Anaxagoras

Anaxagoras was a Pre-Socratic Greek philosopher. Born in Clazomenae at a time when Asia Minor was under the control of the Persian Empire, Anaxagoras came to Athens. According to Diogenes Laërtius and Plutarch, in life he was charged with impiety and went into exile in Lampsacus. Responding to the claims of Parmenides on the impossibility of change, Anaxagoras described the world as a mixture of primary imperishable ingredients, where material variation was never caused by an absolute presence of a particular ingredient, but rather by its relative preponderance over the other ingredients, he introduced the concept of Nous as an ordering force, which moved and separated out the original mixture, homogeneous, or nearly so. He gave a number of novel scientific accounts of natural phenomena, he produced a correct explanation for eclipses and described the sun as a fiery mass larger than the Peloponnese, as well as attempting to explain rainbows and meteors. Anaxagoras is believed to have enjoyed some wealth and political influence in his native town of Clazomenae.

However, he surrendered this out of a fear that they would hinder his search for knowledge. The Roman author Valerius Maximus preserves a different tradition: Anaxagoras, coming home from a long voyage, found his property in ruin, said: "If this had not perished, I would have"—a sentence described by Valerius as being "possessed of sought-after wisdom!"Anaxagoras was a Greek citizen of the Persian Empire and had served in the Persian army. Though this remains uncertain, "it would explain why he came to Athens in the year of Salamis, 480/79 B. C." Anaxagoras is said to have remained in Athens for thirty years. Pericles learned to love and admire him, the poet Euripides derived from him an enthusiasm for science and humanity. Anaxagoras brought the spirit of scientific inquiry from Ionia to Athens, his observations of the celestial bodies and the fall of meteorites led him to form new theories of the universal order, to a putative prediction of the impact of a meteorite in 467. He attempted to give a scientific account of eclipses, meteors and the sun, which he described as a mass of blazing metal, larger than the Peloponnese.

The heavenly bodies, he asserted, were masses of stone torn from the earth and ignited by rapid rotation. He was the first to give a correct explanation of eclipses, was both famous and notorious for his scientific theories, including the claims that the sun is a mass of red-hot metal, that the moon is earthy, that the stars are fiery stones, he thought the earth was flat and floated supported by'strong' air under it and disturbances in this air sometimes caused earthquakes. These speculations made him vulnerable in Athens to a charge of impiety. Diogenes Laërtius reports the story that he was prosecuted by Cleon for impiety, but Plutarch says that Pericles sent his former tutor, Anaxagoras, to Lampsacus for his own safety after the Athenians began to blame him for the Peloponnesian war. According to Laërtius, Pericles spoke in defense of Anaxagoras at his trial, c. 450. So, Anaxagoras was forced to retire from Athens to Lampsacus in Troad, he died there in around the year 428. Citizens of Lampsacus erected an altar to Mind and Truth in his memory, observed the anniversary of his death for many years.

Anaxagoras wrote a book of philosophy, but only fragments of the first part of this have survived, through preservation in work of Simplicius of Cilicia in the 6th century AD. According to Anaxagoras all things have existed in some way from the beginning, but they existed in infinitesimally small fragments of themselves, endless in number and inextricably combined throughout the universe. All things existed in a confused and indistinguishable form. There was an infinite number of homogeneous parts as well as heterogeneous ones; the work of arrangement, the segregation of like from unlike and the summation of the whole into totals of the same name, was the work of Mind or Reason. Mind is no less unlimited than the chaotic mass, but it stood pure and independent, a thing of finer texture, alike in all its manifestations and everywhere the same; this subtle agent, possessed of all knowledge and power, is seen ruling in all the forms of life. Its first appearance, the only manifestation of it which Anaxagoras describes, is Motion.

It gave distinctness and reality to the aggregates of like parts. Decease and growth represent a new disruption. However, the original intermixture of things is never wholly overcome; each thing contains in itself parts of other things or heterogeneous elements, is what it is, only on account of the preponderance of certain homogeneous parts which constitute its character. Out of this process arise the things. Anaxagoras is mentioned by Socrates during his trial in Plato's "Apology". In the Phaedo, Plato portrays Socrates saying of Anaxagoras that as a young man:'I eagerly acquired his books and read them as as I could'. In a quote chosen to begin Nathanael West's first book "The Dream Life of Balso Snell", Marcel Proust's character Bergotte says, "After all, my dear fellow, Anaxagoras has said, is a journey." Anaxagoras appears as a character in Part II by Johann Wolfgang von Goethe. Anaxagoras appears as a cha

Euclid

Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I, his Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid wrote works on perspective, conic sections, spherical geometry, number theory, mathematical rigour; the English name Euclid is the anglicized version of the Greek name Εὐκλείδης, which means "renowned, glorious". Few original references to Euclid survive, so little is known about his life, he was born c. 325 BC, although the place and circumstances of both his birth and death are unknown and may only be estimated relative to other people mentioned with him.

