the Y level, already described. The telescope is attached to a horizontal bar in a similar manner as Troughton's level, but room is just left between the telescope and bar for the compass box. A cross level k is placed upon the telescope at right angles to the principal level 77, by which we are enabled to set up the instrument at once nearly vertical. A mirror m mounted upon a hinge-joint is placed at the end of the level 77, so that the observer, while reading the staff, can at the same time see that the instrument retains its proper position-a precaution by no means unnecessary in windy weather, or on soft spongy ground. The telescope is attached to the main bar by capstan-headed screws B B, as in Troughton's level, by which the line of collimation is set perpendicular to the vertical axis; and the instrument is set upon parallel plates, &c., like the theodolite and the levels already described. This level is much preferred and used by many engineers. (6.) After having treated of the more perfect levelling instruments, it will now be proper to describe the water level, a very simple instrument, adapted to give a rapid delineation of any section of the earth's surface, where very strict accuracy is not required. It can be made by any workman, will cost but a few shillings, and requires no adjustment when using it. It is greatly to be recommended to farmers for determining the levels for draining their lands. "ab is a hollow tube of brass, about half an inch in diameter, and about three feet long; c and d are short pieces of brass tube of larger diameter, into which the long tube is soldered, and are for the purpose of receiving the two small bottles e and f, the ends of which, after the bottoms have been cut off, by tying a piece of string round them when heated, are fixed in their positions by putty or white lead; the projecting short axis g works in a hollow brass cylinder h, which forms the top of a stand: but it may be made in a variety of ways, so as to revolve on any light portable stand. The tube when required use, for is filled with water, coloured with lake or indigo, till it nearly reaches the necks of the bottles, which are then corked for the convenience of carriage. On setting the stand tolerably level with the eye, these corks are both withdrawn, which must be done carefully, and when the tube is nearly level, otherwise the water will be ejected from one of the bottles; and the surface of the water in the bottles, being necessarily on the same level, gives a horizontal line in whatever direction the tube is turned, by which the vane of a levelling staff may be adjusted." LEVELLING STAVES. 6.) The best constructed levelling staff (Gravatt's) consists of three parts sliding one within another, and, when opened out for use, forms a staff 17 feet long, jointed together something after the manner of a fishing rod. The whole length is divided into hundredths of a foot, alternately coloured black and white, and occupying half the breadth of the staff; but for distinctness the lines denoting tenths of feet are continued the whole breadth, every half foot or five tenths being distinguished by a conspicuous black dot on each side, the whole feet being numbered with the figures 1, 2, 3, &c. CORRECTION FOR CURVATURE. B D E (7.) Let B D E be a horizontal line, that is, such as would be given by the line of sight of a level, properly adjusted; BCF an arc of a great circle of the earth, and A its centre. It will at once appear from the figure, that the heights DC, EF, of the apparent level B E, above the true level increase successively from the point B. The height EF of the apparent level above the true, is equal to the square of the distance BE divided by twice the earth's therefore the corrections for curva ture, D C, E F, &c., vary as the squares of the distances BD, BE, &c., since 2AB is a constant quantity. Taking the earth's radius to be 3956 miles, and assuming the *The demonstration of this formula is given in my edition of Nesbit's Surveying, p. 348. = 7913 of a mile = 8·007 distance B D to be 1 mile, then the correction for curvature DC BD÷2 A B = 12 ÷ 7913 inches nearly 8 inches. If the distance BE the correction EF BE÷2 AB=7913 or more than 6 feet. = = = 3 miles, then 72.0637 inches, Let any distance B D = d in miles, and the correction for curvature for 1 mile be taken = 8 inches = of a foot, which it is very nearly; then 2 d2 correction= feet, 3 (8.) The effect of the earth's curvature is modified by another cause, arising from optical deception, namely, refraction; the correction for which varies with the state of the atmosphere, but it may generally be taken at of the correction for curvature, as an average; and since refraction makes objects appear higher than they really are, the correction for it must be deducted from that for curvature. EXAMPLES. 1. Required the correction for curvature and refraction, when the distance of the object is 2 miles. = 4.166 cor. for curvature. of which is....... Difference..... .... 595 cor. for refraction. 3.571 feet, cor. required. 2. Required the correction, as in the last example, when the distance is 60 chains. 602800 = 4.5 cor. for curvature 643 cor. for refraction. Difference... 3.857 inches, cor. required. 3. From a point in the Folkstone road, the top of the keep of Dover Castle was observed to coincide with the horizontal wire of a levelling telescope, when adjusted for observation, and therefore was apparently on the same level; the distance of the instrument from the castle was 44 miles, required the correction for curvature and refraction, that is, the true height of the keep of the castle above the point of observation. See also the tables for these corrections at the end of the book. PRINCIPLES AND PRACTICE OF LEVELLING. To find the differences of the levels of several points on the surface of the earth. (9.) Before entering on this subject, it will be proper to state that the corrections for curvature and refraction, already explained, are seldom applied in the practice of levelling, the spirit level being usually placed midway between the stations, the levels of which are to be observed, hence the resulting corrections for each station are equal, and therefore the difference of the levels at the two stations is as truly shewn by the difference of the readings of the two staves fixed thereon, as if the corrections had been made. Thus the trouble of making these corrections is avoided by simply placing the instrument midway between the two staves. (10.) Let it be required to find the difference of level between the points A and G.-A levelling staff is erected at A, the instrument is set up and adjusted at B, another staff is also erected at C, at the same distance from B that B is from A, as nearly as can be judged by the eye; the readings of the two staves are then noted; the horizontal lines, connecting the staves with the instrument, represent the visual ray or level line of sight. The instrument is then conveyed to D, and the staff that stood at A is now removed to E, the staff C retaining its former position, only its graduated side turned to the instrument, and from being the fore staff at the last observation, it is now the back staff: the reading of the two staves are again noted, and the instrument removed to F, and the staff C to the point G, the staff at E retaining its position, now in its turn becomes the back staff, and so on to the end of the work, which may thus be continued to any extent. The difference of the readings of the staves at A and C will shew the difference of level between the points or stations A and C, because the visual line of the instrument is virtually level, and the same is true with respect to every two consecutive stations. The fall from A to C ............ 1.12 difference. Because when the front reading is the greater the ground falls and vice versa. The rise from C to E............ 4.20 difference. 1.12 The rise from A to E ..... .... 3.08 difference. Because the rise from C to E is greater than the fall from A to C, their difference shews the total rise. Back sight on staff E Fore sight on staff G ......... The fall from E to G ............ 7.62 0.54 difference. This fall taken from the rise from A to E, that is, The difference of the sums of the back and fore readings of the staves, will more readily give the difference of level between A and G: thus, |