The construction of the wonderful canon of logarithms

40 Construction of the Canon.
find the logarithm of this sine now contained in
the table, and then add to it the logarithmic difference
which the short table indicates as required
by the preceding multiplication.
Example.
IT is
378064. Since this sine is outside the limits
required to find the logarithm of the sine
of the Radical table, let it be multiplied by some
proportional number in the foregoing short table,
as by 20, when it will become 7561280. As this
now falls within the Radical table, seek for its
logarithm (by 50) and you will obtain 2795444.9,
to which add 29957311.56, the difference in the
short table corresponding to the proportion of
Where-
twenty to one, and you have 32752756.4.
fore 32752756 is the required logarithm of the
given sine 378064.
55. As half radius is to the sine of half a given arc, so is
the sine of the complement of the half arc to the sine ofthe
whole arc.
Let a b be radius, and a b c its double, on
which as diameter is described a semicircle. On
this lay off the given arc a e, bisect it in d, and
from e in the direction
of c lay off e h,
the complement of
d e, half the given
arc. Then h c is
necessarily equal to
e h, since the quadrant
d e h must equal
i
the remaining quadrant made up of the arcs a d
and h c. Draw e i perpendicular to a i c, then e i
is