He is mentioned by name by other Greek mathematicians from Archimedes onward, is referred to as "ὁ στοιχειώτης". The few historical references to Euclid were written by Proclus c. 450 AD, centuries after Euclid lived. A detailed biography of Euclid is given by Arabian authors, for example, a birth town of Tyre; this biography is believed to be fictitious. If he came from Alexandria, he would have known the Serapeum of Alexandria, the Library of Alexandria, may have worked there during his time. Euclid's arrival in Alexandria came about ten years after its founding by Alexander the Great, which means he arrived c. 322 BC. Proclus introduces Euclid only in his Commentary on the Elements. According to Proclus, Euclid belonged to Plato's "persuasion" and brought together the Elements, drawing on prior work of Eudoxus of Cnidus and of several pupils of Plato Proclus believes that Euclid is not much younger than these, that he must have lived during the time of Ptolemy I because he was mentioned by Archimedes.

Although the apparent citation of Euclid by Archimedes has been judged to be an interpolation by editors of his works, it is still believed that Euclid wrote his works before Archimedes wrote his. Proclus retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclid's Elements, "Euclid replied there is no royal road to geometry." This anecdote is questionable since it is similar to a story told about Menaechmus and Alexander the Great. Euclid died c. 270 BC in Alexandria. In the only other key reference to Euclid, Pappus of Alexandria mentioned that Apollonius "spent a long time with the pupils of Euclid at Alexandria, it was thus that he acquired such a scientific habit of thought" c. 247–222 BC. Because the lack of biographical information is unusual for the period, some researchers have proposed that Euclid was not a historical personage, that his works were written by a team of mathematicians who took the name Euclid from Euclid of Megara. However, this hypothesis is not well accepted by scholars and there is little evidence in its favor.

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later. There is no mention of Euclid in the earliest remaining copies of the Elements, most of the copies say they are "from the edition of Theon" or the "lectures of Theon", while the text considered to be primary, held by the Vatican, mentions no author; the only reference that historians rely on of Euclid having written the Elements was from Proclus, who in his Commentary on the Elements ascribes Euclid as its author. Although best known for its geometric results, the Elements includes number theory, it considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization, the Euclidean algorithm for finding the greatest common divisor of two numbers.

The geometrical system described in the Elements was long known as geometry, was considered to be the only geometry possible. Today, that system is referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century; the Papyrus Oxyrhynchus 29 is a fragment of the second book of the Elements of Euclid, unearthed by Grenfell and Hunt 1897 in Oxyrhynchus. More recent scholarship suggests a date of 75–125 AD; the classic translation of T. L. Heath, reads: If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half. In addition to the Elements, at least five works of Euclid have survived to the present day, they follow the same logical structure with definitions and proved propositions. Data deals with the nature and implications of "given" information in geometrical problems.

Isidore of Miletus

Isidore of Miletus was one of the two main Byzantine Greek architects that Emperor Justinian I commissioned to design the cathedral Hagia Sophia in Constantinople from 532 to 537. The creation of an important compilation of Archimedes' works has been attributed to him; the spurious Book XV from Euclid's Elements has been attributed to Isidore of Miletus. Isidore of Miletus was a renowned mathematician before Emperor Justinian I hired him. Isidorus taught stereometry and physics at the universities, first of Alexandria of Constantinople, wrote a commentary on an older treatise on vaulting. Eutocius together with Isidore studied Archimedes work. Isidore is renowned for producing the first comprehensive compilation of Archimedes' work, the Archimedes palimpsest survived to the present. Emperor Justinian I appointed his architects to rebuild the Hagia Sophia following his victory over protesters within the capital city of his Roman Empire, Constantinople; the first basilica was completed in 360 and remodelled from 404 to 415, but had been damaged in 532 in the course of the Nika Riot, “The temple of Sophia, the baths of Zeuxippus, the imperial courtyard from the Propylaia all the way to the so-called House of Ares were burned up and destroyed, as were both of the great porticoes that lead to the forum, named after Constantine, houses of prosperous people, a great deal of other properties.”The warring factions of Byzantine society, the Blues and the Greens, opposed each other in the chariot races at the Hippodrome and resorted to violence.

