CONSTRUCTION OF THE CANON. 25
and let the differences c d and d e be equal. Let b d, the mean of them, be doubled by producing the line from b beyond e to f, so that b f is double b d. Then b f is equal to both the lines b c of the first logarithm and b e of the third, for from the equals b d and d f take away the equals c d and d e, namely c d from b d and d e from d f, and there will remain b c and e f necessarily equal. Thus since the whole b f is equal to both b e and e f, therefore also it will be equal to both b e and b c, which was to be proved. Whence follows the rule, if of three logarithms you double the given mean, and from this subtract a given extreme, the remaining extreme sought for becomes known; and if you add the given extremes and divide the sum by two, the mean becomes known.
Of the four proportionals, since the ratio between the first and second is that between the third and fourth; therefore of their logarithms (by 36), the difference between the first and second is that between the third and fourth. Hence let such quantities be taken in the line b f as that b a
