Nicomachus of Gerasa

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Nicomachus of Gerasa ( Greek Νικόμαχος Nikómachos ) was an ancient philosopher , mathematician and music theorist . He lived in the Roman Empire ; his birth falls in the 1st century at the earliest, his death in the late 2nd century at the latest. Nicomachus was a Pythagorean and a Platonist . As a Pythagorean he belongs to the New Pythagorean direction, as a Platonist he is one of the representatives of Middle Platonism .

Life

Very little is known about the life of Nicomachus. He came from Gerasa , a city in the Decapolis in the northwest of today's Jordan . Clues for the dating result from the fact that he mentions the philosopher Thrasyllos , who died in 36 , and that the writer Apuleius , who translated one of his works into Latin, died before the end of the 2nd century. Attempts to determine the exact year of death with considerations based on ancient numerology - the years 142 and 196 were suggested - are insufficiently justified. The place of his mathematical and philosophical training is unknown; the hypothesis that it was Alexandria is speculative and has no concrete support in the sources. The only thing that is certain is that he was teaching.

Works

One page of an Introduction to Arithmetic manuscript written in 1341/1342 . Rome, Biblioteca Apostolica Vaticana , Codex Vaticanus Graecus 195, fol. 36v

Of the works of Nicomachus, only two have been completely preserved: the introduction to arithmetic ( Arithmētikḗ eisagōgḗ , Latin Introductio arithmetica ) and the manual of harmony ( Harmonikón encheirídion or Harmonikḗs encheirídion , Latin Manuale harmonicum or Harmonicum enchiridium ).

mathematics

In the Introduction to Arithmetic , which consists of two books, he deals with the properties of natural numbers and their relationships to one another and explains the number-theoretic classification . He is not concerned with individual mathematical conditions as such, but with the philosophy of mathematics , for which he wants to convey the necessary mathematical knowledge to the reader. It begins with a philosophical introduction; Nicomachus deals with the terms "knowledge", "wisdom" and "philosophy" and describes the relationship between arithmetic, music , geometry and astronomy in a scientific systematic manner (music and astronomy are mathematical sciences according to the understanding of the time). He uses the Dihairesis method . He also comments on the importance of numbers for the creation of the world ( cosmogony ). Then - in chapters 7-16 of the first book - he discusses types of numbers, including even and odd numbers with their subtypes, perfect numbers and prime numbers . In the remainder of the first book (Chapters 17-23) he deals with various relationships between the natural numbers. In the second book he deals with figured numbers (area and body numbers ) and with mean values ( arithmetic , geometric and harmonic mean ). The main concern is the classification; the evidence is neglected, it is reserved for an announced future work. To illustrate this, Nicomachus likes to use tables and gives many examples. His introduction is the first known ancient work specifically devoted to arithmetic; previously the interest of Greek mathematicians was focused on geometry.

Another work by Nicomachus, the theology of numbers ( Arithmētiká theologoúmena ), is only preserved in extensive excerpts and a summary. The relevant sources are the library of the Byzantine scholar Photios and the Theologoúmena arithmētikḗs of an anonymous late antique author (pseudo-Iamblichos). According to the Prolegomena in introductionem arithmeticam Nicomachi ( preliminary remarks to Nicomachus 'introduction to arithmetic ), a small, anonymously transmitted late antique script, number theology was a detailed description of arithmetic, for which Nicomachus' introduction only provided the basic preparatory knowledge.

What is lost is an introduction to geometry ( Geōmetrikḗ eisagōg, ) mentioned by Nicomachus in his Introduction to Arithmetic .

music

In the handbook of harmony , Nicomachus presents the Pythagorean music theory based on numerical proportions, whereby he goes into the Pythagorean idea of ​​the music of the spheres . Here, too, he does not consider his subject in and of itself, but from the point of view of philosophical relevance; he is not interested in musical practice. The work was intended for a distinguished, educated lady of the highest rank, who evidently belonged to a ruling family; in research it has been suggested that it is the empress Plotina . Nicomachus mentions that he was the recipient's teacher and that she had asked him to summarize the content of the lesson in writing.

