NICOMACHUS OF GERASA

(fl. ca. A.D. 100)

mathematics, harmonics.

That Nicomachus was from Gerasa probably the city in Palestine, is known from Lucian (Philopatris, 12), from scholia to his commentator Philoponus, and from some manuscripts that contain Nicomachus’ works. The period of his activity is determined by inference. In his Manual of Harmonics Nicomachus mentions Thrasyllus, who died in A.D. 36; Apuleius, born about A.D. 125, is said to have translated the Introduction to Arithmetic into Latin; and a character in Lucian’s Philopatris says, “You calculate like Nicomachus,” Which shows that Lucian, born about A.D. 120, considered Nicomachus a famous man. ¹ Porphyry mentions him, together with Moderatus and others, as a prominent member of the Pythagorean school, and this connection may also be seen in his writings. ² Only two of his works are extant, Manual of Harmonics and Introduction to Arithmetic. He also wrote a Thelogumena arithmeticae, dealing with the mystic properties of numbers, and a larger work on music, some extracts of numbers, and a larger work on music, some extracts of which have survived.³ Other works are ascribed to him, but it is not certain that he wrote any of them.⁴

In the Manual of Harmonics, after an introductory chapter, Nicomachus deals with the musical note in chapters 2–4 and devotes the next five chapters to the octave. Chapter 10 deals with tuning principles based on the stretched string; chapter 11, with the extension of the octave to the two-octave range of the Greater Perfect System in the diatonic genus; and the work ends with a chapter in which, after restating the definitions of note, interval, and system, Nicomachus gives a survey of the Immutable System in the three genera: diatonic, chromatic, and enharmonic. He deals with notes, intervals, systems, and genera, the first four of the seven subdivisions of harmonics recognized by the ancients, but not with keys, modulation, or melodic composition. The treatise exhibits characteristics of both the Aristoxenian and the Pythagorean schools of music. To the influence of the latter must be ascribed Nicomachus’ assignment of number and numerical ratios to notes and intervals, his recognition of the indivisibility of the octave and the whole tone, and his notion that the musical consonances are in either multiple or superparticular ratios. But unlike Euclid, who attempts to prove musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of the lengths of strings. Hence his treatment of consonances and of musical genera, as well as his definition of the note, are Aristoxenian.

The Introduction to Arithmetic is in two books. After six preliminary chapters devoted to the philosophical importance of mathematics, Nicomachus deals with number per se, relative number, plane and solid numbers, and proportions. He enunciates several definitions of number and then discusses its division into even and odd. He states the theorem that any integer is equal to half the sum of the two integers on each side of it and proceeds to give the classification of even numbers (even times even, odd times even, and even times odd), followed by that of odd numbers (prime, composite, and relative prime).⁵ The fundamental relations of number are equality and inequality, and the latter is divided into the greater and the less. The ratios of the greater are multiples, super-particulars, superpartients, multiple super particulars, and multiple superpartients; those of the less are the reciprocal ratios of these. Book I concludes with a general principle whereby all forms of inequality of ratio may be generated from a series of three equal terms.⁶ At the beginning of the second book the reverse principle is given. It is followed by detailed treatments of squares, cubes, and polygonal numbers. Nicomachus divides proportions into disjunct and continuous, and describes ten types. He presents no abstract proofs (as are found in Euclid’s Elements, VII-IX), and he limits himself for the most part to the enunciation of principles followed by examples with specific numbers.⁷ On one occasion this method leads to a serious mistake,⁸ but there are many other mistakes which are independent of the method of exposition—for example, his inclusion of composite numbers, a class which belongs to all numbers, as a species of the odd. Yet despite its notorious shortcomings, the treatise was influential until the sixteenth century and gave its author the undeserved reputation of being a great mathematician.

NOTES

1. For references to modern discussions, see Tarán, Asclepius on Nicomachus, p. 5, n. 3. J. M. Dillon, “A Date for the Death of Nicomachus of Gerasa?” in Classical Review, n.s. 19 (1969), 274–275, conjectures that Nicomachus died in A.D.196, because Proclus, who was born in A.D. 412, is said by Marinus, Vita Procli 28, to have believed that he was a reincarnation of Nicomachus, and because some Pythagoreans believed that reincarnations occur at intervals of 216 years. But Dillon fails to cite any passage in which Proclus would attach particular importance to the number 216 and, significantly enough, this number is not mentioned in Proclus’ commentary on the creation of the soul in Plato’s Timaeus, a passage where one would have expected this number to occur had Dillon’s conjecture been a probable one.

