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In the first case it is usual to say that the angle XOP is a positive angle; in the second case it is a negative angle.

(iii.) The angle XOP 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 mentioned in (i.) and (ii.).

We may express (iii.) thus: XOP is the geometrical representative of XOP+4n right angles, where n is any integer.

6. DEFINITIONS. O is called the origin. The line OX is the initial line. The line OP is the revolving line or radius vector. When referring to the angle XOP the lines OX and OP are called the sides of the angle, and O is called its vertex.

7. To add angle X'OP' to angle XOP, both being positive, revolve OP from its final position when it represents angle XOP,


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FIG. 4.

- 103 right angles.

through an angle equal to angle X'OP'; call this position OP2 Then, Z XOP1 = ≤ XOP + ≤ X'OP'.




Give a geometrical representation of each of the following angles, the initial line being drawn in each case from the origin towards the right:

8.4 right angles. 9. 4 right angles. 10. 4 n right angles.

11. (4 n + 2) right angles.

12. − (4 n + 1) right angles.

13. 1 right angles + 2 right angles.

14. 3 right angles — right angle.

8. Certain propositions which the student has proved while studying plane Geometry will be referred to very frequently, and quoted without proof.

The principal ones are:

a. The Pythagorean theorem.

b. Conditions under which two triangles are similar.

c. Homologous sides of similar triangles are proportional, and homologous angles are equal.

d. I. The ratio

circumference of a circle
diameter of a circle

II. It is an incommensurable number. III. It is 3.14159265+ ·


is a certain fixed number.

9. When we say that this number is incommensurable, we mean that its exact value cannot be stated as an arithmetical fraction.

It also happens that we have no short algebraical expression such as a surd, or combination of surds, which represents it exactly, so that we have no numerical expression whatever, arithmetical nor algebraical, to represent exactly the ratio of the circumference of a circle to its diameter.

Hence the universal custom has arisen, of denoting its exact value by the letter.

Thus stands always for the exact value of a certain incommensurable number, whose approximate value is 3.14159265, and which is the ratio of the circumference of any circle to its diameter.

It cannot be too carefully impressed on the student's memory that stands for this number 3.14159265..., etc., and for nothing else; just as 180 stands for the number one hundred and eighty, and for nothing else.

We may notice that 22


So that 22 and differ by less than a thousandth part of their





10. It is usual to say that we have measured any concrete quantity, when we have found how many times it contains some familiar quantity of the same kind.

We say, for example, that we have measured a line, when we have found how many feet it contains. We say that we have measured a field, when we have found out how many acres or how many square yards it contains.

To know the measurement of any quantity, then, we must have two things. First, we must have a unit, or standard of reference, of the same kind as the thing measured. Secondly, we must have the measure, or the number of times the thing measured contains the unit, or standard quantity.

Hence, the measure of a quantity is the number, and the unit is the concrete quantity, by means of which it is measured.

EXAMPLE 1. A line contains 261 feet; that is, 261 times a foot. Here the measure or number is 261, and the unit a foot.


1. What is the measure of 1 mile when a chain of 66 feet is the unit?

2. What is the measure of an acre when a square whose side is 22 yards is the unit?

3. The length of an Atlantic cable is 2300 miles and the length of the cable from England to France is 21 miles. Express the length of the first in terms of the second as unit.

4. The measure of a certain field is 22 and the unit 1100 square yards: express the area of the field in acres.

5. Find the measure of a miles when b yards is the unit

6. The measure of a certain distance is a when the unit is c feet. Express the distance in yards.

11. Measurement of angles. There are two common methods of measuring angles.

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


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

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 very familiar.

13. The right angle is divided into 90 equal parts, each of which is called a degree; each degree is subdivided into 60 equal parts, each of which is called a minute; and each minute is again subdivided into 60 equal parts, each of which is called a second.

Instruments used for measuring angles are subdivided accordingly; and the size of an angle is known when, with such an instrument, it has been observed that the angle contains a certain number of degrees, and a certain number of minutes beyond the number of complete degrees, and a certain number of seconds beyond the number of complete minutes.

Thus an angle might be recorded as containing 79 degrees +18 minutes +36.4 seconds.

Degrees, minutes, and seconds are indicated respectively by the symbols,,", and the above angle would be written 79° 18' 36.4".

14. An angle given in degrees, minutes, and seconds may be expressed as the decimal of a right angle by the usual method.

EXAMPLE. Express 39° 4' 27" as the decimal of a right angle.

60 27 seconds

60 4.45 minutes

90 39.7416666 etc. degrees

.441574074 etc. right angles

.441574074 of a right angle, Ans.

NOTE. The French proposed to call the 100th part of a right angle a grade (written 38), the 100th part of a grade a minute (written 3'), the 100th part of a minute a second (written 3). So that 1.437275 right angles would be read 143 72 75\\. The decimal method of subdividing the right angles has never been used.


15. DEFINITION. A radian is an angle at the centre of a circle, subtended by an arc equal in length to the radius of the circle. Thus if in the circle RPS, whose centre is O, arc RP = radius OR, then, angle ROP is a radian.

16. We shall now prove that the radian is a constant angle; or stating the same thing differently, we are about to prove that if we take any number of different circles, and S measure on the circumference of each an arc equal in length to its radius, then the angles at the centres of these circles which stand on these arcs respectively, will be all of the same size.

a radian

2 right angles

17. To prove that all radians are equal to one another.

Since the radian at the centre of a circle stands on an arc equal in length to the radius,

and an angle of two right angles at the centre of a circle stands on half the circumference,

Therefore a radian

and since angles at the centre of a circle are to one another as the arcs on which they stand (Geom.),


19. Since

20. Since

and so on.


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3 п



(Art. 8.)

Thus the radian possesses the qualification most essential in a unit; viz. it is always the same.

of 2 right angles,

18. The reasons why a radian is used as a unit are:

(i.) All radians are equal to one another.

(ii) Its use simplifies many formulæ in Theoretical Trigonom



= a certain fixed fraction of 180°.

1 radian

... 1 radian

1 radian

.. radians = 2 rt. = 180°,

radians = 1 rt.

= 90°,


2 rt.




= 57.2957°.

FIG. 5.

2 rt.

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radians 3 rt. 4 = 270°,



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