# Archytas of Taranto

Archytas of Taranto ( Greek Ἀρχύτας Archýtas ; * probably between 435 and 410 BC; † probably between 355 and 350 BC) was an ancient Greek philosopher , mathematician , music theorist , physicist , engineer, statesman and general .

Archytas worked in his hometown, the Greek colony of Taranto in Apulia . As a philosopher he belonged to the direction of the Pythagoreans . He is best known for his friendly relationship with Plato , for the flying dove that he allegedly invented and for a thought experiment with which he wanted to prove the infinity of the universe. Only a few fragments of his writings, which dealt in particular with topics of mathematics and music, have survived.

As a science theorist, Archytas was an optimist. He said that scientific knowledge is easy to come by if you have the right method. He placed particular emphasis on mathematics as a basic science. His most important mathematical achievement was the solution to the problem of doubling the cube . It shows that he had a methodological complexity that was extraordinary for his time. He contributed to the theory of musical harmony with his mathematical theory of harmonic intervals . In optics he tried to find an explanation for the reflection and in acoustics for the different pitches. His scientific achievements, of which little is known due to the scarcity of the information handed down, found recognition in ancient posterity and among modern historians of science.

Archytas played a decisive role politically and militarily as a leading statesman and strategist in his hometown and a league of Greek colonies in southern Italy led by it. His military successes gave him great authority. Domestically, he campaigned for social equilibrium, considering it possible to scientifically justify a concept of justice and thus to bring about a consensus.

## Life

The philosopher's father was probably called Hestiaios. According to other, less credible information, his name was Mnesagoras, Mnasagetes or Mnesarchos. Otherwise nothing is known about the origin of Archytas. His birth can only be roughly dated; it probably falls between 435 and 410 BC. Apparently his family was rich; anecdotal tradition shows him as a large landowner.

### science

Cicero reports that Archytas's philosophical teacher was Philolaus of Croton . That is plausible, but not certain. Philolaos belonged to the class of the Pythagoreans , who invoked the teachings of Pythagoras of Samos . According to the Roman writer Valerius Maximus , Archytas received a long, thorough training in Metapont , a traditional center of Pythagoreanism. There in the 6th century BC BC Pythagoras lived and taught. Archytas fitted himself into the Pythagorean tradition, but that does not mean that he adopted dogmatics from it. In the 4th century BC BC he was primarily perceived as an independent thinker who tied on the themes and ideas of the Pythagoreans. Aristotle made a clear distinction between the views of Archytas and those of the "so-called" Pythagoreans, with whom he dealt separately. - Of the disciples of Archytas only the mathematician Eudoxus of Knidus and an Archedemus whom Plato valued are known by name.

### Political and military activity

Taranto had from around 473 BC. A democratic constitution, but stood in the Peloponnesian War (431–404 BC) on the side of Sparta , whose mixed constitution was characterized by monarchical and aristocratic- oligarchic elements, and thus opposed democratic Athens . Sparta was the mother city of Taranto, which was once founded by Spartan colonists. Syracuse , the predominant power in the Greek-populated part of Sicily, was also one of Athens' ultimately victorious enemies. After the end of the war, Taranto took a friendly stance against Syracuse and stayed out of the military conflicts between the tyrant Dionysius I of Syracuse and a 393 formed league of southern Italian cities. After the tyrant had conquered the leading city of the league, Croton , in 379/378 , Taranto took over the leadership of the league, probably in agreement with Dionysius, and developed into the leading power in the mainland part of Magna Graecia , the Greek-populated parts of Sicily and southern Italian Mainland. Now the city began to flourish. In this period, the seventies and sixties of the 4th century BC. BC, the political and military heyday of Archytas fell, who already found an advantageous starting position and then knew how to use the favorable conditions. From a military point of view, Taranto was a major regional power at the time, it was about as strong as Athens before the outbreak of the Peloponnesian War and had an important fleet. According to information supplied by Strabo , the city was able to muster 30,000 infantrymen and 4,000 cavalrymen.

Archytas' action space. The Greek name Taras is given for Taranto .

The circumstances of Archytas' rise to a leadership position are not documented. In any case, it is certain that he proved himself militarily in an extraordinary way. The imperial historian of philosophy Diogenes Laertios , who probably relied on information from the well-informed philosopher Aristoxenus , praised the fact that Archytas was the only one to be elected general ( strategos ) seven times by his fellow citizens , although the law did not allow re-election after the end of the one-year term of office . What was probably meant was that he belonged to a college of elected generals seven times in a row, although immediate re-election was forbidden, as one wanted to prevent a dangerous concentration of power. Apparently the legal regulation was overridden by a popular resolution especially for Archytas. This special arrangement illustrates the extraordinary confidence that he enjoyed. As a commander, he had special powers associated with his title strategós autokrátor ; he was allowed to make military decisions at his own discretion, but was ultimately subject to the supervision of the people's assembly of his fellow citizens. Once he resigned the general office, allegedly as a result of the machinations of envious opponents; then the Tarentines promptly suffered a defeat, which proved its irreplaceability.

As a leading statesman and strategist in Taranto, Archytas was also supreme commander of the armed forces of the League. Its purpose was mainly the common struggle of the Greek settlers against their traditional opponents, the native Italians , against the Tarent as early as the 5th century BC. Had proceeded with varying degrees of success. Archytas' campaigns against the Italians were all successful.

