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43. Similar rules will hold with respect to the equations for connecting the centesimal and the circular systems, 200 being put in the place of 180: thus

circular measure of an angle containing G grades = G.

π

200

200

number of grades in the angle whose circular measure is 0 = 0. –

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Express in grades, &c. the angles whose circular measures

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44. We shall now give a set of Miscellaneous Examples to illustrate the principles explained in this and the two preceding Chapters.

EXAMPLES.-XII.

1. If the unit of angular measurement be 5o, what is the measure of 2210?

2. If an angle of 421° be represented by 10, what is the unit of measurement ?

3. An angle referred to different units has measures in the ratio 8 to 5; one unit is 2o, what is the other? Express each unit in terms of the other.

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 39°. 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

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Express as decimals of a grade the following angles :

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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

=

two right angles

π

180° 3.14159

= 57°-2958 nearly.

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

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divisions, reckoned from B, will make with

P

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

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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.

R

A

Make angle AOP equal to the unit of circular measure.

Then arc AP = radius 40 (Art. 32).

Now, by Euc. vi. 33,

angle AOR

=

AR

angle AOP AP

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