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Nicomachus of Gerasa

The Greek mathematician Nicomachus of Gerasa (c. 60–c. 100) is credited with breaking Greek arithmetic away from other types of numerological studies and with refining the concepts of the Neo–Pythagorean school of philosophy. His most famous work, the Introduction to Arithmetic, continued to be an important source for the theory of number and calculation well into the medieval period. The Introduction also contained the earliest–known Greek multiplication tables. "During his own lifetime," wrote Professor Martin Luther D'Ooge in the front matter to his translation of the Introduction to Arithmetic, "he enjoyed, apparently, the highest reputation as a mathematician, and after his death he continued to be studied, directly or indirectly, by generation after generation of schoolboys."

Our knowledge of the life of Nicomachus, like that of many other ancient authors, is spotty and inconclusive. Much of what historians know about him comes from references he put into his own works and from brief mentions that appear in works by later authors. Nicomachus in his works described himself as a Gerasene, meaning that he was a citizen of a town called Gerasa. The best known town of that name was in Roman Palestine, about thirty miles southeast of the Sea of Galilee, which was one of the ten cities that made up the union called the Decapolis. The Decapolis was a mutual–defense and trading association that was intended to protect the Greek citizens from the depredations of the local Jews and other indigenous peoples. Other famous towns of the Decapolis included Damascus (the capitol of modern Syria) and Philadelphia (modern Amman, Jordan).

Gerasa, during the time that Nicomachus lived there, was a Greek city–state or polis like those on the Greek mainland, inhabited and ruled by descendants of Greek immigrants. Tradition held that the city was founded by Alexander the Great, who settled a number of his veterans (who were called in ancient Greek gerontes) in the area and the colony of retired soldiers gave the town its name. It was a place where Greek influences predominated over local Jewish and other traditions. The town may have been mentioned in the Bible as the place where Jesus cast out a demon and forced it into a herd of pigs—pigs were a part of the Greek diet, but not part of Jewish traditions, which considered them unclean.

It seems likely that Nicomachus came from a well–to–do family. His family may have been merchants who traded between Rome and Alexandria, Egypt, or imperial officials, or even part of the local landed aristocracy. His advanced training in mathematics suggests that he received part of his education outside his native city, probably in Alexandria. "The choice of that center of learning," wrote D'Ooge, "would also explain the type of his thinking, for in the first century after Christ Alexandria was the most famous seat of Pythagoreanism in the world. There the old doctrines were being revived, and new treatises were being put in circulation under old names; in Alexandria, in short, the Neo–Pythagorean movement received, if not its initial impulse, at least its chief encouragement."

A Prominent Neo – Pythagorean

The Neo–Pythagoreans were a school of philosophical thought based on a synthesis of earlier Greek schools of philosophy performed by Apollonius of Tyana, who lived and worked in the middle of the first century A.D. Apollonius drew, in particular, on theories of number originally promoted by Pythagoras of Croton, a Greek philosopher living in what is now the south of Italy about six centuries earlier. Nicomachus, who both built on Apollonius' work and developed it in different ways, linked the mathematical principles worked out by the original Pythagoras and began incorporating mystical elements into the philosophy. "In his Introduction to Arithmetic," stated a contributor to World of Mathematics, "Nicomachus discusses various kinds of numbers (odd, even, prime, composite, figurate, perfect), but also considers numbers as divine entities with apparently anthropomorphic qualities, such as, for instance, goodness."

Nicomachus believed, and stated in his Theology of Arithmetic, that numbers had arcane powers and could lead to a closer relationship with (or a better understanding of) the supernatural or divine powers. Numbers, the arithmetician declared, were both matter and form: they could be eternal, immaterial and changeless, and at the same time they could be constantly changing and ephemeral. In fact, what Nicomachus created in the Theology of Arithmetic was an understanding of the natural and supernatural worlds derived from the properties of numbers.

Different gods in the Greek pantheon were identified with the properties of different numbers. The monad (the primary element of the number one), for instance, was identified with the primordial chaos, which existed before the gods, but it was also identified with the sun and with Apollo, the god of the sun. The dyad (the primary element of the number two) was associated with deities ranging from Zeus and Demeter to Artemis and Aphrodite. The triad was associated with still other divinities, including Hecate and Athena. And the tetrad was associated with Hermes and Heracles, among others. In addition, each of these numbers also carried mystical connotations, representing sacred ideas: the monad was associated with the power of mind and chaos, the dyad with equality and matter, the triad with marriage, the tetrad with harmony, and so on.

Number, in Nicomachus' universe, was both the foundation of the material world and the underlying principle that allowed humanity to understand the higher powers beyond that world. "Numbers," explained D'Ooge, "are the sources of form and energy in the world; they are dynamic, active even on their own fellows; hence they convey to one another qualities and sometimes take on an almost human character in their capabilities for mutual influence." Arithmetic was the key to understanding both material physics and spiritual metaphysics.

Foreshadowed the Neo – Platonists

Very little is known about Nicomachus' adult career—even less than is guessed about his education as a young man. Ancient authorities credited him with writing several other books besides the Introduction to Arithmetic. His Manual of Harmonics still exists in its entirety, while his Theology of Arithmetic is mostly complete. Nicomachus is also supposed to have written an Introduction to Geometry, a Life of Pythagoras, and another book on music, but they have not survived. These works complement one another; in the Neo–Pythagorean understanding of the universe, numbers and the ratios between them define the disciplines of geometry, music, and even architecture.

The work of Nicomachus and other Neo–Pythagoreans was later incorporated into a more comprehensive school of philosophy known as Neo–Platonism, the last great school of Greco–Roman philosophy. That school, which was most completely elaborated by the third century A.D. philosopher Plotinus, stated that there was a higher level of reality above the material world and that this level of reality could only be understood through the application of intellectual reasoning, not through observation of the material world. Above all was the great Monad, the One, both God before all other gods and first among all other numbers. In the works of Nicomachus the arithmetician, declared D'Ooge, "God contains in himself all the ideal forms, which . . . are the essence of things and secure [for] them and the world in general whatever stability they have."

But we are not even sure if Nicomachus lived the rest of his life in his native city of Gerasa. Evidence from the dedication to his Manual of Harmony suggests that he spent at least some of his adult life as a teacher of the wealthy and that his work involved extensive travel. Perhaps he was called upon to deliver lectures at various private functions throughout the Roman world. His descendants, however, give us a good idea of the esteem that Nicomachus' contemporaries gave the philosopher. "He was reckoned among the 'illustrious men' of the Pythagorean sect," wrote D'Ooge; ". . . we have but to point to the reputation borne by his works and to the number of commentaries that scholars wrote upon them."

Books

Levin, Flora R., The Harmonics of Nicomachus and the Pythagorean Tradition, American Philological Association, 1975.

The Manual of Nicomachus the Pythagorean, translated with a commentary by Flora R. Levin, 1994.

Nicomachus of Gerasa, Introduction to Arithmetic, translated by Martin Luther D'Ooge, Macmillan, 1926.

Notable Mathematicians, Gale Research, 1998.

World Eras, Volume 3: Roman Republic and Empire (264 B.C.E.–476 C.E.), Gale Group, 2001.

World of Mathematics, 2 volumes, Gale Group, 2001.