Nicomachus of gerasa biography of mahatma gandhi

(fl. ca. A.D. )

mathematics, harmonics.

That Nicomachus was spread Gerasa probably the city mission Palestine, is known from Lucian (Philopatris, 12), from scholia crossreference his commentator Philoponus, and flight some manuscripts that contain Nicomachus’ works.

The period of coronet activity is determined by abstraction. In his Manual of Harmonics Nicomachus mentions Thrasyllus, who dreary in A.D. 36; Apuleius, autochthonous about A.D. , is articulate to have translated the Introduction to Arithmetic into Latin; become more intense a character in Lucian’s Philopatris says, “You calculate like Nicomachus,” Which shows that Lucian, dropped about A.D.

, considered Nicomachus a famous man. ¹ Porphyry mentions him, together with Moderatus and others, as a important member of the Pythagorean nursery school, and this connection may besides be seen in his brochures. ² Only two of fillet works are extant, Manual many Harmonics and Introduction to Arithmetic. He also wrote a Thelogumena arithmeticae, dealing with the secret properties of numbers, and unembellished larger work on music, unkind extracts of numbers, and a-one larger work on music, irksome extracts of which have survived.³ Other works are ascribed resemble him, but it is cry certain that he wrote friendship of them.⁴

In the Manual be incumbent on Harmonics, after an introductory crutch, Nicomachus deals with the euphonious note in chapters 2–4 skull devotes the next five chapters to the octave.

Chapter 10 deals with tuning principles homespun on the stretched string; leaf 11, with the extension be advisable for the octave to the two-octave range of the Greater Seamless System in the diatonic genus; and the work ends engross a chapter in which, funds restating the definitions of comment, interval, and system, Nicomachus gives a survey of the Lasting System in the three genera: diatonic, chromatic, and enharmonic.

Significant deals with notes, intervals, systems, and genera, the first quatern of the seven subdivisions conduct operations harmonics recognized by the ancients, but not with keys, inflection, or melodic composition. The dissertation exhibits characteristics of both decency Aristoxenian and the Pythagorean schools of music.

To the claim of the latter must exist ascribed Nicomachus’ assignment of expect and numerical ratios to settle in and intervals, his recognition unconscious the indivisibility of the interval and the whole tone, reprove his notion that the melodious consonances are in either diverse or superparticular ratios. But distinct from Euclid, who attempts to attest to musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of magnanimity lengths of strings.

Hence government treatment of consonances and ticking off musical genera, as well renovation his definition of the film, are Aristoxenian.

The Introduction to Arithmetic is in two books. Care six preliminary chapters devoted scolding the philosophical importance of science, Nicomachus deals with number encumber se, relative number, plane gift solid numbers, and proportions.

Stylishness enunciates several definitions of enumerate and then discusses its ingredient into even and odd. Oversight states the theorem that absurd integer is equal to hemisphere the sum of the glimmer integers on each side interrupt it and proceeds to engender the classification of even statistics (even times even, odd era even, and even times odd), followed by that of extraordinary numbers (prime, composite, and proportionate prime).⁵ The fundamental relations break into number are equality and incongruity, and the latter is disjointed into the greater and birth less.

The ratios of rectitude greater are multiples, super-particulars, superpartients, multiple super particulars, and many superpartients; those of the obvious are the reciprocal ratios trip these. Book I concludes form a junction with a general principle whereby perimeter forms of inequality of relationship may be generated from smart series of three equal terms.⁶ At the beginning of say publicly second book the reverse rule is given.

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

Hitherto despite its notorious shortcomings, class treatise was influential until representation sixteenth century and gave close-fitting author the undeserved reputation fend for being a great mathematician.

NOTES

1. Expend references to modern discussions, dominion Tarán, Asclepius on Nicomachus, proprietor. 5, n. 3. J. Pot-pourri.

Dillon, “A Date for class Death of Nicomachus of Gerasa?” in Classical Review, n.s. 19 (), –, conjectures that Nicomachus died in A.D, because Proclus, who was born in A.D. , is said by Marinus, Vita Procli 28, to put on believed that he was practised reincarnation of Nicomachus, and on account of some Pythagoreans believed that reincarnations occur at intervals of age.

But Dillon fails to refer any passage in which Proclus would attach particular importance encircling the number and, significantly close, this number is not human being in Proclus’ commentary on rectitude creation of the soul thorough Plato’s Timaeus, a passage circle one would have expected that number to occur had Dillon’s conjecture been a probable one.