### Relationship to Plato

Archytas' fame in later times was primarily due to his relationship with Plato. He got to know the later famous Athenian philosopher when he stayed in Taranto on his first trip to Italy (388/387 BC) before he visited Dionysius I in Syracuse. Archytas became the host of the Athenians (xénos) . Plato was probably primarily interested in Archytas' knowledge of mathematics, less in his philosophical views. The hospitality relationship involved mutual obligations for mutual benefit, but was not necessarily associated with close personal friendship. The late tradition according to which Plato hoped for philosophical insights from Archytas or even became his disciple is not trustworthy. Also unbelievable is a late legend according to which Plato was enslaved on the orders of the Syracuse tyrant and bought and released by Archytas.

For further development, Plato's Seventh Letter is the main source. The authenticity of the letter has long been controversial in research, but its description of the course of events is considered credible, even if it does not come from Plato. In any case, the author was familiar with the circumstances. According to the description in the letter, Plato had a friendly relationship between Archytas and during his second stay in Sicily (366-365 BC), when he influenced the tyrant Dionysius II of Syracuse , the son and successor of Dionysius I made by the young Syracuse ruler. Later, after Plato's departure, Archytas visited the tyrant. During his stay in Syracuse, the Tarentine statesman not only cultivated political relations, but apparently also taught Dionysius II philosophically and received a positive impression from him. In the period that followed, both the Tarentines and Dionysius urged Plato, who had returned to Athens, to set off again, despite the resentment that had arisen between him and the tyrant during the second stay. Archytas hoped that Plato's influence on Dionysius would stabilize the good relationship between Taranto and the Syracuse Empire. Urgent requests from Syracuse and Taranto persuaded the old philosopher to go on his third journey to Sicily (361–360 BC). This time, however, he fell out of favor with the tyrant, was involved in political conflicts and his life was in danger. According to the seventh letter, he succeeded in informing Archytas of his plight, whereupon the Tarentines sent an embassy, ​​which intervened in his favor and gave the oppressed philosopher permission to leave.

### death

Archytas probably died between 355 and 350 BC. After his death, Taranto began to decline, which eventually led to the Tarentines settling down from around 340 BC. BC no longer relied on their own military resources, but hired mercenary leaders who from then on had a significant share in the conduct of the war.

In Ode I, 28 ( Te maris et terrae ) by the Roman poet Horace , Archytas is addressed in connection with a shipwrecked man who asks for his burial . Various conclusions were drawn from this that he had died in a shipwreck in the Adriatic . This interpretation is controversial.

Formerly mistakenly identified as Archytas bust in the National Archaeological Museum of Naples

## Pictorial representation

Portraits of a man with a turban-like headgear - a bronze bust in the National Archaeological Museum of Naples and a Roman Herme in the Museo Capitolino in Rome - have been identified as depictions of Archytas because of the peculiar hairstyle, as this hairstyle can also be seen on a correspondingly inscribed coin from Taranto is. However, the coin has been shown to be a modern forgery, which eliminates the basis for identification.

## Works

Of the works of Archytas only four definitely genuine fragments have survived. There is no list of works in the older sources. In later times, many spurious works were in circulation under his name. The original titles of the authentic works are not known, and the information provided by the citing ancient authors is considered unreliable. What is certain is that music and mathematics were covered. For one of the writings, the title on the sciences in different variants ( Perí mathematikón , Perí mathemáton , Peri mathematikés ) is mentioned. Another - or possibly part of On the Sciences - might be called Harmonics . Another work, allegedly entitled Treatises (diatribaí) , may have attempted to give ethics a scientific basis. Archytas may also have written on cosmology, biology, machines, and agriculture.

## philosophy

Although Archytas was a younger contemporary of Socrates , whom he outlived by decades, he is counted among the pre-Socratics because he belonged to an older tradition that was not yet under the influence of Socratic philosophy. This assignment is problematic, however, because his works were only created after the death of Socrates.

Archytas regarded the “numerical science”, which he called logistikē , as the basis of the sciences and also emphasized its priority over geometry. In the appreciation of mathematics, he agreed with Plato. However, while Plato only saw mathematics as a preparation for the study of philosophy and his understanding of education aimed at a purely intellectual grasp of reality, Archytas did not share Plato's disdain for empiricism and also made the sharp Platonic separation between the areas of the spiritually knowable and of the sensually perceptible. For him, arithmetic was also important from a political point of view, because it seemed to offer him the opportunity to find plausible formulas for an amicable, balanced distribution of property among the citizens. Since the application of such formulas was verifiable by everyone, Archytas was convinced that social peace could be established and preserved. This was of the greatest importance in the Greek cities, which were often shaken by bloody power struggles. A balance between the social classes, which should prevent violent conflicts ( stáseis ) in the citizenry, was a central concern of Archytas. He expected the realization from the correct, appropriate “calculation” (logismós) , which verifiably guaranteed that no one would be taken advantage of.

Bruno Snell points to the change in meaning of the word máthema , which in its basic meaning denotes that which has been or can be learned. This expression is first attested to as a term for science in Archytas. For the Tarentine philosopher, mathematics took center stage among the fields of knowledge, but in addition to geometry and arithmetic, astronomy and music also belonged to the mathémata . Archytas called these four sciences "sisterly". Only later was the field of meaning narrowed down to mathematics because only mathematics appeared as a science in the true sense, because only it seemed to meet the requirement that the objects of a science must be recognizable with complete certainty.