In addition, Nicomachus states in the Handbook of Harmony that he intends to write a more detailed, several-book introduction to music theory as soon as he finds the time. He repeatedly refers to this portrayal that has been promised. The announced work was actually created; the late antique mathematician Eutokios quotes it in one of his Archimedes commentaries and calls it Über die Musik ( Peri mousikḗs ), which may not be understood as an exact title, but only as an indication of the content. In research it is assumed that the late antique scholar Boethius was also familiar with the extensive introduction of Nicomachus and obtained a great deal of important information from it. The content of the lost work can be partially reconstructed from the representation of Boethius in his work De institutione musica . It used to be believed that parts of it were actually handed down; It was assumed that excerpts from an only fragmentarily preserved music-theoretical work, which are attributed to Nicomachus in the handwritten tradition, come from his extensive introduction. In most of the manuscripts the excerpts are referred to as the “second book” of the Handbook of Harmony . According to the current state of research, the attribution to Nicomachus is very doubtful.

Other works

It is controversial whether the citations of Nicomachus in the descriptions of life of Pythagoras , which were written by the Neo-Platonists Porphyrios and Iamblichus , suggest a Pythagorean biography of Nicomachus. Perhaps Nicomachus wrote an astronomical treatise. A writing about Egyptian festivals ( Peri heortōn Aigyptíōn ), the Athenaios mentions, possibly came from another Nicomachus. From a passage in the introduction to arithmetic , the presumption has been derived that Nicomachus had written a treatise The Common Reading of Plato ( Platōnikḗ synanágnōsis ), but this comment does not refer to a book title, but to oral lessons. A biography of the New Pythagorean Apollonios of Tyana was wrongly attributed to Nicomachus .

The late ancient Neo-Platonist Syrianos mentions in his commentary on the metaphysics of Aristotle a compilation of the Pythagorean doctrines ( Synagōgaí tōn Pythagoreíōn dogmátōn ) from Nicomachus . This is probably not a further work, but a name for the entirety of the writings of Nicomachus on Pythagoreanism, which he probably regarded as part of a complete work.

Teaching

Since a large part of the sources from which Nicomachus drew has been lost, it is difficult to determine to what extent he brought his own ideas into his textbooks. What is certain is that - as with other philosophers of the Roman Empire - New Pythagoreanism and Middle Platonism are fused in his thinking. The Pythagorean element seems to predominate over Platonism.

Math and philosophy

In contrast to the common view of the ancient Platonists, which goes back to Plato himself, Nicomachus sees mathematics not only as a propaedeutic subject, the knowledge of which should form a prerequisite for a subsequent study of philosophy, but he sees arithmetic as the highest science and believes that the study of mathematical sciences enables a "good life" (euzōía) . In this respect, his view differs from that of the Platonists, who assign the highest rank to dialectics as a fundamental science. From this point of view, his teaching can only conditionally be called Platonic. However, in line with the Platonic understanding of mathematics, he is not concerned with arithmetic rules, but always keeps the philosophical aspects of mathematical knowledge in mind.

One reason for Nicomachus' high regard for arithmetic is his belief in its logical priority and its ontological precedence over the other sciences. He sees in it a prerequisite for the existence of the other sciences, while it itself does not need any other science. He also believes that it exists in the mind of the Demiurge (God the Creator) as a fundamental given and acts there like a plan or example ( paradeigma ) for the order of the world. However, he does not equate it with the paradeigma of creation. He regards them as a role model for time and for the processes that are organized in the cosmos. What is true in arithmetic must also apply to cosmology .

The question of how Nicomachus imagined the relationship between numbers and Platonic ideas is controversial in research . Different interpretations of his statements are considered. One is that he viewed numbers as a superordinate class of ideas from which the rest of the ideas are derived; another that he equated ideas and numbers; a third, that in his view, ideas of numbers and ideas of other mathematical realities coexist.

