means of a small pin i, on the under side of the telescope, slide the eye glass till the spider's lines are seen distinctly; then, with the thumb screw X, which forces out, and draws in, the object glass, adjust this glass to its proper focus, when the object, as well as the spider's lines, will be distinctly seen: after which, by the thumb screws T and L, bring the intersection of the spider's lines exactly upon a well-defined point of the object.

Having done this, revolve the telescope in Y's, half round, when the attached level mn, will come to the upper side. See, if in this position, the horizontal hair appears above or below the point, and in either case, loosen one, and tighten the other, of the two screws that work the horizontal slide, till the horizontal hair has been carried over half the space between its last position and the observed point. Carry the telescope back to its place; direct again the intersection of the spider's lines to the point, and repeat the operation till the horizontal hair neither ascends nor descends, while the telescope is revolved. A similar process will arrange the vertical hair, and the line of collimation is then adjusted. This adjustment is made on the supposition, that the parts of the telescope which come in contact with the Y's, are truly cylindrical. To ascertain if they be so, reverse the telescope in the Y's, turn the vernier plate 180°, and if the intersection of the spider's lines is still directed to the same point, the adjustment may be relied on.

Second adjustment.—To make the axis of the attached level of the upper telescope, parallel to the line of collimation.-Turn the vernier plate, till the telescope comes directly over two of the levelling screws, between the plates DE and FG. Turn these screws contrary ways, keeping them firm against the plate FG, till the bubble of the level mn, stands at the middle of the tube. Then, open the loops, and reverse the telescope. If the bubble still stands in the middle of the tube, the axis of the tube is horizontal; but if not, it is inclined, the bubble being at the elevated end. In that case, by means of the small vertical screws m and n, at the ends of the level, raise the depressed end, or depress the elevated one, half the horizontal . position. Reverse the telescope in the Y's, and make the same correction again, and so on, until the bubble stands in the middle of the tube, in both positions of the telescope: the axis of the level is then horizontal. Let the telescope be now revolved in the Y's. If the bubble continue in the middle of the tube, the axis of the level is not only horizontal, but also parallel to the line of collimation. If, however, the bubble recede from its centre, the axis of the level is inclined to the line of collimation, and must be made parallel to it by means of two small screws, (one of which is seen at p.) which work horizontally. By loosening one of them, and tightening the other, the level is soon brought parallel to the line of collimation, and then, if the telescope be revolved in the Y's, the bubble will continue in the middle of the tube.

It is difficult to make the first part of this adjustment, while the axis of the level is considerably inclined to the line 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 again examined, and proceed thus till the adjustment is completed.

Third adjustment.-To make the limb of the instrument horicontal, or, to make the common axis of the limb and vernier plate truly vertical. 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 them contrary ways, till the upper telescope is horizontal; after which, turn the vernier plate 180°, and if the bubble of the level remain in the middle of the tube, one line of the limb is horizontal. 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. Bring the vernier plate back to its first position, and if the level be not horizontal, make it so, by dividing the error 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.

This adjustment being completed, the levels of the vernier plate are readily made parallel with it, by means of the small screws at their extremities. The three levels being then horizontal, and perpendicular in direction to the axis of the theo. dolite, the bubbles will retain the middle places in the tubes, during an entire revolution of the vernier plate, or of the limb and vernier plate together.

But the levels of the vernier plate may be made parallel with the limb, and the limb truly horizontal, without the aid of the upper level.

Let the upper telescope be placed directly over two of the levelling screws. One of the levels of the vernier plate will then be parallel to the line of these two screws, and the other level will be at right angles to this line, or parallel to the line of the other two levelling screws. In this situation, let the levels, by means of the levelling screws, be made horizontal. Then turn the vernier plate 180°, 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 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 inter

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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 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 admit of being moved laterally, the position of the line of collimation, and the horizontal axis, can then be altered and easily arranged at right angles, if they have not been so placed by the maker.

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 0, are at the opposite extremities of a diameter; which diameter is the intersection of a vertical plane It is important to ascertain the exact arc intercepted on the limb, between its 0 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 0 on the vernier to guide us, the place of the diameter could only be ascertained with certainty to half degrees, there being no means of determining its exact position, when it falls between the lines of division of 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 extremity coincides with a line of the graduated limb, when the line at the other extremity will also coincide with a line of the graduated limb; the sum of the spaces on the vernier being always exactly equal to a given number of spaces on the limb: then count the number of spaces on the limb which the vernier

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covers.

Thirdly.--Examine the limb of the instrument, and ascertain into what parts of a degree it is divided, or the value in minutes of the spaces shown thereon.

Let x represent the value of one of the equal spaces of the vernier, and n their number; then nx is 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 is 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, x= F; m, a, and n being known, x is known, as also the difference between a and x, which we shall show presently to be the smallest certain count of the

ma

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