CHAPTER I.

MEASURE OF ANGLES.

1. Angular Unit.-2. Ratio of Circumference to Diameter.-3. Division of the Circle.

ANGULAR UNIT.

1. AN angle is defined by geometers to be the inclination of one right line to another.

In order to express this kind of magnitude by numbers, it is necessary to select an angular unit, to which all other angles may be referred. As this selection is altogether arbitrary, different units may be proposed; in this Chapter it is not necessary to consider more than those two, which are usually employed.

The angular unit which is commonly used in mathematical treatises may be thus defined:

DEFINITION: The angular unit* is that angle at the centre of a circle which is subtended by an arc equal in length to the radius.

Thus if the arc AU (fig. 1) be taken equal in length to the radius CA, the angle ACU will be equal to the angular unit. Any other angle ACB may be expressed numerically, with reference to this unit, as follows:

Let N be the number which expresses its value; let a be the length of the arc AB, which subtends it; and the radius of the circle. The following proportion is evident:

4 ACB: 4 ACU :: arc AB : arc AU.

* For an account of the other angular unit, vid. p. 5.

But as ACU is the angular unit, and AU equal to the radius, it follows that

That is, The numerical value of an angle, is the number which expresses the ratio of the arc which subtends it, to the radius of the circle.

In equation (1) three quantities, viz. N, a, r, are connected together, from which it follows, that if any two be given, the third may be calculated.

EXAMPLES.

1. If the radius of a circle be 35 feet, calculate the angle at the centre, which is subtended by an arc of 6 feet.

Having reduced the fraction to a decimal fraction, we find

Ans. 0.17142.

2. If the radius be 12 feet 7 inches, and if the arc be 5 inches, find the angle at the centre. Ans. 0.03311. 3. If the radius be 97 feet, and the angle at the centre = 0.734, calculate the length of the subtending arc.

By equation (1), it is evident that 97 must be multiplied by 0.734, in order to obtain the result.

4. If the radius be 10 feet 9 inches, 0.01347, calculate the arc.

Ans. 71.198 feet. and the angle at the centre Ans. 1.73763 inches.

5. An angle at the centre = 1.25 is subtended by an arc whose length is 16 feet, calculate the radius.

In this case it is evident from equation (1) that 16 must be divided by 1.25 to obtain the result. Ans. 12.8 feet.

6. If the angle at the centre be 0.00157, and the subtending arc 6 inches, calculate the radius.

Ans. 318 feet, 5.65 inches.

RATIO OF CIRCUMFERENCE TO DIAMETER.

2. The ratio of the circumference of a circle to its diameter has been determined by geometers to be as follows:

Circumference: Diameter :: 3.14159: 1;

and therefore

Circumference = 3.14159 × Diameter.

As this number, 3.14159, which expresses the ratio of the circumference to the diameter, frequently occurs in mathematical formulæ, it is convenient to represent it by a single letter; that which is invariably used is the Greek letter π. Substituting

this in the last equation, and 2r for the diameter, we have

Circumference = 2πr.

(2)

From this equation may be derived the number which represents a right angle. The arc which subtends a right angle is equal to a fourth part of the circumference, or πr, by equation (2). Dividing this by r we obtain, by equation (1),

The numerical value of a right angle.

The number, or 1.57079, which expresses the value of a right angle, referred to this particular angular unit, signifies that this unit, i. e. the angle ACU (fig. 1), is contained once, and the decimal part •57079 of ACU over, in a right angle. The number π, or 3.14159, which expresses two right angles, signifies that ACU is contained in two right angles, three times, and the decimal part .14159 of ACU

over.

DIVISION OF THE CIRCLE.

3. As the angular unit defined in section 1 is not a submultiple of four right angles, and is, moreover, too large for practical purposes, such as astronomical observations and surveying, another mode of measuring angles, and representing them numerically, has been devised, which may be thus explained:

The circumference of the circle is divided into 360 equal parts; each of these parts subtends at the centre an angle equal to the 360th part of four right angles, this angle is called a degree. The degree is subdivided into 60 equal parts, called minutes; and the minute into 60 equal parts, called seconds. Degrees, minutes, and seconds are represented by the symbols '"; for example, 43° 25′18′′, signifies 43 degrees, 25 minutes, 18 seconds.

In order to reduce an angle expressed in degrees, minutes, and seconds, to its value, referred to the angular unit, and vice versâ, it is necessary to find the number of seconds which are contained in this unit:

Number of seconds in angular unit: number of seconds in four right angles :: arc AU : circumfe

Or,

rence.

Number of seconds in angular unit: 360′′ × 60 × 60

::r: 2πr.

From which it follows, since = 3.14159, that

Number of seconds in angular unit = 206265′′.

If N" be the number of seconds in an angle which is subtended by an arc whose length is a, then

N": 206265":: a: AU (= r);

By equation (3), in which are connected together the radius, the number of seconds in the angle at the centre, and the length of the arc which subtends it, many useful questions may be solved.

EXAMPLES.

1. In a circle of 100 feet radius, calculate the angle in degrees, minutes, and seconds, which is subtended by an arc whose length is 9 feet.

By equation (3), the number of seconds in the angle is 18563′′.85, which, reduced to degrees, minutes, and seconds, gives: Ans. 5° 9′ 23′′.85. 2. A person standing in the centre of a sphere observes that a line on its surface, which he knows to be 6 feet in length, subtends an angle of 3′ 28′′, calculate the radius of the sphere.

Ans. I mile, 223.31 yds. 3. The diameter of the earth is 7926 miles, its distance from the moon is 237,638 miles, calculate the angle which the earth's diameter subtends at the moon. Ans. 1° 54′ 39′′.6.

4. The angle which the moon subtends at the earth is observed to be 31' 7", calculate her diameter in miles. Ans. 2151 miles nearly. 5. It has been ascertained that the angle which the diameter of the earth subtends at the centre of the sun is 17".2. Calculate the sun's distance. Ans. 95, 049, 790 miles. 6. The sun's diameter subtends at the earth an angle of 32′ 3′′. Calculate its diameter. Ans. 886,145 miles.

CHAPTER II.

TRIGONOMETRICAL MAGNITUDES.

1. Definition of Trigonometrical Magnitudes.-2. Relations of Trigonometrical Magnitudes.-3. Complemental and Supplemental Angles.-4. Trigonometrical Tables.

DEFINITION OF TRIGONOMETRICAL MAGNITUDES.

1. THERE are certain magnitudes connected with angles, and entering into every trigonometrical computation, which are defined as follows: