The operation of taking out the differences of level at each reading and entering them in their respective columns, is termed casting out, in doing which the principal thing to remember each time is, that when the backsight is greater than the foresight, the entry is made in the column of Rise; and when the foresight is greatest, then the difference is entered in the column of Fall.

At all times the differences of the totals of backsights and foresights is equal to the differences of the totals of the columns of Rise and Fall.

By referring to the above readings between A and F, it will be found that all the

The above "tottings up" are checks on the correctness of the casting out.

If we now continue our levelling operations on to G, we shall find that G is six feet below A, the descending preponderating over the ascending levels.

=

As yet we have only ascertained the differences of level, and not measured any of the distances as from A to B, to C, &c.; we are not therefore in a condition to make a drawing of our work, that is, "plot the section;" but if at the same time that the levels are taken the distances from staff to staff are measured, we can then lay down the distances at which the differences of level occur; from A to B the distance is 33 chains, and B is 18 feet higher than A; on to C the distance is 33+ 26 59 chains, and the difference of level 32.75; the chaining is carried on simultaneously with the levelling operations, and we get on to D and E; the distance being measured up to every staff, and the difference of level ascertained, we can lay down the distance and corresponding difference of level on paper; up to D we add to the heights all the way; at E there is no difference of level from that at D; but from E we begin to descend, and the differences of level to be deducted all the way to G; or, instead of rising we drop all the way from E to G when plotting the section as well as in the levelling. With the ruled paper in our example, and the figures above given, the reader will have no difficulty in plotting this section; this method, however, is tedious and liable to error in the drawing, and in practice we arrange our figures so as to obtain such results that we are enabled to lay out all the differences of level as heights above a certain point. In our example we have assumed the point A as the point of departure

for distances and relative heights of level, and we may assume a line level with this point throughout the section from which to set off the heights, b B, c D, &c.; but on reaching G we shall have to set off a depth instead of a height: a negative instead of a positive quantity; to avoid this, it is the practice to assume this imaginary horizontal line at such a depth below the starting-point that no such inconvenience arises, and so that all the horizontal distances may be laid down on this line, and from the distances all the levels plotted as heights above this line, which is called Datum, or the given line to the level of which all the levels throughout the section are relative. It may be assumed at 10, 100, 200, or any other convenient depth below a point from which the levels commence, which point should always be of a permanent character, and thoroughly well-defined, so as to admit of no mistake at any time of future reference. The arrangement of the figures which we have mentioned, so that the levels may all refer to this line, is termed reduction of levels, and in our example the whole will be tabulated in the following form, assuming our Datum to be 18 feet below our startingpoint.

In plotting the section, only the first column of distance and the last of reduced levels are referred to-the first for obtaining the horizontal lengths, and the last the vertical heights.

The datum line might be assumed at any height above instead of below the starting-point, but we are not aware that it is ever made so. If it so happens that levelling operations fall in a hilly country, no information about the levels of which is to be obtained, and where perhaps several lines of level are contemplated to ascertain the relative merits of different lines, some at a high and others at a low level, it is often as well to assume

datum as 1000 feet below the starting-point; it is easy to take off 500 feet from the reduced levels if afterwards desired.

We must now make a few observations on the above method of keeping a field-book and the corresponding section.

The distances apart are all a certain number of whole chains, which is not likely to be the case on the ground, where not only differences of level between any given number of points are to be ascertained, but for the preparation of sections, all the undulations along the surface, so that this latter may be truly represented by the line of surface on the section; and in order to make any small differences of level the more apparent on the paper, it is customary to plot the vertical heights to a much larger scale than the horizontal distances; thus a scale of 80 feet to the inch is often used for plotting the vertical lines, whilst one of 6 chains to the inch is used for the horizontals, and again scales of 30 or 40 feet to the. inch vertical against scales of 3 and 4 chains horizontal.

Whilst on this subject, we will refer to the ruled paper on which Fig. 132 is drawn; it is called section-paper, and was introduced by Mr. Brunel some years ago; it is not in general use except in his office; but it certainly has many advantages over plain paper, inasmuch as there is no necessity of drawing a vertical line at right angles every time to the datum on which to plot the reduced levels, because it is already done; neither is there any plotting of distances, as these are also marked on the section-paper according to the scale employed. The principal difficulty there is to contend with in using it is when joining the sheets end to end, to make the horizontal lines coincide, and to get them to lay in straight lines; of course scales are not required with this paper, as it is already done to hand. When instead of the section-paper, the plain ordinary paper mounted on linen is used, the datum line is first drawn in, and from the "cockling" of the paper, often occasions some trouble to get straight; on this line the horizontal lengths are first marked off in miles and furlongs, and the same written in; after this the chain distances where the levelling staves were planted, from which verticals are drawn up from which to plot the reduced levels.