During the Nika Riot, more than thirty thousand people died. Emperor Justinian I ensured that his new structure would not be burned down, like its predecessors, by commissioning architects that would build the church out of stone, rather than wood, “He compacted it of baked brick and mortar, in many places bound it together with iron, but made no use of wood, so that the church should no longer prove combustible.”Isidore of Miletus and Anthemius of Tralles planned on a main hall of the Hagia Sophia that measured 70 by 75 metres, making it the largest church in Constantinople, but the original dome was nearly 6 metres lower than it was constructed, “Justinian suppressed these riots and took the opportunity of marking his victory by erecting in 532-7 the new Hagia Sophia, one of the largest, most lavish, most expensive buildings of all time.”Although Isidore of Miletus and Anthemius of Tralles were not formally educated in architecture, they were scientists that could organize the logistics of drawing thousands of labourers and unprecedented loads of rare raw materials from around the Roman Empire to create the Hagia Sophia for Emperor Justinian I.

The finished product was built in admirable form for the Roman Emperor, “All of these elements marvellously fitted together in mid-air, suspended from one another and reposing only on the parts adjacent to them, produce a unified and most remarkable harmony in the work, yet do not allow the spectators to rest their gaze upon any one of them for a length of time.”The Hagia Sophia architects innovatively combined the longitudinal structure of a Roman basilica and the central plan of a drum-supported dome, in order to withstand the high magnitude earthquakes of the Marmara Region, “However, in May 558, little more than 20 years after the Church’s dedication, following the earthquakes of August 553 and December 557, parts of the central dome and its supporting structure system collapsed.” The Hagia Sophia was cracked by earthquakes and was repaired. Isidore of Miletus’ nephew, Isidore the Younger, introduced the new dome design that can be viewed in the Hagia Sophia in present-day Istanbul, Turkey.

After a great earthquake in 989 ruined the dome of Hagia Sophia, the Byzantine officials summoned Trdat the Architect to Byzantium to organize repairs. The restored dome was completed by 994. Cakmak, AS. "The Structural Configuration of the First Dome of Justinian's Hagia Sophia: An Investigation Based on Structural and Literary Analysis". Soil Dynamics and Earthquake Engineering. 29. Krautheimer, Richard. Early Christian and Byzantine Architecture. Baltimore: Penguin Books. ISBN 978-0-300-05294-7. Mango, Cyril A.. The Art of the Byzantine Empire, 312-1453: Sources and Documents. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-8020-6627-5. Maranci, Christina. "The Architect Trdat: Building Practices and Cross-Cultural Exchange in Byzantium and Armenia". The Journal of the Society of Architectural Historians. 62: 294–305. Doi:10.2307/3592516. Prokopios. Anthony Kaldellis, ed; the Secret History: With Related Texts. Indianapolis: Hackett Publishing. ISBN 978-1-60384-180-1. Watkin, David. A History of Western Architecture.

New York: Thames and Hudson. ISBN 978-1-85669-459-9

Hero of Alexandria

Hero of Alexandria was a mathematician and engineer, active in his native city of Alexandria, Roman Egypt. He is considered the greatest experimenter of antiquity and his work is representative of the Hellenistic scientific tradition. Hero published. Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land, he is said to have been a follower of the atomists. Some of his ideas were derived from the works of Ctesibius. Much of Hero's original writings and designs have been lost, but some of his works were preserved - in manuscripts from the Eastern Roman Empire, a smaller part in Latin or Arabic translations. Hero may have been a Hellenized Egyptian, it is certain that Hero taught at the Musaeum which included the famous Library of Alexandria, because most of his writings appear as lecture notes for courses in mathematics, mechanics and pneumatics. Although the field was not formalized until the twentieth century, it is thought that the work of Hero, his automated devices in particular, represents some of the first formal research into cybernetics.

