**Archytas of Tarentum**was a mathematician, statesman and philosopher who lived in Tarentum in Magna Graecia, an area of southern Italy which was under Greek control in the fifth century BC. The Pythagoreans, who had at one stage been strong throughout Magna Graecia, were attacked and expelled until only the town of Tarentum remained a stronghold for them. Archytas led the Pythagoreans in Tarentum and tried to unite the Greek towns in the area to form an alliance against their non-Greek neighbours. He was commander in chief of the forces in Tarentum for seven years despite there being a law that nobody could hold the post for more than a year. Plato, who became a close friend, made his acquaintance while staying in Magna Graecia. Heath writes in [4]:-

In fact Plato made a number of trips to Sicily and it was on the third of these trips in 361 BC that he was detained by Dionysius II. Plato wrote to Archytas who sent a ship to rescue him. For more details on the relationship between Archytas and Plato consult the interesting article [8].... he is said, by means of a letter, to have saved Plato from death at the hands of Dionysius.

Given the above story and the conclusion that Archytas came after Socrates, it may seem strange to include him in works on pre-socratic philosophers as is done in [3]. This is done, however, because of the style of Archytas's philosophy rather than the strict chronology.

Archytas was a pupil of Philolaus and so was a firm supporter of the philosophy of Pythagoras believing that mathematics provided the path to the understanding of all things. Although Archytas studied many topics, since he was a Pythagorean, mathematics was his main subject and all other disciplines were seen as dependent on mathematics. He claimed that mathematics was composed of four branches, namely geometry, arithmetic, astronomy and music. He also believed that the study of mathematics was important in other respects as a fragment of his writings that has been preserved shows (see [3] or [6]):-

This fragment comes from the preface to one of his works which some claim was entitledMathematicians 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 physics of the universe, they are also likely to have 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, arithmetic, 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[number and magnitude].

*On Mathematics*while others claim that it was entitled

*On Harmonics*. Certainly, coming after this quote, there is a discussion of pitch, frequency and a theory of sound. It does contain some errors but it is still a remarkable piece of work and formed the basis for the theory of sound in the writings of Plato.

Archytas worked on the harmonic mean and gave it that name (it had been called sub-contrary in earlier times). The reason he worked on this was his interest in the problem of duplicating the cube, finding the side of a cube with volume twice that of a given cube. Hippocrates reduced the problem to finding two mean proportionals. Archytas solved the problem with a remarkable geometric solution (not of course a ruler and compass construction).

One interesting innovation which Archytas brought into his solution of finding two mean proportionals between two line segments was to introduce movement into geometry. His method uses a semicircle rotating in three dimensional space and the curve formed by it cutting another three dimensional surface.

We know of Archytas's solution to the problem of duplicating the cube through the writings of Eutocius of Ascalon. In these Eutocius claims to quote the description given in *History of geometry* by Eudemus of Rhodes but the accuracy of the quotation is doubted by the authors of [10].

Another interesting mathematical discovery due to Archytas is that there can be no number which is a geometric mean between two numbers in the ratio (*n*+1) : *n*. The most interesting thing about his proof is that it is close to that given by Euclid many years later, and also that it quotes known theorems which would later appear in Euclid's *Elements* Book VII.

The arguments just given led van der Waerden to claim (see for example [5]) that many of the results which appear in Book VII of the *Elements* predate Archytas. Clearly, he claims, there were some works, written many years before Euclid wrote the *Elements*, which covered the same material. Archytas built on this earlier work and his discoveries are then largely those presented by Euclid in the *Elements* Book VIII. Following these arguments of van der Waerden it is now widely accepted that Euclid borrowed Archytas's work for Book VIII of the *Elements*.

Archytas is sometimes called the founder of mechanics and he is said to have invented two mechanical devices. One device was a mechanical bird [2]:-

Another mechanical device was a rattle for children which was useful, in Aristotle's words (see for example [4]):-The bird was apparently suspended from the end of a pivoted bar, and the whole apparatus revolved by means of a jet of steam or compressed air.

This does seem a remarkably modern thought for an inventor in 400 BC! In fact this interest in applying mathematics is in contrast to the pure mathematical ideas of Plato and this contrast formed the basis for a poem written by the Polish author C K Norwid (1821-1883). This fascinating poem is discussed and given in French translation by Marczewski in [9].... to give to children to occupy them, and so prevent them from breaking things about the house(for the young are incapable of keeping still).

Simplicius, in his *Physics*, quotes Archytas's view that the universe is infinite (in Heath's translation [4]):-

When it came to a philosophy of politics and ethics, again Archytas based his ideas on mathematical foundations. He wrote (see for example [3] or [6]):-If I were at the outside, say at the heaven of the fixed stars, could I stretch my hand or my stick outward or not? To suppose that I could not is absurd: and if I can stretch it out, that which is outside must be either body or space(it makes no difference which it is as we shall see). We may then in the same way get to the outside of that again, and so on, asking on arrival at each new limit the same question; and if there is always a new place to which the stick may be held out, this clearly involves extension without limit. If now what so extends is body, the proposition is proved; but even if it is space, then, since space is that in which body is or can be, and in the case of eternal things we must treat that which potentially is as being, it follows equally that there must be body and space extending without limit.

Finally we quote again from the writings of Archytas about his theory of how to learn. The fragment appears in [3] or [6]:-When mathematical reasoning has been found, it checks political faction and increases concord, for there is no unfair advantage in its presence, and equality reigns. With mathematical reasoning we smooth out differences in our dealings with each other. Through it the poor take from the powerful, and the rich give to the needy, both trusting in it to obtain an equal share...

To become knowledgeable about things one does not know, one must either learn from others or find out for oneself. Now learning derives from someone else and is foreign, whereas finding out is of and by oneself. Finding out without seeking is difficult and rare, but with seeking it is manageable and easy, though someone who does not know how to seek cannot find.

**Article by:** *J J O'Connor* and *E F Robertson*

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