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.0215; for 50 stations, 0.5375; for 10 miles, 60 feet, and a spire Dr treetop apparently level with the instrument at that distance would really be 60 feet above it. Transposing the equation we have D=√A÷0.6. In this form it is applicable to the determination of distances at sea. The Peak of Teneriffe, for example, 16,000 feet high, should be just visible from the sea-level at a distance=√16000 ÷ 0.6

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say 163 miles.

10. TO FIND DIFFERENCES IN ELEVATION BY MEANS OF THE BAROMETER.

Call the required difference D; the barometrical reading at the lower stand, L; that at the upper stand, U.

Then, D = (L− U) ÷ (L + U) × 55000.

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And 0.1226 X 55000=6743, the required difference of elevation in feet.

11. A closer approximation is thought to be attainable by using a thermometer in connection with the mercurial barometer. In that case, having found the difference as above, add

of the result for each degree by which the mean temperature of the air at the two stands exceeds 55°; subtract the like proportion if the mean temperature be below 55°. When the upper thermometer reads highest, for "subtract" say "add,” and vice versa in the foregoing rule.

12. The naked formula, however, will usually be sufficient for the engineer. He can prescribe gradients by it for surveys, which shall develop the ground to be occupied, and can decide between summits well differenced in height. If not so differenced, questions of detour, of approaches, and the like, will contribute to determine the expediency of making an instrumental examination.

13. HEIGHTS BY THE THERMOMETER.

T= the difference, in degrees Fahrenheit, between 212°, the temperature of boiling water at the sea level, and that at th place of observation.

H = the height of place of observation above or below the sea in feet.

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1. Like swallowing, this is more easily done than described. To no detail of field service does the proverb more fitly apply, that "work makes the workman."

2. The problem is, to find where a formation slope of given inclination, beginning at the side of the road-bed, must needs intersect the ground surface. Formation slopes are usually stated in parts horizontal to one part vertical. Thus a slope of 45° is "1 to 1." A slope of "2 to 1" has a horizontal reach of two feet to each foot vertical. The carriages of a stairway with twelve-inch treads and eight-inch risers would have a slope of "1 to 1."

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3. To fix the point where any proposed formation slope must meet the surface on level ground, is quite simple; the distance from the centre line being obviously made up of half the width of road-bed added to the horizontal distance due from the slope, to the depth of cut or height of fill. Thus, with 20 feet road-bed, 9 feet cut, and slope of 1 to 1, the distance out would be 10+ 9 +44 23 feet, as shown in the annexed diagram.

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4. On slant or broken ground, the solution is more difficult: recourse must then be had to the level, with a rodman, a tape

Example No. 1.

5. Let the accompanying figure represent the cross-section at any given point of a proposed excavation; road-bed 20 feet wide, cutting at centre stake 12 feet, and formation slopes 1 to 1.

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6. The first step is to set the level, as at A, commanding, let us suppose, the lower slope, and to ascertain its height above grade at the proposed section. This is usually done by reference to the nearest bench, and pegging from stake to stake as the work progresses. Unless the ground is very steep, and the slope-stakes largely different in elevation, labor will be saved and likelihood of error reduced by levelling over the centre line beforehand, as a separate job, and marking on centre stakes the cuts, fills, and grade points, that is to say, the points where excavation passes into embankment. The rods should be taken carefully at the stakes, and the latter marked on their backs to the nearest tenth, as “grade," "C_12,” signifying cut 12 feet, or "F 6.2," signifying fill 6.2 feet, for example. This being done, each centre stake serves as a benchmark for slope staking at that section, and each cross section can be staked out independently.

7. Instrument height, in the example treated, being by either method fixed at 15.5 above grade, the next step is a guess how far out from the centre stake the formation slope would probably meet the ground surface. The closeness of the guess will correspond to the experience and natural skill of the leveller: the young engineer should not be discouraged if he misses t

S. It is true, that, on a uniform declivity, he might aid conjecture by taking a rod distant half the width of road-bed, or 10 feet, from the centre stake, ascertain thus the slope of the ground surface as well as the cutting at that point; and with these data, knowing also the formation slope, approximate the point sought by solving the terminal triangle of the proposed section, indicated by dotted lines in the figure. But, in practice, he will find it the quicker and better way to approximate the point by vividly imagining the underground forma tion lines; or by vividly imagining a level section, the upper surface of which shall coincide with his instrument height, 15.5 feet above grade. This gives him a point in the air, 10+ 15.5: 25.5 feet out from the centre stake, level with the instrument, as the limit of the imaginary section; and from that point he can pretty well judge where a line corresponding to the formation slope must meet the ground.

9. Suppose him, by either method, or even by random guess. to think that 10 feet for half the road-bed, and 10 more for the slope, looks about right. The formation slope being 1 to 1, this implies a cutting of 10 feet at the side stake, and a rod, therefore, of 15.5 - 10.0 5.5 feet. = Taking a rod accordingly, 20 feet out, measured horizontally from the centre stake, he finds it to be 11.0 instead of 5.5, indicating that he has gone too far down hill. Let him now reason that the rod of 11.0 corresponds to a cutting of 15.5 11.04.5 feet, and that a cutting of 4.5 feet corresponds to a distance out of 10 +4.5 = 14.5 feet. Try, then, a rod 14.5 feet out. It proves to be 9.0, corresponding to a cutting of 15.5 9.0 6.5, instead of 4.5 feet, and a distance out of 16.5 instead of 14.5 feet. Try, next, 16.5 feet out; the rod there, of 10.0 instead of 9.0, shows him again to be in error on the down-hill side of his object; but the lessening error shows also that he is approaching it, and that a few more like trials will reach it.

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10. Recurring to his first error with the 11.0 feet rod, he cannot fail to observe after a little practice, since the ground ascends thence toward the centre line, that the side stake must fall farther out than the point where his second trial was made; that its true position, in fact, divides the distance between those points of observation into two parts which are to one another directly as the inclinations of the formation slope and the ground surface. By degrees he will grow skilful in

observation, will place a slope stake on the second or third trial, without conscious effort of mind.

11. Next, suppose the level at B, 25.5 feet above grade, commanding the upper slope.

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Note the change of ground 11 feet out, and take a rod there, recording the observation. The cutting at that point is 25.5-9.5 = 16 feet, corresponding to a distance out for the side stake of 10+ 16 26 feet, if the ground were level. A trial rod 26 feet out reads 7.8, corresponding to a cutting of 25.57.8= 17.7 feet, and a distance out for the side stake of 10+ 17.7=27.7 feet, showing that the point sought is still beyond. A repetition of such trials will finally fix it; but, as in the case of the lower slope, practice will speedily lessen the number and abridge the labor of them.

12. The foregoing section would be noted in the field book as follows:

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13. In the annexed figure, representing an embankment 14 feet wide on top, with side slopes of 14 to 1, the first thing to attract attention is that the instrument is 1 foot below grade,

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and that, therefore, 1.0 is to be added to all rods, in order to find the height of embankment above the points at which rods are taken.

14. Consider the down-hill side. The engineer, with the

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