Apparently Archytas developed a philosophy of science as a teaching in which he treated the art of searching correctly - the scientific approach - as a prerequisite for success. He professed an epistemological optimism; he believes that discovery is easy and simple with the right method. The details of his method are difficult to determine because of the unfavorable sources. His principle has been handed down that one must first make good distinctions with regard to the nature of the "whole"; if this is successful, the nature of the individual objects can be grasped well. Accordingly, scientific knowledge advances from the more general to the more specific. What exactly he meant by “whole” - such as the general concepts of a particular science - is not clear from the sparse information in the sources. In any case, Archytas was convinced that discovering facts on one's own was superior to adopting existing knowledge. What one has found out for oneself is something one's own (ídion) ; the knowledge that one acquires through learning is something foreign.

In ethics, Archytas placed particular emphasis on the requirement that one should always act according to reason and never act spontaneously out of anger or let one's mind cloud one's mind.

## cosmology

The tradition that Archytas was also active as an astronomer goes back to the Roman poets Horace and Properz , who probably had no reliable information about it. However, his argumentation for the infinity of the universe is authentic and famous. It is a thought experiment that says: If someone who had arrived at an assumed end of the universe would stretch out his hand or a stick there, he would have to come across either a body or empty space, so definitely one Continuation of the universe. Thus the cosmos must be infinitely expanded. This idea was taken up and modified by the Stoics and Epicureans and also by John Locke and Isaac Newton .

## mathematics

### irrationality

Archytas dealt with what was called “over-dividing” in the phrase at the time . These are conditions in which the "surplus" of about te n the part and then: . Archytas found a proof for the sentence “Between two numbers in an overdividing ratio, mean proportions ( geometric means ) can never be found.” In modern terminology this means that there are irrational proportions that cannot be represented as rational numerical ratios ( fractions ) . The square roots are irrational. ${\ displaystyle (n + 1): n}$${\ displaystyle a: b}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle b}$${\ displaystyle a: b = (n + 1): n}$ ${\ displaystyle {\ sqrt {(n + 1): n}}}$

### Curve of Archytas

Curve of Archytas
The process of doubling the cube

Hippocrates of Chios succeeded in reducing the problem of doubling the cube to a relationship problem: It suffices to find the line , the edge of the cube to be doubled, the lines and so - that means: to construct geometrically - that they are in Relationship . Then namely ${\ displaystyle a}$${\ displaystyle u}$${\ displaystyle v}$${\ displaystyle a: u = u: v = v: 2a}$

${\ displaystyle \ left ({\ frac {a} {u}} \ right) ^ {3} = {\ frac {a} {u}} \ times {\ frac {u} {v}} \ times {\ frac {v} {2a}} = {\ frac {a} {2a}} = {\ frac {1} {2}}.}$

So it applies

${\ displaystyle 2 \ times a ^ {3} = u ^ {3}}$

and the cube with the edge is a doubling of the cube with the edge , as desired . ${\ displaystyle u}$${\ displaystyle a}$

It does not, however, succeeded Hippocrates, and for given distances and be so designed that applies even for the only need here special case . This is what later ancient scientists strove to do. The late classical mathematician Eutocius handed in his commentary on the treatise Peri sphaíras kai kylíndrou (About spheres and cylinders) of the Archimedes twelve solutions. Their earliest and best is that of Archytas. He succeeded in doing this with the help of the curve that was named after him . This is the first crooked curve - that is, not contained in any plane - that has been used in the history of mathematics. The construction with which an intersection of three curved surfaces is found is unique in ancient mathematics and especially astonishing for this early stage in the history of mathematics. However, in today's research it is overwhelmingly believed that it actually came from Archytas. ${\ displaystyle u}$${\ displaystyle v}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a: u = u: v = v: b}$${\ displaystyle b = 2a}$

For the solution, Archytas used the surfaces of three bodies: a torus , a cylinder and a cone . In modern representation by means of suitably selected Cartesian coordinates , these surfaces are each given by one of the following equations:

${\ displaystyle {\ begin {matrix} x ^ {2} + y ^ {2} + z ^ {2} = a {\ sqrt {x ^ {2} + y ^ {2}}} \\ x ^ { 2} + y ^ {2} = ax \\ x ^ {2} + y ^ {2} + z ^ {2} = {\ frac {a ^ {2}} {b ^ {2}}} x ^ {2} \ end {matrix}}}$

The torus and cylinder intersect in the curve of Archytas. The intersection of this curve with the cone is a point that satisfies all three equations. So it applies to him if shortened ${\ displaystyle P}$

${\ displaystyle u = {\ sqrt {x ^ {2} + y ^ {2} + z ^ {2}}}}$  and  ${\ displaystyle v = {\ sqrt {x ^ {2} + y ^ {2}}}}$

is written:

${\ displaystyle \ left \ {{\ begin {matrix} u ^ {2} = av \\ v ^ {2} = ax \\ u ^ {2} = {\ frac {a ^ {2}} {b ^ {2}}} x ^ {2} \ end {matrix}} \ right.}$

The first equation says . If in the third equation is substituted for the equivalent according to the second equation , the result is after extraction of the root and conversion . Overall, the desired relationship applies . The distance from to the origin has the length ; so was introduced. So in the case it is the edge of the doubled cube. Stephen Menn gives a reconstruction of Archytas' approach. The construction does not succeed if only compasses and rulers are used; this requirement only became predominant in Greek mathematics after Archytas. ${\ displaystyle a: u = u: v}$${\ displaystyle ax}$${\ displaystyle v ^ {2}}$${\ displaystyle u: v = v: b}$${\ displaystyle a: u = u: v = v: b}$${\ displaystyle P}$${\ displaystyle u}$${\ displaystyle u}$${\ displaystyle b = 2a}$

