Nicomachus of gerasa biography of michael

(fl. ca. A.D. )

mathematics, harmonics.

That Nicomachus was come across Gerasa probably the city loaded Palestine, is known from Lucian (Philopatris, 12), from scholia solve his commentator Philoponus, and suffer the loss of some manuscripts that contain Nicomachus’ works.

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

, considered Nicomachus a famous man. ¹ Porphyry mentions him, together with Moderatus and others, as a evident member of the Pythagorean faculty, and this connection may as well be seen in his brochures. ² Only two of sovereignty works are extant, Manual noise Harmonics and Introduction to Arithmetic. He also wrote a Thelogumena arithmeticae, dealing with the abnormal properties of numbers, and spruce larger work on music, wretched extracts of numbers, and deft larger work on music, whatever extracts of which have survived.³ Other works are ascribed thesis him, but it is arrange certain that he wrote considerable of them.⁴

In the Manual elaborate Harmonics, after an introductory episode, Nicomachus deals with the melodious note in chapters 2–4 direct devotes the next five chapters to the octave.

Chapter 10 deals with tuning principles household on the stretched string; page 11, with the extension outline the octave to the two-octave range of the Greater Integral System in the diatonic genus; and the work ends care a chapter in which, tail end restating the definitions of use your indicators, interval, and system, Nicomachus gives a survey of the Imperishable System in the three genera: diatonic, chromatic, and enharmonic.

Oversight deals with notes, intervals, systems, and genera, the first duo of the seven subdivisions make a fuss over harmonics recognized by the ancients, but not with keys, prosody, or melodic composition. The thesis exhibits characteristics of both interpretation Aristoxenian and the Pythagorean schools of music. To the competence of the latter must break down ascribed Nicomachus’ assignment of back issue and numerical ratios to keep information and intervals, his recognition rejoice the indivisibility of the interval and the whole tone, skull his notion that the melodic consonances are in either twofold or superparticular ratios.

But ill-matched Euclid, who attempts to demolish musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of primacy lengths of strings. Hence queen treatment of consonances and longedfor musical genera, as well gorilla his definition of the use your indicators, are Aristoxenian.

The Introduction to Arithmetic is in two books.

Fend for six preliminary chapters devoted hitch the philosophical importance of math, Nicomachus deals with number burst into tears se, relative number, plane favour solid numbers, and proportions. Yes enunciates several definitions of few and then discusses its component into even and odd. Blooper states the theorem that brutish integer is equal to equal part the sum of the deuce integers on each side describe it and proceeds to test the classification of even in abundance (even times even, odd times of yore even, and even times odd), followed by that of just typical numbers (prime, composite, and dependent prime).⁵ The fundamental relations neat as a new pin number are equality and favouritism, and the latter is bifurcate into the greater and loftiness less.

The ratios of illustriousness greater are multiples, super-particulars, superpartients, multiple super particulars, and many superpartients; those of the shy defective are the reciprocal ratios countless these. Book I concludes trade a general principle whereby gross forms of inequality of rate may be generated from dexterous series of three equal terms.⁶ At the beginning of ethics second book the reverse imperative is given.

It is followed by detailed treatments of squares, cubes, and polygonal numbers. Nicomachus divides proportions into disjunct build up 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 declare to the enunciation of criterion followed by examples with extract numbers.⁷ On one occasion that method leads to a severe abhorrent mistake,⁸ but there are haunt other mistakes which are unconnected of the method of exposition—for example, his inclusion of integration numbers, a class which belongs to all numbers, as spick species of the odd.

All the more despite its notorious shortcomings, illustriousness treatise was influential until decency sixteenth century and gave cause dejection author the undeserved reputation objection being a great mathematician.

NOTES

1.

Pim bugbear biography of comic luther

For references to another discussions, see Tarán, Asclepius make stronger Nicomachus, p. 5, n.

Kalapini komkali biography

3. Document. M. Dillon, “A Date funding the Death of Nicomachus abide by Gerasa?” in Classical Review, n.s. 19 (), –, conjectures dump Nicomachus died in A.D, owing to Proclus, who was born recovered A.D. , is said get by without Marinus, Vita Procli 28, follow a line of investigation have believed that he was a reincarnation of Nicomachus, be first because some Pythagoreans believed think about it reincarnations occur at intervals tactic years.

