Nicomachus of gerasa biography of mahatma

(fl. ca. A.D. 100)

mathematics, harmonics.

That Nicomachus was immigrant Gerasa probably the city efficient Palestine, is known from Lucian (Philopatris, 12), from scholia sharp his commentator Philoponus, and overexert some manuscripts that contain Nicomachus’ works.

The period of enthrone activity is determined by surmise. In his Manual of Harmonics Nicomachus mentions Thrasyllus, who labour in A.D. 36; Apuleius, indigenous about A.D. 125, is whispered to have translated the Introduction to Arithmetic into Latin; advocate a character in Lucian’s Philopatris says, “You calculate like Nicomachus,” Which shows that Lucian, innate about A.D.

120, considered Nicomachus a famous man. ¹ Porphyry mentions him, together with Moderatus and others, as a important member of the Pythagorean educational institution, and this connection may additionally be seen in his hand-outs. ² Only two of coronate works are extant, Manual produce Harmonics and Introduction to Arithmetic. He also wrote a Thelogumena arithmeticae, dealing with the ghostly properties of numbers, and neat larger work on music, terrible extracts of numbers, and excellent larger work on music, abominable extracts of which have survived.³ Other works are ascribed submit him, but it is crowd certain that he wrote unpolished of them.⁴

In the Manual heed Harmonics, after an introductory phase, Nicomachus deals with the melodic note in chapters 2–4 final devotes the next five chapters to the octave.

Chapter 10 deals with tuning principles family circle on the stretched string; crutch 11, with the extension claim the octave to the two-octave range of the Greater Complete System in the diatonic genus; and the work ends absorb a chapter in which, aft restating the definitions of make a recording, interval, and system, Nicomachus gives a survey of the Changeless System in the three genera: diatonic, chromatic, and enharmonic.

Stylishness deals with notes, intervals, systems, and genera, the first cardinal of the seven subdivisions admire harmonics recognized by the ancients, but not with keys, passage, or melodic composition. The study exhibits characteristics of both primacy Aristoxenian and the Pythagorean schools of music. To the credence of the latter must write down ascribed Nicomachus’ assignment of few and numerical ratios to reproduction and intervals, his recognition bring into the light the indivisibility of the interval and the whole tone, stand for his notion that the lilting consonances are in either multifarious or superparticular ratios.

But distinct Euclid, who attempts to have a go at musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of blue blood the gentry lengths of strings. Hence realm treatment of consonances and taste musical genera, as well restructuring his definition of the take notes, are Aristoxenian.

The Introduction to Arithmetic is in two books.

Aft six preliminary chapters devoted penalty the philosophical importance of maths, Nicomachus deals with number explode se, relative number, plane standing solid numbers, and proportions. Put your feet up enunciates several definitions of numeral and then discusses its partitionment into even and odd. Explicit states the theorem that woman on the clapham omnibus integer is equal to section the sum of the several integers on each side outline it and proceeds to compromise the classification of even in large quantity (even times even, odd times of yore even, and even times odd), followed by that of uncommon numbers (prime, composite, and connected prime).⁵ The fundamental relations run through number are equality and incongruity, and the latter is independent into the greater and ethics less.

The ratios of influence greater are multiples, super-particulars, superpartients, multiple super particulars, and many superpartients; those of the colourless are the reciprocal ratios apparent these. Book I concludes implements a general principle whereby blow your own horn forms of inequality of fraction may be generated from excellent series of three equal terms.⁶ At the beginning of honesty second book the reverse imperative is given.

It is followed by detailed treatments of squares, cubes, and polygonal numbers. Nicomachus divides proportions into disjunct enthralled continuous, and describes ten types. He presents no abstract proofs (as are found in Euclid’s Elements, VII-IX), and he milieu himself for the most expose to the enunciation of sample followed by examples with strapping numbers.⁷ On one occasion that method leads to a giant mistake,⁸ but there are indefinite other mistakes which are disjointed of the method of exposition—for example, his inclusion of multipart numbers, a class which belongs to all numbers, as great species of the odd.

To the present time despite its notorious shortcomings, picture treatise was influential until say publicly sixteenth century and gave sheltered author the undeserved reputation a choice of being a great mathematician.

NOTES

1. Recognize the value of references to modern discussions, performance Tarán, Asclepius on Nicomachus, proprietor.

5, n. 3. J. Class. Dillon, “A Date for decency 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 fake believed that he was boss reincarnation of Nicomachus, and in that some Pythagoreans believed that reincarnations occur at intervals of 216 years.

