Archytas

« Archytas (early to mid 4th century BC) Archytas of Tarentum (modern Taranto in southern Italy) was a contemporary and personal acquaintance of Plato, and the last of the famous Pythagoreans in antiquity.

An ancient source (Proclus)chytas with those mathematicians 'who increased the number of theorems and progressed towards a more scientific arrangement of them' and ranks him among the predecessors of Euclid.

His chief contribution in mathematics was to find a solution for the doubling of the cube.

As a Pythagorean philosopher, Archytas gave mathematics universal scope: he viewed the four cardinal branches of Greek scientific knowledge - arithmetic, geometry, astronomy and music - as 'sister sciences' since they could be formulated mathematically.

In both mathematics and music he emphasized the study of mean proportionals.

He also conducted empirical investigations in acoustics and invented simple technical devices by which to illustrate the application of mathematical principles to mechanics.

Archytas was able to combine his philosophical-scientific interests with an active political career; he was a leading statesman of Tarentum and served as a successful general.

1 Life When Plato, on his third trip to Sicily, was forcibly detained in Syracuse by Dionysius II, he requested the help of Archytas and other friends in Tarentum, who sent a ship to rescue him.

This took place in 361/360 BC, which locates Archytas' activity in the first half of the fourth century BC.

(The label 'Presocratic', often used of him, describes Archytas' intellectual, rather than his strictly chronological, position.) Tarentum had continued as a stronghold of Pythagoreanism (§1) after the general dispersal of the Pythagoreans and their emigration from southern Italy.

Archytas, who supposedly studied with Philolaus, devoted his interests chiefly to mathematics, musical theory and mechanics.

His stature as a leading Pythagorean thinker was such that Aristotle wrote three separate treatises about him.

In addition to his philosophical-scientific enterprises, Archytas achieved renown as a military commander of Tarentum, earning continual reappointments beyond the usual terms of office.

2 Mathematics and music Archytas is famous for his geometrical solution to the problem of doubling the cube.

This long-standing problem, which may originally have arisen among the Greeks from architectural consideration of how to double a solid body while retaining its shape, had been reduced by Hippocrates of Chios to that of finding the two mean proportionals.

Building on Hippocrates' insight, Archytas solved the problem by means of moving, threedimensional constructions (half-cylinders and cones), thus also introducing the concept of movement into geometry (previous Pythagoreans had not concerned themselves with motion).

This has earned him a place in the annals of mathematics.

But Archytas is of no less significance to historians of philosophy for at least two reasons: first, because he makes evident the high degree of sophistication that mathematics had attained by the fourth century BC, against which background the mathematical activities of Plato and his associates in the Academy must be understood; second, and more generally, because although his mathematics far surpasses in complexity the number speculation of early Pythagoreanism, Archytas none the less gives the clearest expression of the Pythagorean view that mathematics provides the philosophical key for the understanding of all of nature.

The cosmic application of mathematics appears more pronounced in Archytas than in his fellow, non-Pythagorean Greek mathematicians (for example, Hippocrates of Chios, Eudoxus and, later, Euclid).

The following fragment reveals the universal importance that Archytas assigned to mathematical insight: Mathematicians seem to me to have excellent discernment, and it is not at all strange that they should think correctly about the particulars that are; for inasmuch as they can discern excellently about the nature (physis) of the universe, they are also likely to have an excellent perspective on the particulars that are.

Indeed, they have transmitted to us a keen discernment about the velocities of the stars and their risings and settings, and about geometry, numbers [arithmetic], sphericity [astronomy], and, not least of all, music.

These seem to be sister sciences, for they concern themselves with the first two related forms of being [that is, number and magnitude].

(fr.

1) Mathematics is thus foundational for correct thinking about being. The view of number as an all-powerful explanatory concept for the orderly arrangement of the universe originated from the Pythagoreans' discovery of the numerical ratios governing musical harmony and seemed to them to be corroborated by the observation that figures and shapes could be expressed arithmetically: for example, the line by two, the triangle by three, the pyramid by four points (see Pythagoreanism §2).

That even three-dimensional bodies could in one way be accounted for by an aggregate of points led to a certain fusion of mathematical and physical properties in Pythagorean philosophy, allowing Aristotle to say of the Pythagoreans that 'they construct the whole universe out of numbers - only not numbers consisting of abstract units; they suppose the units to have spatial magnitude' (Metaphysics 1080b18-20).

Archytas reports of another Pythagorean, Eurytus, that he arranged a number of pebbles to represent the figures of man or horse, and after counting the pebbles declared that such was the number of man and such of horse.

Pythagorean number theory, even in the crude form it assumed in the pebble arithmetic of Eurytus, comes to a legitimate fruition with Archytas.

He viewed geometry, arithmetic, astronomy and music - the classic quadrivium of medieval authors - collectively as 'sister sciences' (as later Plato would call astronomy and harmonics, expressly citing Pythagorean precedent (Republic 530d)), since they all have to do with number or numerical relations: geometry and arithmetic for obvious reasons, astronomy because it was treated mathematically, notably when the properties of the sphere were studied as a geometrical model to explain the movements of the celestial sphere, and music because it involved numerically expressed harmonic proportions.

His high regard for these sciences was governed by the conviction - still held in modern science, whose theses often take the form of equations and formulas - that number supplies the precise, quantitative measures by which to comprehend the world.

Fragment 1, which concerns the universality of mathematical insight, forms the introduction to a work that is variously entitled On Mathematics or On Harmonics.

Indeed, Archytas immediately continues to discuss the relation between pitch and frequency, citing a series of examples to show that swift movements produce high notes and slow movements low notes (for example, a stick moved at different speeds produces variations in the pitch of the sound, or the air emitted from the upper holes of a pipe yields higher notes than that from the lower holes).

While Archytas' account also contains some inaccurate conclusions from his observations (the speed of the motion that produces a sound was confused with the speed of the sound itself), his acoustic theories are none the less informative about the type of rudimentary empirical investigation that characterized Presocratic science.

Moreover, Archytas' acoustics provide an interesting footnote to Plato: discussing physical. »