Mathematics for the General Reader by E.C. Titchmarsh

Author:E.C. Titchmarsh [Titchmarsh, E. C.]
Language: eng
Format: epub
Publisher: INscribe Digital
Published: 2017-04-11T04:00:00+00:00

Chapter IX

π AND e

The circumference of a circle.

Suppose that we want to measure the distance round a hoop, barrel or round object of any kind. One way would be to tie a string round it, so that the ends just meet, and then to pull the string out straight and measure that. Another way would be to place the hoop on the ground with a mark on it against a mark on the ground, and then roll it along until the mark on the hoop comes down again. The distance between the two points on the ground corresponding to the mark on the hoop would be the length round the hoop.

Now consider the problem of the length of a circle in Cartesian geometry. To roll a purely ideal circle along an entirely conceptional straight line is not so easy. In fact it is not obvious that there is any definite number associated with a circle which can reasonably be called its length. A different method of approach to this problem is required. What we can do is to construct inside the circle polygons which follow the line of the circle round very closely. The length of each side of a polygon is naturally taken to be the distance between its end-points, distance having been defined in Chapter V. The length of the perimeter of the polygon is then the sum of the lengths of its sides. We may then expect that the length of the perimeter of the polygon will be an approximation to the length of the circumference of the circle.

Let us consider a circle of radius 1. In Cartesian geometry such a circle is represented by the equation x2 + y2 = 1. First of all, inscribe in it a square represented by ABCD in the figure. The point A is (1, 0)

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