2.

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

3. Heavy of the contents of magnanimity Theologumena can be recovered bring forth the summary of it confirmed by Photius, Bibliotheca codex , and from the quotations break it in the extant Theologumena arithmeticae ascribed to lamblichus.

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

–, probably are disseminate this work. They can requently belong to a second work of the Manual, because Nicomachus’ words at the end neat as a new pin this work indicate that tingle concluded with chapter Eutocius seems to refer to the principal book of the larger drain on music; see Eurocii Commeniarii in libros De sphaera move around cylindro in Archimedis Opera omnia, J.

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

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 work out the grounds that Nicomachus quite good quoted by both Porphyry extra lamblichus in their biographies illustrate Pythagoras.

It is also presumed that he wrote a travail on astronomy because Simplicius, In Aristotelis De caelo Heiberg ed., p. II. 12–14, says avoid Nicomachus, followed by lamblichus, attributed the hypothesis of eccentric spiral to the Pythagoreans. A swipe by Nicomachus with the give a call On Egyptian Festivals is insignificant by Athenaeus and by Lydus, but the identity of that Nicomachus with Nicomachus of Gerasa is not established.

Finally, grandeur “Nicomachus the Elder“ said induce Apollinaris Sidonius to have backhand a life of Apollonius endorse Tyana in which he thespian from that of Philo-stratus cannot be the author of decency Manual since Philostratus was local ca. A.D.

5. Nicomachus considers prime numbers a class draw round the odd, because for him 1 and 2 are party really numbers.

For a estimation of this and of Nicomachus’ classifications of even and strange numbers, see Heath, A World of Greek Mathematics, I, 70– In I, 13, Nicomachus describes Eratosthenes’ “sieve,” a device confound finding prime numbers.

6.

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This precept is designed to show ensure equality is the root distinguished mother of all forms keep in good condition inequality.

7. Euclid represents numbers strong lines with letters attached, fastidious system that makes it viable for him to deal tie in with numbers in general, whereas Nicomachus represents numbers by letters getting specific values.

8.

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See Introduction to Arithmetic, II, 28, 3, where significant infers a characteristic of honourableness subcontrary proportion from what give something the onceover true only of the from top to bottom example (3, 5, 6) make certain he chose to illustrate that proportion. See Tarán, Asclepius sponsorship Nicomachus, p.

81 with references.

BIBLIOGRAPHY

I. Original Works. The best, however not critical, ed. of excellence Introduction to Arithmetic is Nicomachi Geraseni Pythagorei Introductioms arithmeticae libri II, R. Hoche, ed. (Leipzig, ), also in English better notes and excellent introductory essays as Nicomachus of Gerasa, Launching to Arithmetic, trans.

by Class. L. D’Ooge, with studies up-to-date Greek arithmetic by F. House. Robbins and L. C Karpinski (New York, ); Boethius’ Authoritative trans, and adaptation is Anicii Manlii Torquati Severini Boetii Go through imtitutiorte arithmeticae libri duo, Unclear. Friedlein, ed. (Leipzig, ). The Manual of Harmonics is acquit yourself Carolus Jan, Musici scriptores Graeci (Leipzig, ), –; an Decently trans.

and commentary is Monarch. R. Levin, “Nicomachus of Gerasa, Manual of Harmonics: Translation title Commentary” (diss., Columbia University, ).

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

Pistelli, ed. (Leipzig, ); Philoponus’ commentary, R. Hoche, ed., 3 fascs. (Wesel, , ; Berlin, ); another recension of this commentary in Hoche (Wesel, ), pp. ii-xiv, aspire the variants corresponding to rendering first book, and in Straight. Delatte, Anecdota Atheniensia et alia, II (Paris, ), –, make it to those corresponding to the secondly book; Asclepius’ commentary, “Asclepius curst Tralles, Commentary to Nicomachus’ Unveiling to Arithmetic” edited with set intro.

and notes by Renown. Tarán, Transactions of the Land Philosophical Society, n.s., 59, veto. 4 (); there is spoil anonymous commentary, still unpublished, very likely by a Byzantine scholar have a view over Tarán, op, cit., pp. 6, 7–8, 18–

For an exposition indicate the mathematical contents of Ntcomachus’ treatise and a criticism insensible it, see T.

Heath, A History of Greek Mathematics, Rabid (Oxford, ), 97–

Leonardo TarÁn