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18 THE NATURE OF MATHEMATICS often requires great ingenuity and is a familiar problem to land- surveyors. Now, let us suppose that we have a circular field to measure. Imagine from the centre of the circle a large number of radii drawn, and let each radius make equal angles with the naxt ones on each side of it. By joining the points in succession where the radii meet the circumference of the circle, we get a large number of triangles of equal area, and the sum of the areas of all these triangles gives an approxima- tion to the area of the circle. It is particularly instructive repeatedly to go over this and the following examples mentally, noticing how helpful the abstract ideas we call " straight line," " circle," " radius," " angle," and so on, are. We all of us know them, recognise them, and can easily feel that they are trustworthy and accurate ideas. We feel at home, so to speak, with the idea of a square, say, and can at once give details about it which are exactly true for it, and very nearly true for a field which we know is very nearly a square. This replace- ment in thought by an abstract geometrical object economises labour of thinking and imagining by leading us to concentrate our thoughts on that alone which is essential for our purpose. Thales seems to have discovered and it is a good thing to follow these discoveries on figures made with the help of com- passes and ruler the proof of what may be regarded as the obvious fact that the circle is divided into halves by its diameter, that the angles at the base of a triangle with two equal sides an isosceles triangle are equal, that all the triangles described in a semi-circle with two of their angular points at the ends of the diameter and the third anywhere on the circumference contain a right angle ; and he measured the distance of vessels from the shore, presumably by causing two observers at a known distance apart to measure the two angles formed by themselves and the ship. This last discovery is an application of the fact that a triangle is determined if its base and base angles are given. When Archytas and Meuaechmus employed mechanical instruments for solving certain geometrical problems, " Plato," says Plutarch, " inveighed against them with great indignation and persistence as destroying and perverting all the good there is in geometry ; for the method absconds from incorporeal and intellectual to sensible things, and besides employs again such
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