17. Since the circumference varies as the radius, the ratio

circumference

is the same for all circles, and therefore the ratio

radius circumference diameter

is the same for all circles.

18. Def. The ratio

circumference
diameter

is denoted by the sym

bol T.

19. The value of this numerical quantity

cannot be determined exactly, but it has been approximately determined by various methods.

If we take a piece of string which will exactly go round a penny, and another piece which will exactly stretch across the diameter of the penny: if we then set off along a straight line seven lengths of the first string, and on another straight line by the side of the first we set off twenty-two lengths of the second string, we shall find that the two lines are very nearly equal. Hence 22 diameters are nearly equal to 7 circumferences, circumference 22 nearly, or in other words

that is the ratio

diameter

22
7

7

the fraction is a rough approximation to the value of π.

355
113

The fraction gives a closer approximation.

The accurate value of the ratio to 5 places of decimals is 3·14159.

20. Suppose we call the radius of a circle r: then the dia

In the following examples the value of may be taken as

22 7'

(1) The diameter of a circle is 5 feet, what is its circumference?

(2) The circumference of a circle is 542 ft. 6 in., what is its radius ?

(3) The driving-wheel of a locomotive-engine of diameter 6 feet makes 2 revolutions in a second: find approximately the number of miles per hour at which the train is going.

(4) Supposing the earth to be a perfect sphere whose circumference is 25000 miles, what is its diameter ?

(5) The diameter of the Sun is 883220 miles, what is its circumference?

(6) The circumference of the moon is 6850 miles, what is its radius ?

(9) The circumference of a circle is 150 feet, what is the side

of a square inscribed in it?

(10) The circumference of a circle is 200 feet, what is the side of a square inscribed in it?

(11) A water-wheel, whose diameter is 12 feet, makes 30 revolutions per minute. Find approximately the number of miles per hour traversed by a point on the circumference of the wheel.

(12) A mill-sail, whose length is 21 feet, makes 15 revolutions per minute. How many miles per hour does the end of the sail

traverse?

21. The angle subtended at the centre of a circle by an arc equal to the radius of the circle is the same for all circles.

B

Let O be the centre of a circle, whose radius is r;

AB the arc of a quadrant, and therefore AOB a right angle;

AP an arc equal to the radius AO.

Thus the magnitude of the angle AOP is independent of r and

is therefore the same for all circles.

CHAPTER II.

ON THE MEASUREMENT OF ANGLES.

22. EUCLID defines a plane rectilineal angle as the inclination of two straight lines to each other, which meet, but are not in the same straight line. Hence an angle in Geometry must be less than two right angles.

In Trigonometry the term angle is used in a more extended sense, the magnitude of angles in this science being unlimited.

23. An angle in Trigonometry is defined in the following

manner.

Let WQE be a fixed straight line, and QP a line which revolves about the fixed point Q, and which at first coincides with QE.

Then when QP is in the position represented in the figure, we say that it has described the angle PQE.

The advantage of this definition is that it enables us to consider angles not only greater than two right angles, but greater than four right angles, viz. such as are described by the revolving line when it makes more than one complete revolution.

24. In speaking of a trigonometrical angle we must take into account the position from which the line that has described the angle started.