2. In Eusebius of Caesarea, Historia ecclesiastica, VI, xix, 8.

3. Some of the contents of the Theologumena can be recovered from the summary of it given by Photius, Bibliotheca codex 187, and from the quotations from it in the extant Theologumena arithmeticae ascribed to lamblichus.

In his Manual of Harmonics, I, 2, Nicomachus promises to write a longer and complete work on the subject; and the extracts in some MSS, published by Jan in Musici scriptores Gracci, pp. 266–282, probably are from this work. They can hardly belong to a second book of the Manual, because Nicomachus’ words at the end of this work indicate that it concluded with chapter 12. Eutocius seems to refer to the first book of the larger work on music; see Eurocii Commeniarii in libros De sphaera et cylindro in Archimedis Opera omnia, J. L. Heiberg, ed., Ill (Leipzig, 1915), 120, II. 20–21.

4. In his Introduction to Arithmetic, II, 6, I, Nicomachus refers to an Introduction to Geometry. Some scholars attribute to him a Life of Pythagoras on the grounds that Nicomachus is quoted by both Porphyry and lamblichus in their biographies of Pythagoras. It is also conjectured that he wrote a work on astronomy because Simplicius, In Aristotelis De caelo Heiberg ed., p. 507. II. 12–14, says that Nicomachus, followed by lamblichus, attributed the hypothesis of eccentric circles to the Pythagoreans. A work by Nicomachus with the title On Egyptian Festivals is cited by Athenaeus and by Lydus, but the identity of this Nicomachus with Nicomachus of Gerasa is not established. Finally, the “Nicomachus the Elder“ said by Apollinaris Sidonius to have written a life of Apollonius of Tyana in which he drew from that of Philo-stratus cannot be the author of the Manual since Philostratus was born ca. A.D. 170.

5. Nicomachus considers prime numbers a class of the odd, because for him 1 and 2 are not really numbers. For a criticism of this and of Nicomachus’ classifications of even and odd numbers, see Heath, A History of Greek Mathematics, I, 70–74. In I, 13, Nicomachus describes Eratosthenes’ “sieve,” a device for finding prime numbers.

6. This principle is designed to show that equality is the root and mother of all forms of inequality.

7. Euclid represents numbers by lines with letters attached, a system that makes it possible for him to deal with numbers in general, whereas Nicomachus represents numbers by letters having specific values.

8. See Introduction to Arithmetic, II, 28, 3, where he infers a characteristic of the subcontrary proportion from what is true only of the particular example (3, 5, 6) that he chose to illustrate this proportion. See Tarán, Asclepius on Nicomachus, p. 81 with references.

BIBLIOGRAPHY

I. Original Works. The best, but not critical, ed. of the Introduction to Arithmetic is Nicomachi Geraseni Pythagorei Introductioms arithmeticae libri II, R. Hoche, ed. (Leipzig, 1866), also in English with notes and excellent introductory essays as Nicomachus of Gerasa, Introduction to Arithmetic, trans. by M. L. D’Ooge, with studies in Greek arithmetic by F. E. Robbins and L. C Karpinski (New York, 1926); Boethius’ Latin trans, and adaptation is Anicii Manlii Torquati Severini Boetii De imtitutiorte arithmeticae libri duo, G. Friedlein, ed. (Leipzig, 1867). The Manual of Harmonics is in Carolus Jan, Musici scriptores Graeci (Leipzig, 1895), 235–265; an English trans. and commentary is F. R. Levin, “Nicomachus of Gerasa, Manual of Harmonics: Translation and Commentary” (diss., Columbia University, 1967).

II. Secondary Literature. Ancient commentaries are an anonymous “Prolegomena” in P. Tannery, ed., Diophanti Opera omnia,II (Leipzig, 1895), 73–76; lamblichus’ commentary, lamblichi in Nicomaehi Arithmeticam introductionem liber, H. Pistelli, ed. (Leipzig, 1894); Philoponus’ commentary, R. Hoche, ed., 3 fascs. (Wesel, 1864, 1865; Berlin, 1867); another recension of this commentary in Hoche (Wesel, 1865), pp. ii-xiv, for the variants corresponding to the first book, and in A. Delatte, Anecdota Atheniensia et alia, II (Paris, 1939), 129–187, for those corresponding to the second book; Asclepius’ commentary, “Asclepius of Tralles, Commentary to Nicomachus’ Introduction to Arithmetic” edited with an intro. and notes by L. Tarán, Transactions of the American Philosophical Society, n.s., 59, pt. 4 (1969); there is an anonymous commentary, still unpublished, probably by a Byzantine scholar see Tarán, op, cit., pp. 6, 7–8, 18–20.