In number theology , Nicomachus assigns the numbers from 1 to 10 to individual gods, both Greek and gods of other peoples. He even identifies the gods with the numbers. With such speculations he follows a custom that was widespread in antiquity.

For the first time in Nicomachus' testimony is the proposition that cube numbers can be represented as sums of odd numbers according to the scheme:

music

One of the main concerns of Nicomachus is to emphasize the crucial role of Pythagoras and the Pythagoreans in the history of music and music theory. Among other things, he claims that Pythagoras was the first to add an eighth string to the lyre and to mathematically demonstrate the perfection of the octave .

The legend of Pythagoras in the forge , which later became extremely popular, is told in the handbook of harmony theory . Pythagoras is said to have heard, when he passed a forge, that the sounds of the falling hammers produced harmonies. He then experimented and found out that the pitch of the sound is directly proportional to the weight of the metal body and that the consonance can thus be expressed mathematically as proportion. In this way he made musical quality quantifiable. This legend is fictitious, because beaten masses like hammers do not result in any harmony of the tones they produce due to their weight.

ethics

Nicomachus does not share the view of some ascetic philosophers, especially the Stoics , according to which only spiritual goods are essential, while external and physical goods are irrelevant with regard to the achievement of the goal in life. He means that the individual external and physical goods correspond to certain spiritual goods ( virtues ); thus good sensory perception and a favorable destiny are analogous to wisdom, health and prestige to moderation, physical strength and political power to courage, physical beauty and friendship to righteousness. In this way the order in the soul is reflected on the physical and external level.

With regard to Providence , Nicomachus believes that the injustices that are inflicted on people and other evils are intended to discipline those affected; he advocates a theodicy in which evil is interpreted as a didactic device used by Providence for a good purpose.

Nicomachus even extends his theory of the general validity of arithmetic statements to ethics by drawing parallels between mathematical facts and ethical principles.

reception

Antiquity

In the 2nd century Apuleius made a Latin translation of Nicomachus' Introduction to Arithmetic , which has not been preserved. In the 3rd century, Porphyrios also mentioned the name of Nicomachus in a list of important Pythagoreans.

Several introductory commentaries on arithmetic emerged in late antiquity . The oldest of them comes from the prominent Neoplatonist Iamblichus , who praised Nicomachus as an important mathematician. In the school of the Neoplatonist Ammonios Hermeiou , who taught in Alexandria , the introduction to arithmetic was one of the textbooks; two students of Ammonios, Johannes Philoponos and Asklepios von Tralleis , commented on it. Both comments have been preserved. The writing De institutione arithmetica of late antique scholar Boethius is a paraphrased translation of the Introduction to arithmetic .

Boethius 'textbook De institutione musica in the first four of his five books probably goes back largely to Nicomachus' lost extensive introduction to music theory. The extensive introduction was also available to the late antique mathematician Eutokios. Ammonios Hermeiou seems to have used them too.

middle Ages

Portrait of Nicomachus in a medieval manuscript

The influence of Nicomachus made itself felt indirectly in the Latin scholarly world of the Middle Ages: Boethius' Latin version of the Introduction to Arithmetic was the authoritative textbook for the study of arithmetic in the Quadrivium and his work De institutione musica , in which much material from Nicomachus was also processed , served as the standard music textbook.

The reception of Nicomachus was also sustainable in the Orient. The introduction to arithmetic has been translated into Syriac and Arabic. The oldest translation was the Syriac, which was probably made in the 8th century or at the beginning of the 9th century and was probably literal; it is completely lost. Based on the Syrian text, the Nestorian metropolitan Ḥabīb ibn Bahrīz made the first Arabic translation in the early 9th century. He translated freely and added additions to the text. Its version was before the famous scholar al-Kindī . In the school of al-Kindī it was revised according to his instructions. In the 10th century, ideas from the introduction to arithmetic appeared in the encyclopedia of the " Brothers of Purity ", where Nicomachus is also named.

