the arc. minute.

= 1

Thirty spaces on the vernier are equal to twenty nine on The reading is therefore to of 30 minutes The zero of the vernier stands between 41° 30' and 42°. On looking along the vernier, it is seen that the fifth and sixth lines coincide about equally well. The vernier therefore reads 41° 35' 30"

179. Reading backwards. Sometimes it is required to read backwards from the zero point on the limb. When this is done, the numbers on the vernier must be read in reverse, the highest being called zero, and the zero the highest.

3

10

2

Fig. 61.

360

Thus, in Fig. 61, the zero of the vernier standing to the right of 360 on the limb, between 1° 30' and 2°, and the division marked with an arrow-head being in line, the angle is 1° 41'. This mode of reading is needful when using the theodolite to take angles of depression, and also when using the transit to trace a line that bends backwards and forwards, the angle of deflection being then generally taken, and recorded to the right or to the left, as the case may be.

180. Double Verniers. To avoid the inconvenience of reading backwards, a double vernier is frequently made. It consists of two direct verniers having the same zero point, as shown in Fig. 62.

The arc in this figure is divided into degrees, and eleven spaces on the arc are equal to twelve on the vernier: the reading is therefore to 5 minutes. When the figures on the arc increase to the right, the right-hand vernier is used, and vice versa. The reading on the figure is 2° 45′ to the left.

181. Another form of double vernier is shown in Fig. 63.

In the figure, the vernier reads to minutes. When the zero of the vernier is to the left of that on the limb, the figures begin at the zero and increase towards the left to 15'; they then pass to the right-hand extremity, and again proceed to the left; that is, they stop at A and commence again at B. The upper figures of each half are the continuation of the lower figures of the other half. The reading in Fig. 63 is 1°8′ to the left.

In Fig. 64 the reading is 3° 19' to the right.

182. If the preceding descriptions have been thoroughly understood, the student will have no difficulty in reading the arc on any limb, however it may be divided. He should study the different positions until he can determine the angle with readiness, however the index may be placed. For this purpose, as before remarked, he should make for himself verniers with different scales, so that they can be placed in various positions.

Fig. 65.

E

The construction of such verniers is very simple. Suppose, for example, it is desired to divide the arc into degrees and subdivide it by the vernier so as to read to 5 minutes: twelve spaces on the vernier must equal eleven on the arc, or one space on the vernier will equal of a space on the arc. Let (Fig. 65) E be the centre and AB a portion of the limb, which, for the purpose intended, should not be of less radius. than ten or twelve inches, and let CD be the vernier; with some other radius EG, which should be greater than EB, describe an arc GF; take EI: EG:: number A of divisions on the vernier : the number F that occupies the same space on the arc, H -in this case, as 12 to 11. Take from the table of chords the chord of 1° or 10, as the case may be, and multiply it by the length of EG; lay off the product on GF, thus determining the points 1, 2, 3, &c., and lay off the same length on IH, determining the points a, b, c, &c.; stick a fine needle in the centre E; then, resting the ruler against the needle, bring it so as to coincide with I, and draw the

3. 2, 1.

C

a

B

I

division on AB; then, keeping it pressed against the needle, bring it successively to the other points on GF, and draw the corresponding divisions on AB. The arc will then be divided. In the same way, resting the ruler against the needle, and bringing it successively to the points on IH, the vernier may be divided. The reason of this process is, that since ab 1.2, the degrees of ab will be to the degrees of 1.2 as the radius of GF is to the radius of HI, as 11 to 12. Hence each division of the vernier is of one division of the arc.

=

By this means the divisions may be made with facility and accuracy.

183. Adjustments. In order that the theodolite and transit may give correct results when used, it is necessary that the different parts should bear the precise relations to each other that they are intended to have. By the term adjustment is meant the due relation of the parts to each other: when it is said an instrument is in adjustment, it is meant that every part bears to every other precisely its proper relations, so that the instrument is in perfect working order.

Before making any observations with a new instrument, it should be carefully examined to verify the adjustment. If the parts are not found to be properly adjusted, they must be rectified.

184. For measuring horizontal angles, the following conditions are necessary :

1. The levels should be parallel to the plates, so that when the bubbles are in the middle of their run, the plates shall be horizontal.

2. The axes of the two horizontal plates should be perfectly parallel and perpendicular to the plane of the plates. 3. The line of collimation should be perpendicular to the horizontal axis.

4. The horizontal axis should be parallel to the plane of the plates, so that when they are horizontal it may be so likewise.

185. First Adjustment. The levels should be parallel to the horizontal plates.

Verification. Clamp the two plates together; loosen the clamp C, (Figs. 51, 52;) bring the telescope directly over one pair of levelling screws, and level the plates as directed in Art. 170. Turn the plates half round: if the bubbles retain their position, the plane of the levels is perpendicular to the axis on which the lower plate turns. If either of them inclines to one end of its tube, it is out of adjustment, and requires rectification.

To rectify the fault, bring the bubble half way back to the middle by means of the capstan screw attached to one end, and the other half by the levelling screws. Again reverse the position of the plate: if the bubble now remains in the middle, the rectification is complete; if not, the operation must be repeated. When both levels have been so arranged that the bubbles retain their position in the middle of their run when the plates are turned all round, the adjustment is perfect, and the axis is perpendicular to the plane of the levels.

186. Second Adjustment. plates should be parallel.

The axes of the horizontal

Verification. Level the plates, as directed in last article. Clamp the lower plate, and loosen the vernier-plate. Turn it half round: if both bubbles still retain their position the axes are parallel. If the plates move freely over each other without binding in any position, they are perpendicular to the axes, or, at least, the upper one is so.

If any defects be found in either of these particulars, the instrument should be returned to the maker to be rectified.

187. Third Adjustment. The line of collimation of the telescope of the theodolite should be parallel to the common axis of the cylinders on which it rests in its Y's.