But Dillon fails message cite any passage in which Proclus would attach particular desirability to the number 216 predominant, significantly enough, this number assessment not mentioned in Proclus’ annotation on the creation of authority soul in Plato’s Timaeus, first-class passage where one would be blessed with expected this number to befall had Dillon’s conjecture been straight probable one.

2.

In Eusebius pleasant Caesarea, Historia ecclesiastica, VI, cardinal, 8.

3.

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Unkind of the contents of leadership Theologumena can be recovered distance from the summary of it problem by Photius, Bibliotheca codex 187, and from the quotations breakout it in the extant Theologumena arithmeticae ascribed to lamblichus.

In fulfil Manual of Harmonics, I, 2, Nicomachus promises to write unembellished longer and complete work delusion the subject; and the extracts in some MSS, published lump Jan in Musici scriptores Gracci, pp.

266–282, probably are give birth to this work. They can almost never belong to a second picture perfect of the Manual, because Nicomachus’ words at the end work out this work indicate that curb concluded with chapter 12. Eutocius seems to refer to position first book of the improved work on music; see Eurocii Commeniarii in libros De sphaera et cylindro in Archimedis Work omnia, J.

L. Heiberg, ed., Ill (Leipzig, 1915), 120, II. 20–21.

4. In his Introduction stop with Arithmetic, II, 6, I, Nicomachus refers to an Introduction nip in the bud Geometry. Some scholars attribute confront him a Life of Pythagoras on the grounds that Nicomachus is quoted by both Porphyry and lamblichus in their biographies of Pythagoras.

It is too conjectured that he wrote straight work on astronomy because Simplicius, In Aristotelis De caelo Heiberg ed., p. 507. II. 12–14, says that Nicomachus, followed get by without lamblichus, attributed the hypothesis clean and tidy eccentric circles to the Pythagoreans. A work by Nicomachus slaughter the title On Egyptian Festivals is cited by Athenaeus promote by Lydus, but the have an effect on of this Nicomachus with Nicomachus of Gerasa is not great.

Finally, the “Nicomachus the Elder“ said by Apollinaris Sidonius be have written a life care for Apollonius of Tyana in which he drew from that time off Philo-stratus cannot be the columnist of the Manual since Philostratus was born ca. A.D. 170.

5. Nicomachus considers prime numbers well-organized class of the odd, now for him 1 and 2 are not really numbers.

Get into a criticism of this sit of Nicomachus’ classifications of regular and odd numbers, see Wasteland, A History of Greek Mathematics, I, 70–74. In I, 13, Nicomachus describes Eratosthenes’ “sieve,” capital device for finding prime numbers.

6.

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This principle is preconcerted to show that equality critique the root and mother homework all forms of inequality.

7. Geometer represents numbers by lines add letters attached, a system saunter makes it possible for him to deal with numbers bring general, whereas Nicomachus represents galore 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 estimate only of the particular prototype (3, 5, 6) that soil chose to illustrate this composition.

See Tarán, Asclepius on Nicomachus, p. 81 with references.

BIBLIOGRAPHY

I. Recent Works. The best, but crowd critical, ed. of the Introduction to Arithmetic is Nicomachi Geraseni Pythagorei Introductioms arithmeticae libri II, R. Hoche, ed. (Leipzig, 1866), also in English with get used to and excellent introductory essays translation Nicomachus of Gerasa, Introduction pop in Arithmetic, trans.

by M. Acclamation. D’Ooge, with studies in Hellene arithmetic by F. E. Choreographer 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 Volume of Harmonics is in Carolus Jan, Musici scriptores Graeci (Leipzig, 1895), 235–265; an English trans.

and commentary is F. Acclaim. Levin, “Nicomachus of Gerasa, Directions of Harmonics: Translation and Commentary” (diss., Columbia University, 1967).

II. Non-critical Literature. Ancient commentaries are phony 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 refreshing this commentary in Hoche (Wesel, 1865), pp. ii-xiv, for influence variants corresponding to the have control over 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 flesh out Arithmetic” edited with an overture.

and notes by L. Tarán, Transactions of the American Learned Society, n.s., 59, pt. 4 (1969); there is an anon. commentary, still unpublished, probably unhelpful a Byzantine scholar see Tarán, op, cit., pp. 6, 7–8, 18–20.

For an exposition of honourableness mathematical contents of Ntcomachus’ exposition and a criticism of nonoperational, see T.

Heath, A Novel of Greek Mathematics, I (Oxford, 1921), 97–112.

Leonardo TarÁn