:

levels be made horizontal, by means of the levelling screws. Then turn the vernier plate 180o, and if they both continue horizontal, the limb is truly level. But if both, or either of them, be changed from a horizontal position, let the error be divided between the level and the limb; and repeat the operation until the levels will continue horizontal during an entire revolution: the limb is then horizontal, and the axis of the instrument truly vertical.

FOURTH ADJUSTMENT. - To make the axis of the vertical limb truly horizontal, or perpendicular to the axis of the instru

ment.

Bring the intersection of the spider's lines of the upper telescope upon a plumb line, or any well-defined vertical object, and move the telescope with the thumb-screw Z: if the intersection of the spider's lines continue on the vertical line, the axis is horizontal.

Or, the adjustment may be effected thus: Direct the intersection of the spider's lines to a well-defined point that is considerably elevated: then turn the vertical limb, until the axis of the telescope rests on some other well-defined point, upon or near the ground: reverse the telescope, and turn the vernier plate 180°; now, if in elevating and depressing the telescope, the line of collimation passes through the two points before noted, the axis is horizontal. If it be found, by either of the above methods, that the axis is not horizontal, it must be made so by the screws which fasten the frame-work to the vernier plate.

There are two important lines of the theodolite, the positions of which are determined with great care by the maker, and fixed permanently. First, the axis of the instrument is placed exactly at right angles with the limb and vernier plate; and unless it have this position, the vernier plate will not revolve at right angles to the axis, as explained in the third adjustment. Secondly, the line of collimation of the upper telescope, is fixed at right angles to the horizontal axis of the vertical limb. We can ascertain whether these last lines are truly at right angles, by directing the intersection of the spider's lines to a well-defined point; then removing the caps which consine the horizontal axis in its supports, and reversing the axis: if the intersection of the spider's lines can be made to cover exactly the same point, without moving the vernier plate, the line of collimation is at right angles to the axis.

If the theodolite be so constructed that either of the Y's admits of being moved laterally, so as to vary the angle between the horizontal axis and the line of collimation, these lines may be adjusted at right angles to each other, if they have not been so placed by the maker.

The lower telescope being used merely as a guard, requires no adjustment, although it is better to make the axis, about which its vertical motions are performed, horizontal, or perpendicular to the axis of the instrument; and this is easily effected by means of the two small screws k and l, which work into the slide A', that is connected with the horizontal axis.

The theodolite being properly adjusted, the particular uses of its several parts, and the manner of measuring angles, are now to be explained.

There are two verniers on the vernier plate, and the points of them marked o, are at the opposite extremities of a diameter; which diameter is the intersection of a vertical plane passed through the line of collimation, with the vernier plate. It is important to ascertain the exact arc intercepted on the limb, between its o point, (this being the point from which the degrees are numbered), and this diameter, for any position which it may assume. The limb being divided to half degrees, if we had only the line marked o on the vernier, to guide us, the place of the extremity of the diameter could only be ascertained with certainty to half degrees, as there would be no means of determining its exact position, when it falls between the lines of division on the limb. But the vernier affords results much more accurate. As most instruments for the measurement of angles have verniers, it will perhaps be best to explain their use generally.

First. Count carefully the number of spaces into which the vernier is divided: this number is one less than the number of lines which limit them.

Secondly. Turn the vernier till the line at one extremite coincides with a line of the graduated limb, when the line at the other extremity will also coincide with a line of thy

1.

always exactly equal to a given number of spaces on the limb; then count the number of spaces on the limb which the vernier covers.

Thirdly. Examine the limb of the instrument, and ascertain into what parts of a degree it is divided, and express one of those equal parts in minutes.

Let x represent the value of one of the equal spaces of the vernier, and n their number; then nx will be equal to the space covered by the vernier. Let a represent the smallest equal space into which the limb is divided, and m the number of such spaces covered by the vernier; then ma will be equal to the space on the limb covered by the vernier, which is also equal to nx.

The equation nx=ma is called the equation of the instrument. In this equation,

m, a, and n, being known, a can be found, as also the difference between a and 2, which we shall show presently, to be the smallest certain count of the instrument.

