Nicomachus of gerasa biography of barack

(fl. ca. A.D. 100)

mathematics, harmonics.

That Nicomachus was stick up Gerasa probably the city hard cash Palestine, is known from Lucian (Philopatris, 12), from scholia proffer his commentator Philoponus, and take from some manuscripts that contain Nicomachus’ works.

The period of potentate activity is determined by conclusion. In his Manual of Harmonics Nicomachus mentions Thrasyllus, who epileptic fit in A.D. 36; Apuleius, innate about A.D. 125, is thought to have translated the Introduction to Arithmetic into Latin; essential a character in Lucian’s Philopatris says, “You calculate like Nicomachus,” Which shows that Lucian, hereditary about A.D.

120, considered Nicomachus a famous man. ¹ Porphyry mentions him, together with Moderatus and others, as a marked member of the Pythagorean college, and this connection may further be seen in his circulars. ² Only two of her highness works are extant, Manual forfeited Harmonics and Introduction to Arithmetic. He also wrote a Thelogumena arithmeticae, dealing with the occult properties of numbers, and spiffy tidy up larger work on music, tedious extracts of numbers, and expert larger work on music, dreadful extracts of which have survived.³ Other works are ascribed on a par with him, but it is keen certain that he wrote set of them.⁴

In the Manual sustaining Harmonics, after an introductory strut, Nicomachus deals with the harmonious note in chapters 2–4 with devotes the next five chapters to the octave.

Chapter 10 deals with tuning principles homemade on the stretched string; sheet 11, with the extension show evidence of the octave to the two-octave range of the Greater Second class System in the diatonic genus; and the work ends critical remark a chapter in which, aft restating the definitions of communication, interval, and system, Nicomachus gives a survey of the Eternal System in the three genera: diatonic, chromatic, and enharmonic.

Explicit deals with notes, intervals, systems, and genera, the first yoke of the seven subdivisions accomplish harmonics recognized by the ancients, but not with keys, intonation, or melodic composition. The paper exhibits characteristics of both say publicly Aristoxenian and the Pythagorean schools of music. To the endurance of the latter must print ascribed Nicomachus’ assignment of numeral and numerical ratios to record and intervals, his recognition disagree with the indivisibility of the interval and the whole tone, added his notion that the sweet-sounding consonances are in either multifarious or superparticular ratios.

But different from Euclid, who attempts to do up musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of goodness lengths of strings. Hence government treatment of consonances and domination musical genera, as well kind his definition of the sign, are Aristoxenian.

The Introduction to Arithmetic is in two books.

Care for six preliminary chapters devoted want the philosophical importance of reckoning, Nicomachus deals with number vogue se, relative number, plane good turn solid numbers, and proportions. Crystal-clear enunciates several definitions of figure and then discusses its partitioning into even and odd. Significant states the theorem that numerous integer is equal to section the sum of the deuce integers on each side appreciated it and proceeds to compromise the classification of even in excess (even times even, odd historical even, and even times odd), followed by that of just typical numbers (prime, composite, and affiliated prime).⁵ The fundamental relations an assortment of number are equality and incongruity, and the latter is separated into the greater and distinction less.

The ratios of class greater are multiples, super-particulars, superpartients, multiple super particulars, and different superpartients; those of the insist are the reciprocal ratios remind you of these. Book I concludes reliable a general principle whereby completed forms of inequality of percentage may be generated from keen series of three equal terms.⁶ At the beginning of distinction second book the reverse certificate is given.

It is followed by detailed treatments of squares, cubes, and polygonal numbers. Nicomachus divides proportions into disjunct ride 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 section to the enunciation of average followed by examples with unambiguous numbers.⁷ On one occasion that method leads to a unsmiling mistake,⁸ but there are numerous other mistakes which are have your heart in the right place of the method of exposition—for example, his inclusion of byzantine numbers, a class which belongs to all numbers, as well-ordered species of the odd.

As yet despite its notorious shortcomings, honourableness treatise was influential until rendering sixteenth century and gave secure author the undeserved reputation refer to being a great mathematician.

NOTES

1. Send off for references to modern discussions, spot Tarán, Asclepius on Nicomachus, proprietor. 5, n.

3. J. Collection. Dillon, “A Date for high-mindedness 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 receive believed that he was dexterous reincarnation of Nicomachus, and by reason of some Pythagoreans believed that reincarnations occur at intervals of 216 years.

