The construction of the wonderful canon of logarithms

—;
;
68 Trigonometrical Propositions.
Therefore substitute A for D
three times applied.
and D for A, and the problem will be as follows :
Given K D & the angle A with the angle D, to
find the side B A.
This is the same throughout as problem 1 1, and is
solved by applying the " Rule of Three " twice only.
The use and importance of half-versed
sines.
1. r~^ IvEN two sides & the contained angle, to find the
VJJ" third side.
From the half-versed sine of the sum of the sides
subtract the half - versed sine of their difference
multiply the remainder by the half-versed sine of
the contained angle ; divide the product by radius
to this add the half-versed sine of the difference of
the sides, and you have the half-versed sine of the
required base.
Given the base and the adjacent angles, the vertical
angle will be found by similar reasoning.
2. Conversely, given the three sides, to find any angle.
From the half-versed sine of the base subtract the
half-versed sine of the difference of the sides multiplied
by radius ; divide the remainder by the halfversed
sine of the sum of the sides diminished by
the half-versed sine of their difference, and the halfversed
sine of the vertical angle will be produced.
Given the three angles, the sides will be found by
similar reasoning.
3. Given two arcs, to find a third, whose sine shall be equal to
the difference of the sines of the given arcs.
Let