Theodorus ofCyrene (Ancient Greek:Θεόδωρος ὁ Κυρηναῖος,romanized: Theódōros ho Kyrēnaîos;fl.c. 450 BC) was an ancientGreek mathematician. The only first-hand accounts of him that survive are in three ofPlato's dialogues: theTheaetetus, theSophist, and theStatesman. In the first dialogue, he posits a mathematical construction now known as theSpiral of Theodorus.
Little is known as Theodorus' biography beyond what can be inferred from Plato's dialogues. He was born in the northern African colony ofCyrene, and apparently taught both there and inAthens.[1] He complains of old age in theTheaetetus, the dramatic date of 399 BC of which suggests his period of flourishing to have occurred in the mid-5th century. The text also associates him with thesophistProtagoras, with whom he claims to have studied before turning to geometry.[2] A dubious tradition repeated among ancient biographers likeDiogenes Laërtius[3] held that Plato later studied with him inCyrene, Libya.[1]This eminent mathematician Theodorus was, along withAlcibiades and many other of Socrates' companions (many of whom would be associated with theThirty Tyrants), accused of distributing the mysteries at a symposium, according toPlutarch, who himself was priest of the temple atDelphi.
Theodorus' work is known through a sole theorem, which is delivered in the literary context of theTheaetetus and has been argued alternately to be historically accurate or fictional.[1] In the text, his studentTheaetetus attributes to him the theorem that the square roots of the non-square numbers up to 17 are irrational:
Theodorus here was drawing some figures for us in illustration of roots, showing that squares containing three square feet and five square feet are not commensurable in length with the unit of the foot, and so, selecting each one in its turn up to the square containing seventeen square feet and at that he stopped.[4]
The square containingtwo square units is not mentioned, perhaps because the incommensurability of its side with the unit was already known.) Theodorus's method of proof is not known. It is not even known whether, in the quoted passage, "up to" (μέχρι) means that seventeen is included. If seventeen is excluded, then Theodorus's proof may have relied merely on considering whether numbers are even or odd. Indeed, Hardy and Wright[5]and Knorr[6] suggest proofs that rely ultimately on the following theorem: If is soluble inintegers, and is odd, then must becongruent to 1modulo 8 (since and can be assumed odd, so their squares are congruent to 1modulo 8.
That one cannot prove the irrationality the square root of 17 by considerations restricted to the arithmetic of the even and the odd has been shown in one system of the arithmetic of the even and the odd in[7] and,[8] but it is an open problem in a stronger natural axiom system for the arithmetic of the even and the odd[9]
A possibility suggested earlier byZeuthen[10] is that Theodorus applied the so-calledEuclidean algorithm, formulated in Proposition X.2 of theElements as a test for incommensurability. In modern terms, the theorem is that a real number with aninfinitecontinued fraction expansion is irrational. Irrational square roots haveperiodic expansions. The period of the square root of 19 has length 6, which is greater than the period of the square root of any smaller number. The period of √17 has length one (so does √18; but the irrationality of √18follows from that of √2).
The so-called Spiral of Theodorus is composed of contiguousright triangles withhypotenuse lengths equal √2, √3, √4, …, √17; additional triangles cause the diagram to overlap.Philip J. Davisinterpolated the vertices of the spiral to get a continuous curve. He discusses the history of attempts to determine Theodorus' method in his bookSpirals: From Theodorus to Chaos, and makes brief references to the matter in his fictionalThomas Gray series.
That Theaetetus established a more general theory of irrationals, whereby square roots of non-square numbers are irrational, is suggested in the eponymous Platonic dialogue as well as commentary on, andscholia to, theElements.[11]
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