A History of Greek Mathematics Volume 1 by Sir Thomas Heath

Author:Sir Thomas Heath
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2013-06-26T16:00:00+00:00

The squaring of the circle.

There is presumably no problem which has exercised such a fascination throughout the ages as that of rectifying or squaring the circle; and it is a curious fact that its attraction has been no less (perhaps even greater) for the non-mathematician than for the mathematician. It was naturally the kind of problem which the Greeks, of all people, would take up with zest the moment that its difficulty was realized. The first name connected with the problem is Anaxagoras, who is said to have occupied himself with it when in prison.6 The Pythagoreans claimed that it was solved in their school, ‘as is clear from the demonstrations of Sextus the Pythagorean, who got his method of demonstration from early tradition’7; but Sextus, or rather Sextius, lived in the reign of Augustus or Tiberius, and, for the usual reasons, no value can be attached to the statement.

The first serious attempts to solve the problem belong to the second half of the fifth century B.C. A passage of Aristophanes’s Birds is quoted as evidence of the popularity of the problem at the time (414 B.C.) of its first representation. Aristophanes introduces Meton, the astronomer and discoverer of the Metonic cycle of 19 years, who brings with him a ruler and compasses, and makes a certain construction ‘in order that your circle may become square’.8 This is a play upon words, because what Meton really does is to divide a circle into four quadrants by two diameters at right angles to one another; the idea is of streets radiating from the agora in the centre of a town; the word ττράγωνος then really means ‘with four (right) angles’ (at the centre), and not ‘square’, but the word conveys a laughing allusion to the problem of squaring all the same.

We have already given an account of Hippocrates’s quadratures of lunes. These formed a sort of prolusio, and clearly did not purport to be a solution of the problem; Hippocrates was aware that ‘plane’ methods would not solve it, but, as a matter of interest, he wished to show that, if circles could not be squared by these methods, they could be employed to find the area of some figures bounded by arcs of circles, namely certain lunes, and even of the sum of a certain circle and a certain lune.

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