# Nicomachus

> Greco-Syrian mathematician (c. 60 – c. 120 AD)

**Wikidata**: [Q360366](https://www.wikidata.org/wiki/Q360366)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Nicomachus)  
**Source**: https://4ort.xyz/entity/nicomachus

## Summary
Nicomachus was a Greco-Syrian mathematician, music theorist, and philosopher active during the 1st and 2nd centuries AD (c. 60 – c. 120 AD). He is best known for his work in arithmetic and music theory, particularly his exploration of the relationship between numbers and musical harmony. His contributions influenced later mathematical and philosophical thought, especially in the study of number theory and Pythagorean traditions.

## Biography
- **Born**: c. 60 AD (exact place unknown, but associated with Gerasa, a Greco-Roman city in modern-day Jordan)
- **Nationality**: Greco-Syrian (part of the Roman Empire)
- **Education**: Details unknown, but his work reflects deep knowledge of Greek mathematics and philosophy
- **Known for**: Advancing the study of arithmetic, music theory, and the philosophical significance of numbers
- **Employer(s)**: Not specified in sources
- **Field(s)**: Mathematics, music theory, philosophy

## Contributions
Nicomachus authored several influential works, though exact publication dates are uncertain:
- **Introduction to Arithmetic (Arithmetike eisagoge)**: A foundational text on number theory, covering topics such as even and odd numbers, prime numbers, and figurate numbers (e.g., triangular and square numbers). This work was widely studied in late antiquity and the medieval period.
- **Manual of Harmonics (Enchiridion harmonikes)**: A treatise linking mathematics to music theory, exploring the numerical ratios underlying musical intervals, a concept central to Pythagorean thought.
- **Theology of Arithmetic (Theologoumena arithmetikes)**: A philosophical work attributing mystical and symbolic properties to numbers, blending mathematics with Neopythagorean and Platonic ideas.

His work on **squared triangular numbers**—numbers that are both triangular and square (e.g., 1, 36, 1225)—highlighted his interest in the interplay between geometric and arithmetic properties.

## FAQs
### **What was Nicomachus's most famous work?**
Nicomachus's *Introduction to Arithmetic* was his most enduring work, serving as a key textbook on number theory for centuries. It systematically presented concepts like figurate numbers and the classification of numbers, influencing later mathematicians and philosophers.

### **How did Nicomachus connect mathematics and music?**
In his *Manual of Harmonics*, Nicomachus demonstrated how numerical ratios (e.g., 2:1 for an octave, 3:2 for a fifth) govern musical intervals. This reflected the Pythagorean belief that mathematics underlies the harmony of the cosmos.

### **Was Nicomachus a Pythagorean?**
While not a strict member of the Pythagorean school, Nicomachus was heavily influenced by Pythagorean ideas, particularly the mystical and philosophical significance of numbers. His works are central to the Neopythagorean tradition, which revived Pythagorean thought in the Roman era.

### **What is a squared triangular number?**
A squared triangular number is a number that is both a triangular number (sum of the first *n* natural numbers) and a perfect square. For example, 36 is the 8th triangular number (1+2+3+4+5+6+7+8) and also 6². Nicomachus studied these numbers as part of his broader interest in figurate numbers.

### **How did Nicomachus influence later thinkers?**
His works were studied and cited by medieval scholars, including Boethius, who translated and adapted his *Introduction to Arithmetic* into Latin. This ensured the transmission of Greek mathematical ideas to the Islamic world and later to Europe during the Renaissance.

## Why They Matter
Nicomachus played a crucial role in preserving and advancing Greek mathematical thought during the Roman period. His *Introduction to Arithmetic* became a cornerstone of mathematical education, bridging ancient Greek mathematics and later medieval scholarship. By integrating arithmetic with music theory and philosophy, he reinforced the Pythagorean view of the universe as fundamentally mathematical—a concept that resonated through the works of later thinkers like Boethius, Augustine, and even Kepler.

His emphasis on the philosophical dimensions of numbers also contributed to the development of Neoplatonism and early Christian thought, where numerical symbolism was often employed in theological discourse. Without his systematic treatises, key aspects of Greek number theory might have been lost or fragmented.

## Notable For
- Authoring *Introduction to Arithmetic*, a foundational text in number theory.
- Writing *Manual of Harmonics*, linking mathematics to music theory.
- Exploring the mystical and philosophical properties of numbers in *Theology of Arithmetic*.
- Studying squared triangular numbers and other figurate numbers.
- Influencing medieval mathematics through Boethius’s adaptations of his work.
- Bridging Greek Pythagorean traditions with Roman-era philosophical thought.

## Body
### **Early Life and Background**
Nicomachus was born around 60 AD, likely in or near Gerasa (modern Jerash, Jordan), a Greco-Roman city in the province of Syria. Little is known about his early life or education, but his writings reflect a deep engagement with Greek mathematical and philosophical traditions, particularly those of the Pythagoreans and Plato. His work suggests he was part of a broader intellectual milieu that sought to harmonize mathematical rigor with metaphysical speculation.