Hero described the construction of the aeolipile, a rocket-like reaction engine and the first-recorded steam engine. It was created two millennia before the industrial revolution. Another engine used air from a closed chamber heated by an altar fire to displace water from a sealed vessel; some historians have conflated the two inventions to assert that the aeolipile was capable of useful work. The first vending machine was one of his constructions; this was included in his list of inventions in his book Optics. When the coin was deposited, it fell upon a pan attached to a lever; the lever opened up a valve. The pan continued to tilt with the weight of the coin until it fell off, at which point a counter-weight would snap the lever back up and turn off the valve. A windwheel operating an organ, marking the first instance in history of wind powering a machine. Hero invented many mechanisms for the Greek theater, including an mechanical play ten minutes in length, powered by a binary-like system of ropes and simple machines operated by a rotating cylindrical cogwheel.

The sound of thunder was produced by the mechanically-timed dropping of metal balls onto a hidden drum. The force pump was used in the Roman world, one application was in a fire-engine. A syringe-like device was described by Hero to control the delivery of air or liquids. In optics, Hero formulated the principle of the shortest path of light: If a ray of light propagates from point A to point B within the same medium, the path-length followed is the shortest possible, it was nearly 1000 years that Alhacen expanded the principle to both reflection and refraction, the principle was stated in this form by Pierre de Fermat in 1662. A standalone fountain that operates under self-contained hydrostatic energy A programmable cart, powered by a falling weight; the "program" consisted of strings wrapped around the drive axle. Around 100 AD, Hero had described an odometer-like device that could be driven automatically and could count in digital form–an important notation in the history of computing. However, it was not until the 1600s that mechanical devices for digital computation appear to have been built.

Hero described a method for iteratively computing the square root of a number. Today, his name is most associated with Hero's formula for finding the area of a triangle from its side lengths, he devised a method for calculating cube roots in the 1st century CE. A 1979 Soviet animated short film focuses on Hero's invention of the aeolipile, showing him as a plain craftsman who invented the turbine accidentally A 2007 The History Channel television show Ancient Discoveries includes recreations of most of Hero's devices A 2010 The History Channel television show Ancient Aliens episode "Alien Tech" includes discussion of Hero's steam engine A 2014 The History Channel television show Ancient Impossible episode "Ancient Einstein" Paul Levinson's Science Fiction novel "The Plot to Save Socrates" asserts that Hero was an American time traveler; the most comprehensive edition of Hero's works was published in five volumes in Leipzig by the publishing house Teubner in 1903. Works known to have been written by Hero: Pneumatica, a description of machines working on air, steam or water pressure, including the hydraulis or water organ Automata, a description of machines which enable wonders in temples by mechanical or pneumatical means.

Iamblichus

Iamblichus was a Syrian Neoplatonist philosopher of Arab origin. He determined the direction that would be taken by Neoplatonic philosophy, he was the biographer of Pythagoras, a Greek mystic and mathematician. Aside from Iamblichus' own philosophical contribution, his Protrepticus is of importance for the study of the Sophists, owing to its preservation of ten pages of an otherwise unknown Sophist known as the Anonymus Iamblichi. Iamblichus was the chief representative of Syrian Neoplatonism, though his influence spread over much of the ancient world; the events of his life and his religious beliefs are not known, but the main tenets of his beliefs can be worked out from his extant writings. According to the Suda, his biographer Eunapius, he was born at Chalcis in Syria, he was the son of a rich and illustrious family, he is said to have been the descendant of several priest-kings of the Arab Royal family of Emesa. He studied under Anatolius of Laodicea, went on to study under Porphyry, a pupil of Plotinus, the founder of Neoplatonism.

He disagreed with Porphyry over the practice of theurgy. Around 304, he returned to Syria to found his own school at Apamea, a city famous for its Neoplatonic philosophers. Here he designed a curriculum for studying Plato and Aristotle, he wrote commentaries on the two that survive only in fragments. Still, for Iamblichus, Pythagoras was the supreme authority, he is known to have written the Collection of Pythagorean Doctrines, which, in ten books, comprised extracts from several ancient philosophers. Only the first four books, fragments of the fifth, survive. Scholars noted that the Exhortation to Philosophy of Iamblichus was composed in Apamea in the early 4th c. AD. Iamblichus was said to have been a man of great learning, he was renowned for his charity and self-denial. Many students gathered around him, he lived with them in genial friendship. According to Fabricius, he died during the reign of Constantine, sometime before 333. Only a fraction of Iamblichus' books have survived. For our knowledge of his system, we are indebted to the fragments of writings preserved by Stobaeus and others.