For an exposition of the mathematical contents of Ntcomachus’ treatise and a criticism of it, see T. Heath, A History of Greek Mathematics, I (Oxford, 1921), 97–112.

Leonardo TarÁn

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.

Nicomachus of Gerasa

c. 60-c. 100

Roman Syrian Mathematician and Philosopher

In a play by the satirist Lucian, one character says to another, "You calculate like Nicomachus." The latter, a philosopher of the neo-Pythagorean school, is best remembered for his Arithmetike eisagoge, or Introduction to Arithmetic, a highly influential if rather unusual mathematical text.

Gerasa is now the town of Jerash, Jordan, then part of Roman Syria, and it is likely that Nicomachus was ethnically related to the peoples of this area while being thoroughly Romanized in terms of culture and speech. It can be inferred that he studied in a school that espoused the ideas of Pythagoras (c. 580-c. 500 b.c.), because those ideas pervade his Introduction to Arithmetic.

In the latter text, Nicomachus examined odd, even, prime, composite, and perfect numbers. He also presented an interesting theorem in which he showed that by adding consecutive odd numbers, successively including one additional number, it was possible to produce a series of the sums of all cubed numbers. Thus 1³ = 1; 2³ = 3 + 5; 3³ = 7 + 9 + 11; and so on.

On the other hand, Nicomachus sometimes made false statements that seemed to be true because the information he furnished supported his original assertion. Hence his statement that all perfect numbers—ones in which the divisors add up to the number itself (e.g., 1 + 2 + 3 = 6)—end in 6 or 8 alternately. This is not the case, but it seemed to be so, due to the fact that the only perfect numbers he knew were 6, 28, 496, and 8,128.

The book also shows the strange, highly unscientific, side of Pythagorean mathematics, which for instance assigned personalities to numbers. In discussing an abundant number (one in which the sum of its divisors is greater than the number itself), Nicomachus wrote that these called to mind an animal "with ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands...." The opposite of an abundant number was a deficient number, whose divisors add up to less than the number, and these resembled a creature "with a single eye...one-armed, or one of his hands has less than five fingers...."

Pappus (fl. c. a.d. 320) and other mathematicians of the late ancient world despised Nicomachus's book, and Boethius (c. 480-524) perhaps revealed himself as a true medieval with his admiration of it. He turned it into a school book, and despite—or perhaps because of—its peculiarities, the Introduction to Arithmetic became the standard arithmetic text of the Middle Ages. Not until the Crusades (1095-1291), when western Europeans increasingly became exposed to Arab versions of more significant ancient works, was it replaced.

In addition to the Introduction to Arithmetic, Nicomachus wrote a book of musical theory, the Manual of Harmonics. Here, too, he applied Pythagorean notions, though in this case he was on much firmer ground with the relation of music to mathematics.

JUDSON KNIGHT

Nicomachus of Gerasa

Flourishing 100 c.e.

Mathematician, astronomer

Sources

Pythagorean. Nicomachus was a famous writer on mathematics and astronomy, but practically nothing is known of his life. The Gerasa associated with his birth is likely the city by that name in Palestine. The philosophical lecturer Lucian of Samosata, a fortified city on the Euphrates, said, “You calculate like Nicomachus.” Lucian (born circa 120) indicates that when he wrote this line, Nicomachus was a famous man. A specialist on Pythagorean theory, Nicomachus wrote Introduction to Arithmetic, a text that influenced scholars in the Middle Ages. He also contributed to the study of harmonics, and several of his other mathematical works are lost.

Sources

Flora R. Levin, The Harmonics of Nicomachus and the Pythagorean Tradition (University Park, Pa.: American Philological Association, 1975).

G. J. Toomer, “Nicomachus” in The Oxford Classical Dictionary, edited by Simon Hornblower and Antony Spawforth, third edition (Oxford: Oxford University Press, 1999), p. 1042.