Nicomachus
Encyclopedia
Nicomachus was an important mathematician in the ancient world and is best known for his works Introduction to Arithmetic
(Arithmetike eisagoge) and Manual of Harmonics in Greek
. He was born in Gerasa, in the Roman province of Syria
(now Jerash
, Jordan
), and was strongly influenced by Aristotle. He was a Pythagorean
.
and that he came from Gerasa. The age in which he lived (c. 100 AD) is only known because he mentions Thrasyllus
in his Manual of Harmonics, and because his Introduction to Arithmetic was apparently translated into Latin
in the mid 2nd century by Apuleius
. His Manual of Harmonics was addressed to lady of noble birth, at whose request Nicomachus wrote the book, which suggests that he was a respected scholar of some status. He mentions his intent to write a more advanced work, and how the journeys he frequently undertakes leaves him short of time.
, the lesser work on arithmetic. As a Neo-Pythagorean, Nicomachus was often more interested in the mystical properties of numbers rather than their mathematical properties. He distinguishes between the wholly conceptual immaterial number, which he regards as the 'divine number', and the numbers which measure material things, the 'scientific' number. He writes extensively on number
s, especially on the significance of prime numbers and perfect numbers and argues that arithmetic
is ontologically prior to the other mathematical sciences (music
, geometry
, and astronomy
), and is their cause. Boethius' De institutione arithmetica is in large part a Latin translation of this work.
treatise since the time of Aristoxenus
and Euclid
. It provides the earliest surviving record of the story of Pythagoras
's epiphany outside 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 unites Aristoxenian and Pythagorean concerns, normally regarded as antitheses. In the midst of theoretical discussions, Nicomachus also describes the instruments
of his time, also providing a valuable resource. In addition to the Manual, ten extracts survive from what appear to have originally been a more substantial work on music.
Introduction to Arithmetic
Introduction to Arithmetic was written by Nicomachus almost two thousand years ago, and contains both philosophical prose and very basic mathematical ideas. Nicomachus refers to Plato quite often, and wrote about how philosophy can only be possible if one knows enough about mathematics. This is...
(Arithmetike eisagoge) and Manual of Harmonics in Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
. He was born in Gerasa, in the Roman province of Syria
Syria (Roman province)
Syria was a Roman province, annexed in 64 BC by Pompey, as a consequence of his military presence after pursuing victory in the Third Mithridatic War. It remained under Roman, and subsequently Byzantine, rule for seven centuries, until 637 when it fell to the Islamic conquests.- Principate :The...
(now Jerash
Jerash
Jerash, the Gerasa of Antiquity, is the capital and largest city of Jerash Governorate , which is situated in the north of Jordan, north of the capital Amman towards Syria...
, Jordan
Jordan
Jordan , officially the Hashemite Kingdom of Jordan , Al-Mamlaka al-Urduniyya al-Hashemiyya) is a kingdom on the East Bank of the River Jordan. The country borders Saudi Arabia to the east and south-east, Iraq to the north-east, Syria to the north and the West Bank and Israel to the west, sharing...
), and was strongly influenced by Aristotle. He was a Pythagorean
Pythagoreanism
Pythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...
.
Life
Little is known about the life of Nicomachus except that he was a PythagoreanPythagoreanism
Pythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...
and that he came from Gerasa. The age in which he lived (c. 100 AD) is only known because he mentions Thrasyllus
Thrasyllus of Mendes
Thrasyllus of Mendes, whose full name was Tiberius Claudius Thrasyllus , was an Egyptian Greek grammarian and literary commentator from Mendes, Egypt...
in his Manual of Harmonics, and because his Introduction to Arithmetic was apparently translated into Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
in the mid 2nd century by Apuleius
Apuleius
Apuleius was a Latin prose writer. He was a Berber, from Madaurus . He studied Platonist philosophy in Athens; travelled to Italy, Asia Minor and Egypt; and was an initiate in several cults or mysteries. The most famous incident in his life was when he was accused of using magic to gain the...
. His Manual of Harmonics was addressed to lady of noble birth, at whose request Nicomachus wrote the book, which suggests that he was a respected scholar of some status. He mentions his intent to write a more advanced work, and how the journeys he frequently undertakes leaves him short of time.
Introduction to Arithmetic
Introduction to ArithmeticIntroduction to Arithmetic
Introduction to Arithmetic was written by Nicomachus almost two thousand years ago, and contains both philosophical prose and very basic mathematical ideas. Nicomachus refers to Plato quite often, and wrote about how philosophy can only be possible if one knows enough about mathematics. This is...
