The construction of the wonderful canon of logarithms

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 ofthe 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.
V T c d e S
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.
"of T, towards S, let
Again, on the other side
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, tjie difference of the
sines