1 foot
... diameter=

= x 1 foot
= inches

=3.8 inches, nearly. If a greater degree of accuracy be desired, we must use 13 instead of 2 We then get, the diameter=3.7699 inches,

=3•77 inches very nearly.


In the answers of the first 12 of the following examples ?? is used for .

(1) Find the circumference of a circle whose diameter is one yard.

(2) Find the circumference of a circle whose radius is 4 feet. (3) Find the circumference of a 48 inch bicycle wheel.

(4) The circumference of a circle is 10 feet; find its diameter.

(5) What must be the diameter of a locomotive driving wheel, that it may make 220 revolutions


mile? (6) How many revolutions does a 36 inch bicycle wheel make per mile?

(7) How many more revolutions per mile does a 50 inch bicycle wheel make than one of 52 inches ?

(8) A locomotive whose driving wheel is 5 feet high has an instrument to record the number of revolutions made. What number will the instrument record in running 100 miles ?

(9) If the instrument in Question 8 indicates 3 revolutions per second, how many miles per hour is the engine running?

(10) What is the diameter of the driving wheel of a locomotive engine which makes 4 revolutions per second when the engine is going at the rate of 60 miles per hour ?

(11) The large hand of the Westminster clock is 11 feet long; how many yards per day does its extremity travel? How far does the extremity move in a minute ?

(12) The diameter of the whispering gallery in St Paul's is 108 feet; what is its circumference?

(13) Find the number of inches of wire necessary to construct a figure consisting of a circle with a regular hexagon inscribed in it, one of whose sides is 3 feet.

(14) How many inches of wire would be necessary in a figure similar to that in Question (13), if the circumference of the circle were ten feet?

(15) Find how many inches of wire are necessary to make a figure consisting of a circle and a square inscribed in it, when each side of the square is 2 feet.

(16) How many inches of wire are necessary for a figure similar to that in Question (15), when the circumference of the circle is 12 feet?

(17) Find the length of string necessary to string the handle of a cricket bat; having given the diameter of the handle=11 in., the length of the handle = 12 in., the diameter of the string= lith of an inch.



38. In elementary Geometry (Euclid I.–VI.) the angles considered are each always less than two right angles.

For example, in speaking of the angle ROP in Euclid we should always mean the angle less than two right angles,


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R not an angle measured in the opposite direction greater than two right angles.

39. In Trigonometry, by the angle ROP is meant, not the present inclination of the two lines OR, OP but, the amount of turning which OP has gone through when, starting from the position OR, it has turned about O into the position OP.

Example. Suppose a race run round a circular course. The position of any one of the competitors would be known, if we remark that he has described a certain angle about the centre of the course. Thus, if the distance to be run is three times round, the ļine joining each competitor to the centre would have to describe an angle of 12 right angles.

When we remark that a competitor has described an angle of 63 right angles, we record not only his present position, but the total distance he has gone. He would in such a case have gone a little more than one and a half times round the course.

40. DEFINITION. The angle between two lines OR, OP is the amount of turning about the point 0 which one of



0 the lines OP has gone through in turning from the position OR into the position OP.

41. The angle ROP may be the geometrical representative of an unlimited number of Trigonometrical angles.

(i) The angle ROP may represent the angle less than two right angles as in Euclid.

In this case OP has turned from the position OR into the position OP by turning about in the direction contrary to that of the hands of a watch.

(ü) The angle ROP may represent the angle described by OP in turning from the position OR into the position OP in the same direction as the hands of a watch.

In the first case it is usual to say that the angle ROP is described in the positive direction, in the second that the angle is described in the negative direction.

(iii) The angle ROP may be the geometrical representation of any of the Trigonometrical angles formed by any number of complete revolutions in the positive or in the negative direction, added to either of the first two angles. (We shall return to this subject in Chapter IX.)

EXAMPLES. VI. Give a geometrical representation of each of the following angles, the starting line being drawn in each case from the turning point towards the right. 1. +3 right angles.

7. - 10} right angles. 2. +5 right angles.

8. + 4 right angles. 3. +4} right angles.

9. - 4 right angles. 4. +77 right angles.

10. 4n right angles. 5. - 1 right angle.

11. (4n+2) right angles. 6. 103 right angles.

12. – (4n+ }) right angles.

42. There are two methods of measuring angles.

(i) The rectangular measure.
(ü) The circular measure.


43. Angles are always measured in practice with the right angle (or part of the right angle) as unit.

44. The reasons why the right angle is chosen for a unit are :

(i) All right angles are equal to one another.
(ii) A right angle is practically easy to draw.
(iii) It is an angle whose size is


familiar. 45. The right angle is a large angle, and it is therefore subdivided for practical purposes. It is usual to explain two methods of subdivision,

(i) The sexagesimal method.
(ü) The centesimal, or decimal, method *.

• See Art. 55.

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