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Suppose for instance that QP, starting from the position QE and revolving in a direction contrary to that in which the hands of a watch revolve, has come into the position indicated in the figure. It has then described an angle EQP greater than two right angles.

25. The magnitudes of angles are represented by numbers expressing how many times the given angles contain a certain angle fixed upon as the unit of angular measure.

When we speak of an angle ✪ we mean an angle which contains the unit of angular measurement

times.

26. There are three modes of measuring angles, called

I. The Sexagesimal or English method,

II. The Centesimal or French method,

III. The Circular Measure,

which we now proceed to describe in order.

I. The Sexagesimal Method.

27. In this method we suppose a right angle to be divided into 90 equal parts, each of which parts is called a degree, each degree to be divided into 60 equal parts, each of which is called a minute, and each minute to be divided into 60 equal parts, each of which is called a second. Then the magnitude of an angle is expressed by the number of degrees, minutes and seconds, which it contains. Degrees, minutes and seconds are marked respectively by the symbols °, ', ": thus, to represent 14 degrees, 9 minutes, 37.45 seconds, we write 14°. 9'.37"-45.

28. We can express the measure of an angle (expressed in degrees, minutes and seconds) in degrees and decimal parts of a degree by the following process.

Let the given angle be 39o. 5'. 33′′,

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Express as the decimal of a degree the following angles.

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29. In this method we suppose a right angle to be divided into 100 equal parts, each of which parts is called a grade, each grade to be divided into 100 equal parts, each of which is called a minute, and each minute to be divided into 100 equal parts, each of which is called a second. Then the magnitude of an angle is expressed by the number of grades, minutes and seconds, which it contains. Grades, minutes and seconds are marked respectively by the symbols ", ', ": thus, to represent 35 grades, 56 minutes, 84.53 seconds, we write 35o. 56. 84"-53.

The advantage of this method is that we can write down the minutes and seconds as the decimal of a grade by inspection.

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30. If the number expressing the minutes or seconds has only one significant digit, we must prefix a cipher to occupy the place of tens before we write down the minutes and seconds as the decimal of a grade.

and

Thus

25%. 9. 54". = 25%. 09'. 54"

=25-0954 grades,

36%. 8.4". = 36%. 08. 04“

=36-0804 grades.

EXAMPLES.-V.

Express as decimals of a grade the following angles :

(1) 258. 14.25",

(2) 38.4.15",

(3) 214. 3.7“,

(4) 15.7" 45,

(5)

4258. 13.5-54,

(6) 28. 2. 2-22.

31. The Centesimal Method was introduced by the French Mathematicians in the 18th century. The advantages that would have been obtained by its use were not considered sufficient to counterbalance the enormous labour which must have been spent on the re-arrangement of the Mathematical Tables then in use.

III. The Circular Measure.

32. In this method, which is chiefly used in the higher branches of Mathematics, the unit of angular measurement may be described as

(1) The angle subtended at the centre of a circle by an arc equal to the radius of the circle,

or, which is the same thing, as we proved in Art. 21, as

(2) The angle whose magnitude is the 7th part of two right angles.

33. It is important that the beginner should have a clear conception of the size of this angle, and this he will best obtain by considering it relatively to the magnitude of that angular unit which we call a degree.

Now the unit of circular measure

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Now if BC be the quadrant of a circle, and if we suppose the arc BC to be divided into 90 equal parts,

the right angle BAC will be divided by

the radii which pass through these points into 90 equal angles, each of which is called a degree.

A radius AP meeting the arc at a certain point between the 57th and 58th divisions, reckoned from B, will make with

AB an angle equal in magnitude to the unit of circular measure.

B

Hence an angle whose circular measure is 2 contains rather more than 114 degrees, and one whose circular measure is 3 contains about 172 degrees, or rather less than two right angles.

2 right angles,

34. Again, since the unit of circular measure =
π times the unit of circular measure = 2 right angles.
Hence

π

an angle whose circular measure is is equal to 2 right angles,

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35. To shew that the circular measure of an angle is equal to a fraction which has for its numerator the arc subtended by that angle at the centre of any circle, and for its denominator the radius of that circle.

Let EOD be any angle.

About O as centre and with any radius, describe a circle cutting OE in A, and OD in R.

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Make angle AOP equal to the unit of circular measure.

Then arc AP = radius AO (Art. 32).

Now, by Euc. vi. 33,

angle AOR

=

AR

angle AOP AP

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