spectively added to, or subtracted from, the assumed distance of the datum line, and the next rise or fall again added to, or subtracted from, the sum or difference :-thus 2.15, being a fall, is subtracted from 100 (the assumed distance of the datum line) leaving 97.85 feet, the height of the ground at b; the next fall 6.75 is then subtracted from 97.85, leaving 91 10 feet for the height at c; and so on to 3.53, which is the last fall :-the next 6.57, being a rise, is added, as well as 9.93;-thus the last reduced level is 90.79 feet, which taken from the datum 100 leave 9.21 feet, agreeing with the differences of the sums of the back and fore sights, and of the sums of the rises and falls, and showing the work of casting to be correct. Thus are obtained a series of vertical heights to be set off perpendicularly to the datum line, through the upper extremities of which the sectional line must be drawn. PRACTICAL LEVEL BOOK. (Datum line 100 feet below the bench mark at A.) In laying down the sectional line from the above columns of reduced levels and distances, the former are always taken from a much larger scale than the latter, otherwise the undulations on the surface of the ground would in many cases be hardly perceptible. Draw the horizontal line A G, setting off the distances A B, A C, &c., as in the column of distances, that is A B = 4.60 = chains, A C 7.80, &c., then draw A a = 100 feet, perpendicular to A G and parallel to A a draw B b, C c,. &c., setting off their heights 97.85, 91.10, &c., respectively, from the column of reduced levels, and through the points a, b, c, &c., draw the required sectional line a g. NOTE. The above operations, though extremely simple, require great care, otherwise, in extensive works of this kind, errors creep in imperceptibly, to check which the agreement of the differences in the level book is essential. LEVELS FOR THE FORMATION OF A SECTION. In this case it is required to take the levels of a line of country, where the ground plan is already made, and the line of section determined upon, and marked out on the plan. Here, in addition to what is required in running or check levels, the distances to the several stations of the levelling staves from the starting point must be measured. Two additional assistants are required in this case to measure the distances of the stave stations along the lines while the operation of levelling goes on, which is the same in every respect as that already described, excepting that, in this case, the operation is conducted upon a line, on the surface plan, a copy of which must be in the surveyor's possession to direct him, and the distances of the several stave stations must be noted in the level book, in the column marked "Distances." The following is the level book of an example, showing the manner of keeping it, and also the method of reducing the levels, to obtain the actual heights of each station above the datum line, which is placed 100 feet below the starting point, for convenience of drawing the section. The whole operation being similar to that already given at page 123, excepting that here we give the particular manner of performing the several parts of the field work, in order that it may be clearly understood by those who are unacquainted with the subject, as it is presumed that, in a short time, railways will become the common means The several differences of the sums of the back and fore sights, of the sums of the rises and falls, and of the last reduced level and the datum, exactly agreeing, proves the accuracy of the arithmetical operation in the preceding level book, all these differences being 49.33 feet, which is the height of the last station above the first. It is advisable for the surveyor to reduce the levels in the field as he proceeds, as it will occupy very little time, and can be easily done while the staffman is taking a new position. The surveyor will thus be enabled to detect with the eye if he is committing any glaring error; for instance, inserting a number in the column of rises, when it ought to be in that of falls, the surface of the ground at once reminding him that he is going downward instead of ascending. It is seldom the case in practice that the instrument can be placed precisely equi-distant from the back and fore staves, on account of the inequalities of the ground, ponds, &c.; it would appear, therefore, to be necessary, to make our results perfectly correct, to apply to each observation the correction for curvature and refraction as explained at page 118: this, we believe, is seldom done unless in particular cases, where the utmost possible accuracy is required, on account of the smallness of such correction, as may be seen by referring to the table at the end of the book, where this correction for 11 chains is shown to be no more than part of a foot; and as the difference in the distances between the instrument and the fore and back staves can in no case equal that sum, it is evident that such correction may be safely disregarded in practice. Besides, it is not necessary to have the level placed directly between the staves while making observations, as it is frequently inconvenient to do so, for reasons just given, nor does a deviation from a line of the staves, in this respect, in the least affect the accuracy of the result. The distances in the sixth column of the level book are assumed to be horizontal distances, and in measuring them, care should be taken that they are as nearly such as possible, or they must be afterwards reduced thereto, otherwise the section will be longer than it ought to be. For the purpose of assisting the surveyor in making the necessary reduction from the hypothenusal to the horizontal measure, when laying down the section, a table is given in Baker's Land and Engineering Surveying, page 146, shewing the reduction to be made on each chain's length for the several quantities of rise, as shewn by the reading of the staves. NOTE. For extensive information on this subject see Baker's Land and Engineering Surveying, where an engraved plan and section, adapted to this example, are given at the end of the work. THE METHOD OF LAYING OUT RAILWAY CURVES ON THE GROUND. In railway practice, the curve adopted is always an arc of a circle, to which the straight portions of the railway are tangents at each extremity of the arc. Sometimes the curve consists of two, three, or more circular arcs with their concavities turned in the same or different directions, as in the compound and serpentine curves. PROBLEM I. To lay out a railway curve on the ground by the common method. CASE I.—Let H A, 14 15 be the tangental portions of a railway, the extremities A and 14 of which are required to be united by the circular curve A 4, to which H A, 95 94 shall be tangents; the radius of the curve being supposed to be previously determined. = 1 =pq, which set off at right angles to A p, thus giving the first point in the curve. In the direction A q, measure q P2 chain, and set off P2 = twice pq 4.95 × 2 = 9.9 inches, at right angles to q P2; then 7, is the second point in the curve. This last operation must be repeated till the curve shall have been set out to the point q4. Lastly 94 P5 being measured = 1 chain, in the direction 93 94, the offset p5 95 will be found 4.95 inches the first offset p q, thus proving the accuracy of the work. In this manner the operation is conducted, whatever be the length of the curve. CASE II. Let A Or, and 8 = A p = q p2 &c., which may be either less or greater than one chain; then the general 8 2 length of the first and last offsets P 9, P5 P5 is and the " 2r length of each of the other offsets is 82 or twice the first or , |