Still referring to Fig. 132, it will be seen that the distances A B and B C are respectively 33 and 26 chains in length; and that the backsight 1 A 25 chains long, almost too great a distance to read with accuracy, and requiring the instrument to be in a very perfect state of adjustment; besides this, however, where correct levelling is required, it will be necessary to take the correction for curvature and refraction into consideration.

INTERMEDIATE SIGHTS.

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In the tables for corrections of curvature and refraction, 20·00 chains is the highest figure in the column of distances for chains, the correction being as the square of the distance; take the correction due to 5 chains, which is 00224, and multiply by 5225; the product will be '056, which will be the correction to apply to the reading 18:00 from 1 to B, which will reduce it to 1794; the same method of proceeding may be applied to the backsight 2 B.

But by always setting up the instrument about midway between the staves sufficient compensation for practical purposes is made, without any other reference to the subject; by this midway setting up, any little mechanical defects in the level, either from construction or from its being out of order, is remedied so far that we need not be troubled by them; by this, however, it is not to be understood, that by midway we mean exactly half the distance between the staves, because any such close approximation would in practice be out of the question; what is really meant is, that when the distance between two staves is very great, it is not desirable to set up the instrument so that there shall be an extreme disproportion in the distances. between it and the staves.

Intermediate Sights.--Instead of taking the levels along the previous surface A, B, C, D, . . . . to G, let us consider it as required to take the levels for a section along the line A, B, c, d, e, f, g, h, i, F. The level of B we have already found to be 36 feet above datum; we set up the level at 11, and the staff is planted at B, and we get for backsight 1.50; the staff is moved on to c, and we read for foresight 7.50, which gives a fall of 6 feet, and distance 61.00. Now to obtain the difference of level between c and d, we might set up the level between, and so consecutively along the broken ground between c' and i; but we may save much time under such circumstances by following another course. Now let it be carefully observed, that what is called backsight may just as well be termed first sight, and that foresight may equally well be termed second sight, for they mean one as well as the other; therefore, instead of moving the level from 11, after reading the foresight c, let the staff be moved on to d, and the reading taken from 11, which will be 13.50; the difference between the readings at c and d is equal to 6:00, which is the difference in the levels as may be seen on the section; then the reading, 7.50, which was formerly fore or first sight, has become back or second sight; let it be entered as such in the column foresight of the level-book, and the difference is a fall, for the foresight is greater than the backsight. Let the staff be moved on to e, and read off still from 11; this

reading will be 4.50; but 13:50, the last foresight, now in its turn becomes backsight, and the difference, which is a rise-for the foresight reading is less than the backsight-is equal to 9 feet. We may proceed exactly in the same manner on as far as i, after which the level may be fresh set up where required, between and F, to continue the levels onwards; from A the level-book will read in the following manner.

The above is the theory and practice of "Intermediate sights;" it is merely necessary to remember that the figures last read as foresight become the backsight to the next reading, as c the foresight to B becomes backsight to d, and d the foresight to c becomes the backsight to e, and so on as far as i, or as long as the levelling instrument is not fresh planted. In practice it is of eminent service, as in taking levels over broken ground, across brooks, canals, rivers, or to obtain the details of the surface of a road, or indeed any other details as to levels when they are not far removed from each other; by following it out, the time saved in fresh setting up of the instrument is often very considerable, and the student will do well first to impress the theory on his mind and then practise it.

Bench Marks.-This is another subject which must be well understood before any levelling operations are undertaken. In Fig. 132, and according to what we have said before, we have commenced our levels from Low-water mark, and if in doing so we had no other defined mark than the surface of the water, we might never probably be able to find it again for future reference, as for instance, to check the accuracy of our levels, or to level from the same point for another section, for low-water may and does alter from day to day; but if we have started from a point permanently fixed, as for instance, a particular landing, or any other permanently established level, known as low-water