Page:The Construction of the Wonderful Canon of Logarithms.djvu/50

26 CONSTRUCTION OF THE CANON,

ferences a c and e g equal, so that d placed in the middle of c e is of necessity also placed in the middle of a g. Then the sum of b c the second and b e the third is equal to the sum of b a the first and b g the fourth. For (by 37) the double of b d, which is b f, is equal to b c and b e together, because their differences from b d, namely c d and d e, are equal; for the same reason the same b f is also equal to b a and b g together, because their differences from b d, namely a d and d g, are also equal. Since, therefore, both the sum of b a and b g and the sum of b c and b e are equal to the double of b d, which is b f, therefore also they are equal to each other, which was to be proved. Whence follows the rule, of these four logarithms if you subtract a known mean from the sum of the known extremes, there is left the mean sought for; and if you subtract a known extreme from the sum of the known means, there is left the extreme sought for.

39.The difference of the logarithms of two sines lies between two limits; the greater limit being to radius as the difference of the sines to the less sine, and the less limit being to radius as the difference of the sines to the greater sine.

Let T S be radius, d S the greater of two given sines, and e S the less. Beyond S T let the distance T V be marked off by the point V, so that S T is to T V as e S, the less sine, is to d e, the difference of the sines, Again, on the other side of T, towards S, let the distance T c be marked off by the point c, so that T S is to T c as d S, the greater sine, is to d e, the difference of the

sines