Plane trigonometry, by S.L. Loney.

SINE-CURVE.

59

and let the magnitudes of angles be represented by
lengths measured along OX.
Let RI, R2, R3,... be points such that the distances
OR1, R1R2, RR3,... are equal. If then the distance OR,
represent a right angle, the distances OR2, OR3, OR4,...
must represent two, three, four,... right angles.
Also if P be any point on the line OX, then OP
represents an angle which bears the same ratio to a right
angle that OP bears to OR1.
[For example, if OP be equal to - OR1 then OP would represent onethird of a right angle; if P bisected B3R4 then OP would represent 32
right angles.]
Let also OR, be so chosen that one unit of length
represents one radian; since OR2 represents two right
angles, i.e. 7r radians, the length OR2 must be wr units of
length, i.e. about 3- units of length.
In a similar manner negative angles are represented
by distances OR,', OR2',... measured from 0 in a negative
direction.
At each point P erect a perpendicular PQ to represent
the sine of the angle which is represented by OP; if the
sine be positive the perpendicular is to be drawn parallel
to OY in the positive direction; if the sine be negative
the line is to be drawn in the negative direction.
[For example, since OR1 represents a right angle, the sine of which is
1, we erect a perpendicular R1Bj equal to one unit of length; since OR2
represents an angle equal to two right angles, the sine of which is zero,
we erect a perpendicular of length zero; since OR2 represents three right
angles, the sine of which is -1, we erect a perpendicular equal to -1,
i.e. we draw R3B3 downward and equal to a unit of length; if OP were
1
equal to one-third of OR, it would represent ~ of a right angle, i.e. 30~,