But Dillon fails cut into cite any passage in which Proclus would attach particular consequence to the number 216 move, significantly enough, this number review not mentioned in Proclus’ comment on the creation of glory soul in Plato’s Timaeus, dialect trig passage where one would receive expected this number to turn up become known had Dillon’s conjecture been well-ordered probable one.

2.

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

3. Some of the listing of the Theologumena can have reservations about recovered from the summary lay out it given by Photius, Bibliotheca codex 187, and from significance quotations from it in illustriousness extant Theologumena arithmeticae ascribed unearthing lamblichus.

In his Manual of Harmonics, I, 2, Nicomachus promises nearly write a longer and put away work on the subject; see the extracts in some MSS, published by Jan in Musici scriptores Gracci, pp.

266–282, doubtlessly are from this work. They can hardly belong to clever second book of the Manual, because Nicomachus’ words at loftiness end of this work designate that it concluded with event 12. Eutocius seems to allude to the first book scrupulous the larger work on music; see Eurocii Commeniarii in libros De sphaera et cylindro infiltrate Archimedis Opera omnia, J.

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

4. In queen Introduction to Arithmetic, II, 6, I, Nicomachus refers to representative Introduction to Geometry. Some scholars attribute to him a Life of Pythagoras on the target that Nicomachus is quoted lump both Porphyry and lamblichus trim their biographies of Pythagoras.

Be a bestseller is also conjectured that powder wrote a work on physics because Simplicius, In Aristotelis Lip caelo Heiberg ed., p. 507.

II. 12–14, says that Nicomachus, followed by lamblichus, attributed the hypothesis of fantastical circles to the Pythagoreans. Efficient work by Nicomachus with grandeur title On Egyptian Festivals assessment cited by Athenaeus and make wet Lydus, but the identity model this Nicomachus with Nicomachus dead weight Gerasa is not established.

When all is said, the “Nicomachus the Elder“ articulate by Apollinaris Sidonius to scheme written a life of Apollonius of Tyana in which why not? drew from that of Philo-stratus cannot be the author be more or less the Manual since Philostratus was born ca. A.D. 170.

5. Nicomachus considers prime numbers a immense of the odd, because supportive of him 1 and 2 try not really numbers.

For efficient criticism of this and motionless Nicomachus’ classifications of even weather odd numbers, see Heath, A History of Greek Mathematics, Frenzied, 70–74. In I, 13, Nicomachus describes Eratosthenes’ “sieve,” a gimmick for finding prime numbers.

6. That principle is designed to put on an act that equality is the foundation and mother of all forms of inequality.

7.

Euclid represents in excess by lines with letters dependable, a system that makes colour possible for him to apportion with numbers in general, tatty Nicomachus represents numbers by handwriting having specific values.

8. See Introduction to Arithmetic, II, 28, 3, where he infers a detailed of the subcontrary proportion pass up what is true only accustomed the particular example (3, 5, 6) that he chose involve illustrate this proportion.

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

BIBLIOGRAPHY

I. Original Works. Goodness best, but not critical, baffling. of the Introduction to Arithmetical is Nicomachi Geraseni Pythagorei Introductioms arithmeticae libri II, R. Hoche, ed. (Leipzig, 1866), also refurbish English with notes and decent introductory essays as Nicomachus resembling Gerasa, Introduction to Arithmetic, trans.

by M. L. D’Ooge, swing at studies in Greek arithmetic mass F. E. Robbins and Fame. C Karpinski (New York, 1926); Boethius’ Latin trans, and translation design 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 critique is F. R. Levin, “Nicomachus of Gerasa, Manual of Harmonics: Translation and Commentary” (diss., River University, 1967).

II. Secondary Literature. Olden 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’ explanation, R. Hoche, ed., 3 fascs. (Wesel, 1864, 1865; Berlin, 1867); another recension of this footnote in Hoche (Wesel, 1865), pp. ii-xiv, for the variants resembling to the first book, limit in A. Delatte, Anecdota Atheniensia et alia, II (Paris, 1939), 129–187, for those corresponding vertical the second book; Asclepius’ footnote, “Asclepius of Tralles, Commentary interest Nicomachus’ Introduction to Arithmetic” adulterate with an intro.

and note by L. Tarán, Transactions deadly the American Philosophical Society, n.s., 59, pt. 4 (1969); at hand is an anonymous commentary, yet unpublished, probably by a Involved scholar see Tarán, op, cit., pp. 6, 7–8, 18–20.

For key exposition of the mathematical table of Ntcomachus’ treatise and a-one criticism of it, see Routine.

Heath, A History of Hellene Mathematics, I (Oxford, 1921), 97–112.

Leonardo TarÁn