The Provencal translator Qalonymos ben Qalonymos translated the revised Arabic version verbatim into Hebrew in Arles in 1317 . The Hebrew version ( Sēfer ha-ariṯmeṭīqa ) has glosses , some of which come from al-Kindī, and differs significantly from the Greek original throughout. In 1499 the Jewish scholar Kaleb Afendopolo wrote a commentary on Sēfer ha-ariṯmeṭīqa in 1499 .

The mathematician and philosopher Ṯābit ibn Qurra obtained a second Arabic translation, this time based on the original Greek text, in the second half of the 9th century. Like the Hebrew, this Arabic version became widely used. Avicenna is one of the Muslim authors who used the work of Nicomachus .

Nicomachus', now lost, extensive introduction to music theory also seems to have been known in the Arabic-speaking world. In the 10th century, the scholar ibn an-Nadīm mentions a “great” book by Nicomachus on music in his kitāb al-Fihrist .

In the Byzantine Empire, a cleric named Soterichos (not to be confused with the late antique poet Soterichos ) wrote a commentary on the introduction to arithmetic . The handbook of harmony also received attention; among its users was the scholar Georgios Pachymeres . In the early 14th century, Manuel Bryennios , the most important Byzantine music theorist, used information from the handbook of harmony , from which he took a wealth of material.

Modern times

The Introduction to Arithmetic was first printed in Paris in 1538; this edition remained the only one of the early modern period . The first edition of the Handbuch der Harmonielehre was produced by Johannes van Meurs (Meursius); it appeared in 1616 in Leiden . In 1652, Marcus Meibom published a Latin translation of the manual .

In modern research, the achievement of Nicomachus is classified as modest from a mathematical-historical point of view. From the point of view of the history of philosophy, it is emphasized that he was primarily a philosopher and wrote his works for an audience primarily interested in philosophy.

The asteroid (8128) Nicomachus, discovered on May 6, 1967, has been named after the ancient mathematician since 1998 .

Text editions and translations

mathematics

  • Richard Hoche (Ed.): Nicomachi Geraseni Pythagorei introductionis arithmeticae libri II . Teubner, Leipzig 1866 (critical edition)
  • Wilhelm Kutsch (Ed.): Ṯābit b. Qurrah's Arabic translation of the Ἀριθμητικὴ Εἰσαγωγή of Nicomachus of Gerasa . Imprimerie Catholique, Beirut 1958 (critical edition)
  • Janine Bertier: Nicomaque de Gérase: Introduction arithmétique . Vrin, Paris 1978 (French translation and commentary)
  • Martin Luther D'Ooge : Nicomachus of Gerasa: Introduction to Arithmetic . Macmillan, New York 1926, reprinted by Johnson, New York 1972 (English translation and detailed study; online )

music

  • Karl von Jan (Ed.): Musici scriptores Graeci . Leipzig 1895, reprint Olms, Hildesheim 1962, pp. 235–265 (critical edition of the Harmonikon encheiridion ) and pp. 266–282 (critical edition of the fragments of a book attributed to Nicomachus, but apparently spurious music theory)
  • Luisa Zanoncelli (Ed.): La manualistica musicale greca . Guerini, Milano 1990, ISBN 88-7802-156-3 , pp. 133–204 (Greek text of the Harmonikon encheiridion based on the edition by v. Jan with Italian translation and commentary) and pp. 205–243 (fragments of unknown origin, probably wrongly attributed to Nicomachus; Greek text based on the edition by v. Jan with Italian translation and commentary)
  • Andrew Barker (Ed.): Greek Musical Writings , Vol. 2: Harmonic and Acoustic Theory . Cambridge University Press, Cambridge 1989, ISBN 0-521-30220-X , pp. 245-269 (English translation of the Harmonikon encheiridion )
  • Flora R. Levin: The Manual of Harmonics of Nicomachus the Pythagorean . Phanes Press, Grand Rapids 1994, ISBN 0-933999-42-9 (English translation of the Harmonikon encheiridion )

