![]() ![]() Meton: Ah! These are my special rods for measuring the air. Peisthetaerus (eyeing Meton's instruments ): And what are these for? Famous throughout the Hellenic world - you must have heard of my hydraulic clock at Colonus? Peisthetaerus: Good lord, who do you think you are? Meton: I propose to survey the air for you: it will have to be marked out in acres. ), Aristophanes, Birds (London, 1978) or for a shorter quote ):. ![]() Two characters are speaking, Meton is the astronomer (see D Barrett (trs. Now the problem must have become quite popular shortly after this, not just among a small number of mathematicians, but quite widely, since there is a reference to it in a play Birds written by Aristopenes in about 414 BC. Anaxagoras, indeed, wrote on the squaring of the circle while in prison. There is no place that can take away the happiness of a man, nor yet his virtue or wisdom. Plutarch, in his work On Exile which was written in the first century AD, says :. The first mathematician who is on record as having attempted to square the circle is Anaxagoras. The ancient Greeks, however, did not restrict themselves to attempting to find a plane solution (which we now know to be impossible ), but rather developed a great variety of methods using various curves invented specially for the purpose, or devised constructions based on some mechanical method. This is really asking whether squaring the circle is a 'plane' problem in the terminology of Pappus given above (we shall often refer to a 'plane solution' rather than use the more cumbersome 'solutions using ruler and compass" ). Now we usually think of the problem of squaring the circle to be a problem which has to be solved using a ruler and compass. For the construction in these cases curves other than those already mentioned are required, curves having a more varied and forced origin and arising from more irregular surfaces and from complex motions. There remain the third type, the so-called 'linear' problem. ![]() For it is necessary in the construction to use surfaces of solid figures, that is to say, cones. Those problems that are solved by the use of one or more sections of the cone are called 'solid' problems. Those that can be solved with straight line and circle are properly called 'plane' problems, for the lines by which such problems are solved have their origin in a plane. There are, we say, three types of problem in geometry, the so-called 'plane', 'solid', and 'linear' problems. Pappus, writing in his work Mathematical collection at the end of the period of Greek development of geometry, distinguishes three types of methods used by the ancient Greeks (see for example ):. The methods one was allowed to use to do this construction were not entirely clear, for really the range of methods used in geometry by the Greeks was enlarged through attempts to solve this and other classical problems. The problem was, given a circle, to construct geometrically a square equal in area to the given circle. The problem of squaring the circle in the form which we think of it today originated in Greek mathematics and it is not always properly understood. This is quite a good approximation, corresponding to a value of 3. Although this is not really a geometrical construction as such it does show that the problem of constructing a square of area equal to that of a circle goes back to the beginnings of mathematics. It is a scroll about 6 metres long and 1 3 \large\frac\normalsize 9 1 off the circle's diameter and to construct a square on the remainder. One of the oldest surviving mathematical writings is the Rhind papyrus, named after the Scottish Egyptologist A Henry Rhind who purchased it in Luxor in 1858. From the oldest mathematical documents known up to the mathematics of today the problem and related problems concerning π have interested both professional mathematicians and amateur mathematicians. One of the fascinations of this problem is that it has been of interest throughout the whole of the history of mathematics. The present article studies what has become the most famous for these problems, namely the problem of squaring the circle or the quadrature of the circle as it is sometimes called. Although these are closely linked, we choose to examine them in separate articles. These problems were those of squaring the circle, doubling the cube and trisecting an angle. There are three classical problems in Greek mathematics which were extremely influential in the development of geometry.
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