242 OF f»LANK I III G ON 0 MET lit.
6. The versed sine of an arch is the part of the dia
meter intercepted between the arch and its sine : so AF
is the versed sine of AB, and DF of DB.
7. The co-sine of an arch is the part of the diameteT
intercepted between the centre and the sine; and is equal
to the sine of the complement of that arch. Thus CF is
the co-sine of the arch AB, and is equal to BI, the sine
of its complement HB.
8. The tangent of an arch, is a right line touching
the circle in one extremity of that arch, continued from
thence to meet a line drawn from the centre through the
other extremity; which line is called the secant of the
same arch : thus AG is the tangent, and CG the secant
of the arch A B.
9. The co-tangent and co-secant of an arch are the
tangent and secant of the complement of that arch;
thus HK and CK. are the co-tangent and co-secant of
the arch AB.
10. A trigonometrical canon is a table exhibiting
the length of the sine, tangent, See. to every degree
and minute of the quadrant, with respect to the radius
which is supposed unity, and conceived to be divided
into 10000000 or more decimal parts. Upon this table
the numerical solution of the several cases in trigono
metry depend; it will therefore be proper to begin with
its construction.
mioposiTioN r*
The number of degrees and minutes, Sc. in an arch
being given; to find both its sine and co-sine.
This problem is resolved, by having the ratio of the
circumference to the diameter, and by means of the
known series for the sine and co-sine (hereafter de
monstrated). For, the semi-circumference of the circle,
whose radius is unity, being 3,141592653589793 &c.
it will therefore be, as the number of degrees or mi
nutes in the whole semi-circle is to the degrees or
minutes in the arch proposed, so is 3,14159265358 &c.
to the length of the said arch; which let be denoted by
a; then, by the series above quoted, it$ siue will be ex-