APPENDIX. 49
correspond to each of them in order. Accordingly, between a tenth and unity, or between ten and unity (adding for the purpose of calculation as many cyphers as you wish, say twelve), find four mean proportionals, or rather the least of them, by extracting the fifth root, which for ease in demonstration call A. Similarly, between A and unity, find the least of four mean proportionals, which call B. Between B and unity find four means, or the least of them, which call C. And thus proceed, by the extraction of the fifth root, dividing the interval between that last found and unity into five proportional intervals, or into four means, of all which let the fourth or least be always noted down, until you come to the tenth least mean; and let them be denoted by the letters D, E, F, G, H, I, K.
When these proportionals have been accurately computed, proceed also to find the mean proportional between K and unity, which call L. Then find the mean proportional between L. and unity, which call M. Then in like manner a mean between M and unity, which call N. In the same way, by extraction of the square root, may be formed between each last found number and unity, the rest of the intermediate proportionals, to be denoted by the letters O, P, Q, R, S, T, V.
To each of these proportionals in order corresponds its Logarithm of the first series. Whence 1 will be the Logarithm of the number V, whatever it may turn out to be, and 2 will be the Logarithm of the number T, and 4 of the number S, and 8 of the number R, 16 of the number Q, 32 of the number P, 64 of the number O, 128 of the number N, 256 of the number M, 512 of the number L, 1024 of the number K; all of which is manifest from the above construction.
From these, once computed, there may then be formed both the proportionals of other Logarithms and the Logarithms of other proportionals.
