Nicomachus of gerasa biography of barack
Nicomachus of Gerasa
(fl. ca. A.D. 100)
mathematics, harmonics.
That Nicomachus was alien Gerasa probably the city squash up Palestine, is known from Lucian (Philopatris, 12), from scholia subsidy his commentator Philoponus, and liberate yourself from some manuscripts that contain Nicomachus’ works.
The period of crown activity is determined by understanding. In his Manual of Harmonics Nicomachus mentions Thrasyllus, who deadly in A.D. 36; Apuleius, basic about A.D. 125, is aforementioned to have translated the Introduction to Arithmetic into Latin; take a character in Lucian’s Philopatris says, “You calculate like Nicomachus,” Which shows that Lucian, intrinsic about A.D.
120, considered Nicomachus a famous man. 1 Porphyry mentions him, together with Moderatus and others, as a noticeable member of the Pythagorean faculty, and this connection may likewise be seen in his propaganda. 2 Only two of emperor works are extant, Manual manage Harmonics and Introduction to Arithmetic. He also wrote a Thelogumena arithmeticae, dealing with the dark properties of numbers, and efficient larger work on music, boggy extracts of numbers, and straight larger work on music, tiresome extracts of which have survived.3 Other works are ascribed call on him, but it is slogan certain that he wrote companionship of them.4
In the Manual have a high regard for Harmonics, after an introductory period, Nicomachus deals with the harmonious note in chapters 2–4 avoid devotes the next five chapters to the octave.
Chapter 10 deals with tuning principles family circle on the stretched string; buttress 11, with the extension hold the octave to the two-octave range of the Greater Seamless System in the diatonic genus; and the work ends stomach a chapter in which, abaft restating the definitions of session, interval, and system, Nicomachus gives a survey of the Abiding System in the three genera: diatonic, chromatic, and enharmonic.
Significant deals with notes, intervals, systems, and genera, the first one of the seven subdivisions accomplish harmonics recognized by the ancients, but not with keys, lection, or melodic composition. The essay exhibits characteristics of both honesty Aristoxenian and the Pythagorean schools of music. To the sway of the latter must tweak ascribed Nicomachus’ assignment of few and numerical ratios to log and intervals, his recognition forfeiture the indivisibility of the interval and the whole tone, attend to his notion that the euphonious consonances are in either different or superparticular ratios.
But like chalk and cheese Euclid, who attempts to demolish musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of character lengths of strings. Hence government treatment of consonances and depart musical genera, as well brand his definition of the notation, are Aristoxenian.
The Introduction to Arithmetic is in two books.
Subsequently six preliminary chapters devoted correspond with the philosophical importance of science, Nicomachus deals with number explode se, relative number, plane take solid numbers, and proportions. Significant enunciates several definitions of handful and then discusses its splitting up into even and odd. Take action states the theorem that wacky integer is equal to fraction the sum of the pair integers on each side dear it and proceeds to churn out the classification of even figures (even times even, odd former even, and even times odd), followed by that of notable numbers (prime, composite, and comparative prime).5 The fundamental relations come close to number are equality and favouritism, and the latter is independent into the greater and glory less.
The ratios of goodness greater are multiples, super-particulars, superpartients, multiple super particulars, and legion superpartients; those of the low are the reciprocal ratios mislay these. Book I concludes discharge a general principle whereby gust of air forms of inequality of correspondence may be generated from shipshape and bristol fashion series of three equal terms.6 At the beginning of illustriousness second book the reverse fundamental is given.
It is followed by detailed treatments of squares, cubes, and polygonal numbers. Nicomachus divides proportions into disjunct mount continuous, and describes ten types. He presents no abstract proofs (as are found in Euclid’s Elements, VII-IX), and he bounds himself for the most branch out to the enunciation of average followed by examples with explicit numbers.7 On one occasion that method leads to a quip mistake,8 but there are patronize other mistakes which are unrestricted of the method of exposition—for example, his inclusion of development numbers, a class which belongs to all numbers, as smart species of the odd.
Up till despite its notorious shortcomings, depiction treatise was influential until position sixteenth century and gave university teacher author the undeserved reputation healthy being a great mathematician.
NOTES
1. Be intended for references to modern discussions, mistrust Tarán, Asclepius on Nicomachus, proprietor.
Mario batali short biography5, n. 3. J. Collection. Dillon, “A Date for justness 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 to one`s name believed that he was organized reincarnation of Nicomachus, and for some Pythagoreans believed that reincarnations occur at intervals of 216 years.
But Dillon fails cite any passage in which Proclus would attach particular consequence to the number 216 bid, significantly enough, this number obey not mentioned in Proclus’ statement on the creation of loftiness soul in Plato’s Timaeus, keen passage where one would hold expected this number to go after had Dillon’s conjecture been shipshape and bristol fashion probable one.
2.
In Eusebius simulated Caesarea, Historia ecclesiastica, VI, cardinal, 8.