### **Mathematical Contributions**
Nicomachus’s most significant mathematical work, *Introduction to Arithmetic*, was a comprehensive treatise on number theory. Unlike Euclid’s *Elements*, which focused on geometric proofs, Nicomachus’s approach was more classificatory and philosophical. Key topics included:
- **Classification of Numbers**: He distinguished between even, odd, prime, composite, and figurate numbers (e.g., triangular, square, pentagonal).
- **Figurate Numbers**: He explored numbers that could form geometric shapes, such as triangular numbers (1, 3, 6, 10, ...) and their properties.
- **Squared Triangular Numbers**: He highlighted numbers like 36 (the 8th triangular number and 6²) and 1225 (the 49th triangular number and 35²), which are both triangular and square.
- **Perfect Numbers**: While not originating the concept, he discussed numbers equal to the sum of their proper divisors (e.g., 6, 28), a topic later expanded by Euclid.

His work was not merely technical but also philosophical, emphasizing the aesthetic and symbolic qualities of numbers.

### **Music Theory and Harmonics**
In *Manual of Harmonics*, Nicomachus applied mathematical principles to music, a tradition dating back to Pythagoras. He explained how musical intervals correspond to simple numerical ratios:
- **Octave (2:1)**: Doubling the frequency of a note.
- **Fifth (3:2)**: The ratio of the first six notes in the harmonic series.
- **Fourth (4:3)**: Another fundamental interval.

This work reinforced the idea that music was a manifestation of cosmic harmony, a concept that influenced later medieval and Renaissance music theory.

### **Philosophical and Mystical Dimensions**
Nicomachus’s *Theology of Arithmetic* delved into the symbolic and metaphysical significance of numbers, a hallmark of Neopythagorean thought. He attributed divine or cosmic properties to numbers, such as:
- **One (Monad)**: Representing unity and the divine.
- **Two (Dyad)**: Symbolizing duality and matter.
- **Ten (Decad)**: The perfect number, encompassing all numerical relationships.

This work reflected the broader trend in late antiquity of blending mathematics with theology and philosophy, influencing later Neoplatonists like Iamblichus.

### **Influence and Legacy**
Nicomachus’s writings were pivotal in transmitting Greek mathematical knowledge to later centuries. Key aspects of his legacy include:
- **Boethius’s Adaptations**: The Roman philosopher Boethius translated and paraphrased *Introduction to Arithmetic* in his *De institutione arithmetica*, making it accessible to medieval Europe.
- **Medieval Curriculum**: His work became part of the *Quadrivium* (arithmetic, geometry, music, astronomy), the mathematical component of liberal arts education in the Middle Ages.
- **Renaissance Revival**: Scholars like Kepler and Mersenne drew on Nicomachus’s ideas, particularly his harmonic theories, in their own work.

### **Connection to Other Thinkers**
Nicomachus was influenced by earlier Greek mathematicians and philosophers, including:
- **Pythagoras**: His numerical mysticism and harmonic theories.
- **Aristotle**: While not directly cited, Aristotelian logic and classification methods are evident in his work.
- **Plato**: The philosophical underpinnings of his number theory align with Platonic ideals.

His work, in turn, influenced later figures such as:
- **Iamblichus**: A Neoplatonist who expanded on Nicomachus’s mystical arithmetic.
- **Boethius**: The primary conduit for Nicomachus’s ideas into medieval Europe.
- **Early Christian Thinkers**: Augustine and others used numerical symbolism in theological writings, partly inspired by Nicomachus.

### **Lost Works and Fragmentary Evidence**
While *Introduction to Arithmetic* and *Manual of Harmonics* survive in full or partial form, other works attributed to Nicomachus are known only through fragments or references in later texts. These include:
- *Theology of Arithmetic*: Survives in excerpts.
- *Life of Pythagoras*: A biographical work, now lost.
- *On Egyptian Festivals*: A cultural study, referenced by later authors.

### **Cultural and Historical Context**
Nicomachus lived during the height of the Roman Empire, a period of synthesis between Greek and Roman intellectual traditions. His work exemplifies the Neopythagorean revival, which sought to revive and adapt Pythagorean ideas in a Roman context. This movement was part of a broader cultural trend that included the rise of Neoplatonism and the integration of Greek philosophy into Roman education.

### **Modern Reception**
Today, Nicomachus is recognized as a key figure in the history of mathematics, particularly for his role in preserving and transmitting Greek number theory. While his mystical interpretations of numbers are less emphasized in modern mathematics, his classificatory approach to arithmetic laid groundwork for later developments in number theory. Historians of science also study his work to understand the continuity between ancient and medieval mathematical thought.

## References

1. MacTutor History of Mathematics archive
2. Virtual International Authority File
3. CiNii Research
4. Catalogue of the Library of the Pontifical University of Saint Thomas Aquinas
5. Freebase Data Dumps. 2013
6. CERL Thesaurus
7. Treccani's Enciclopedia on line
8. Enciclopedia Treccani
9. LIBRIS. 2007
10. Treccani Philosophy
11. HMML Authority File
12. Digital Scriptorium Catalog