, the lesser work on arithmetic. As a Neo-Pythagorean, Nicomachus was often more interested in the mystical properties of numbers rather than their mathematical properties. He distinguishes between the wholly conceptual immaterial number, which he regards as the 'divine number', and the numbers which measure material things, the 'scientific' number. He writes extensively on number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s, especially on the significance of prime numbers and perfect numbers and argues that arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
is ontologically prior to the other mathematical sciences (music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
), and is their cause. Boethius' De institutione arithmetica is in large part a Latin translation of this work.
Manual of Harmonics
Manual of Harmonics . This is the first important music theoryMusic theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
treatise since the time of Aristoxenus
Aristoxenus
Aristoxenus of Tarentum was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musical treatise, Elements of Harmony, survives incomplete, as well as some fragments concerning rhythm and...
and Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
. It provides the earliest surviving record of the story of Pythagoras
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
's epiphany outside a smithy
Blacksmith
A blacksmith is a person who creates objects from wrought iron or steel by forging the metal; that is, by using tools to hammer, bend, and cut...
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
Musica universalis
Musica universalis is an ancient philosophical concept that regards proportions in the movements of celestial bodies—the Sun, Moon, and planets—as a form of musica . This 'music' is not usually thought to be literally audible, but a harmonic and/or mathematical and/or religious concept...
." Nicomachus's discussion of the governance of the ear and voice in understanding music unites Aristoxenian and Pythagorean concerns, normally regarded as antitheses. In the midst of theoretical discussions, Nicomachus also describes the instruments
Musical instrument
A musical instrument is a device created or adapted for the purpose of making musical sounds. In principle, any object that produces sound can serve as a musical instrument—it is through purpose that the object becomes a musical instrument. The history of musical instruments dates back to the...
of his time, also providing a valuable resource. In addition to the Manual, ten extracts survive from what appear to have originally been a more substantial work on music.
Lost Works
The works which are lost are:- Art of Arithmetic , the larger work on arithmetic, mentioned by Photius.
- A larger work on music, promised by Nicomachus himself, and apparently referred to by Eutocius in his comment on the sphere and cylinder of ArchimedesArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
. - An Introduction to Geometry, referred to by Nicomachus, although whether it was his work in unclear.
- Theology of Arithmetic , on the Pythagorean mystical properties of numbers in two books 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.
- A Life of Pythagoras, one of the main sources used by PorphyryPorphyry (philosopher)Porphyry of Tyre , Porphyrios, AD 234–c. 305) was a Neoplatonic philosopher who was born in Tyre. He edited and published the Enneads, the only collection of the work of his teacher Plotinus. He also wrote many works himself on a wide variety of topics...
and Iamblichus, for their (extant) Lives of Pythagoras. - A collection of Pythagorean dogmata, referred to by Iamblichus.
- On Egyptian festivals , is mentioned by AthenaeusAthenaeusAthenaeus , of Naucratis in Egypt, Greek rhetorician and grammarian, flourished about the end of the 2nd and beginning of the 3rd century AD...
, but whether by this Nicomachus is uncertain.
See also
- Monad
- DyadDyad (symbol)The Dyad is a title used by the Pythagoreans for the number two, representing the principle of "twoness" or "otherness".Numenius said that Pythagoras gave the name of Monad to God, and the name of Dyad to matter...
- TriadTriad (symbol)The Triad is a Pythagorean title for the number three. According to Priya Hemenway they considered it the most beautiful number, as it is the only number to equal the sum of all the terms below it, and the only number whose sum with those below equals the product of them and itself.-See also:*...
- PentadPentadThe pentad was a Pythagorean term for the quantity of five.-See also:*Related greek numerical terms**Monad **Dyad **Triad **Tetrad **Decad *Other...
- TetradTetrad (symbol)The tetrad or number four is the first number formed by the addition and multiplication of equals. To the Pythagoreans, this symbol and number represented justice as it is the first number that is divisible every way into equal parts.-See also:*Monad*Dyad...
- DecadDecadPythagorus regarded the number 10, or "decad" as being a perfect number, which was often represented by the tetractys, which embraced the whole nature of number....
- Nicomachus's theorem
External links
- Nicomachus' "Introduction to Arithmetic" translated by Martin Luther D'ooge.
- Nicomachus' "Introduction to Arithmetic" from the Million Books Project (Greek)