## music

The theorem, proven by Archytas, that there can be no geometrical mean between the numbers and , which are in a “divided” relationship, has the consequence for harmony that it is impossible to define the fundamental harmonic intervals - the octave (2: 1 ), the fifth (3: 2), the fourth (4: 3) and the whole tone (9: 8) - divided into two equal parts by an average proportional. Therefore, Archytas divided the fifth and fourth using the arithmetic and harmonic means. Based on these relationships, he developed a mathematical theory of harmonic intervals for all three tetrachords used at the time (the enharmonic, the chromatic and the diatonic tones). He determined the numerical proportions of the intervals within the tetrachords, beginning with the highest tone, for the diatonic tone gender as 9: 8, 8: 7, 28:27, for the chromatic as 32:27, 243: 224; 28:27 and for the enharmonic as 5: 4, 36:35, 28:27. In music practice, it was about the two inner strings of a four-stringed instrument, the "movable" strings, which were tuned differently in the three tone genders. ${\ displaystyle n + 1}$${\ displaystyle n}$

The work of Archytas was continued in the further course of ancient music theory research. One result of these efforts were the findings in Euclid's Sectio canonis . The contextual connection between the studies of Archytas and the Sectio canonis has led some modern researchers to assume that Euclid's work essentially reproduces a text by the Tarentine. Against this assumption, however, it is argued that the differences are more important than the similarities. At least Archytas can be regarded as a forerunner.

## physics

### optics

The imperial writer Apuleius , who presumably relied on credible information from Archimedes , mentioned in his Apologia an optical theory by Archytas to explain the reflections. The phenomenon of mirroring was interpreted differently in antiquity. According to one of the hypotheses, everything constantly sends out atoms which, when they reach the eyes of the sighted, convey to them the shape of the object from which they originate. Accordingly, reflection is based on the fact that the reflecting surface throws the atoms back. According to an alternative hypothesis, the eye emits rays when seeing, which make contact with objects in the outside world and are reflected when they are reflected. Archytas, like Plato, was one of the proponents of the second declaration. But his conception differs considerably from that of Plato. While Plato believed that the rays emanating from the eye can only create an optical impression in conjunction with external light, Archytas believed that they did not need such support. How he could explain on this assumption that nothing is visible in outer darkness is unknown; he must have assumed some external hindrance.

For the Pythagoreans, mathematical optics was one of the most important fields of research. They wanted to describe the visual process through geometric relationships. Who of them founded mathematical optics is unknown; in research into the history of science it is assumed that Archytas played a pioneering role.

### Acoustics

Archytas presented a theory of acoustics , relying on results of unnamed earlier researchers, which he considered conclusive. Apparently he was referring to a research consensus that existed at the time. He based his argumentation on "experiments" as well as on experiences with the human voice and with the functioning of musical instruments. By experiments, he understood not only experimental arrangements in the modern sense, but also general observations of all kinds, especially everyday experiences. In the acoustic phenomena he did not see products of the hearing apparatus of perception, but objective facts that are also given when nobody is listening. Thus, according to his understanding, a noise is also to be understood as such if it is outside the range of perception of the listener, for example because it is too quiet. Apparently following a consensus at the time, he named the effect of moving things that collide with one another as a condition for the creation of noises. For the fact that some noises can be heard and others not, he gave three reasons which, according to him, lie in human nature: If you cannot hear a sound, either the collision of the producing things is too weak or the distance to the source of the noise too big or the sound is so loud that it cannot be heard due to its strength. The latter occurs when the sound is so voluminous that it does not fit in the narrow ear canal and therefore cannot penetrate the ear. Such noises are therefore in principle inaudible. Apparently Archytas viewed sound as a kind of matter that moves through space and also gets into the ear if it fits. It is unclear whether he meant the harmony of the spheres - the tones supposedly produced by the heavenly bodies - by the extremely loud and therefore inaudible sounds .

Archytas' explanation of the different pitches of audible sounds is as follows: High notes are those that hit the ear at relatively high speed; the slower a note is when it arrives, the deeper it appears to the listener. This theory has not been attested before Archytas, it probably originated from him. This is supported by the fact that he gave detailed reasons for his hypothesis, apparently because he considered it necessary to convince the audience of a new idea. He compared the loud and quiet tones with more or less powerfully fired or thrown weapons, which only fly far and hit with force if they have received a strong impulse. To illustrate this, Archytas also pointed out that when speaking or singing loudly, one has to make an effort to produce the desired volume with a powerful voice. He also stated that the sound produced by a wind instrument is deeper if the sound has to travel a longer distance through the pipe with the same force of blowing. He attributed this effect to a weakening of the movement due to the greater distance. However, his theory still lacks a distinction between the independent causes of pitch and volume; the aspects he mentions - the speed and force of impact - have the same cause. The differentiation of causality was only achieved by later ancient research. - The idea of ​​Archytas that the pitch depends on the varying speed of the sound became generally accepted in ancient acoustics. However, Theophrastus contradicted her .

In modern terminology, the statement of Archytas means that the volume, the amplitude , depends on the energy communicated to the sound from its source, and that the distance the sound travels until it is attenuated to inaudibility, the greater the amplitude is longer. The approach of reducing pitch to speed was correct, except that Archytas mistakenly assumed it was the speed at which sound travels. In reality it is the frequency , the speed of the oscillation, that is, the number of back and forth movements per unit of time.

