.. circumference

D; where D is the diameter, or 2πr, where r is the radius of the circle.

πη

Hence the length of the are of a quadrant is; of a semi

circle, or 180°, is πr; and of 270°, or three quadrants, is 2

Now if any arc a subtend an angle of 4o, then since

πλ

2

subtends 90°, and that by Euclid vI. 33, angles are proportional to the arcs which subtend them,

From this expression any one of the quantities may be

found when the others are given.

Ex. 1. Find the length of an arc of 45° of a circle whose radius is 10 feet;

45°

π

180° a
10

5. Most modern writers on Trigonometry take also for the unit of angular measure the number of degrees in an angle, subtended by an arc equal to the radius*. If U represent that angle, then by equation (1),

* If ACB be an angle at the centre of a circle, subtended by an arc equal to the radius of the circle, then, since by the 33rd Proposition of the 6th Book of Euclid, the angles at the centre of a circle are to each other as the arcs on which they stand, Angle ACB: four right angles :: arc AB: cir cumference, but AB is an arc equal to the radius, .. Angle ACB: four right angles :: r: 2 r' :: 1:2,

.. ACB

D

B

four right angles,

which, being independent of r, is constant

2π

for any circle; it may therefore be used to measure other angles.

a

r

which is called the circular measure of the angle.

From equation (2) we see that the measuring unit, U°,

а

must be multiplied by the fraction to find the angle;

r

Now, suppose we take an angle of 22° 27′ 39′′, then this is put into decimals at once by the centesimal division, without putting down any work on paper, it being 22°.2739; whereas, by the sexagesimal, we must proceed in the following manner: 60/ 39

60) 27.65

22.4608

If we wish to find how many grades and minutes are contained in this angle, here

Find the number of degrees and minutes in 46° 56′ 36′′.

(1) If F' and F", E' and E" represent the magnitude of a French and English minute and second respectively, shew that

E' 50

3.32

2.52°

Also, F" 1 French second;

.. F" x 100 x 100 x 100 = a quadrant,

E1 English second;

.. E x 60 × 60 × 90 a quadrant

hence, F" x 1000000

=

E" × 324000,

E" X 81;

E" 250 2.53

(2)- Compare the interior angles of a regular octagon and dodecagon:

360°

In a polygon of n sides, Æ° = 180° n

360°

In octagon 180°

8

4

(3) The earth being supposed a sphere, of which the diameter is 7980 miles, find the length of an arc of 1°.

(4) Find the diameter of a globe when an arc of 25° of the meridian measures 4 feet.

(5) Find the number of degrees in a circular arc 30 feet in length, of which the radius is 25 feet.

(6) Find the number of degrees in an angle of which the circular measure is 7854, the value of being 3.1416.

(7) The interior angles of a rectilineal figure are in arithmetical progression, the least angle is 120°, and the common difference 5; required the number of sides.

The last is the congruent value of n, since no angle can be so great as 180°; .. the figure has nine sides.

(8) One regular polygon has two sides more than another, and each of its angles exceeds each angle of the other polygon by 15°; find the number of sides in each.