Nicomachus of Gerasa ( ‹See Tfd› Greek : Νικόμαχος ; c. 60 – c. 120 AD ) was an Ancient Greek Neopythagorean philosopher from Gerasa , in the Roman province of Syria (now Jerash , Jordan ). Like many Pythagoreans , Nicomachus wrote about the mystical properties of numbers, best known for his works Introduction to Arithmetic and Manual of Harmonics , which are an important resource on Ancient Greek mathematics and Ancient Greek music in the Roman period . Nicomachus' work on arithmetic became a standard text for Neoplatonic education in Late antiquity , with philosophers such as Iamblichus and John Philoponus writing commentaries on it. A Latin paraphrase by Boethius of Nicomachus's works on arithmetic and music became standard textbooks in medieval education.
35-458: Nicomachus (c. 60 – c. 120) was a mathematician and Pythagorean philosopher from Gerasa. Nicomachus may also refer to: Nicomachus Little is known about the life of Nicomachus except that he was a Pythagorean who came from Gerasa . 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
70-418: A History of Philosophy ( Philosophos Historia ) with vitae of philosophers that included a life of his teacher, Plotinus. His life of Plato from book iv exists only in quotes by Cyril of Alexandria . His book Vita Pythagorae on the life of Pythagoras is not to be confused with the book of the same name by Iamblichus . His commentary on Ptolemy's Harmonics ( Eis ta Harmonika Ptolemaiou hypomnēma )
105-480: A distinction between the intelligible world of Forms and the sensible world, however, he also makes more Pythagorean distinctions, such as between Odd and even numbers. Unlike many other Neopythagoreans, such as Moderatus of Gades , Nicomachus makes no attempt to distinguish between the Demiurge , who acts on the material world, and The One which serves as the supreme first principle . For Nicomachus, God as
140-540: 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 first n cubes equals the sum of the first n ( n + 1 ) / 2 {\displaystyle n(n+1)/2} odd numbers, that is, the odd numbers from 1 to n ( n + 1 ) − 1 {\displaystyle n(n+1)-1} . The average of these numbers
175-424: A more advanced work, and how the journeys he frequently undertakes leave him short of time. 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. He mentions Thrasyllus in his Manual of Harmonics , and his Introduction to Arithmetic was apparently translated into Latin in
210-664: A partial translation of the Introduction to Arithmetic . The Manual of Harmonics also became the basis of the Boethius' Latin treatise titled De institutione musica . The work of Boethius on arithmetic and music was a core part of the Quadrivium liberal arts and had a great diffusion during the Middle Ages . At the end of Chapter 20 of his Introduction to Arithmetic , Nicomachus points out that if one writes
245-540: A very short work often considered to be a commentary on Aristotle 's Categories , hence the title. According to Barnes 2003 , however, the correct title is simply Introduction (Εἰσαγωγή Isagoge ), and the book is an introduction not to the Categories in particular, but to logic in general, comprising as it does the theories of predication, definition, and proof. The Introduction describes how qualities attributed to things may be classified, famously breaking down
280-436: Is also known as an opponent of Christianity and defender of Paganism ; his precise contribution to the philosophical approach to traditional religion may be discovered in the fragments of Philosophy from Oracles (Περὶ τῆς ἐκ λογίων φιλοσοφίας; De Philosophia ex Oraculis Haurienda ), which was originally three books in length. There is debate as to whether it was written in his youth (as Eunapius reports ) or closer in time to
315-520: Is an accepted version of this page Porphyry of Tyre ( / ˈ p ɔːr f ɪr i / ; Koinē Greek : Πορφύριος , romanized: Porphýrios ; c. 234 – c. AD 305 ) was a Neoplatonic philosopher born in Tyre , Roman Phoenicia during Roman rule . He edited and published the Enneads , the only collection of the work of Plotinus , his teacher. He wrote original works in
350-748: Is an important source for the history of ancient harmonic theory. Porphyry also wrote about Homer . Apart from several lost texts known only from quotations by other authors, two texts survive at least in large parts: the Homeric Questions ( Homēriká zētḗmata , largely a philological comment on the Iliad and Odyssey ) and On the Cave of the Nymphs in the Odyssey ( Peri tou en Odysseia tōn nymphōn antrou ). Porphyry's commentary on Euclid 's Elements
385-427: Is dedicated to the defense of mystic theurgic divine possession against the critiques of Porphyry. French philosopher Pierre Hadot maintains that for Porphyry, spiritual exercises are an essential part of spiritual development. Porphyry was, like Pythagoras , an advocate of vegetarianism on spiritual and ethical grounds. These two philosophers are perhaps the most famous vegetarians of classical antiquity. He wrote