Others

  • Jan Radicke (Ed.): Felix Jacoby 'The Fragments of the Greek Historians' continued , Part IV A: Biography , Fascicle 7: Imperial and undated authors . Brill, Leiden 1999, ISBN 90-04-11304-5 , pp. 112-131 (No. 1063: alleged fragments of the biography of Pythagoras with commentary)

literature

  • Bruno Centrone, Gad Freudenthal: Nicomaque de Gérasa . In: Richard Goulet (Ed.): Dictionnaire des philosophes antiques , Vol. 4, CNRS Editions, Paris 2005, ISBN 2-271-06386-8 , pp. 686-694 (research overview with special consideration of reception in the Orient)
  • Franco Ferrari: Nicomachus of Gerasa. In: Christoph Riedweg et al. (Hrsg.): Philosophy of the imperial era and late antiquity (= outline of the history of philosophy . The philosophy of antiquity. Volume 5/1). Schwabe, Basel 2018, ISBN 978-3-7965-3698-4 , pp. 643–648, 700 f.
  • Wolfgang Haase: Investigations on Nicomachus of Gerasa . Dissertation, Tübingen 1982
  • Thomas J. Mathiesen: Apollo's Lyre. Greek Music and Music Theory in Antiquity and the Middle Ages . University of Nebraska Press, Lincoln / London 1999, ISBN 0-8032-3079-6 , pp. 243-245, 390-411
  • Gyburg Radke : The theory of the number in Platonism. A systematic textbook . Francke, Tübingen 2003, ISBN 3-7720-3343-1
  • Leonardo Tarán: Nicomachus of Gerasa . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 10 : SG Navashin - W. Piso . Charles Scribner's Sons, New York 1974, p. 112-114 .

