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This section is from "The American Cyclopaedia", by George Ripley And Charles A. Dana. Also available from Amazon: The New American Cyclopædia. 16 volumes complete..
Circle, a plane figure bounded by a line every part of which is equally distant from its centre. This line is called the circumference, and in popular language the word circle is sometimes used for circumference. The diameter of a circle is a straight line which passes through its centre and is terminated at both ends by the circumference. The ratio of the diameter to the circumference was shown by Archimedes to be about as 7 to 22, and by Metius as still more nearly 113 to 355, which, though slightly too small, is as accurate as is needed for practical use. The exact ratio cannot be expressed in numbers, although many algebraic expressions can be obtained for it, of which we will give but one. If the number 4 be divided by 1, 5, 9, 13, and every 4th number in succession, and afterward by 3, 7, 11, and every 4th number in succession, the difference between the sum of the first set of quotients and that of the second set will be equal to this ratio; or approximately in decimals, 3.14159265. The ratio of 7 to 22, decimally expressed, is 3.14285714; that of 113 to 355 is 3.14159292.
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