Nicomachus of Gerasa : Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician best known for his works Introduction to Arithmetic and Manual of Harmonics in Greek. He was born in Gerasa, in the Roman province of Syria (now Jerash, Jordan). He was a Neopythagorean, who wrote about the mystical properties of numbers.(Greek
Little is known about the life of Nicomachus except that he was a Pythagorean who came from Gerasa. Historians consider him a Neopythagorean based on his tendency to view the numbers having mystical properties.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 a 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 leave him short of time.
Introduction to Arithmetic (Ἀριθμητικὴ εἰσαγωγή, Arithmetike eisagoge), the lesser work on arithmetic. As a Neopythagorean, Nicomachus was often more interested in the mystical properties of numbers rather than their mathematical properties.According to Henrietta O. Midonick (1965), 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 numbers, 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. However Introduction of Arithmetic does contain quite elementary errors which show that Nicomachus chose not to give proofs of his results because he did not in general have such proofs. Many of the results were known by Nicomachus to be true since they appeared with proofs in Euclid, although in a geometrical formulation. Sometimes Nicomachus states a result which is simply false and then illustrates it with an example that happens to have the properties described in the result. We can deduce from this that some of the results are merely guesses based on the evidence of the numerical examples.
Although he was preceded by the Babylonians and the Chinese,Nicomachus provided one of the earliest Greco-Roman multiplication tables, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the British Museum).
Manuale Harmonicum (Ἐγχειρίδιον ἁρμονικῆς, Encheiridion Harmonikes). This is the first important music theory treatise since the time of Aristoxenus and Euclid. It provides the earliest surviving record of the legend 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.
The works which are lost are:
Diophantus of Alexandria was an Alexandrian mathematician, who was the author of a series of books called Arithmetica, many of which are now lost. His texts deal with solving algebraic equations. Diophantine equations and of Diophantine approximations are important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers or defined as generalizations of the integers.
Pythagoras of Samos was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gem-engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated for complete vegetarianism.
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 meter. The Elements is the chief source of our knowledge of ancient Greek music.
Philolaus who was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pythagorean tradition and the most outstanding figure in the Pythagorean school. Pythagoras developed a school of philosophy that was dominated by both mathematics and mysticism. Most of what is known today about the Pythagorean astronomical system is derived from Philolaus's views. He may have been the first write about Pythagorean doctrine. According to August Böckh (1819), who cites Nicomachus, Philolaus was the successor of Pythagoras.
Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spread throughout Magna Graecia.
Aesara of Lucania was a Pythagorean philosopher who wrote On Human Nature, of which a fragment is preserved by Stobaeus.
Iamblichus was a Syrian Neoplatonist philosopher of Arab origin. He determined the direction that would later be taken by Neoplatonic philosophy. He was also the biographer of the Greek mystic, philosopher and mathematician Pythagoras.
Hippasus of Metapontum was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer.
Theano of Crotone was a 6th-century BC Pythagorean philosopher. She has been called the wife or student of Pythagoras, although others see her as the wife of Brontinus. Her place of birth and the identity of her father are uncertain as well. Some authors have suggested that there was more than one person whose details have become merged. Theano is considered by some to be the first known woman mathematician. She may have worked on The Golden Mean and The Golden Rectangle.
Thomas Taylor was an English translator and Neoplatonist, the first to translate into English the complete works of Aristotle and of Plato, as well as the Orphic fragments.
Neopythagoreanism was a school of Hellenistic philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced neoplatonism. It originated in the 1st century BC and flourished during the 1st and 2nd centuries AD. The Encyclopædia Britannica Eleventh Edition describes neopythagoreanism as "a link in the chain between the old and the new" within Hellenistic philosophy. Central to neopythagorean thought was the concept of a soul and its inherent desire for a unio mystica with the divine.
Greek mathematics refers to mathematics texts written during and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by Greek culture and the Greek language. The word "mathematics" itself derives from the Ancient Greek: μάθημα, romanized: máthēmaAttic Greek: [má.tʰɛː.ma]Koine Greek: [ˈma.θi.ma], meaning "subject of instruction". The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is an important difference between Greek mathematics and those of preceding civilizations.
The tetractys, or tetrad, or the tetractys of the decad is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary motions and music.
According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. According to Nicomachus in his 2nd century CE Enchiridion harmonices Pythagoras noticed that hammer A produced consonance with hammer B when they were struck together, and hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be "singing" the same note. Pythagoras rushed into the blacksmith shop to discover why, and found that the explanation was in the weight ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 9 and 8 pounds. Their ratios with hammer A were and. The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval.
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.
The book Introduction to Arithmetic is the only extant work on mathematics by Nicomachus.
Thymaridas of Paros was an ancient Greek mathematician and Pythagorean noted for his work on prime numbers and simultaneous linear equations.
Domninus of Larissa was an ancient Hellenistic Syrian mathematician.
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