Nicomachus of Gerasa | |
|---|---|
 Plato (left) and Nicomachus (right), as inventors of music, from a 12th-century manuscript | |
| Born | c. 60 |
| Died | c. 120 |
| Philosophical work | |
| Era | Ancient Roman philosophy |
| School | Neopythagoreanism |
| Main interests | Arithmetic,Music |
| Notable works | Introduction to Arithmetic Manual of Harmonics |
| Notable ideas | Multiplication tables |
Nicomachus of Gerasa (Ancient Greek:Νικόμαχος;c. 60 – c. 120 AD) was anAncient GreekNeopythagorean philosopher fromGerasa, in theRoman province of Syria (nowJerash,Jordan). Like manyPythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his worksIntroduction to Arithmetic andManual of Harmonics, which are an important resource onAncient Greek mathematics andAncient Greek music in theRoman period. Nicomachus' work on arithmetic became a standard text for Neoplatonic education inLate antiquity, with philosophers such asIamblichus andJohn Philoponus writing commentaries on it. A Latin paraphrase by Boethius of Nicomachus's works on arithmetic and music became standard textbooks in medieval education.
Little is known about the life of Nicomachus except that he was aPythagorean who came fromGerasa.[1] HisManual of Harmonics was addressed to a lady of noble birth, at whose request Nicomachus wrote the book, which suggests that he was a respected scholar of some status.[2] He mentions his intent to write a more advanced work, and how the journeys he frequently undertakes leave him short of time.[2]The approximate dates in which he lived (c. 100 AD) can only be estimated based on which other authors he refers to in his work, as well as which later mathematicians who refer to him.[1] He mentionsThrasyllus in hisManual of Harmonics, and hisIntroduction to Arithmetic was apparently translated intoLatin in the mid 2nd century byApuleius,[2]while he makes no mention at all of eitherTheon of Smyrna's work on arithmetic orPtolemy's work on music, implying that they were either later contemporaries or lived in the time after he did.[1]
Historians consider Nicomachus aNeopythagorean based on his tendency to view numbers as havingmystical properties rather than their mathematical properties,[3][4] citing an extensive amount of Pythagorean literature in his work, including works byPhilolaus,Archytas, andAndrocydes.[1] He writes extensively onnumbers, especially on the significance ofprime numbers andperfect numbers and argues thatarithmetic is ontologically prior to the other mathematical sciences (music,geometry, andastronomy), and is theircause. Nicomachus distinguishes between the wholly conceptual immaterial number, which he regards as the 'divine number', and the numbers which measure material things, the 'scientific' number.[2] Nicomachus provided one of the earliest Greco-Romanmultiplication tables; the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in theBritish Museum).[5]
Although Nicomachus is considered a Pythagorean,John M. Dillon says that Nicomachus's philosophy "fits comfortably within the spectrum ofcontemporary Platonism."[6] In his work on arithmetic, Nicomachus quotes fromPlato'sTimaeus[7] to make a distinction between the intelligible world ofForms and the sensible world, however, he also makes more Pythagorean distinctions, such as betweenOdd and even numbers.[6] Unlike many other Neopythagoreans, such asModeratus of Gades, Nicomachus makes no attempt to distinguish between theDemiurge, who acts on the material world, andThe One which serves as the supremefirst principle.[6] For Nicomachus,God as the supreme first principle is both the demiurge and the Intellect (nous), which Nicomachus also equates to being themonad, thepotentiality from which all actualities are created.[6]
Two of Nicomachus' works, theIntroduction to Arithmetic and theManual of Harmonics are extant in a complete form, and two others, a work onTheology of Arithmetic and aLife of Pythagoras survive in fragments, epitomes, and summaries by later authors.[1] TheTheology of Arithmetic (Ancient Greek:Θεολογούμενα ἀριθμητικῆς), on the Pythagorean mystical properties of numbers in two books is mentioned by Photius. There is an extant work sometimes attributed to Iamblichus under this title written two centuries later which contains a great deal of material thought to have been copied or paraphrased from Nicomachus' work.Nicomachus'sLife of Pythagoras was one of the main sources used byPorphyry andIamblichus, for their (extant)Lives of Pythagoras.[1] AnIntroduction to Geometry, referred to by Nicomachus himself in theIntroduction to Arithmetic,[8] has not survived.[1] Among his known lost work is another larger work on music, promised by Nicomachus himself, and apparently[citation needed] referred to byEutocius in his comment on the sphere and cylinder ofArchimedes.
Introduction to Arithmetic (Ancient Greek:Ἀριθμητικὴ εἰσαγωγή,Arithmetike eisagoge) is the only extant work on mathematics by Nicomachus. The work contains both philosophical prose and basic mathematical ideas. Nicomachus refers toPlato quite often, and writes thatphilosophy can only be possible if one knows enough aboutmathematics. Nicomachus also describes hownatural numbers and basic mathematical ideas are eternal and unchanging, and in anabstract realm. The work consists of two books, twenty-three and twenty-nine chapters, respectively.
Nicomachus's presentation is much less rigorous thanEuclid centuries earlier. Propositions are typically stated and illustrated with one example, but not proven through inference. In some instances this results in patently false assertions. For example, he states that from(a−b) ∶ (b−c) ∷ c ∶ a it can be concluded thatab=2bc, only because this is true for a=6, b=5 and c=3.[9]
Boethius'De institutione arithmetica is in large part a Latin translation of this work.
Manuale Harmonicum (Ἐγχειρίδιον ἁρμονικῆς,Encheiridion Harmonikes) is the first importantmusic theory treatise since the time ofAristoxenus andEuclid. It provides the earliest surviving record of the legend ofPythagoras's epiphany outside of a smithy that pitch is determined by numeric ratios. Nicomachus also gives the first in-depth account of the relationship between music and the ordering of the universe via the "music of the spheres." Nicomachus's discussion of the governance of the ear and voice in understanding music unitesAristoxenian and Pythagorean concerns, normally regarded as antitheses.[10] In the midst of theoretical discussions, Nicomachus also describes theinstruments of his time, also providing a valuable resource. In addition to theManual, ten extracts survive from what appear to have originally been a more substantial work on music.
TheIntroduction to Arithmetic of Nicomachus was a standard textbook in Neoplatonic schools, and commentaries on it were written byIamblichus (3rd century) andJohn Philoponus (6th century).[1]
TheArithmetic (in Latin:De Institutione Arithmetica) of Boethius was aLatinparaphrase and a partial translation of theIntroduction to Arithmetic.[11] TheManual of Harmonics also became the basis of the Boethius' Latin treatise titledDe institutione musica.[12]
The work of Boethius on arithmetic and music was a core part of theQuadrivium liberal arts and had a great diffusion during theMiddle Ages.[13]
At the end of Chapter 20 of hisIntroduction to Arithmetic, Nicomachus points out that if one writes a list of the odd numbers, the first is the cube of 1, the sum of the next two is the cube of 2, the sum of the next three is the cube of 3, and so on. He does not go further than this, but from this it follows that the sum of the firstn cubes equals the sum of the first odd numbers, that is, the odd numbers from 1 to. The average of these numbers is obviously, and there are of them, so their sum is Many early mathematicians have studied and provided proofs of Nicomachus's theorem.[14]
