Nicomachus

Nicomachus of Gerasa (Greek Νικόμαχος Nikómachos ) was an ancient philosopher, mathematician and music theorist. He lived in the Roman Empire; his birth falls earliest in the 1st century, his death no later than the late 2nd century. Nicomachus was Pythagoreans and Platonists. As Pythagoreans, he belongs to the Neo-Pythagorean direction, as a Platonist, he is one of the representatives of the Mittelplatonismus.

• 3.1 Mathematics and Philosophy
• 3.2 Music
• 3.3 Ethics

• 4.1 antiquity
• 4.2 Middle Ages
• 4.3 Modern Times

Life

Very little is known about the life of Nicomachus. He came from Jerash, a city in the Decapolis in the northwest of present-day Jordan. Evidence for the dating result from the fact that he mentions the deceased in the year 36 Thrasyllos philosophers and that the writer Apuleius, who translated one of his works into Latin, died before the end of the 2nd century. Trials, the year of death with considerations emanating from the ancient numerology to determine exactly - the years have been suggested 142 and 196 - are insufficiently substantiated. The place of his mathematical and philosophical formation is unknown; the hypothesis that it was Alexandria is speculative and finds in the sources no concrete support. The only certainty is that he has pursued a teaching career.

Works

From the works of Nicomachus are only two remained intact: the "Introduction to Arithmetic" ( Arithmētikḗ eisagōgḗ, Latin Introductio arithmetica ) and the "Manual of Harmony " ( Harmonikón Encheiridion or Harmonikḗs Encheiridion, latin manuals harmonicum or Harmonicum enchiridium ).

Mathematics

In the "Introduction to Arithmetic", which consists of two books, he looks at each other with properties of natural numbers and their relationships, and explains the number-theoretic classification. In this he is not single mathematical conditions as such, but about the philosophy of mathematics, for which he wants to convey to the reader the necessary mathematical skills. It all begins with a philosophical introduction; Nicomachus is made with the terms " knowledge ", " wisdom " and " philosophy " apart and describes science systematically the relationship between arithmetic, music, geometry and astronomy ( music and astronomy are claimed damaligem understanding mathematical sciences). His approach is the method of Dihairesis. He also commented on the significance of the figures for the origin of the world ( cosmogony ). Then - in chapters 7-16 of the first book - he discussed number of species, including odd and even numbers with their sub- types, perfect numbers and primes. In the remaining part of the first book (chapters 17-23 ) he discusses various relations of natural numbers to each other. In the second book he deals inter alia with figurate numbers (surface and body numbers) and with means (arithmetic, geometric and harmonic mean ). The main interest is the classification; the evidence is neglected, it remains subject to a pre-announced future work. To illustrate Nicomachus is like a table and brings many examples. His "Introduction " is the first known work of antiquity, which is specifically devoted to arithmetic; before the interest of the Greek mathematicians concentrated on the geometry.

Another signature of Nicomachus, the " number of theology " ( Arithmetica theologoumena ), obtained only in extensive extracts and a summary. The relevant sources are the library of the Byzantine scholar Photius and the theologoumena arithmētikḗs an anonymous late antique author (pseudo - Iamblichus ). According to the Prolegomena in introductionem arithmeticam Nicomachi ( " Preface to Nicomachus ' Introduction to Arithmetic" ), a small, anonymous handed down late antique font, it was in the "Number theology " a detailed description of the arithmetic to the Nicomachus ' "Introduction" only the preliminary basic knowledge delivered.

Lost an introduction to the geometry ( Geōmetrikḗ eisagōgḗ ), which Nicomachus in his mentioned "Introduction to Arithmetic".

Music

In "Manual of Harmony " provides Nicomachus is the building on numerical proportions of the Pythagorean theory of music, which he responds to the Pythagorean notion of the music of the spheres. Again, it does not consider his subject and of itself, but from the standpoint of philosophical relevance; musical practice does not interest him. The plant was designed for an elegant, educated lady of the highest rank, which apparently belonged to a royal family; in research has been suggested that it is the Empress Plotina. Nicomachus mentioned that he had been the teacher of the recipient and that they had asked him to summarize in writing the content of the lesson.

In addition, Nicomachus divides in the "Manual of harmony " with that he intended to write a more detailed, several books comprehensive introduction to the theory of music as soon as he finds time for it. He repeatedly refers to this in view Asked representation. The announced work is actual; the late antique mathematician Eutocius quoted it in one of his Archimedes comments and calls it " about the music " ( Peri Mousikes ), but this may not be construed as exact title statement, but only as an indication of the content. In research, it is believed that even the late antique scholars Boethius knew the extensive introduction of Nicomachus and removed her numerous important information. From the representation of Boethius in his De musica institutione the contents of the lost work can be reconstructed partially. Previously it was believed that parts are even handed down directly; it was assumed that extracts, only originate from a fragmentary theory of music writing, which are attributed in the manuscript tradition Nicomachus from his extensive introduction. In most manuscripts the extracts are referred to as " second book " of the "Handbook of harmony ". According to the current state of research, the attribution to Nicomachus, however, is very doubtful.

Other works

Whether, suggesting a biography of Pythagoras, Nicomachus of Nicomachus quotes in the biographies of Pythagoras, which were written by the Neo-Platonists Porphyry and Iamblichus, is controversial. Perhaps Nicomachus wrote an astronomical treatise. A document mentions " About Egyptian festivals " ( Peri heortōn Aigyptíōn ), which Athenaeus, possibly came from another Nicomachus. From a point in the "Introduction to Arithmetic" the presumption has been derived, Nicomachus had written a treatise " The common Plato - reading " ( Platōnikḗ synanágnōsis ), but this remark does not refer to a book title, but on oral teaching. Wrongly Nicomachus was formerly attributed to a biography of Apollonius of Tyana Neupythagoreers.

The late antique Neoplatonist Syrianos mentioned in his commentary on the Metaphysics of Aristotle, a certificate from Nicomachus ' compilation of Pythagorean doctrines " ( Synagōgaí TON of Pythagorean dogmátōn ). This is probably not for another work, but a name for the body of the writings of Nicomachus the Pythagorean, which he probably regarded as parts of an overall plant.

Teaching

Since a large part of the sources from which Nicomachus scooped, is lost, can be determined hard, how far he has introduced its own ideas in his textbooks. One thing is certain - are fused in his thinking and neo-Pythagoreanism Mittelplatonismus - as well as with other philosophers of the Roman Empire. The Pythagorean element seems to predominate over the Platonism.

Mathematics and Philosophy

Contrary to the popular conception of the ancient Platonists, which goes back to Plato himself, Nicomachus considered mathematics not only as a propaedeutic specialist whose knowledge is to form a prerequisite for a subsequent study of philosophy, but he sees in arithmetic, the highest science and says, that the study of mathematical sciences a "good life " ( euzōía ) enabled. In this respect, his view from that of the Platonists, that assign the highest rank of the dialectic as a fundamental science is different. From this point of his teaching can only be conditionally referred to as platonic. However, he is doing - in the spirit of Platonic mathematics understanding - in arithmetic calculation rules not to, but he always has the philosophical aspects of mathematical knowledge in mind.

One reason for Nicomachus ' high estimation of arithmetic is his conviction of their logical priority and its ontological priority over the other sciences. He sees it as a prerequisite for the existence of other sciences, while they themselves needed no other science. He also thinks that it exists in the mind of the Demiurge ( Creator God ) as a fundamental condition and there like a plan or model ( paradeigma ) for the order of the world am working. However, he does not put them with the paradeigma of creation absolutely equal. He regards them as a model for the time and for the orderly processes taking place in the cosmos. What is true in arithmetic, must also apply to cosmology.

In research controversial is the question of how Nicomachus, the relationship between numbers and Platonic ideas presented. Different interpretations of the statements should be considered. One is that he looked at the numbers as a parent class of ideas, of the other ideas are derived; another, that he equated ideas and figures; a third, that ideas of numbers and other mathematical ideas realities that, in his view side by side.

In the "Number theology " Nicomachus assigns the numbers 1 to 10 individual gods, both Greek and gods of other peoples. He identifies even the gods with the numbers. With such a widespread speculation he follows in the ancient custom.

Testified for the first time in Nicomachus is the rate that cubes can be represented as sums of odd numbers according to the scheme:

Music

One of the main concerns of Nicomachus is the emphasis on the essential role of Pythagoras and the Pythagoreans in the history of music and music theory. Among other things, he claims that Pythagoras 've added an eighth string as the first of the lyre and the perfection of the octave demonstrated mathematically.

In "Manual of Harmony " which later extremely popular legend of Pythagoras is told in the forge. Pythagoras is said to have as he passed a forge, heard the sounds of the falling hammers were harmonies. However, he had experimented and found that the sound level the weight of the metal body is directly proportional and thus the consonance is mathematically expressible as a proportion. So he made musical quality quantifiable. This legend is fictitious, because whipped masses such as hammers found no induced by its weight harmony of the sounds produced by them.

Ethics

Nicomachus does not share the opinion of some ascetic oriented philosophers, especially the Stoics, that only spiritual goods essential to external and physical goods, however, are irrelevant with regard to the achievement of the goal in life. He thinks that the single external and bodily goods correspond to each particular mental goods ( virtues ); so are good sensory perception and a favorable fate of wisdom analog, health and reputation of moderation, physical strength and political power to the courage, physical beauty and friendship of justice. Thus, the ruling in the soul -range order on the physical and on the external plane mirrors.

With regard to the providence is Nicomachus the view that the injustice that happens to people, and other evils serve the purpose, to discipline those affected; he represents a theodicy in which the evil is interpreted as a to good purpose employed by Providence didactic tool.

His doctrine of the universality of arithmetic statements expands Nicomachus even on the ethics of, by drawing parallels between mathematical situations and ethical principles.

