Abstract
I found my method on the known principle, that the geometric mean between two quantities is also a geometric mean between the arithmetic and harmonic means of those quantities. We may therefore approximate to the geometric mean of two quantities in this way:—Take their arithmetic and harmonic means; then take the arithmetic and harmonic means of those means; then of these last means again, and so on, as far as we please. If the ratio of the original quantities lies within the ratio of 1 : 2, the approximation proceeds with extraordinary rapidity, so that, in obtaining a fraction nearly equal to √2 by this method, we obtain a result true to eleven places of decimals at the fourth mean. I name this merely to show the rate of approximation. The real application of the method is to the integration of functions embracing a radical of the square root.
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