dial to be subdivided into six equal parts, each of which is called a degree, and the measurement of an angle consists in finding how many of these degrees are included in the opening between its sides. At one o'clock the angle between the hands of a watch is thirty degrees, which is usually written 30°, at three o'clock it is 90°, at six o'clock 180°, etc.

A watch may be used to measure angles. How? But a more convenient instrument is the protractor, which is shown in Fig. 1, applied to the angle A B  C and showing that A B C = 85° as nearly as the protractor scale can be read.

The student should have and use a protractor, such as is furnished with this book, for the numerous exercises which are to follow.

Fig. 1.—A protractor.

Exercise 2.—Draw neatly a triangle with sides about 100 millimeters long, measure each of its angles and take their sum. No matter what may be the shape of the triangle, this sum should be very nearly 180°—exactly 180° if the work were perfect—but perfection can seldom be attained and one of the first lessons to be learned in any science which deals with measurement is, that however careful we may be in our work some minute error will cling to it and our results can be only approximately correct. This, however, should not be taken as an excuse for careless work, but rather as a stimulus to extra effort in order that the unavoidable errors may be made as small as possible. In the present case the measured angles may be improved a little by adding (algebraically) to each of them one third of the amount by which their sum falls short of 180°, as in the following example:

This process is in very common use among astronomers, and is called "adjusting" the observations.

Fig. 2.—Triangulation.

3. Triangles.—The instruments used by astronomers for the measurement of angles are usually provided with a telescope, which may be pointed at different objects, and with a scale, like that of the protractor, to measure the angle through which the telescope is turned in passing from one object to another. In this way it is possible to measure the angle between lines drawn from the instrument to two distant objects, such as two church steeples or the sun and moon, and this is usually called the angle between the objects. By measuring angles in this way it is possible to determine the distance to an inaccessible point, as shown in Fig. 2. A surveyor at A desires to know the distance to C, on the opposite side of a river which he can not cross. He measures with a tape line along his own side of the stream the distance A B = 100 yards and then, with a suitable instrument, measures the angle at A between the points C and B, and the angle at B between C and A, finding B A C = 73.4°, A B C = 49.3°. To determine the distance A C he draws upon paper a line 100 millimeters long, and marks the ends a and b; with a protractor he constructs at a the angle b a c = 73.4°, and at b the angle a b c = 49.3°, and marks by c the point where the two lines thus drawn meet. With the millimeter scale he now measures the distance a c = 90.2 millimeters, which determines the distance A C across the river to be 90.2 yards, since the triangle on paper has been made similar to the one across the river, and millimeters on the one correspond to yards on the other. What is the proposition of geometry upon which this depends? The measured distance A B in the surveyor's problem is called a base line.

Exercise 3.—With a foot rule and a protractor measure a base line and the angles necessary to determine the length of the schoolroom. After the length has been thus found, measure it directly with the foot rule and compare the measured length with the one found from the angles. If any part of the work has been carelessly done, the student need not expect the results to agree.

Fig. 3.—Finding the moon's distance from the earth.

In the same manner, by sighting at the moon from widely different parts of the earth, as in Fig. 3, the moon's distance from us is found to be about a quarter of a million miles. What is the base line in this case?

4. The horizon—altitudes.—In their observations astronomers and sailors make much use of the plane of the horizon, and practically any flat and level surface, such as that of a smooth pond, may be regarded as a part of this plane and used as such. A very common observation relating to the plane of the horizon is called "taking the sun's altitude," and consists in measuring the angle between the sun's rays and the plane of the horizon upon which they fall. This angle between a line and a plane appears slightly different from the angle between two lines, but is really the same thing, since it means the angle between the sun's rays and a line drawn in the plane of the horizon toward the point directly under the sun. Compare this with the definition given in the geographies, "The latitude of a point on the earth's surface is its angular distance north or south of the equator," and note that the latitude is the angle between the plane of the equator and a line drawn from the earth's center to the given point on its surface.

A convenient method of obtaining a part of the plane of