APPENDIX, 51
Also if from a given number there be extracted the second, third, fifth, &c., roots; and the Logarithm of the given number be divided by two, three, five, &c., there will be produced the Logarithms of these roots.
Finally any common number being formed from other common numbers by multiplication, division, [raising to a power] or extraction [of a root]; its Logarithm is correspondingly formed from their Logarithms by addition, subtraction, multiplication, by 2, 3, &c. [or division by 2, 3, &c.]: whence the only difficulty is in finding the Logarithms of the prime numbers; and these may be found by the following general method.
For finding all Logarithms, it is necessary as the basis of the work that the Logarithms of some two common numbers be given or at least assumed; thus in the fore-going first method of construction, 0 or a cypher was assumed as the Logarithm of the common number one, and 10,000,000,000 as the Logarithm of one-tenth or of ten. These therefore being given, the Logarithm of the number 5 (which is a prime number) may be sought by the following method. Find the mean proportional between 10 and 1, namely , also the arithmetical mean between 10,000,000,000 and 0, namely 5,000,000,000; then jind the geometrical mean between 10 and , namely , also the arithmetical mean between 10,000,000,000 and 5,000,000,000, namely 7,500,000,000;.....
In all continuous proportionals.
AS the sum of the means and one or other of the extremes to the same extreme; so ts the difference of the extremes to the difference of the same extreme and the nearest mean.