### mechanics

Archytas has long been considered the founder of mechanics . But there is no proof of this if one associates the concept of engineering as an applied science with the term mechanics. According to Diogenes Laertios, Archytas was the first to methodically treat mechanics using mathematical principles. Accordingly, as a pioneer, he made a contribution to theory; this does not mean anything for the application. Plutarch claimed that Archytas and Eudoxus of Knidos were the first scientists to study mechanics and technology. With mechanical equipment, they had solved problems like doubling the cube, which were difficult to solve theoretically and graphically. However, Plato criticized this approach as non-mathematical and put an end to it. However, according to the current state of research, this does not apply to Archytas, because his method of doubling the cube was purely abstract, no instruments were used. Thus there is no evidence of his role as the founder of practical engineering. Of the devices whose invention Archytas was ascribed to, only two actually come from him: the flying "Dove of Archytas" and probably also a rattle . But the rattle is just a toy, and the dove does not make Archytas the founder of applied engineering. It was only one of the important precursors that created the conditions for the development of this branch of knowledge. There is no evidence to support the claims that he invented the pulley system and devised war machines. Although Archytas was apparently fundamentally interested in the application of mathematical knowledge to physical objects, it cannot be shown that he went beyond abstract considerations.

#### Dove of Archytas

Aulus Gellius , a Roman writer of the 2nd century, reports about the dove . He refers to a now lost Greek script of his older contemporary Favorinus , which he quotes, and to information from other authors who are not named. Based on this representation, Archytas constructed a wooden replica of a pigeon that could fly by means of a mechanism he had devised. In addition, Gellius remarks that this seems unbelievable, but should be taken to be true. The construction was balanced by counterweights (libramenta) . The pigeon was set in motion by a hidden, trapped air current. However, according to the Favorinus quote, she was unable to get up again after landing.

Gellius and Favorinus Archytas of Taranto describe the inventor of the device. In research, however, the possibility is considered that there is a mix-up with an author of the same name who later wrote a treatise on mechanics.

In the specialist literature, various explanatory hypotheses have been discussed, the starting point of which is a model presented by Wilhelm Schmidt in 1904. According to this, the pigeon did not fly free, but was part of a larger apparatus for which roles were required. A string connected them to the counterweight that was suspended in the air. The wooden bird was hollow, filled with compressed air and provided with a hidden valve through the opening of which air could escape. This reduced the weight, and the counterweight, which was as heavy as the pigeon and the compressed air, became overweight and lowered, so that the pigeon was shot up. Carl A. Huffman has presented a modified version of the model that does not require compressed air in the pigeon; According to his interpretation, the required air flow was generated outside the bird. Karin Luck-Huyse suspects “a kind of jet propulsion using compressed air”.

#### Rattle

The second device that Archytas apparently invented is a rattle (Greek platagḗ ). Aristotle reports that the "rattle of Archytas" is intended for restless children; their purpose is to keep young children busy and prevent them from doing undesirable activities. The expression became proverbial: A person who cannot keep himself calm was said to need a rattle of Archytas. According to an anecdotal lore, Archytas was particularly interested in children and liked to play with them. Hence it seems plausible that he actually devised such a device. But it is also possible that the inventor was an architect of the same name. What the rattle looked like is unknown; it was probably a device in the style of castanets .

## reception

### Ancient and Middle Ages

Aristotle dealt intensively with the philosophy of Archytas. He treated them in a special script from three books. He also wrote a comparison of Plato's dialogue Timaeus and the writings of Archytas. Both works of Aristotle are lost today. His student Aristoxenus, who came from Archytas' hometown of Taranto, wrote a biography of his famous compatriot, which is also lost. Aristoxenus, who as a younger contemporary of Archytas was well informed, utilized anecdotal material and gave a benevolent account. A large part of the later biographical and doxographical tradition, including the brief Archytas biography of Diogenes Laertios, is based on this biography .

A number of treatises and fragments as well as two letters in the Doric dialect have survived under the name of Archytas , which are certainly not from him, but from various unknown authors. They belong to the pseudepigraphic (spread under false author names) philosophical literature, whose anonymous authors ascribed their writings to well-known Pythagoreans of the past in order to draw attention to their literary fictions. Among the Pythagoreans, Archytas is the one under whose name most of such works circulated in antiquity. The dating of the pseudo-archyteic writings is controversial; According to an older research opinion ( Holger Thesleff ), most of them belong to the early Hellenistic period. According to the dating approaches that are predominant today, some are in the 1st century BC. BC or in the 1st century AD. Questions of logic , epistemology , metaphysics , ethics and state theory are dealt with .

In ancient times, a number of anecdotes circulated that shaped the image of the famous philosopher and statesman for the educated public. The question of a historical core of the narrative remains open. Striking motifs were Archytas' child-friendliness as well as his self-control and demand for rationality; It was said that he refused to punish in anger and preferred to forego the punishment that was actually necessary than to carry it out under the influence of emotion. He was also credited with a statement about the social character of man, handed down by Cicero: If someone could ascend to heaven and perceive the nature and beauty of the cosmos from there, he would be amazed, but could not enjoy this admiration of the world, if there was no one to share it with. Cicero used this alleged Archytas quote for his praise of friendship.

Cicero had the Roman statesman Marcus Porcius Cato Censorius appear in his literary dialogue Cato maior de senectute . In the fictitious, in the year 150 BC Cato reports from his stay in Taranto in 209 BC. At that time, as a guest of a local Pythagorean, he heard about a speech of Archytas, the content of which his host informed him according to a local oral tradition. The dialogue character Cato finds that Archytas is "to be reckoned with in the first place among the great, outstanding men".

