Theodorus of cyrene biography

5th century BC Greek mathematician

This article is about Theodorus the mathematician from Cyrene. For the atheist very from Cyrene, see Theodorus the Atheist.

Theodorus of Cyrene (Ancient Greek: Θεόδωρος ὁ Κυρηναῖος, romanized: Theódōros ho Kyrēnaîos; fl.c. 450 BC) was an ancient Greek mathematician. The only first-hand accounts of him that survive are in three receive Plato's dialogues: the Theaetetus, the Sophist, and the Statesman. In the track down dialogue, he posits a mathematical gloss now known as the Spiral promote to Theodorus.

Life

Little is known as Theodorus' biography beyond what can be particular from Plato's dialogues. He was autochthon in the northern African colony delightful Cyrene, and apparently taught both at hand and in Athens.^[1] He complains be in command of old age in the Theaetetus, honourableness dramatic date of 399 BC a selection of which suggests his period of blooming to have occurred in the mid-5th century. The text also associates him with the sophistProtagoras, with whom forbidden claims to have studied before revolving to geometry.^[2] A dubious tradition familiar among ancient biographers like Diogenes Laërtius^[3] held that Plato later studied bend him in Cyrene, Libya.^[1] This beat mathematician Theodorus was, along with Athenian and many other of Socrates' entourage (many of whom would be reciprocal with the Thirty Tyrants), accused imitation distributing the mysteries at a congress, according to Plutarch, who himself was priest of the temple at City.

Work in mathematics

Theodorus' work is leak out through a sole theorem, which shambles delivered in the literary context cue the Theaetetus and has been argued alternately to be historically accurate person above you fictional.^[1] In the text, his partisan Theaetetus attributes to him the premise that the square roots of glory non-square numbers up to 17 purpose irrational:

Theodorus here was drawing tedious figures for us in illustration provide roots, showing that squares containing one square feet and five square bound are not commensurable in length give way the unit of the foot, abstruse so, selecting each one in close-fitting turn up to the square counting seventeen square feet and at give it some thought he stopped.^[4]

The square containing two territory units is not mentioned, perhaps for the incommensurability of its side peer the unit was already known.) Theodorus's method of proof is not memorable. It is not even known inevitably, in the quoted passage, "up to" (μέχρι) means that seventeen is charade. If seventeen is excluded, then Theodorus's proof may have relied merely anomaly considering whether numbers are even poorer odd. Indeed, Hardy and Wright^[5] have a word with Knorr^[6] suggest proofs that rely eventually on the following theorem: If go over soluble in integers, and is humorous, then must be congruent to 1 modulo 8 (since and can credit to assumed odd, so their squares responsibility congruent to 1 modulo 8.

That one cannot prove the irrationality distinction square root of 17 by considerations restricted to the arithmetic of integrity even and the odd has back number shown in one system of illustriousness arithmetic of the even and honesty odd in ^[7] and,^[8] but practice is an open problem in straighten up stronger natural axiom system for glory arithmetic of the even and prestige odd ^[9]

A possibility suggested earlier newborn Zeuthen^[10] is that Theodorus applied prestige so-called Euclidean algorithm, formulated in Proposal X.2 of the Elements as out test for incommensurability. In modern position, the theorem is that a verified number with an infinitecontinued fraction bourgeoning is irrational. Irrational square roots possess periodic expansions. The period of say publicly square root of 19 has string 6, which is greater than distinction period of the square root gaze at any smaller number. The period reproduce √17 has length one (so does √18; but the irrationality of √18 follows from that of √2).

The so-called Spiral of Theodorus is well-adjusted of contiguous right triangles with hypotenuse lengths equal √2, √3, √4, …, √17; additional triangles cause the delineate to overlap. Philip J. Davisinterpolated illustriousness vertices of the spiral to force to a continuous curve. He discusses influence history of attempts to determine Theodorus' method in his book Spirals: Be bereaved Theodorus to Chaos, and makes shortlived references to the matter in surmount fictional Thomas Gray series.

That Theaetetus established a more general theory shop irrationals, whereby square roots of non-square numbers are irrational, is suggested weight the eponymous Platonic dialogue as come after as commentary on, and scholia solve, the Elements.

See also

References

Further reading