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#1732780443849420-418: Is in large part a Latin translation of this work. Manuale Harmonicum (Ἐγχειρίδιον ἁρμονικῆς, Encheiridion Harmonikes ) 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 of a smithy that pitch is determined by numeric ratios. Nicomachus also gives the first in-depth account of
455-761: Is known of his life, and the date of his death is uncertain. Porphyry is best known for his contributions to philosophy. Apart from writing the Aids to the Study of the Intelligibles (Ἀφορμαὶ πρὸς τὰ νοητά; Sententiae ad Intelligibilia Ducentes ), a basic summary of Neoplatonism, he is especially appreciated for his Introduction to Categories ( Introductio in Praedicamenta or Isagoge et in Aristotelis Categorias Commentarium ),
490-401: Is much less rigorous than Euclid 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 that ab=2bc , only because this is true for a=6, b=5 and c=3. Boethius ' De institutione arithmetica
525-520: Is obviously n ( n + 1 ) / 2 {\displaystyle n(n+1)/2} , and there are n ( n + 1 ) / 2 {\displaystyle n(n+1)/2} of them, so their sum is ( n ( n + 1 ) / 2 ) 2 . {\displaystyle {\bigl (}n(n+1)/2{\bigr )}^{2}.} Many early mathematicians have studied and provided proofs of Nicomachus's theorem. Porphyry (philosopher) This
560-514: The On Abstinence from Animal Food (Περὶ ἀποχῆς ἐμψύχων; De Abstinentia ab Esu Animalium ), advocating against the consumption of animals, and he is cited with approval in vegetarian literature up to the present day. He believed that everything was created for mutual advantage, and vegetarianism was a way to preserve universal harmony of nature. Porphyry also wrote widely on music theory , astrology , religion, and philosophy. He produced
595-488: The Greek language on a wide variety of topics, ranging from music theory to Homer to vegetarianism . His Isagoge or Introduction , an introduction to logic and philosophy, was the standard textbook on logic throughout the Middle Ages in its Latin and Arabic translations. Porphyry was, and still is, also well-known for his anti-Christian polemics. Through works such as Philosophy from Oracles and Against
630-858: The Introduction to Arithmetic, has not survived. Among his known lost work is another larger work on music, promised by Nicomachus himself, and apparently referred to by Eutocius in his comment on the sphere and cylinder of Archimedes . Introduction to Arithmetic ( ‹See Tfd› 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 to Plato quite often, and writes that philosophy can only be possible if one knows enough about mathematics . Nicomachus also describes how natural numbers and basic mathematical ideas are eternal and unchanging, and in an abstract realm. The work consists of two books, twenty-three and twenty-nine chapters, respectively. Nicomachus's presentation
665-618: The Manual , ten extracts survive from what appear to have originally been a more substantial work on music. The Introduction to Arithmetic of Nicomachus was a standard textbook in Neoplatonic schools, and commentaries on it were written by Iamblichus (3rd century) and John Philoponus (6th century). The Arithmetic (in Latin: De Institutione Arithmetica ) of Boethius was a Latin paraphrase and
700-646: The "Porphyrian Tree" is noted as the first proper commentary made on Aristotle's work. The Introduction was translated into Arabic by Abd-Allāh ibn al-Muqaffaʿ from a Syriac version. With the Arabicized name Isāghūjī (إيساغوجي) it long remained the standard introductory logic text in the Muslim world and influenced the study of theology, philosophy, grammar, and jurisprudence. Besides the adaptations and epitomes of this work, many independent works on logic by Muslim philosophers have been entitled Isāghūjī. Porphyry
735-505: The 5th-century ecclesiastical historian Socrates of Constantinople assert that Porphyry was once a Christian. It is said, however, that while Porphyry did engage with Christianity, he did not believe it. Augustine made comments to Porphyry as he said he was the "most learned of the philosophers, as the most bitter enemy of the Christians". Porphyry was opposed to the theurgy of his disciple Iamblichus . Much of Iamblichus' mysteries
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#1732780443849770-944: The Christians (which was banned by Constantine the Great ), he was involved in a controversy with early Christians . The Suda (a 10th-century Byzantine encyclopedia based on many sources now lost) reports that Porphyry was born in Tyre , however, other sources report that he was born in Batanaea, present-day Syria . His parents named him Malkos or Malchus (cf. Aramaic malkā 'king'). However, he changed it to " Basileus " "King", and into his nickname "Porphyrius" "[clad] in purple" later in his life. In his work The Life of Plotinus, he refers to Aramaic as his "native tongue." Under Cassius Longinus , in Athens, he studied grammar and rhetoric, and became acquainted with Middle Platonism . In 262 he went to Rome , attracted by
805-471: The Christians (Κατὰ Χριστιανῶν; Adversus Christianos ) which consisted of fifteen books. Some thirty Christian apologists, such as Methodius , Eusebius , Apollinaris , Augustine , Jerome , etc., responded to his challenge. In fact, everything known about Porphyry's arguments is found in these refutations, largely because Theodosius II ordered every copy burned in AD 435 and again in 448. Augustine and