Horace praised Archytas' scientific achievements in one of his odes. In the poem, the Tarentine appears as the one "who measured land and sea and sand in an uncountable amount", who explored heavenly spaces and in spirit flew through the universe. With the measurement the geometry is meant, with the exploration of the universe the argument for the infinity of the cosmos.

Archytas was not associated with the religious aspect of the Pythagorean tradition until late antiquity. In the Middle Ages he was portrayed as one of the great sages of antiquity and also as a magician.

### Early modern age

In the 16th and 17th centuries, Gellius 'report on Archytas' flying dove fascinated scholars who hoped for new technical achievements. The ancient scientist was hailed as a pioneer of mechanics. Athanasius Kircher and Gaspar Schott , who, like their contemporary René Descartes, viewed animals as machines, thought it was fundamentally possible to construct a flying pigeon. However, their attempts to find out the secret of Archytas were unsuccessful.

Archytas appears as an exemplary sage and statesman in the first version of Christoph Martin Wieland's novel Geschichte des Agathon (1766–1767). After an eventful career, the Athens-born protagonist Agathon went to Syracuse, where he was defeated in a political battle and thrown into prison. His old friendship with Archytas' son Critolaus will save him. Archytas, who as a wise statesman and lawmaker rules the Republic of Taranto, ensures that Agathon is released, and Critolaus moves him to move to Taranto. Wieland describes the Tarentine republic as an idyllic state, as the paradisiacal, utopian-looking world of Archytas, which preserves its ideal social condition “more through the power of morals than through the respect of the law”. Archytas, now a venerable old man, has, thanks to his measured, sensible way of life, preserved the “liveliness of all forces”, which is something rare in his age. Wielands Archytas owes his success and his authority as a universally admired and beloved state leader above all to his balanced, harmonious nature. He has always kept his distance from the tyranny of passions, the “aberrations of the mind and heart”, and he leaves metaphysical speculations that go beyond the limits of human understanding to his friend Plato. His philosophy is very practical; it is limited to the truths "which general feeling can reach" and which reason affirms. With the coherence of her life, the fictional character Archytas realizes an important concern of Wieland, but Wieland expressly distances himself from his own idyll: He appears as an innocent editor who only took the story told from an old Greek manuscript and criticizes the fictional author who In the last part of his work I got lost in a wonderland, "the land of beautiful souls and the utopian republics". - In the third, revised version from 1794, Wieland expanded the final part of the novel, placing particular emphasis on the role of Archytas.

### Modern

In modern research into the history of science, Archytas' achievements have received great recognition, although many of the assumptions emphasize that they are uncertain because of the unfavorable sources. Pierre Wuilleumier , who published an extensive monograph on the history and culture of the Greek Taranto in 1939, described Archytas as an innovative genius, as a dominant figure in politics and the intellectual life of his hometown and its region and as "the first and most beautiful example of a philosopher in power" . Maria Timpanaro Cardini (1962) praised the “modernity” of his attitude, the breadth of his spectrum of interests, the clarity of his thinking and his consistently scientific approach. Walter Burkert (1962) stated that Archytas had taken the path to a general number theory such as that which Euclid then presented. He created his number theory, which had grown out of music theory, "using the methods of proof of highly developed geometry from speculative theory of numbers and music" and expanded music theory using number theory. Burkert described the doubling of the dice as groundbreaking. Myles Frederic Burnyeat (2005) saw Archytas as a brilliant mathematician and the founder of mathematical optics, Leonid Zhmud (2013) praised him as "the rare example of an outstanding mathematician and original thinker who was also a successful statesman".

The judgment of Bartel Leendert van der Waerden (1956) was more critical . He praised the versatility and wealth of ideas of the ancient scientist, whose thinking was entirely kinematic, and emphasized "how lively his spatial conception and his ideas of movement were". On the other hand, van der Waerden criticized the verbosity of the explanations and noted a "strange contrast between his ingenious ideas, his creative imagination, his great mastery of geometrical methods on the one hand and his deficient logic, his inability to express himself precisely and clearly, his thinking errors and inconveniences on the other ". This assessment contradicted Carl A. Huffman, who in 2005 presented a large monograph on Archytas with an edition of the fragments and testimony. In particular, he rejected the claim that Archytas' style was unclear and that his discourse was logically flawed. Huffman's standard work, the first monograph on the Tarentine since 1840, has shaped the course of research since its publication.

The lunar crater Archytas and the asteroid (14995) Archytas are named after the ancient scientist.