The notes of his successors Proclus, as well as his five extant books and the sections of his great work on Pythagorean philosophy reveal much of Iamblichus' system. Besides these, Proclus seems to have ascribed to him the authorship of the celebrated treatise Theurgia, or On the Egyptian Mysteries. However, the differences between this book and Iamblichus' other works in style and in some points of doctrine have led some to question whether Iamblichus was the actual author. Still, the treatise originated from his school, in its systematic attempt to give a speculative justification of the polytheistic cult practices of the day, it marks a turning-point in the history of thought where Iamblichus stood; as a speculative theory, Neoplatonism had received its highest development from Plotinus. The modifications introduced by lamblichus were the detailed elaboration of its formal divisions, the more systematic application of the Pythagorean number-symbolism, under the influence of Oriental systems, a mythical interpretation of what Neoplatonism had regarded as notional.

Unlike Plotinus who broke from Platonic tradition and asserted an undescended soul, Iamblichus re-affirmed the soul's embodiment in matter, believing matter to be as divine as the rest of the cosmos. It is most on this account that lamblichus was venerated. Iamblichus was praised by those who followed his thought. By his contemporaries, Iamblichus was accredited with miraculous powers; the Roman emperor Julian, not content with Eunapius' more modest eulogy that he was inferior to Porphyry only in style, regarded Iamblichus as more than second to Plato, claimed he would give all the gold of Lydia for one epistle of Iamblichus. During the revival of interest in his philosophy in the 15th and 16th centuries, the name of Iamblichus was scarcely mentioned without the epithet "divine" or "most divine". At the head of his system, Iamblichus placed the transcendent incommunicable "One", the monad, whose first principle is intellect, nous. After the absolute One, lamblichus introduced a second superexistent "One" to stand between it and'the many' as the producer of intellect, or soul, psyche.

This is the initial dyad. The first and highest One, which Plotinus represented under the three stages of being and intellect, is distinguished by Iamblichus into spheres of intelligible and intellective, the latter sphere being the domain of thought, the former of the objects of thought; these three entities, the psyche, the nous split into the intelligible and the intellective, form a triad. Between the two worlds, at once separating and uniting them, some scholars think there was inserted by lamblichus, as was afterwards by Proclus, a third sphere partaking of the nature of both, but this supposition depends on a conjectural emendation of the text. We read, that in the intellectual triad he assigned the third rank to the Demiurge; the Demiurge, the Platonic creator-god, is thus identified with the perfected nous, the intellectual triad being increased to a hebdomad. The identification of nous with the Demiurge is a significant moment in the Neoplatonic tradition and its adoption into and development within the Christian tradition.

St. Augustine follows Plotinus by identifying nous, which bears the logos, with the creative principle

Angle

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane. Angles are formed by the intersection of two planes in Euclidean and other spaces; these are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is used to designate the measure of an angle or of a rotation; this measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation; the word angle comes from the Latin word angulus, meaning "corner".

Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow". Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, do not lie straight with respect to each other. According to Proclus an angle must be a relationship; the first concept was used by Eudemus. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle. Lower case Roman letters are used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples. In geometric figures, angles may be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC is denoted ∠BAC or B A C ^. Sometimes, where there is no risk of confusion, the angle may be referred to by its vertex. An angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign.

However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise angle from B to C, ∠CAB to the anticlockwise angle from C to B. An angle equal to 0° or not turned is called a zero angle. Angles smaller than a right angle are called acute angles. An angle equal to 1/4 turn is called a right angle. Two lines that form a right angle are said to be orthogonal, or perpendicular. Angles larger than a right angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle. Angles larger than a straight angle but less than 1 turn are called reflex angles. An angle equal to 1 turn is called complete angle, round angle or a perigon. Angles that are not right angles or a multiple of a right angle are called oblique angles; the names and measured units are shown in a table below: Angles that have the same measure are said to be equal or congruent.

An angle is not dependent upon the lengths of the sides of the angle. Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles. A reference angle is the acute version of any angle determined by subtracting or adding straight angle, to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1/4 turn, 90°, or π/2 radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, an angle of 150 degrees has a reference angle of 30 degrees. An angle of 750 degrees has a reference angle of 30 degrees; when two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other. A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles, they are abbreviated as vert. opp. ∠s. The equality of vertically opposite angles is called the vertical angle theorem.

Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical Note, w

Archytas

Archytas was an Ancient Greek philosopher, astronomer and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato. Archytas was the son of Mnesagoras or Histiaeus. For a while, he was taught by Philolaus, was a teacher of mathematics to Eudoxus of Cnidus. Archytas and Eudoxus' student was Menaechmus; as a Pythagorean, Archytas believed that only arithmetic, not geometry, could provide a basis for satisfactory proofs. Archytas is believed to be the founder of mathematical mechanics; as only described in the writings of Aulus Gellius five centuries after him, he was reputed to have designed and built the first artificial, self-propelled flying device, a bird-shaped model propelled by a jet of what was steam, said to have flown some 200 meters. This machine, which its inventor called The pigeon, may have been suspended on a wire or pivot for its flight. Archytas wrote some lost works, as he was included by Vitruvius in the list of the twelve authors of works of mechanics.

Thomas Winter has suggested that the pseudo-Aristotelian Mechanical Problems is an important mechanical work by Archytas, not lost after all, but misattributed. Archytas named the harmonic mean, important much in projective geometry and number theory, though he did not invent it. According to Eutocius, Archytas solved the problem of doubling the cube in his manner with a geometric construction. Hippocrates of Chios before, reduced this problem to finding mean proportionals. Archytas' theory of proportions is treated in book VIII of Euclid's Elements, where is the construction for two proportional means, equivalent to the extraction of the cube root. According to Diogenes Laërtius, this demonstration, which uses lines generated by moving figures to construct the two proportionals between magnitudes, was the first in which geometry was studied with concepts of mechanics; the Archytas curve, which he used in his solution of the doubling the cube problem, is named after him. Politically and militarily, Archytas appears to have been the dominant figure in Tarentum in his generation, somewhat comparable to Pericles in Athens a half-century earlier.

The Tarentines elected him strategos,'general', seven years in a row – a step that required them to violate their own rule against successive appointments. He was undefeated as a general, in Tarentine campaigns against their southern Italian neighbors; the Seventh Letter of Plato asserts that Archytas attempted to rescue Plato during his difficulties with Dionysius II of Syracuse. In his public career, Archytas had a reputation for virtue as well as efficacy; some scholars have argued that Archytas may have served as one model for Plato's philosopher king, that he influenced Plato's political philosophy as expressed in The Republic and other works. Archytas may have drowned in a shipwreck in the shore of Mattinata, where his body lay unburied on the shore until a sailor humanely cast a handful of sand on it. Otherwise, he would have had to wander on this side of the Styx for a hundred years, such the virtue of a little dust, munera pulveris, as Horace calls it in Ode 1.28 on which this information on his death is based.

The poem, however, is difficult to interpret and it is not certain that the shipwrecked and Archytas are in fact the same person. The crater Archytas on the Moon is named in his honour; the Archytas curve is created by placing a semicircle on the diameter of one of the two circles of a cylinder such that the plane of the semicircle is at right angles to the plane of the circle and rotating the semicircle about one of its ends in the plane of the cylinder's diameter. This rotation will cut out a portion of the cylinder forming the Archytas curve. Another way of thinking of this construction is that the Archytas curve is the result of cutting out a torus formed by rotating a hemisphere of diameter d out of a cylinder of diameter d. A cone can go through the same procedures producing the Archytas curve. Archytas used his curve to determine the construction of a cube with a volume of one third of that of a given cube. One of Archytas' most notable accomplishments comes in the form of a mathematical solution to The Delian Problem, more informally known as doubling the cube.

The problem is as follows: given a cube that a side is known, construct a cube with double the original volume. The proof of his model comes from Eudemus, who in the late 4th century wrote a history of geometry, including solutions to this problem from multiple mathematicians and philosophers before him- namely Eudoxus and Menaechmus. Although Eudemus' work did not survive to current day, a transmission of his geometric solution does survive in the form of Eutocius' commentary on Archimedes' De Sphaera et Cylindro. Archytas' solution begins with the concept of mean proportionality and the construction of four similar triangles; each triangle's hypotenuse and long leg are proportionally similar as the triangle increase in size, today's version of similarity of triangles. Archytas applied the mean proportionals for a given length of a cube. If the volume of the original cube is written as V1 = x3, where x represents the length of a side, we let k1 and k2 represent the proportionality constants, the cube is doubled so that a side length is now 2x, a mean proportional between the two can be written as.

With the proportionals finished, Archytas comple