of collimation; for, if the level were truly horizontal in one' position of the telescope, when the telescope is reversed, the bubble would not stand in the middle of the tube, except in one position of the level. This suggests the necessity of making the first part of the adjustment with tolerable accuracy; then, having made the second with care, let the first be re-examined, and proceed thus till the adjustment is completed. THIRD ADJUSTMENT.-To make the axes of the levels, on the limb, perpendicular to the axis of the instrument. This adjustment is effected, partly by the levelling 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 the screws 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 horizontal. But if the bubble recedes 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 take place when the level revolves around a vertical line, it follows that the limb will then be horizontal, and the axis of the instrument vertical. bring the bubbles to the centres, and the axes of the levels will 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 continues 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°; then, 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 adjust ment. 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 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 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 in the measurement of angles. VERNIERS. 25. Before explaining the vernier, as applied to the theodolite, we shall discuss the general theory of verniers. A VERNIER is a contrivance for measuring smaller arcs than those into which the limb of an instrument is divided. 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 limb or scale of equal the principle, however, is the same in all, and may be illustrated 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, say 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 one-tenth 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. READINGS. 26. The true reading of an 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 limb, 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 b, and the reading will become 9 b. If we again move the vernier till the division 2 coincides with the division 11 of the limb, the 0 point will have advanced an additional distance, equal to b, and the reading becomes 9+b; when 3 coincides with division 12, the reading will become 9+ b, and so on, till 10 3 10 finally, when the point 10 coincides with 19 of the limb, 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 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 readings is the reading of the instrument. In the theodolite described, the limb is divided into halfdegrees, 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. TO MEASURE A HORIZONTAL ANGLE WITH THE THEODOLITE. 27. 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 centre of 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 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 L and Z, bring the intersection |