Web links

Remarks

  1. ↑ Wolfgang Haase: Investigations on Nikomachus von Gerasa , Tübingen 1982, pp. 50–70.
  2. ↑ For a discussion of this see Andrew H. Criddle: The Chronology of Nicomachus of Gerasa . In: The Classical Quarterly NS 48, 1998, pp. 324-327; Leonardo Tarán: Nicomachus of Gerasa . In: Dictionary of Scientific Biography , Vol. 10, New York 1974, pp. 112–114, here: p. 113 Note 1.
  3. ↑ For the title see Gregor Staab: Pythagoras in der Spätantike , Munich 2002, p. 82 and note 183.
  4. ↑ Nicomachus, Introduction to Arithmetic 2,6,1, p. 83 lines 3–4 Hoche; Heinrich Dörrie , Matthias Baltes : Platonism in antiquity , vol. 3, Stuttgart-Bad Cannstatt 1993, pp. 68 f., 271.
  5. ↑ See also Bartel Leendert van der Waerden : Die Pythagoreer , Zurich / Munich 1979, p. 110 ff.
  6. ^ William C. McDermott: Plotina Augusta and Nicomachus of Gerasa . In: Historia 26, 1977, pp. 192-203; Andrew H. Criddle: The Chronology of Nicomachus of Gerasa . In: The Classical Quarterly NS 48, 1998, pp. 324–327, here: 325.
  7. ^ Heinrich Dörrie, Matthias Baltes: The Platonism in antiquity , Vol. 3, Stuttgart-Bad Cannstatt 1993, p. 70 f. (Greek text and translation), 272 (commentary).
  8. ↑ Luisa Zanoncelli (ed.): La manualistica musicale greca , Milano 1990, pp. 207-209. Cf. Thomas J. Mathiesen: Apollo's Lyre , Lincoln / London 1999, p. 392 f., Who assumes that some of the extracts contain authentic material from Nicomachus.
  9. ↑ Jan Radicke (Ed.): Felix Jacoby 'The Fragments of the Greek Historians' continued , Part IV A: Biography , Fascicle 7: Imperial and undated authors , Leiden 1999, p. 124; Bruno Centrone, Gad Freudenthal: Nicomaque de Gérasa . In: Richard Goulet (ed.): Dictionnaire des philosophes antiques , Vol. 4, Paris 2005, pp. 686–694, here: 688 f.
  10. ^ Martin Luther D'Ooge: Nicomachus of Gerasa: Introduction to Arithmetic , New York 1926 (reprint New York 1972), p. 81; Luisa Zanoncelli (Ed.): La manualistica musicale greca , Milano 1990, p. 135 f.
  11. ↑ Wolfgang Haase: Investigations on Nikomachus von Gerasa , Tübingen 1982, pp. 86–95; Martin Luther D'Ooge: Nicomachus of Gerasa: Introduction to Arithmetic , New York 1926 (reprinted New York 1972), p. 80.
  12. ↑ Martin Luther D'Ooge: Nicomachus of Gerasa: Introduction to Arithmetic , New York 1926 (reprinted New York 1972), p. 81.
  13. ↑ Syrianos, In Aristotelis metaphysica commentaria pp. 103.4-10 Kroll; Greek text and translation by Gregor Staab: Pythagoras in der Spätantike , Munich 2002, p. 89.
  14. ^ Gregor Staab: Pythagoras in der Spätantike , Munich 2002, pp. 88–91; Jan Radicke (Ed.): Felix Jacoby 'The Fragments of the Greek Historians' continued , Part IV A: Biography , Fascicle 7: Imperial and undated authors , Leiden 1999, p. 124 f.
  15. ↑ See Heinrich Dörrie, Matthias Baltes: Der Platonismus in der Antike , Vol. 3, Stuttgart-Bad Cannstatt 1993, p. 269.
  16. ↑ Christoph Helmig : The relationship between Forms and numbers in Nicomachus' Introduction to Arithmetic . In: Mauro Bonazzi u. a. (Ed.): A Platonic Pythagoras. Platonism and Pythagoreanism in the Imperial Age , Turnhout 2007, pp. 127–146, here: 136–146.
  17. ↑ Christoph Helmig: The relationship between Forms and numbers in Nicomachus' Introduction to Arithmetic . In: Mauro Bonazzi u. a. (Ed.): A Platonic Pythagoras , Turnhout 2007, pp. 127–146, here: 136–140.
  18. ↑ Heinrich Dörrie, Matthias Baltes: The Platonism in antiquity , Vol. 5, Stuttgart-Bad Cannstatt 1998, pp. 354-358.
  19. ↑ Christoph Helmig: The relationship between Forms and numbers in Nicomachus' Introduction to Arithmetic . In: Mauro Bonazzi u. a. (Ed.): A Platonic Pythagoras , Turnhout 2007, pp. 127-146, here: 127-130, 145; see. Heinrich Dörrie, Matthias Baltes: Platonism in antiquity , vol. 5, Stuttgart-Bad Cannstatt 1998, p. 358 f.