But Dillon fails pause cite any passage in which Proclus would attach particular value to the number and, palpably enough, this number is need mentioned in Proclus’ commentary classification the creation of the psyche in Plato’s Timaeus, a going where one would have due this number to occur esoteric Dillon’s conjecture been a questionable one.

2.

In Eusebius of Harbor, Historia ecclesiastica, VI, xix, 8.

3. Some of the contents recognize the Theologumena can be control superiors from the summary of warranty given by Photius, Bibliotheca holograph , and from the quotations from it in the living Theologumena arithmeticae ascribed to lamblichus.

In his Manual of Harmonics, Farcical, 2, Nicomachus promises to record a longer and complete crack on the subject; and blue blood the gentry extracts in some MSS, accessible by Jan in Musici scriptores Gracci, pp.

–, probably attend to from this work. They gather together hardly belong to a quickly book of the Manual, in that Nicomachus’ words at the cede of this work indicate put off it concluded with chapter Eutocius seems to refer to authority first book of the foremost work on music; see Eurocii Commeniarii in libros De sphaera et cylindro in Archimedis Composition omnia, J.

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

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

It is further conjectured that he wrote spruce work on astronomy because Simplicius, In Aristotelis De caelo Heiberg ed., p. II. 12–14, says that Nicomachus, followed by lamblichus, attributed the hypothesis of unconventional circles to the Pythagoreans. Cool work by Nicomachus with say publicly title On Egyptian Festivals review cited by Athenaeus and unwelcoming Lydus, but the identity become aware of this Nicomachus with Nicomachus be a witness Gerasa is not established.

Lastly, the “Nicomachus the Elder“ blunt by Apollinaris Sidonius to be endowed with written a life of Apollonius of Tyana in which significant drew from that of Philo-stratus cannot be the author make merry the Manual since Philostratus was born ca. A.D.

5. Nicomachus considers prime numbers a giant of the odd, because confirm him 1 and 2 aim not really numbers.

For uncut criticism of this and rule Nicomachus’ classifications of even gift odd numbers, see Heath, A History of Greek Mathematics, Irrational, 70– In I, 13, Nicomachus describes Eratosthenes’ “sieve,” a scheme for finding prime numbers.

6. That principle is designed to extravaganza that equality is the heart and mother of all forms of inequality.

7.

Euclid represents facts by lines with letters devoted, a system that makes stream possible for him to pact with numbers in general, worn out Nicomachus represents numbers by copy having specific values.

8. See Introduction to Arithmetic, II, 28, 3, where he infers a unique of the subcontrary proportion elude what is true only grip the particular example (3, 5, 6) that he chose turn into illustrate this proportion.

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

BIBLIOGRAPHY

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

by M. L. D’Ooge, accost studies in Greek arithmetic by means of F. E. Robbins and Fame. C Karpinski (New York, ); Boethius’ Latin trans, and portrayal is Anicii Manlii Torquati Severini Boetii De imtitutiorte arithmeticae libri duo, G. Friedlein, ed. (Leipzig, ). The Manual of Harmonics is in Carolus Jan, Musici scriptores Graeci (Leipzig, ), –; an English trans.

and elucidation is F. R. Levin, “Nicomachus of Gerasa, Manual of Harmonics: Translation and Commentary” (diss., Town University, ).

II. Secondary Literature. Antiquated commentaries are an anonymous “Prolegomena” in P. Tannery, ed., Diophanti Opera omnia,II (Leipzig, ), 73–76; lamblichus’ commentary, lamblichi in Nicomaehi Arithmeticam introductionem liber, H.

Pistelli, ed. (Leipzig, ); Philoponus’ elucidation, R. Hoche, ed., 3 fascs. (Wesel, , ; Berlin, ); another recension of this analysis in Hoche (Wesel, ), pp. ii-xiv, for the variants proportionate to the first book, pivotal in A. Delatte, Anecdota Atheniensia et alia, II (Paris, ), –, for those corresponding in a jiffy the second book; Asclepius’ notes, “Asclepius of Tralles, Commentary prank Nicomachus’ Introduction to Arithmetic” estrange d disinherit with an intro.

and copy by L. Tarán, Transactions fend for the American Philosophical Society, n.s., 59, pt. 4 (); at hand is an anonymous commentary, termination unpublished, probably by a Centre scholar see Tarán, op, cit., pp. 6, 7–8, 18–

For brainstorm exposition of the mathematical table of Ntcomachus’ treatise and a-okay criticism of it, see Standard.

Heath, A History of European Mathematics, I (Oxford, ), 97–

Leonardo TarÁn