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Extreme accuracy in the contours need not be attempted. Note the courses of streams, ravines, and ridges, the average slopes at frequent intervals, and, on irregular ground, make illustrative sketches to aid in utilizing the other notes. Practice gradually teaches how to observe critical points intelligently, and to record them briefly. In valleys or plains, where the location indicated is made up of long tangents and easy curves, little detail is required; but on bluffy, tortuous ground, with unavoidable divides to overcome, and long reaches of maximum gradient to be fitted, the method by contours is not only the simplest and clearest way of compiling necessary information, but is an aid to the engineer in projecting the right line, which no substitute can fully replace.

5. The writer is forced by the strong constraint of experience to differ on this subject with Mr. Trautwine. The difference, however, is a permissible one, and implies no lack of grateful respect for that veteran engineer, whose books are our handy-books, and to whose genius we are all debtors.

6. Having made the map, with ten-foot contours, suppose, for example, that a continuous gradient five miles long is to be located. Spread the dividers to 500 feet by the scale, start at the foot of the ascent, and step up, complying with the general trend of the ground, to the summit. This needful preliminary gives about the distance you have to work on, which cannot in many cases be derived from the experimental line directly. The profile furnishes the height to be overcome; and you are thus prepared to assume a summit cut, and determine the gradient. Having adopted one, say, of 66 feet per mile, observe that this rises five feet in 400 feet. Spread the dividers, then, to 400 feet by scale, and stand one leg on or near the summit, at a point corresponding to a five or ten unit in the elevation of the gradient. That is to say, if the grade elevation at the summit be 362, for instance, stand the leg of the dividers a little beyond or a little short of the summit, at a point where the grade elevation is 365 or 360. Thence, exercising good judgment to conform in a general way to what the location ought to be, and to make no angular indirections which cannot be closed with the maximum curvature, step forward down the incline. Name each step mentally as it is made, 355, 350, 345, 340, &c., and spot at the same time with a penciling to it in elevation. Connect the pencil-marks with a faint dotted line.

7. Were the ground a straight, regular hillside, the steps would be made directly from contour to half-space, thence to the next contour below, and the dotted line would mark out a tangent conforming exactly to the ground surface. On devious slopes, rounding within the limit of the sharpest permissible curve, the same exact conformity could be obtained, if desired, and a grade-line laid down which should require the least possible expense in building. On irregular, winding ground, an approximation only to the dotted line can be made: it is nevertheless a guide to go by; and, the more nearly the location project approaches it, the lighter will the work of construction be. The dotted line, in short, is analogous to a profile; and the engineer can prescribe his cuts and fills with reference to it, by means of curve or tangent, just as on the profile he does the same by means of grade-lines. A fairly correct map will enable him to construct a profile from the project, and to amend its errors without the trouble and expense of tentative field-work. The writer's habitual practice has been to base his preliminary estimates on a profile thus deduced from the map; and he recommends the practice to others. They will be surprised to observe the likeness between such a profile, tolerably well done, and that of the subsequent location.

8. It is a good custom, and one which cannot prudently be neglected where long reaches of maximum gradient are encountered, to “slacken grade on the curves. In making this adjustment, the contour map is exceedingly useful. An approximate project is first required, in order to determine the curvature, and, from that, the varying gradient. The location can then be laid down on the map with satisfactory precision. Opinions differ as to the right allowance per degree of curvature, and no experiments seem to have been made from which to deduce an authoritative rule. Some say 0.025 per degree per 100 feet; others, 0.05; others, variously between the two. Probably 0.05 is the safer rate. This amounts to 2.64 feet on a mile of continuous one-degree curve, and makes a nine-degree curve, about the curve of double resistance at ordinary passenger speeds.

9. In projecting locations, the better way generally is to 10. The following tables may be of assistance. It was needful, calculating them at all, to calculate them right; but of course such exactness as the figures would seem to indicate is unattainable in practice.

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11. TABLE SHOWING THE DISTANCE, D, IN FEET, AT WHICH

A STRAIGHT LINE MUST PASS FROM THE NEAREST
POINT OF

ANY CURVE, STRUCK WITH RADIUS 1, IN
ORDER THAT A TERMINAL BRANCH HAVING RADIUS
R=2 r, AND CONSUMING A GIVEN ANGLE, X, MAY
MERGE IN SAID STRAIGHT LINE.

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If R= 1} r, use half the tabular distance; if R=3 r, use twice the tabular distance; if R = 4r, use three times the tabular distance, and so on.

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12. TABLE SHOWING THE DISTANCE, d, IN FEET, AT WHICH

CURVES OF CONTRARY FLEXURE MUST BE PLACED
ASUNDER IN ORDER THAT THE CONNECTING TANGENT,
T, MAY BE 300 FEET LONG.

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Examples. A 7o and 49 should be 19.9 feet asunder; a 5o and go should be 25.07 feet asunder.

As approximations, for a connecting tangent 400 feet long, take twice the tabular distance: for a connecting tangent 200

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