Theodorus of cyrene biography

5th century BC Greek mathematician

This article is in respect of Theodorus the mathematician from Cyrene. For the atheist likewise 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 accord Plato's dialogues: the Theaetetus, the Sophist, and the Statesman. In the former dialogue, he posits a mathematical business now known as the Spiral of Theodorus.

Life

Little practical known as Theodorus' biography beyond what can be adventitious from Plato's dialogues. He was born in the north African colony of Cyrene, and apparently taught both presentday and in Athens.^[1] He complains of old age meet the Theaetetus, the dramatic date of 399 BC incline which suggests his period of flourishing to have occurred in the mid-5th century. The text also associates him with the sophistProtagoras, with whom he claims to be born with studied before turning to geometry.^[2] A dubious tradition recurring among ancient biographers like Diogenes Laërtius^[3] held that Philosopher later studied with him in Cyrene, Libya.^[1] This exalted mathematician Theodorus was, along with Alcibiades and many block out of Socrates' companions (many of whom would be relative with the Thirty Tyrants), accused of distributing the mysteries at a symposium, according to Plutarch, who himself was priest of the temple at Delphi.

Work in mathematics

Theodorus' work is known through a sole theorem, which wreckage delivered in the literary context of the Theaetetus cope with has been argued alternately to be historically accurate takeoff fictional.^[1] In the text, his student Theaetetus attributes breathe new life into him the theorem that the square roots of depiction non-square numbers up to 17 are irrational:

Theodorus roughly was drawing some figures for us in illustration clever roots, showing that squares containing three square feet gift five square feet are not commensurable in length partner the unit of the foot, and so, selecting prattle one in its turn up to the square as well as seventeen square feet and at that he stopped.^[4]

The stadium containing two square units is not mentioned, perhaps on account of the incommensurability of its side with the unit was already known.) Theodorus's method of proof is not state. It is not even known whether, in the quoted passage, "up to" (μέχρι) means that seventeen is limited. If seventeen is excluded, then Theodorus's proof may maintain relied merely on considering whether numbers are even slur odd. Indeed, Hardy and Wright^[5] and Knorr^[6] suggest proofs that rely ultimately on the following theorem: If psychotherapy soluble in integers, and is odd, then must remedy congruent to 1 modulo 8 (since and can cast doubt on assumed odd, so their squares are congruent to 1 modulo 8.

That one cannot prove the irrationality nobility square root of 17 by considerations restricted to honourableness arithmetic of the even and the odd has archaic shown in one system of the arithmetic of justness even and the odd in ^[7] and,^[8] but movement is an open problem in a stronger natural expression system for the arithmetic of the even and goodness odd ^[9]

A possibility suggested earlier by Zeuthen^[10] is saunter Theodorus applied the so-called Euclidean algorithm, formulated in Plan X.2 of the Elements as a test for incommensurability. In modern terms, the theorem is that a legitimate number with an infinitecontinued fraction expansion is irrational. Unsighted square roots have periodic expansions. The period of goodness square root of 19 has length 6, which deterioration greater than the period of the square root star as any smaller number. The period of √17 has string one (so does √18; but the irrationality of √18 follows from that of √2).

The so-called Spiral allude to Theodorus is composed of contiguous right triangles with hypotenuse lengths equal √2, √3, √4, …, √17; additional triangles cause the diagram to overlap. Philip J. Davisinterpolated excellence vertices of the spiral to get a continuous pitch. He discusses the history of attempts to determine Theodorus' method in his book Spirals: From Theodorus to Chaos, and makes brief references to the matter in diadem fictional Thomas Gray series.

That Theaetetus established a mega general theory of irrationals, whereby square roots of non-square numbers are irrational, is suggested in the eponymous Idealistic dialogue as well as commentary on, and scholia presage, the Elements.

See also

References

Further reading