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All the meridians passing through a survey of moderate extent, are considered as straight lines parallel to each other.

The bearing or course of a line is the angle which it makes with a meridian passing through one end; and it is reckoned from the north or south point of the horizon, toward the east or west.

Thus, if NS represent a meridian, and the angle NAB is 40°, then the bearing of AB from the point A is 40° to the west of north, and is written N. 40° W., and read north forty degrees west.

С The reverse bearing of a line is the bearing taken from the other end of the line. The forward bearing and reverse bearing

А of a line are equal angles, but lie between directly opposite points. Thus, if the bearing of AB from A is N. 40° W., the bearing of the same line from B is S. 40° E.

(138.) For measuring vertical angles, the instrument commonly used is

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A QUADRANT. It consists of a quarter of a circle, usually made of brass, and its limb, AB, is divided into

А. degrees and minutes, numbered from A up to 90°. It is furnished either with a pair of plain sights or with a telescope, CD, which is to be directed toward the object observed. A plumb line, CE, is suspended from the center of the quadrant, and indicates when the radius CB is B brought into a vertical position.

To measure the angle of elevation, for example, of the top of a tower, point the telescope, CD, toward the tower, keeping the radius, CB, in a vertical position by means of the plumb line, CE. Move the telescope until the given object is seen in the middle of the field of view. The center of the field is indicated by two wires placed in the focus of the object-glass of

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the telescope, one wire being vertical and the other horizontal. When the horizontal wire is made to coincide with the summit of the tower, the angle of elevation is shown upon the are AB by means of an index which moves with the telescope.

As the arc is not commonly divided into parts smaller than half degrees, when great accuracy is required, some contrivance is needed for obtaining smaller fractions of a degree. This is usually effected by a vernier.

(139.) A Vernier is a scale of small extent, graduated in such a manner that, being moved by the side of a fixed scale; we are enabled to measure minute portions of this scale. The length of this movable scale is equal to a certain number of parts of that to be subdivided, but it is divided into parts one more or one less than those of the primary scale taken for the length of the vernier. Thus, if we wish to measure hundredths of an inch, as in the case of a barometer, we first divide an inch into ten equal parts. We then construct a vernier equal in length to 11 of these divisions, but divide it into 10 equal parts, by which means each division on the vernier is 'oth longer than a division of the primary scale.

Thus, let AB be the upper end of a barometer tube, the mercury standing at the point C; the scale is divided into inches and tenths of an inch, and the middle piece, numbered from 1 to 9, is the vernier that slides up and down, having 10 of its divisions equal to 11 divisions of the scale, that is, to izths of an inch. Therefore, each division of the vernier is Voths of an inch; or one division of the vernier exceeds one divi

-29 sion of the scale by both of an inch. Now, as the sixth division of the vernier (in the figure) coincides with a division B of the scale, the fifth division of the vernier will stand onth of an inch above the nearest division of the scale; the fourth division oths of an inch, and the top of the vernier will be tooths of an inch above the next lower division of the scale; i. e., the top of the vernier coincides with 29,66 inches upon the scale. In practice, therefore, we ob

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serve what division of the vernier coincides with a division of the scale; this will show the hundredths of an inch to be added to the tenths next below the vernier at the top.

A similar contrivance is applied to graduated circles, to obtain the value of an arc with greater accuracy. If a circle is graduated to half degrees, or 30', and we wish to measure single minutes by the vernier, we take an arc equal to 31 divisions upon the limb, and divide it into 30 equal parts. Then each division of the vernier will be equal to 3ths of a degree, while each division of the scale is a ths of a degree. That is, each space on the vernier exceeds one on the limb by 1'.

In order, therefore, to read an angle for any position of the vernier, we pass along the vernier until a line is found coinciding with a line of the limb. The number of this line from the zero point indicates the minutes which are to be added to the degrees and half degrees taken from the graduated circle Sometimes a vernier is attached to the common surveyor's compass.

(140.) An instrument in common use for measuring both horizontal and vertical angles is

THE THEODOLITE. The theodolite has two circular brass plates, C and D (see fig. next page), the former of which is called the vernier plate, and the latter the graduated limb. Both have a horizontal motion about the vertical axis, E. This axis consists of two parts, one external, and the other internal; the former secured to the graduated limb, D, and the latter to the vernier plate, C, so that the vernier plate turns freely upon the lower. The edge of the lower plate is divided into degrees and half degrees, and this is subdivided by a vernier attached to the upper plate into single minutes. The degrees are numbered from 0 to 360.

The parallel plates, A and B, are held together by a ball which rests in a socket. Four screws, three of which, a, a, a, are shown in the figure, turn in sockets fixed to the lower plate, while their heads press against the under side of the upper plate, by which means the instrument is leveled for observation. The whole rests upon a tripod, which is firmly attached to the body of the instrument.

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To the vernier plate, two spirit-levels, C, C, are attached at right angles to each other, to determine when the graduated limb is horizontal. A compass, also, is placed at F. Two frames, one of which is seen at N, support the pivots of the horizontal axis of the vertical semicircle KL, on which the tel. escope, GH, is placed. One side of the vertical arc is divided into degrees and half degrees, and it is divided into single minutes by the aid of its vernier. The graduation commences at the middle of the arc, and reads both ways to 90°. Under and parallel to the telescope is a spirit-level, M, to show when the telescope is brought to a horizontal position. To enable us to direct the telescope upon an object with precision, two lines called wires are fixed at right angles to each other in the focus of the telescope.

To measure a Horizontal Angle with the Theodolite. (141.) Place the instrument exactly over the station from which the angle is to be measured; then level the instrument by means of the screws, a, a, bringing the telescope over each pair alternately until the two spirit-levels on the vernier plate retain their position, while the instrument is turned entirely round upon its axis. Direct the telescope to one of the objects to be observed, moving it until the cross-wires and object coincide. Now read off the degrees upon the graduated limb, and the minutes indicated by the vernier. Next, release the upper plate (leaving the graduated limb undisturbed), and move it round until the telescope is directed to the second object, and make the cross-wires bisect this object, as was done by the first. Again, read off the vernier ; the difference between this and the former reading will be the angle required.

The magnetic bearing of an object is determined by simply reading the angle pointed out by the compass-needle when the object is bisected.

To measure an Angle of Elevation with the Theodolite.

(142.) Direct the telescope toward the given object so that it

may be bisected by the horizontal wire, and then read off the arc upon the vertical semicircle. After observing the object with the telescope in its natural position, it is well to revolve the telescope in its supports until the level comes uppermost, and repeat the observation. The mean of the two meas, ures may be taken as the angle of elevation.

By the aid of the instruments now described, we may determine the distance of an inaccessible object, and its height above the surface of the earth.

HEIGHTS AND DISTANCES.

PROBLEM I.

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(143.) To determine the height of a vertical object situated on a horizontal plane.

Measure from the object to any convenient distance in a straight line, and then take the angle of elevation subtended by the object.

If we measure the distance DE, and the angle of elevation CDE, there will be given, in the right-angled triangle CDE, the base and the angles, to find the perpendicular CE (Art. 46). To this we must add the height of the instrument, to obtain the entire height of the object al ove the plane AB.

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