Reception

Antiquity

In the 2nd century Apuleius made ​​a Latin translation of Nicomachus ' Introduction to Arithmetic, which has not survived. In the 3rd century Porphyry resulted in a list of important Pythagoreans also the name of Nicomachus.

In late antiquity resulted in several comments for "Introduction to Arithmetic". The oldest of them dates from the prominent Neo-Platonist Iamblichus, of Nicomachus boasts a famous mathematician. In the school of teaching in Alexandria, Neoplatonist Ammonius Hermeiou the "Introduction to Arithmetic" was one of the textbooks; two students of Ammonius, John Philoponus and Asclepius of Tralles, she commented. Both comments are preserved. The treatise De institutione arithmetica of late antique scholars Boethius is a paraphrased translation of the "Introduction to Arithmetic".

Boethius ' De textbook institutione musica goes to the first four of his five books probably largely on Nicomachus ' back lost extensive introduction to music theory. The extensive introduction was also the late ancient mathematicians Eutocius available. Ammonius Hermeiou seems to have used as well.

Middle Ages

In the Latin -speaking scholarly world of the Middle Ages, the influence of Nicomachus indirect reference made ​​: Boethius ' Latin version of "Introduction to Arithmetic" has been a primary textbook for the arithmetic studies in the quadrivium and his treatise De institutione musica, in which also a lot of Nicomachus material derived was processed, served as the standard textbook of music.

Sustainable Nicomachus the reception was also in the Orient. The "Introduction to Arithmetic" was translated into Syriac and into Arabic. The oldest translation was the Syrian, which probably originated in the 8th century or early 9th century and was probably word for word; it is entirely lost. Based on the Syrian text of the Nestorian Metropolitan Habib ibn Bahrīz made ​​in the early 9th century to the first Arabic translation. He translated freely, adding additives in the text. His version was before the famous scholar al -Kindi. In school, al - Kindi was revised according to its instructions. In the 10th century ideas emerged from the "Introduction to Arithmetic" in The Encyclopedia of the " Brethren of Purity " on where Nicomachus is also named.

The revised Arabic version rendered the Provençal translator Qalonymos ben Qalonymos in 1317 in Arles verbatim into Hebrew. The Hebrew version ( Sefer ha - ariṯmeṭīqa ) is equipped with glosses, some of which come from al -Kindi, and differs considerably throughout the Greek original. 1499 wrote the Jewish scholar Caleb Afendopolo 1499 a commentary on the Sefer ha - ariṯmeṭīqa.

A second Arabic translation, this time to the original Greek text, concerned the mathematician and philosopher Thabit ibn Qurra in the second half of the 9th century. This Arabic version became just like the Hebrew widespread. The Muslim authors who used the work of Nicomachus, a part of Avicenna.

Also Nicomachus ' now lost extensive introduction to music theory seems to have been known in the Arabic-speaking world. In the 10th century, the scholar Ibn an- Nadim mentions in his al - Fihrist kitāb a "big " book of Nicomachus about music.

In the Byzantine Empire, wrote a cleric named Soterichos ( not to be confused with the late Roman poet Soterichos ) a commentary on the "Introduction to Arithmetic". Also, the "Manual of Harmony " attracted attention; to its users belonged to the scholar Georgios Pachymeres. In the early 14th century recycled Manuel Bryennios, the most important Byzantine music theorist, information from the "Manual of Harmony ", which he took from a wealth of material.

Modern Times

The "Introduction to Arithmetic" was first printed in 1538 in Paris; this edition was the only one of the early modern period. The first edition of the "Handbook of harmony " worried Johannes van Meurs ( Meursius ); she appeared in 1616 in Leiden. 1652 published a Latin translation of Marcus meibomian of the "Handbook ".

In modern research, the performance of Nicomachus of mathematics historical perspective is considered modest. Under philosophiegeschichtlichem aspect is emphasized that he was a philosopher in the first place and wrote his works for a primarily philosophical interested audience.

Text editions and translations

• Richard Hoche (ed.): Nicomachi Geraseni Pythagorean introductionis arithmeticae libri II Teubner, Leipzig 1866 ( critical edition )
• Wilhelm Kutsch (ed.): Thabit b. Qurra '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, reprint: Johnson, New York, 1972 ( English translation and in-depth investigation )

• Karl from Jan (ed.): Musici Scriptores Graeci. Leipzig 1895, reprint Olms, Hildesheim 1962, pp. 235-265 ( critical edition of Harmonikon Encheiridion ) and p 266-282 ( critical edition of the fragments of a Nicomachus attributed, but apparently spurious music theory text)
• Luisa Zanoncelli (ed. ): La Greca manualistica musicale. Guerini, Milano 1990, ISBN 88-7802-156-3, pp. 133-204 (Greek text of Harmonikon Encheiridion after the issuance of v. Jan. In Italian translation and commentary ), and p 205-243 (fragments of unknown origin, probably wrongly attributed to Nicomachus; Greek Text after the issuance of v. Jan. In 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 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 Harmonikon Encheiridion )

• January Radicke (ed.): Felix Jacoby ' The fragments of 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: putative fragments of the Life of Pythagoras with comment )