In the theodolite, m=29, a=30′ and n=30 hence;

the excess of a space on the limb over a space on the vernier. Fig. 2. Let AB be a portion of the limb of the instrument, and CED the vernier in one of its positions, its 0 point coinciding with the line marked 10 on the limb. Now, since each space of the vernier is less by 1' than each space of the limb, the first line on the left of o, will be 1' to the right of the first line on the left of the 10 on the limb; and if the vernier plate be moved 1' towards the left, these lines will coincide, and the second line from o will then be 1' to the right of the second line from 10; if the vernier be moved another minute, these last lines will coincide. The vernier would then show 10° 2'.

If the vernier plate be turned still farther, till the third, fourth, fifth, &c. lines coincide, it is plain, that the o point of the veinier will have passed the line 10 on the limb, by as many minutes as there are lines of the vernier which shall have coincided with lines of the limb. When the last line of the vernier coincides with a line of the limb, the vernier will have been moved 30', or half a degree; and the 0 point will at the same time coincide with a line of the limb, and show 10°30′.

The general rule for reading the angle for any position of the vernier may now be stated.

When the o line of the vernier coincides with a line of the limb, the arc is easily read from the limb; but when it falls between two lines, note the degrees and half degrees up to the line on the right; then pass along the vernier till a line is found coinciding with a line of the limb: the number of this line from the o point, indicates the minutes which are to be added to the degrees and half degrees, for the entire angle.

To measure a horizontal angle with the theodolite.

Place the axis of the instrument directly over the point at which the angle is to be measured. This is effected by means of a plumb, suspended from the plate which forms the upper end of the tripod.

Having made the limb truly level, place the o of the ver-. nier at 0 or 360° of the limb, and fasten the clamp-screw S of the vernier plate. Then, facing in the direction between the lines which subtend the angle to be measured, turn the limb with the outer spindle, until the telescope points to the object on the left, very nearly. Clamp the limb with the clamp-screw K, and by means of the tangent screws Land 7, bring the intersection of the spider's lines to coincide exactly with the object.

Having loosened the clamp-screw of the lower telescope MN, direct it with the thumb-screw P to the same object at which the upper telescope is directed; then tighten the clampscrew Q. This being done, loosen the clamp-screw S of the vernier plate, and direct the telescope to the other object: the arc passed over by the 0 point of the vernier, is the measure of the angle sought.

The lower telescope having been made fast to the limb, will indicate any change of its position, should any have taken place; and, as the accuracy of the measurements depends on the fixedness of the limb, the lower telescope ought to be often examined, and if its position has been altered, the limb must be brought back to its place by the tangent-screw L.

It is not necessary to place the o point of the vernier at the o point of the limb, previously to commencing the measurement of the angle, but convenient merely; for, whatever be the position of this point on the limb, it is evident that the arc which it passes over is the true measure of the horizontal angle. If, therefore, its place be carefully noted for the first direction, and also for the second, the difference of these two readings will be the true angle, unless the vernier shall have passed the o point of the limb, in which case the greater reading must be subtracted from 360o, and the remainder added to the less.

To measure a vertical angle.

In Fig. 3, AB represents a view of the vertical limb opposite the thumb-screw Z, and ED is the vernier. The o point of this vernier is at the middle division line, and fifteen spaces lie on each side of it. The relation which exists between the spaces on the limb and those of the vernier, is the same as that between the divisions of the horizontal limb and its vernier, and the degrees and half degrees are read in the same manner: the angles of elevation being read from the o of the limb towards the right, and those of depression in the contrary direction. For the minutes, we pass along the vernier in the direction in which the degrees are counted, and if we reach the extreme line, which is the fifteenth, without. finding a coincidence, we must then pass to the other extremity of the vernier, and look along towards the o point till two lines are found to coincide: the number of the line on the vernier will show the minutes. The lines of the vernier are numbered both ways from the o point, and marked 5, 10, 15, to one extremity, and correspondingly from the other extremity 15, 20 and 25, to the 0 point again. The upper range shows the minutes for angles of elevation, and the lower range for those of depression. The vernier in Fig. 3 stands at 2o 15' of depression. Had the 15th line at the left, passed the short line with which it now coincides, we should pass to the line 15, on the lower range to the right, and then count towards the o to the left.

The first thing to be done, is to ascertain the point of the vertical limb at which the 0 point of the vernier stands, when the line of collimation of the upper telescope, together with