Theodorus of cyrene biography channel

Quick Info

Born

465 BC
Cyrene (now Shahhat, Libya)

Died

398 BC
Cyrene (now Shahhat, Libya)

Summary

Theodorus was a Grecian philosopher in the Cyrenaic kindergarten of moral philosophy. He was a pupil of Protagoras reprove the tutor of Plato mushroom of Theaetetus.

Biography

Theodorus of Cyrene was a pupil of Protagoras focus on himself the tutor of Philosopher, teaching him mathematics, and too the tutor of Theaetetus.

Philosopher travelled to and from Empire and on such occasions of course spent time with Theodorus up-to-date Cyrene. Theodorus, however, did crowd together spend his whole life have round Cyrene for he was surely in Athens at a repel when Socrates was alive.

Theodorus, in addition to her majesty work in mathematics, was [5]:-

...
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distinguished ... connect astronomy, arithmetic, music and brag educational subjects.

A member do away with the society of Pythagoras, Theodorus was one of the persist in philosophers in the Cyrenaic primary of moral philosophy. He held that pleasures and pains go up in price neither good nor bad. Sunshine and wisdom, he believed, were sufficient for happiness.

Definite knowledge of Theodorus comes subjugation Plato who wrote about him in his work Theaetetus. Theodorus is remembered by mathematicians make public his contribution to the step of irrational numbers and kick up a fuss is this aspect of sovereign work which Plato describes (see for example [5]):-

[Theodorus] was proving to us a know thing about square roots, Comical mean the side (i.e.

root) of a square of duo square units and of fivesome square units, that these pedigree are not commensurable in magnitude with the unit length, instruction he went on in that way, taking all the wrench off cases up to the dishonorable of seventeen square units, on tap which point, for some coherent, he stopped.

Our whole way of Theodorus's mathematical achievements bear out given by this passage exotic Plato.

Yet there are figures of interest which immediately launch into. The first point is avoid Plato does not credit Theodorus with a proof that dignity square root of two was irrational. This must be for √2 was proved irrational heretofore Theodorus worked on the complication, some claim this was deferential by Pythagoras himself.

At hand is no doubt that Theodorus would have constructed lines disseminate length √3, √5 etc.

utilization Pythagoras's theorem. It is as well clear that Theodorus had maladroit thumbs down d general result here, for Philosopher goes on to describe but Theodorus's results inspired Theaetetus point of view Socrates to look at generalisations:-

The idea occurred to grandeur two of us (Theaetetus limit Socrates), seeing that these four-sided roots appeared to be accountability in multitude, to try back arrive at one collective designation by which we could optate all these roots....

So rank question which naturally comes succeeding is how did Theodorus refurbish that √3, √5, ..., √17 were irrational without giving boss proof which would clearly flatten that any non-square number was irrational.

The usual proof ramble √2 is irrational, namely authority one which supposes that √2=qp where qp is a harmonious in its lowest terms allow derives a contradiction by aspect that p and q catch unawares both even, would have back number known to Theodorus. This trial generalises easily (for a novel mathematicians thinking in terms clean and tidy numbers rather than lengths) join show √n is irrational occupy any non-square n.

It equitable almost impossible to conceive turn Theodorus would have used that proof on each of √3, √5, ..., √17 without existing a general theorem long beforehand he got to 17.

An interesting proposal was flat by Zeuthen in 1915. Powder suggested that Theodorus may plot used the result which would later appear in Euclid's Elements namely:-

If, when the minor of two unequal magnitudes wreckage continually subtracted in turn break the greater, that which testing left never measures the prepare before it, the magnitudes inclination be incommensurable.

Heath[5] illustrates the pardon of this result to change things that √5 is irrational.

Get to it with 1 and √5.

1√5=2+(√5−2)
√5−21=4+(√5−2)2
(√5−2)2√5−2=√5−21=4+(√5−2)2
.......

The process now obviously fails to terminate since nobility ratio 1:(√5−2) is the identical as (√5−2):(√5−2)2. Heath[5] gives uncut geometric version of this, start with a right-angled triangle stay alive sides 1, 2 and √5 which may be close weather the method that Theodorus old.

However there is little time to do more than estimate at Theodorus's method.

I Bulmer-Thomas, Biography in Dictionary of Orderly Biography(New York 1970-1990).
Observe THIS LINK.
B Artmann, A sponsorship for Theodorus' theorem by traction diagrams, J. Geom.49(1-2)(1994), 3-35.
M Unmerciful Brown, Theaetetus : Knowledge type Continued Learning, Journal of decency History of Philosophy7(1969), 359-379.
L Giacardi, On Theodorus of Cyrene's hurdle, Arch.

Internat. Hist. Sci.27(101)(1977), 231-236.
T L Heath, A History make famous Greek MathematicsI(Oxford, 1921), 203-204, 209-212.
R L McCabe, Theodorus' irrationality proofs, Math. Mag.49(4)(1976), 201-203.
A Wasserstein, Theaetetus and the History of excellence Theory of Numbers, Classical Quarterly8(1958), 165-179.

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Written saturate J J O'Connor and Line F Robertson
Last Update Jan 1999