## Text output

• Carl A. Huffman: Archytas of Tarentum. Pythagorean, Philosopher and Mathematician King . Cambridge University Press, Cambridge 2005, ISBN 0-521-83746-4 (basic study; contains edition of the fragments with English translation and detailed commentary and a compilation of all other source references)

## literature

Commons : Archytas of Taranto  - Collection of images, videos and audio files

## Remarks

1. See also Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, p. 6; Bernard Mathieu: Archytas de Tarente, Pythagoricien et ami de Plato. In: Bulletin de l'Association Guillaume Budé , vol. 1987, pp. 239-255, here: 240.
2. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, p. 18; George C. Brewer: Taras. Its History and Coinage , New Rochelle 1986, p. 45 f.
3. Cicero, De oratore 3,34,139.
4. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 7, 34, 46; Monte Ransome Johnson: Sources for the Philosophy of Archytas. in: Ancient Philosophy 28, 2008, pp. 173–199, here: 181.
5. Valerius Maximus, Facta et dicta memorabilia 4,1, ext. 1.
6. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 6-8. On the independence of Archytas as a philosopher, see Monte Ransome Johnson: Sources for the Philosophy of Archytas. in: Ancient Philosophy 28, 2008, pp. 173–199, here: 176 f.
7. George C. Brewer: Taras. Its History and Coinage , New Rochelle 1986, p. 27 f. and note 5; Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, p. 17 f.
8. George C. Brewer: Taras. Its History and Coinage , New Rochelle 1986, pp. 31, 43-45; Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 5, 9-11; Pierre Wuilleumier: Tarente des origines à la conquête romaine , Paris 1939, pp. 62–66.
9. Diogenes Laertios 8.79.
10. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 10-14; Pierre Wuilleumier: Tarente des origines à la conquête romaine , Paris 1939, pp. 68–71.
11. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 9-14; Pierre Wuilleumier: Tarente des origines à la conquête romaine , Paris 1939, pp. 70–73.
12. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 32-35, 37, 41; Bernard Mathieu: Archytas de Tarente, Pythagoricien et ami de Plato. In: Bulletin de l'Association Guillaume Budé , vol. 1987, pp. 239-255, here: 246 f .; Monte Ransome Johnson: Sources for the Philosophy of Archytas. in: Ancient Philosophy 28, 2008, pp. 173–199, here: 182 f.
13. Alice Swift Riginos: Platonica , Leiden 1976, p. 90 f.
14. Seventh Letter 338c – 340a. See also Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 32–42; Geoffrey Lloyd : Plato and Archytas in the Seventh Letter. In: Phronesis 35, 1990, pp. 159-174, here: 162 f., 165-168, 172 f.
15. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, p. 5 f .; Pierre Wuilleumier: Tarente des origines à la conquête romaine , Paris 1939, p. 77 f. Cf. George C. Brauer: Taras. Its History and Coinage , New Rochelle 1986, pp. 61-63.
16. ^ Ross S. Kilpatrick: Archytas at the Styx (Horace Carm. 1. 28). In: Classical Philology 63, No. 3 (1968), pp. 201-206; Gerhard Fink (ed. And translator): Q. Horatius Flaccus. Odes and Epodes . Tusculum Collection , Artemis & Winkler, Düsseldorf / Zurich 2002, ISBN 978-3-11-036002-8 , p. 382 f. (accessed via De Gruyter Online).
17. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 19-21; Pierre Wuilleumier: Tarente des origines à la conquête romaine , Paris 1939, p. 74 f.
18. ^ Richard Neudecker : The Sculpture Equipment of Roman Villas in Italy (= Contributions to the Development of Hellenistic and Imperial Sculpture and Architecture. Volume 9). Philipp von Zabern, Mainz 1988, p. 148 No. 14.3; Bruno Centrone, Marie-Christine Hellmann : Archytas de Tarente . In: Richard Goulet (ed.): Dictionnaire des philosophes antiques , Vol. 1, Paris 1989, pp. 339–342, here: 342; Gisela MA Richter : The Portraits of the Greeks , Vol. 2, London 1965, p. 179 (cf. Vol. 1, London 1965, p. 79).
19. A directory is provided by Holger Thesleff: An Introduction to the Pythagorean Writings of the Hellenistic Period , Åbo 1961, pp. 8–11.
20. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 30–32, 187 f., 228–232; Andrew Barker: Archytas Unbound. In: Oxford Studies in Ancient Philosophy 31, 2006, pp. 297–321, here: 299 f .; Monte Ransome Johnson: Sources for the Philosophy of Archytas. in: Ancient Philosophy 28, 2008, pp. 173–199, here: 179 f.
21. Andrew Barker: Archytas Unbound. In: Oxford Studies in Ancient Philosophy 31, 2006, pp. 297–321, here: 297.
22. ^ Christoph Riedweg : Pythagoras , 2nd, revised edition, Munich 2007, p. 146; Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 68-76, 190-193; Bernard Mathieu: Archytas de Tarente, Pythagoricien et ami de Plato. In: Bulletin de l'Association Guillaume Budé , vol. 1987, pp. 239-255, here: 253; Andrew Barker: Archytas Unbound. In: Oxford Studies in Ancient Philosophy 31, 2006, pp. 297–321, here: 309–312.
23. Bruno Snell: The expressions for the concept of knowledge in pre-Platonic philosophy , 2nd edition, Hildesheim / Zurich 1992, pp. 76-80.
24. ^ Leonid Zhmud: Archytas from Taranto (DK 47) . In: Hellmut Flashar et al. (Ed.): Early Greek Philosophy (= Outline of the History of Philosophy. The Philosophy of Antiquity , Volume 1), Half Volume 1, Basel 2013, pp. 425–428, here: 427 f.
25. ^ Leonid Zhmud: The Origin of the History of Science in Classical Antiquity , Berlin 2006, p. 68.
26. Andrew Barker: Archytas Unbound. In: Oxford Studies in Ancient Philosophy 31, 2006, pp. 297–321, here: 302–309.
