screws, and partly by the thumb-screw Z. Turn the vernier plate, until the upper telescope comes directly over two of the levelling screws, then turn them contrary ways, till the upper telescope is horizontal; after which, turn the vernier plate 180°, and if the bubble of the level remains. in the middle of the tube, one line of the limb is horizon tal. But if the bubble recede from the centre of the level, raise the lower, or depress the upper end, one-half by the levelling screws, the other by the thumb-screw Z, till it is brought into a horizontal position. Turn the vernier plate again 180°, and if the level be not then horizontal, make it so, by dividing the error as before, and repeat the operation until the line of the limb is truly horizontal. Then turn the vernier plate 90°, and level as before. The limb ought now to be truly horizontal; but lest the first horizontal line may have been changed, in obtaining the second, it is well to bring the telescope and level two or three times over the levelling screws, until an entire revolution can be made without displacing the bubble from the middle of the tube. As this can only be the case when the level revolves around a vertical line, it follows that the limb will then be horizontal, and the axis of the instrument vertical. Then, by means of the small screws at the ends of the levels, bring the bubbles to the centres, and the axes of the levels will then be perpendicular to the axis of the instrument.

FOURTH ADJUSTMENT. To make the axis of the vertical limb perpendicular to the axis of the instrument.

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, ard turn the vernier plate 180°; now, if in elevating

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 welldefined point; then removing the caps which confine 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 7, which work into the slide A', that is connected with the horizontal axis.

Having explained the methods of properly adjusting the theodolite, we will now explain the particular uses of its several parts, and the manner of measuring angles.

VERNIERS.

21. Before explaining the vernier, as applied to the tne

A Vernier is a contrivance for measuring parts of the equal spaces marked off on a given scale or limb.

It is a graduated scale, so arranged, as to cover an exact number of equal spaces on the primary scale or limb, to which it is applied. It is divided into a number of equal parts, greater by one than the number of equal spaces which it covers on the limb.

The vernier may be applied to any scale of equal parts. The modes of its application are extremely various; the principle, however, is the same in all, and may be illus trated by a simple diagram.

Let AB be any limb or scale of equal parts, one of which let us suppose equal to b. Let CD be a vernier, equal in length to nine of these parts, and itself divided into ten equal spaces, each one of which is then equal to nine-tenths of b. The difference between a space on the limb and a space on the vernier, is therefore equal to onetenth of b or b. This is the least space that can be measured by means of the vernier, and is called the least count; hence,

The least count of a vernier is equal to one of the equal divisions of the limb divided by the number of spaces on the

vernier.

22. The true reading of the instrument, for any position of the vernier, expresses the distance from the point where the graduation on the limb begins, marked 0, to the 0 point of the vernier. In the diagram, that distance is expressed by nine units of the scale, or 9.

If, now, the vernier be moved till the division 1 coincides with the division 10 of the limb, the 0 point will have advanced along the limb a distance equal to, and the reading will become 9+b. If we again move the vernier till the division 2 coincides with the division 11 of the scale, the 0 point will have advanced an

9+2b; when 3 coincides with division 12, the reading will become 9+13, and so on, till finally, when the point 10 coincides with 19 of the scale, the distance 9 will have been increased by 18b, and will become 10, as it should, since, in that case, the 0 point will have been moved a whole space, and will coincide with the division 10 of the limb. Hence, the following rule for reading an instrument which has a vernier.

Read the limb in the direction of the graduation up to the division line next preceding the 0 point of the vernier; this is called the reading on the limb. Look along the vernier till a dividing line is found to coincide with a line of the limb: multiply the number of this first line by the least count of the vernier: this is the reading on the vernier: the sum of these two realings is the reading of the instrument.

23. In the theodolite described, the limb is divided into half degrees, and 30 spaces on the vernier cover 29 spaces on the limb. Hence, the least count of this instrument is of a half degree or 1'. Fig. 2, Plate 1, exhibits the vernier of the horizontal limb, and Fig. 3 the vernier of the vertical limb.

1

TO MEASURE A HORIZONTAL ANGLE WITH THE THEODOLITE.

24. Place the axis of the instrument directly over the point at which the angle is to be measured. This is ef fected 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 0 of the vernier at 0 or 360° of the limb, and fasten the clampscrew S of the vernier plate. Then, facing in the direc tion 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 L and Z, bring the intersection of the spider's lines to coincide exactly with the object.

Having loosened the clamp-screw of the lower tele

same object at which the upper telescope is directed; then tighten the clamp-screw 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 the position of the limb, should any have taken place; and, as the accuracy of the mea surements 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 0 point of the vernier at the 0 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 0 point of the vernier shall have passed the 0 point of the limb, in which case the greater reading must be subtracted from 360°, and the remainder added to the less.

TO MEASURE A VERTICAL ANGLE.

25. We shall first explain the method of determining the index error. Having levelled the horizontal limb, direct the telescope to some distinctly marked object as the top of a chimney, and read the instrument. Reverse the telescope in the Y's, and turn the vernier plate 180°, and having directed the telescope to the same object, again read the instrument. If the two readings are the same, the limb is adjusted; that is, the 0 of the limb coincides with the 0 of its vernier, when the axis of the telescope is parallel to the horizontal limb.

When the reading found with the eye end of the telescope nearest the vernier, is greater than that obtained in