The construction of the wonderful canon of logarithms

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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,
38. Offour geometricalproportionals, as the product of the
means is equal to the product of the extremes ; so of their
logarithms, the sum, of the means is equal to the sum of the
extremes. Whence any three of these logarithms being
given, the fourth 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
b a c d e g f
I 1—
1
1—
may represent the first logarithm, b c the second,
b e the third, and b g the fourth, making the dif-
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ferences
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