This is exactly what obtains in the practice of levelling; in ten chains, for instance, the combined allowance for curvature and refraction is only nine-thousandths of a foot, and ten chains is a tolerably long sight. As, however, instances may occur when it would be useful to apply it, we will give one example for doing so:-Let E in the last figure be a spot where a level is set up and adjusted; on making an observation on a house seven miles off, the centre of the diaphragm intersects a window on the first floor of the house; from the table of differences between the apparent and true levels for miles, the tabular number is found to be 28.01 equal to the difference of level, and the window which appeared on the same level as the eye of the observer at the telescope, is in fact 28:01 feet higher; but to find the difference of level between the ground on which the observer is standing and the window, the height of the instrument above the ground must be added to the above correction; let this be four feet, then 28.01 + 4 = 32.01 equal to the difference of level between the ground at the observer's station and the window in question. In the absence of tables, a convenient rule to remember is that two-thirds of the square of the distance in miles, is equal to the correction in feet.

The practice of levelling is a very simple operation, but it is very certain that it is one of the most delicate the engineer has to perform, and demands the greatest caution and attention.

We will first describe the levelling instrument, that is the "level," and shall confine ourselves to that so well known as Gravatt's "Dumpy Level," being the one used by at least ninetynine persons out of every hundred. It is shown at Fig. 130A. As much of what has been said on the subject of the theodolite is applicable to this instrument also, our remarks will be shorter than they otherwise would be. As generally constructed for packing in its case, the "Dumpy Level" consists of the telescope, the tube above carrying the spirit-bubble, or rather, the airbubble in the spirit, with a small mirror and cross level attached, and of the compass-box under the telescope; this is screwed on to the axis of the instrument with the parallel plates P PPP, and these again on to the tripod staff, the same as the theodolite. The spirit-level carries a diaphragm consisting of two vertical wires and one horizontal wire. The object of the small mirror, placed on the telescope and mounted on a hinge-joint when it is in use, is for the observer to be able to ascertain, at the same time that he is reading the levelling-staff through the telescope, that the instrument has remained steady in its position, by noticing in the mirror any alteration in the position of the bubble

THE DUMPY LEVEL.

267

with regard to the divisions engraved on the glass tube; the small cross level is to assist in setting up the instrument approximatively by means of the legs only, by which a great deal of time is saved, as well as wear and tear of the parallel plate screws. This cross level is always made much too short, the one we have attached to our level is six inches long, and we find it very useful in quickly setting up the instrument; a steady motion, when adjusting to distinct vision, is secured by the mill-headed screw in the side of the telescope. The little mirror we have mentioned is seldom used, but is nevertheless very useful in going over soft springy ground or in windy weather; in boggy ground it is particularly so, for here the slightest motion of the observer may disturb the instrument, which, by keeping the mirror in sight whilst observing, will detect the disturbance. The instrument is so constructed that the diaphragm is adjusted by means of the capstan-headed screw near the eye end of the telescope. The little lens attached to the compass-box is for reading off the graduations on the silver ring.

As we suppose the reader has already studied the construction of the theodolite in former pages of this work, and will therefore have no difficulty in fully understanding that of the level, we shall now give Mr. Gravatt's method of adjusting his level.

"To Examine and Correct the Collimation.

"On a tolerably level piece of ground drive in three stakes at intervals of about four or five chains, calling the first stake α, the second b, and the third c.

"Place the instrument half way between the stakes a and b, and read the staff A, placed on the stake a, and also the staff B, placed on the stake b; call the two readings A' and B'; then, although the instrument be out of adjustment, yet the points read off will be equidistant from the earth's centre, and consequently level.

"Now remove the instrument to a point half way between b and c. Again read off the staff B, and read also a staff placed on the stake c, which call staff C (the one before, called A, being removed into that situation). Now by adding the difference of the readings on B (with its proper sign) to the reading on C, we get three points, say A', B', and C', equidistant from the earth's centre, or in the same true level.

"Place the instrument at any short distance, say half a chain, beyond A, and, using the bubble merely to see that you do not disturb the instrument, read all three staffs, or, to speak more

correctly, get a reading from each of the stakes a, b, c: call these three readings A" B" C". Now, if the stake b be half way between a and c, then ought C"-C′-(A"-A') be equal 2 [B”B'-(A"-A')]; but if not, alter the screw which adjusts the diaphragm, and consequently the horizontal spider-line, or wire, until such be the case; and then the instrument will be adjusted for collimation.

"To adjust the spirit-bubble, without removing the instrument, read the staff A; say it reads A", then adding (A”—A') with its proper sign B' we get a value, say B".

"Adjust the instrument by means of the parallel plate-screws,

to read B" on the staff B.

"Now, by the screw attached to the bubble-tube, bring the bubble into the centre of its run.

