In Sir Thomas Heath’s entrancing Manual of Greek Mathematics it is stated on p. 52 that the Babylonians discovered “a most perfect or musical proportion” between two numbers, which Pythagoras introduced into Greece. This is the ratio a ½(a + b):: 2ab/(a + b) b. The two middle terms are the arithmetic and harmonic means of the extreme terms. Not knowing of this when considering the fundamental principle of the logarithm, it seemed to me that the use of these two means should assist in evaluating the natural logarithm of a ratio in which the numerator and denominator are the first and last terms. In this way I found the surprisingly exact but approximate expression for the numerical value of such a logarithm which appeared in Nature of 14th March, 1931.