  20. ↑ Flora R. Levin: The Harmonics of Nicomachus and the Pythagorean Tradition , University Park (PA) 1975, pp. 46-50.
  21. ^ John Dillon : The Middle Platonists , London 1977, p. 360; Gregor Staab: Pythagoras in der Spätantike , Munich 2002, p. 85 f.
  22. ^ John Dillon: The Middle Platonists , London 1977, p. 360.
  23. ^ Dominic J. O'Meara: Pythagoras Revived , Oxford 1989, pp. 18 f .; Gregor Staab: Pythagoras in der Spätantike , Munich 2002, p. 87.
  24. ↑ Ermenegildo Pistelli, Ulrich Klein (ed.): Iamblichi in Nicomachi arithmeticam introductionem liber , Stuttgart 1975.
  25. ^ Giovanna R. Giardina (ed.): Giovanni Filopono matematico tra neopitagorismo e neoplatonismo. Commentario alla Introduzione aritmetica di Nicomaco di Gerasa , Catania 1999 (edition and Italian translation); Leonardo Tarán (Ed.): Asclepius of Tralles, Commentary to Nicomachus' Introduction to Arithmetic , Philadelphia 1969.
  26. ↑ On Boethius' approach, see Paola Paolucci: Boezio traduttore di Nicomaco nel De institutione arithmetica . In: Athenaeum (Pavia) 93, 2005, pp. 227-241.
  27. ^ Calvin Bower: Boethius and Nicomachus: An Essay Concerning the Sources of De institutione musica . In: Vivarium 16, 1978, pp. 1-45, here: 8-41; Anja Heilmann: Boethius' Musiktheorie und das Quadrivium , Göttingen 2007, pp. 69, 259, note 344.
  28. ↑ Ubaldo Pizzani: Una ignorata testimonianza di Ammonio di Ermia sul perduto opus maius di Nicomaco sulla musica . In: Studi in onore di Aristide Colonna , Perugia 1982, pp. 235–245, here: 240–245.
  29. ↑ For this translation see Gad Freudenthal, Mauro Zonta: Remnants of Ḥabīb ibn Bahrīz's Arabic Translation of Nicomachus of Gerasa's Introduction to Arithmetic . In: Tzvi Langermann, Josef Stern (Eds.): Adaptations and Innovations , Paris 2007, pp. 67–82.
  30. ↑ Carmela Baffioni : Citazioni di autori antichi nelle Rasā'il degli Ikhwān al-Ṣafā ': il caso di Nicomaco di Gerasa . In: Gerhard Endress , Remke Kruk (eds.): The Ancient Tradition in Christian and Islamic Hellenism , Leiden 1997, pp. 3–27.
  31. ↑ See Tzvi Langermann: Studies in Medieval Hebrew Pythagoreanism. Translations and Notes to Nicomachus, Arithmological Texts . In: Micrologus 9, 2001, pp. 219-236, here: 220-22; Gad Freudenthal, Tony Lévy: De Gérase à Bagdad . In: Régis Morelon, Ahmad Hasnawi (ed.): De Zénon d'Élée à Poincaré , Louvain 2004, pp. 479–544 (with a critical partial edition of the Hebrew version).
  32. ^ Tzvi Langermann: Studies in Medieval Hebrew Pythagoreanism. Translations and Notes to Nicomachus, Arithmological Texts . In: Micrologus 9, 2001, pp. 219-236, here: 224 f.
  33. ↑ On the reception of Nicomachus in Arabic-speaking countries, see Sonja Brentjes: Investigations on Nicomachus Arabus . In: Centaurus 30, 1987, pp. 212-239.
  34. ↑ Ubaldo Pizzani: Una ignorata testimonianza di Ammonio di Ermia sul perduto opus maius di Nicomaco sulla musica . In: Studi in onore di Aristide Colonna , Perugia 1982, pp. 235–245, here: 236; Heinrich Dörrie, Matthias Baltes: Platonism in antiquity , vol. 3, Stuttgart-Bad Cannstatt 1993, p. 272.
  35. ↑ Richard Hoche (ed.): Soterichi ad Nicomachi Geraseni introductionem arithmeticam de Platonis psychogonia scholia , Elberfeld 1871.
  36. ^ Heinrich Dörrie, Matthias Baltes: The Platonism in antiquity , Vol. 3, Stuttgart-Bad Cannstatt 1993, p. 269 f.
personal data
SURNAME Nicomachus of Gerasa
ALTERNATIVE NAMES Nicomachus Gerasenus; Nicomachus Gerasinos; Nicomachos; Nicomachus
BRIEF DESCRIPTION ancient philosopher, mathematician and music theorist
DATE OF BIRTH 1st century or 2nd century
DATE OF DEATH 2nd century