaterial, exact entities, Nicomachus sees arithmetic as an important step toward philosophical understanding, preparing the mind for metaphysics and theology.

Historically, Nicomachus exerted significant influence in several directions:

1. Late Antique Platonism and Neopythagoreanism
   Figures such as Iamblichus and later Neoplatonists took up his arithmetical classifications and symbolic interpretations, integrating them into elaborate metaphysical systems. The close association of number with divine principles in late antique theologies of number owes much to Nicomachian models.

2. Boethius and the Latin Middle Ages
   Through Boethius’ adaptation, Nicomachus’ arithmetic and harmonics became canonical authorities in the medieval quadrivium (arithmetic, geometry, music, astronomy). Medieval scholars studied versions of his classifications of numbers, his account of perfect numbers, and his conception of music as a science of proportion.

3. Continuity of Pythagorean themes
   Nicomachus helped preserve and transmit Pythagorean ideas about harmony, proportion, and the symbolic value of numbers. These themes surface repeatedly in later Western thought, from musical theory to speculative cosmology.

Modern historians of mathematics often evaluate Nicomachus ambivalently. On one hand, critics point out his lack of rigorous proof, occasional arithmetical inaccuracies, and dependence on earlier Greek mathematicians. On the other, his work documents a non-Euclidean tradition of arithmetic, one more concerned with classification, symbolism, and pedagogy than with axiomatics. Proponents of a broader cultural history of mathematics regard him as a key witness to how numbers were understood in the philosophical and educational practices of the early Roman Empire.

In contemporary scholarship, Nicomachus is thus studied not primarily as a “great mathematician” in the Euclidean sense, but as an important intellectual mediator. His writings show how Greek arithmetic and Pythagorean cosmology were recombined, interpreted, and ultimately passed on to shape later conceptions of the rational, numerically ordered universe.