Measured angles. Correction. Corrected angles. ° ° ° A 73.4 + 0.1 73.5 B 49.3 + 0.1 49.4 C 57.0 + 0.1 57.1 ----- ----- Sum 179.7 180.0 Defect + 0.3 This process is in very common use among astronomers, and is called "adjusting" the observations. [Illustration: 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. [Illustration: 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?