OF THE UNIT OF ANGULAR MAGNITUDE

Excerpts

OF THE UNIT OF ANGULAR MAGNITUDE
The angle of easiest construction is the angle of an equilateral triangle, which is also two-thirds of a right angle.
OF THE UNIT OF ANGULAR MAGNITUDE
It is called the *unit of circular measure*, and the ratio of any angle to this unit is called the *circular measure* of the angle.
OF THE UNIT OF ANGULAR MAGNITUDE
For such reasons perhaps the sexagesimal scale, which has prevailed since the time of Ptolemy, was originally adopted.
OF THE UNIT OF ANGULAR MAGNITUDE
Two angles are said to be complements, each of the other, when their sum is a right angle.
OF THE UNIT OF ANGULAR MAGNITUDE
The $60$th part of a degree is a minute, denoted by $1'$, $\therefore 1° = 60'$.

Equations

OF THE UNIT OF ANGULAR MAGNITUDE
\therefore 1° = 60'
A degree is sixty minutes of arc.
OF THE UNIT OF ANGULAR MAGNITUDE
\therefore 1' = 60''
A minute of arc is sixty seconds of arc.
OF THE UNIT OF ANGULAR MAGNITUDE
1'' = 60'''
In older books a second of arc is divided sexagesimally into sixtieths, the third-order subdivision.
OF THE UNIT OF ANGULAR MAGNITUDE
1''' = 60^\text{iv}
In older books the sixtieth of a third of arc is the fourth-order subdivision.
OF THE UNIT OF ANGULAR MAGNITUDE
1^\text{iv} = 60^\text{v}
In older books the sixtieth of a fourth-order subdivision is the fifth-order subdivision.
OF THE UNIT OF ANGULAR MAGNITUDE
\theta = \frac{AB}{AC} = \frac{AB}{R}
The circular measure of an angle equals the arc subtending it at the centre divided by the radius, since the unit angle subtends an arc equal to the radius.
OF THE UNIT OF ANGULAR MAGNITUDE
AB = R\theta
The arc subtending an angle at the centre equals the radius times the circular measure of the angle.
OF THE UNIT OF ANGULAR MAGNITUDE
\frac{AB}{R} = \frac{A'B'}{R'}
The ratio of an arc to its radius is the same for any circle, because the same angle subtends both arcs.
OF THE UNIT OF ANGULAR MAGNITUDE
\frac{\frac{1}{4} \text{ circumference}}{R} = \frac{\pi}{2}
The circular measure of a right angle is a quarter of the circumference divided by the radius, which equals pi over two.
OF THE UNIT OF ANGULAR MAGNITUDE
= \frac{180°}{\pi}
The number of degrees in one unit of circular measure is 180 degrees divided by pi.
OF THE UNIT OF ANGULAR MAGNITUDE
\theta > \dfrac{\pi}{2}
When the angle exceeds a right angle in circular measure, its complement is negative and measured in the negative direction.
OF THE UNIT OF ANGULAR MAGNITUDE
\theta > \pi
When the angle exceeds two right angles in circular measure, its supplement is negative and measured in the negative direction.

Problems

No exercises in this chapter.