portional dimensions. But when the observations are numerous, they should be kept in a tabular form, such as that which is given below. The names of the points, or "Stations," whose heights are demanded, are placed in the first column; and their heights, as finally ascertained, in reference to the first point, in the last column. The heights above the starting point are marked +, and those below it are marked The back-sight to any station is placed on the line below the point to which it refers. When a back-sight exceeds a fore-sight, their difference is placed in the column of "Rise;" when it is less, their difference is a “Fall.” The following table represents the same observations as the last figure, and their careful comparison will explain any obscurities in either. The above table shows that B is 4 feet below A; that C is 5 feet below A; that E is 1 foot above A; and so on. To test the calculations, add up the back-sights and fore-sights. The difference of the sums should equal the last "total height." Another form of the levelling field-book is presented below. It refers to the same stations and levels, noted in the previous form, and shown in Fig. 420. Stations. Distances. Back-sights. Ht. Inst. above Datum. Fore-sights. Total Heights. In the above form it will be seen that a new column is introduced, containing the Height of the Instrument (i. e., of its line of sight), not above the ground where it stands, but above the Datum, or starting-point, of the levels. The former columns of "Rise" and "Fall" are omitted. The above notes are taken thus: The height of the starting-point or "Datum," at A, is 0.00. The instrument being set up and levelled, the rod is held at A. The back-sight upon it is 2.00; therefore the height of the instrument is also 2.00. The rod is next held at B. The fore-sight to it is 6.00. That point is therefore 6.00 below the instrument, or 2.00 — 6.00 ——4.00 below the datum. The instrument is now moved, and again set up, and the back-sight to B, being 3.00, the Ht. Inst. is 4.00 +3.00=- 1.00; * In the figure, the limits of the page have made it necessary to contract the horizontal distances to one-tenth of their proper proportional size. and so on the Ht. Inst. being always obtained by adding the back-sight to the height of the peg on which the rod is held, and the height of the next peg being obtained by subtracting the fore-sight to the rod held on that peg, from the Ht. Inst. The level lines given by these instruments are all lines of apparent level, and not of true level, which should curve with the surface of the earth. These level lines strike too high; but the difference is very small in sights of ordinary length, being only one-eighth of an inch for a sight of one-eighth of a mile, and diminishing as the square of the distance; of the distance; and it may be completely compensated by setting the instrument midway between the points whose difference of level is desired; a precaution which should always be taken, when possible. It may be required to show on paper the ups and downs of the line which has been levelled; and to represent, to any desired scale, the heights and distances of the various points of a line, its ascents and descents, as seen in a side-view. This is called a “Profile." It is made thus. Any point on the paper being assumed for the first station, a horizontal line is drawn through it; the distance to the next station is measured along it, to the required scale; at the termination of this distance a vertical line is drawn; and the given height of the second station above or below the first is set off on this vertical line. The point thus fixed determines the second station, and a line joining it to the first station represents the slope of the ground between the two. The process is repeated for the next station, &c. But the rises and falls of a line are always very small in proportion to the distances passed over; even mountains being merely as the roughnesses of the rind of an orange. If the distances and the heights were represented on a profile to the same scale, the latter would be hardly visible. To make them more apparent it is usual to "exaggerate the vertical scale" ten-fold, or more; i. e., to make the representation of a foot of height ten times as great as that of a foot of length, as in Fig. 420, in which one inch represents one hundred feet for the distances, and ten feet for the heights. The preceding Introduction to Levelling has been made as brief as possible; but by any of the simple instruments described in it, and either of its tabular forms, any person can determine with sufficient precision whether a distant spring is higher or lower than his house, and how much; as well as how deep it would be necessary to cut into any intervening hill to bring the water. He may in like manner ascertain whether a swamp can be drained into a neighboring brook; and can cut the necessary ditches at any given slope of so many inches to the rod, &c., having thus found a level line; or he can obtain any other desired information which depends on the relative heights of two points. To explain the peculiarities of the more elaborate levelling instruments, the precautions necessary in their use, the prevention and correction of errors, the overcoming of difficulties, and the various complicated details of their applications, would require a great number of pages. This will therefore be reserved for another volume, as announced in the Preface. ANALYTICAL TABLE OF CONTENTS. PART I. GENERAL PRINCIPLES AND FUNDAMENTAL METHODS. CHAPTER I. Definitions and Methods. (21) Pins 19 30) Rods. 24 (22) Staves 24 (23) How to chain 19 25 } (24) Tallies 25 19 32) Measuring-wheel (33) Measuring Angles CHAPTER III. Drawing the Map. CHAPTER III. Surveying by Perpendiculars. To set out Perpendiculars. Diagonals and Perpendiculars. Offsets. 69 114 1) Taking offsets. 75 (123) Field work. Erroneous rules Reducing to one triangle CHAPTER V. Obstacles to Measurement in Chain Surveying. (138) The obstacles to Alinement and Measurement. Problems on Perpendiculars. (140) PROBLEM 1. To erect a perpendicular at any point of a line......... (143) 2. when the point is at or near the 96 96 97 98 4. To let fall a perpendicular from a given point to a given line 99 (160) PROBLEM 1. To run a line from a given point parallel to a given line. 102 (165) 2. Do. 103 |