"The instrument will now be in complete practical adjustment, for level, curvature, and horizontal refraction, for any distance not exceeding ten chains, the maximum error being only of a foot.

th

"Example. The instrument being placed half way between two stakes, a and b, (at one chain from each,) the staff on a or A' read 6:53, and staff on b or B' read 334, placing the instrument half way between the stakes b and c (three chains from each), the staff on b read 401, and the staff on c read 5.31. "Hence, taking stake a as the datum, we have

"The instrument being now placed at d (say five feet from a, but the closer the better), the staff on a or A" read 4:01, on b or B", 103, and on c or C", 3:07. Now, had the instrument been in complete adjustment (under which term curvature and refraction are included) when the reading on staff a was 401, the readings on b and c should have been respectively 0.82 and

2.12.

"The instrument therefore points upwards, the error at b being 0-21, and the error at c, 0.95: now, were the bubble only in error (as is supposed in all other methods of adjustment), the error at c ought to be four times as great as at b, but 4 × 0.21 = 0.84 only, there is an error, therefore, of 0.95 −0·84 = 0·11 not due to the bubble.

"For the purpose of correcting this error (and be it remembered contrary to former practice, for this purpose only), we must use the capstan-headed screw at the eye end of the telescope,

and neglecting the actual error of level, we are only to make the error at b one-fourth that of c.

"After a few trials, whilst the reading at a continued 4.01, the reading on b became 075, and that on c, 1.84.

"Now 0.82-075 0.07, and 2.12-1.84 0.28.

=

=

0.28, the telescope is now adjusted for

"All that remains to be done, is to raise the object end of the telescope by means of the parallel plate-screws, until the staff at c reads 2.12, and then to bring the bubble into the centre of its

run.

"The operation of collimating, when once performed upon levels on Mr. Gravatt's construction, will scarcely ever need being repeated."

The instrument may also be collimated by means of a pool of still water, if large enough to have a couple of stakes, three or four chains apart, driven exactly level with the surface of the water; the instrument should be set up about half a chain beyond one of the stakes; if the line of collimation is correct, the readings of the staff placed alternately on each of the stakes will be exactly the same; if different, the instrument must be collimated in the manner already described.

We must not omit to state that, in observing through the Dumpy Level objects are inverted, that is, everything is seen upside down. At first this has a disturbing effect on many persons; but after a few hours it passes entirely away, and we should now feel at least equally disturbed on looking through a level to perceive any other result; it is merely to read the staff downwards instead of upwards.

In almost all instruments it is very difficult, and sometimes, practically speaking, almost impossible to make the bubble maintain the centre of its run, whilst the instrument is turned completely round on its axis; that is, to use the technical expression, to make the telescope reverse; a tighter or slacker connexion between the telescope and the parallel plate portions of the instrument, will often be the cause of it. For this reason, therefore, and to prevent accidents, we must particularly impress on the mind of the student the necessity of always turning the telescope to the right when reversing it; if it were often turned to the left, not only will it reverse all the more incorrectly, but the telescope may get screwed off and fall to the ground. When the instrument will not reverse correctly, a mere touch on one of the parallel plate screws will bring the bubble back to the centre of its run.

Parallax is also to be carefully attended to when levelling; it

exists when on looking through the telescope at a distant object, and moving the eye sideways, or up or down, the wires appear to move. Generally a motion of the mill-headed screw in the side of the telescope, one way or the other, rectifies this, at the same time that it produces distinct vision; at the same time it brings out the wires clear and distinct against the divisions on the levelling staff.

In looking at our drawing of the Dumpy Level, it will be perceived that one of the parallel plate screws rests in a notch or groove on the lower plate.

When setting up the level for use, we prefer to have this particular screw over one of the legs of the tripod, and this leg we keep nearest to us in planting the instrument on the ground, at the same time we bring the eye end of the telescope towards us; so that in fact, it is at right-angles to the position shown in the drawing. We obtain with the legs an approximation of the bubbles to the centres of their runs; we then fix the two legs furthest from us firm in the ground, and by means of the third leg moved backward or forwards, or sideways, as the bubbles may appear to require, we get them so nearly in the centres of the tubes, that a touch or two on the parallel plate screws brings the whole into complete adjustment, as far as the bubbles are concerned. The next adjustment is for distinct vision, and to prevent parallax, which we need not repeat here.

The tripod we always prefer to have made of fir, instead of mahogany; stout, and six feet in length when closed for carriage; this length enables us to get a firm wide spread when set up for use, and with such a stand there is less to apprehend from the wind causing vibration of the instrument when observing; when high, however, its effects will be felt more or less; therefore in windy weather we avoid taking long sights as much as possible, ten or twelve chains for instance, or else watch for a lull in the wind to do so.

Levelling Staves.-The different descriptions and modifications of levelling staves introduced at different times since the old vane-staff has become obsolete, and which we shall therefore not describe, have been almost innumerable; that in use now was introduced by Mr. William Gravatt, and a description of its principle of construction will enable the reader to make any modification of it he may please. This staff varies in length from 14to 17 feet; it consists of three joints, and when stowed away for carriage, they are thrust down into each other, but drawn out when the staff is in use; the bottom joint is about four inches wide, the whole length is divided into feet, tenths, and hundredths; the two first are numbered, but not the latter,