The construction of the wonderful canon of logarithms

;
Construction of the Canon, 41
is the sine of the arc a d e. Draw a e ; its half,
f e, is the sine of the arc d e, the half of the arc
a d e. Draw e c ; its half, e g, is the sine of the
arc e h, and is therefore the sine of the complement
of the arc d e. Finally, make a k half the
radius a b. Then as a k is to e f, so is e g to e i.
For the two triangles c e a and c i e are equiangular,
since i c e or a c e is common to both
and c i e and c e a are each a right angle, the
former by hypothesis, the latter because it is in
the circumference and occupies a semicircle.
Hence a c, the hypotenuse of the triangle c e a, is
to a e, its less side, as e c, the hypotenuse of the
triangle c i e, is to e i its less side. And since
a c, the whole, is to a e as e c, the whole, is to e i,
it follows that a b, half of a c, is to a e as e g, half
of e c, is to e i. And now, finally, since a b, the
whole, is to a e, the whole, as e g is to e i, we
necessarily conclude that a k, half of a b, is to f e,
half of a e, as e g is to e i.
56. Double the logarithm of an arc of /^^ degrees is the
logarithm of half radius.
Referring to the preceding figure, let the case
be such that a e
and e c are equal.
e
In that case i will
fall on b, and e i
will be radius ; also
e f and e g will be
equal, each of them
being the sine of 45
*
degrees. Now (by
55) the ratio of a k, half radius, to e f, a sine of 45
degrees, is likewise the ratio of e g, also a sine of
F 45