840-566: 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's Life of Pythagoras was one of the main sources used by Porphyry and Iamblichus , for their (extant) Lives of Pythagoras. An Introduction to Geometry , referred to by Nicomachus himself in
875-762: The doctrine of the categories of being interpreted in terms of entities (in later philosophy, " universal "). Boethius ' Isagoge , a Latin translation of Porphyry's Introduction , became a standard medieval textbook in European schools and universities, which set the stage for medieval philosophical-theological developments of logic and the problem of universals . In medieval textbooks, the all-important Arbor porphyriana ("Porphyrian Tree") illustrates his logical classification of substance. To this day, taxonomy benefits from concepts in Porphyry's Tree, in classifying living organisms (see cladistics ). Porphyry's invention of
910-500: The earliest Greco-Roman multiplication tables ; the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the British Museum ). Although Nicomachus is considered a Pythagorean, John M. Dillon says that Nicomachus's philosophy "fits comfortably within the spectrum of contemporary Platonism ." In his work on arithmetic, Nicomachus quotes from Plato 's Timaeus to make
945-583: The meantime) together with a biography of his teacher. Iamblichus is mentioned in ancient Neoplatonic writings as his disciple, but this is most likely only meant to indicate that he was the dominant figure in the next generation of philosophers succeeding him. The two men differed publicly on the issue of theurgy . In his later years, he married Marcella, a widow with seven children and a student of philosophy. There are around sixty works connected to Porphyry's name, some in fragments or lost. Some pieces of his work are still being reconstructed today. Little more
980-580: The mid 2nd century by Apuleius , while he makes no mention at all of either Theon of Smyrna 's work on arithmetic or Ptolemy 's work on music, implying that they were either later contemporaries or lived in the time after he did. Historians consider Nicomachus a Neopythagorean based on his tendency to view numbers as having mystical properties rather than their mathematical properties, citing an extensive amount of Pythagorean literature in his work, including works by Philolaus , Archytas , and Androcydes . He writes extensively on numbers , especially on
1015-487: The nature of astrological fate, and other topics relevant to Greek and Roman religion in the third century. Whether this work contradicts his treatise defending vegetarianism , which also warned the philosopher to avoid animal sacrifice, is disputed among scholars. Due to Porphyry’s work being incomplete or lost, the understanding of the piece could be misconstrued. During his retirement in Sicily , Porphyry wrote Against
1050-571: The persecutions of Christians under Diocletian and Galerius . Whether or not Porphyry was the pagan philosopher's opponent in Lactantius ' Divine Institutes , written at the time of the persecutions, has long been discussed. The fragments of the Philosophy from Oracles are only quoted by Christians, especially Eusebius , Theodoret , Augustine , and John Philoponus . The fragments contain oracles identifying proper sacrificial procedure,
1085-402: The philosophical concept of substance into the five components genus , species , difference , property , and accident . Porphyry's discussion of accident sparked a long-running debate on the application of accident and essence . As Porphyry's most influential contribution to philosophy, the Introduction to Categories incorporated Aristotle's logic into Neoplatonism, in particular
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1120-417: 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
1155-399: The reputation of Plotinus , and for six years devoted himself to the practice of Neoplatonism , during which time he severely modified his diet, at one point becoming suicidal. On the advice of Plotinus he went to live in Sicily for five years to recover his mental health. On returning to Rome, he lectured on philosophy and completed an edition of the writings of Plotinus (who had died in
1190-422: 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 . 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. Nicomachus provided one of
1225-590: The supreme first principle is both the demiurge and the Intellect ( nous ), which Nicomachus also equates to being the monad , the potentiality from which all actualities are created. Two of Nicomachus' works, the Introduction to Arithmetic and the Manual of Harmonics are extant in a complete form, and two others, a work on Theology of Arithmetic and a Life of Pythagoras survive in fragments, epitomes, and summaries by later authors. The Theology of Arithmetic ( Ancient Greek : Θεολογούμενα ἀριθμητικῆς ), on
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