27. See Andrew Barker: Archytas Unbound. In: Oxford Studies in Ancient Philosophy 31, 2006, pp. 297–321, here: 312.
28. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 24, 283-290, 323-337.
29. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 22-24, 541-550; Monte Ransome Johnson: Sources for the Philosophy of Archytas. in: Ancient Philosophy 28, 2008, pp. 173–199, here: 186 f.
30. Bartel Leendert van der Waerden: Die Pythagoreer , Zurich 1979, pp. 373, 406.
31. See also Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 457-470.
32. Stephen Menn: How Archytas Doubled the Cube. In: Brooke Holmes, Klaus-Dietrich Fischer (eds.): The Frontiers of Ancient Science , Berlin 2015, pp. 407–435, here: 407 f .; Monte Ransome Johnson: Sources for the Philosophy of Archytas. in: Ancient Philosophy 28, 2008, pp. 173–199, here: 184 f. Luc Brisson , however, has a different opinion : Archytas and the duplication of the cube. In: Gabriele Cornelli et al. (Ed.): On Pythagoreanism , Berlin 2013, pp. 203–233, here: 213–222.
33. Stephen Menn: How Archytas Doubled the Cube. In: Brooke Holmes, Klaus-Dietrich Fischer (ed.): The Frontiers of Ancient Science , Berlin 2015, pp. 407–435, here: 409–434.
34. Bartel Leendert van der Waerden: Die Pythagoreer , Zurich 1979, p. 16 f .; Leonid Zhmud: Archytas from Taranto (DK 47) . In: Hellmut Flashar et al. (Ed.): Early Greek Philosophy (= Outline of the History of Philosophy. The Philosophy of Antiquity , Volume 1), Half Volume 1, Basel 2013, pp. 425–428, here: 427.
35. ^ Leonid Zhmud: Archytas from Taranto (DK 47) . In: Hellmut Flashar et al. (Ed.): Early Greek Philosophy (= Outline of the History of Philosophy. The Philosophy of Antiquity , Volume 1), Half Volume 1, Basel 2013, pp. 425–428, here: 427; Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, p. 130.
36. Apuleius, Apologia 15.
37. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 550-556.
38. ^ Myles Frederic Burnyeat: Archytas and Optics. In: Science in Context 18, 2005, pp. 35-53. Cf. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, p. 567 f.
39. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 129-138.
40. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 138-148; Alan C. Bowen: The Foundations of Early Pythagorean Harmonic Science: Archytas, Fragment 1. In: Ancient Philosophy 2, 1982, pp. 79-104, here: 92 f.
41. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 140, 144.
42. Diogenes Laertios 8.83.
43. ^ Plutarch, Marcellus 14.
44. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 77-83.
45. Aulus Gellius, Noctes Atticae 10, 12, 9 f.
46. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, p. 571.
47. ^ Wilhelm Schmidt: From ancient mechanics . In: New Yearbooks for Classical Antiquity 13, 1904, pp. 329–351, here: 349–351.
48. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 570-579.
49. Karin Luck-Huyse: The dream of flying in antiquity , Stuttgart 1997, p. 133.
50. Aristotle, Politics 1340b25–31.
51. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 302-307.
52. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 3–5.
53. Bruno Centrone: Pseudo-Archytas offers overviews . In: Richard Goulet (ed.): Dictionnaire des philosophes antiques , Vol. 1, Paris 1989, pp. 342-345 and Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 595-609.
54. Cicero, Laelius de amicitia 23.88. Cf. Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 18 f., 283-290, 293-296.
55. Cicero, Cato maior de senectute 12.39. On the question of the origin of the content, see Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 324–331; Federico Russo: L'incontro tra Archita, Platone e Ponzio Sannita in Cic. Cato 12, 39-41. In: Mediterraneo Antico 10, 2007, pp. 433-445.
56. Horace, Odes 1.28. See Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 21-24.
57. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 4, 25.
58. ^ Anthony Grafton : Conflict and Harmony in the Collegium Gellianum. In: Leofranc Holford-Strevens, Amiel Vardi (ed.): The Worlds of Aulus Gellius , Oxford 2004, pp. 318–342, here: 338–342.
59. Christoph Martin Wieland: History of the Agathon. First version , edited by Fritz Martini , Stuttgart 1985, pp. 558-567.
60. Christoph Martin Wieland: History of the Agathon. First version , edited by Fritz Martini, Stuttgart 1985, pp. 552-557.
61. Walter Erhart: "History of the Agathon". In: Jutta Heinz (Ed.): Wieland-Handbuch , Stuttgart / Weimar 2008, pp. 259–274, here: 262 f., 266–272.
62. Pierre Wuilleumier: Tarente des origines à la conquête romaine , Paris 1939, pp. 67, 584.
63. ^ Maria Timpanaro Cardini (ed.): Pitagorici. Testimonianze e frammenti , Volume 2, Florence 1962, p. 262.
64. ^ Walter Burkert: Wisdom and Science , Nuremberg 1962, pp. 364, 423, 425.
65. ^ Myles Frederic Burnyeat: Archytas and Optics. In: Science in Context 18, 2005, pp. 35–53, here: 33.
66. ^ Leonid Zhmud: Archytas from Taranto (DK 47) . In: Hellmut Flashar et al. (Ed.): Early Greek Philosophy (= Outline of the History of Philosophy. The Philosophy of Antiquity , Volume 1), Half Volume 1, Basel 2013, pp. 425–428, here: 425.
67. Bartel Leendert van der Waerden: Awakening Science , Basel / Stuttgart 1956, pp. 247–249, 252 f.
68. ^ Carl A. Huffman: Archytas of Tarentum , Cambridge 2005, pp. 468-470.
 This article was added to the list of excellent articles in this version on February 26, 2020 .