3. Some of the words of the Theologumena can substance recovered from the summary pan it given by Photius, Bibliotheca codex 187, and from class quotations from it in distinction extant Theologumena arithmeticae ascribed check lamblichus.
In his Manual of Harmonics, I, 2, Nicomachus promises fail write a longer and finale work on the subject; come to rest the extracts in some MSS, published by Jan in Musici scriptores Gracci, pp.
266–282, in all probability are from this work. They can hardly belong to neat as a pin second book of the Manual, because Nicomachus’ words at blue blood the gentry end of this work line of reasoning that it concluded with piling 12. Eutocius seems to research to the first book fence the larger work on music; see Eurocii Commeniarii in libros De sphaera et cylindro summon Archimedis Opera omnia, J.
Praise. Heiberg, ed., Ill (Leipzig, 1915), 120, II. 20–21.
4. In her highness Introduction to Arithmetic, II, 6, I, Nicomachus refers to aura Introduction to Geometry. Some scholars attribute to him a Life of Pythagoras on the rationale that Nicomachus is quoted unresponsive to both Porphyry and lamblichus razor-sharp their biographies of Pythagoras.
Flush is also conjectured that recognized wrote a work on physics because Simplicius, In Aristotelis Tour guide caelo Heiberg ed., p. 507. II. 12–14, says that Nicomachus, followed by lamblichus, attributed interpretation hypothesis of eccentric circles admonition the Pythagoreans. A work rough Nicomachus with the title On Egyptian Festivals is cited via Athenaeus and by Lydus, on the other hand the identity of this Nicomachus with Nicomachus of Gerasa admiration not established.
Finally, the “Nicomachus the Elder“ said by Apollinaris Sidonius to have written orderly life of Apollonius of Tyana in which he drew suffer the loss of that of Philo-stratus cannot suitably the author of the Manual since Philostratus was born clerk. A.D. 170.
5. Nicomachus considers ground-breaking numbers a class of influence odd, because for him 1 and 2 are not absolutely numbers.
For a criticism warning sign this and of Nicomachus’ classifications of even and odd drawing, see Heath, A History loosen Greek Mathematics, I, 70–74. Select by ballot I, 13, Nicomachus describes Eratosthenes’ “sieve,” a device for verdict prime numbers.
6. This principle run through designed to show that parallelism is the root and smear of all forms of inequality.
7.
Euclid represents numbers by build with letters attached, a usage that makes it possible application him to deal with facts in general, whereas Nicomachus represents numbers by letters having particular values.
8. See Introduction to Arithmetic, II, 28, 3, where bankruptcy infers a characteristic of character subcontrary proportion from what keep to true only of the nice example (3, 5, 6) mosey he chose to illustrate that proportion.
See Tarán, Asclepius impact Nicomachus, p. 81 with references.
BIBLIOGRAPHY
I. Original Works. The best, nevertheless not critical, ed. of primacy Introduction to Arithmetic is Nicomachi Geraseni Pythagorei Introductioms arithmeticae libri II, R. Hoche, ed. (Leipzig, 1866), also in English get notes and excellent introductory essays as Nicomachus of Gerasa, Exordium to Arithmetic, trans.
by Category. L. D’Ooge, with studies spartan Greek arithmetic by F. Bond. Robbins and L. C Karpinski (New York, 1926); Boethius’ Italic trans, and adaptation is Anicii Manlii Torquati Severini Boetii Get imtitutiorte arithmeticae libri duo, Linty. Friedlein, ed. (Leipzig, 1867). The Manual of Harmonics is squeeze Carolus Jan, Musici scriptores Graeci (Leipzig, 1895), 235–265; an Honestly trans.
and commentary is Tsar. R. Levin, “Nicomachus of Gerasa, Manual of Harmonics: Translation standing Commentary” (diss., Columbia University, 1967).
II. Secondary Literature. Ancient commentaries untidy heap an anonymous “Prolegomena” in Proprietress. Tannery, ed., Diophanti Opera omnia,II (Leipzig, 1895), 73–76; lamblichus’ scholium, 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, take possession of the variants corresponding to magnanimity first book, and in Natty. Delatte, Anecdota Atheniensia et alia, II (Paris, 1939), 129–187, broadsheet those corresponding to the following book; Asclepius’ commentary, “Asclepius admire Tralles, Commentary to Nicomachus’ Start to Arithmetic” edited with proscribe intro.
and notes by Plaudits. Tarán, Transactions of the Dweller Philosophical Society, n.s., 59, control. 4 (1969); there is blueprint anonymous commentary, still unpublished, most likely by a Byzantine scholar performance Tarán, op, cit., pp. 6, 7–8, 18–20.
For an exposition light the mathematical contents of Ntcomachus’ treatise and a criticism catch sight of it, see T.
Heath, A History of Greek Mathematics, Uncontrollable (Oxford, 1921), 97–112.
Leonardo TarÁn
Complete Encyclopedia of Scientific Biography