tributaries at first descending into a lower country than that at their sources, encounter rising ground, which prevents their progress in that direction, and turns them off towards their principals.

These examples will suffice to point out the nature of the investigation to which an ordinary map should be subjected. A good topographical map, exhibiting contour of ground, would of course be a far better guide. Conflicting points of passage of a ridge may be compared by means of an altitude and azimuth instrument or theodolite; by first levelling the instrument, elevating the telescope till the line of sight passes through one of the points in question, and then turning the instrument in azimuth till the line of sight passes by the other point in question; if it passes above, the latter point is lower than the former, and vice versâ.

Next, a personal reconnoissance of the country should be made, and information sought from the inhabitants as to the nature of the ground, convenience for obtaining constructing materials, the mineral and agricultural wealth of the region, etc., etc. Three or four routes may thus be selected, one or other of which is to be finally decided upon by a rough survey of them all. This survey is conducted with the compass and chain, and level. The former instruments furnish a plot of the route by the method pointed out at p. 238, and the latter, a continuous profile or longitudinal section of the ground along the whole route, as seen at p. 251. A comparison of the compass plots of the different routes wil! determine which is the shortest, in a horizontal direction, and a comparison of the profiles will show which presents most elevation and depression to be overcome. The route being thus finally selected, it must be surveyed with care, another column being added to the field book, entitled grade, the numbers in which express the height of the roadway or bottom of the canal above the plane of reference at the same points of the route, for which the column of heights expresses those of the natural ground. The numbers in the column of grade will depend upon the elevation or depression of the natural ground, and the slope which the construction permits, that of a common road being greater than that of a railroad, and the latter being greater than that of a canal. The determination of these numbers will require an exercise of eye and judgment. A prime object to be had in view is the equalization of excavation and embankment; i. e., the grade should be so adjusted to the natural slope of the ground as to cut off as much earth as would be required to fill the depression adjoining, up to the level of the grade. The survey being finished, a double profile must be made, one of the natural ground in black ink, the other of the grade in red, upon the same base line, and with the

same abscissas or horizontal distances between the ordinates or lines of heights. An inspection of this, and a computation, if necessary, of areas between the black and red lines of section, will serve to show how well the excavation and embankment have been equalized, and the result may require a modification of the grade to adapt it better for the purpose in question, to the natural ground; cross-sections of the route must also be surveyed at all points of change in the longitudinal or latitudinal slope, and more frequently, if these occur at long intervals. The amount of excavation and embankment may be then obtained with sufficient accuracy for an estimate of expense, by computing the areas of the cross-sections of the work as it will be when completed, and multiplying half the sum of the areas of two of them by the longitudinal distance between them. The cross-sections, when the work is in embankment, will be of the form exhibited below, and the same turned upside down when the work is in

excavation. These are easily drawn from the field book. The cross-section bac being plotted, a corresponding to the point where it intersects the longitudinal section of ground, af will be the difference of the numbers in the columns of height and grade, df, fe, each equal to half the breadth fixed upon for the top of the road or bottom of the canal, dc, eb are then drawn at the proper slopes, for common earth 1 base to 1 in height. These sections may be regarded as quadrilaterals, and each divided into two triangles, for the purpose of obtaining their area.* section of ground is parallel to the top of the road, canal, they are trapezoids, and the area will then be

Where the cross

or bottom of the the sum of the

To compute the area of the cross-section from the numbers in the field book, conceive parallels to be drawn from the points b and c in the diagram, to af, meeting de, produced both ways; a trapezoid will be formed, from which, if two right angled triangles be deducted, the area of the sections will be obtained. If d denote the difference of level between b and f, d will be one parallel side of the trapezoid, and the altitude of one of the triangles, the base of which if the slope be 13 will be 2 d. And if d' denote the difference of level between c and ƒ, similar expressions will be had for the other side and altitude; and the expressions for the area of the section will be, b being the breadth of the roadway,

↓ (d + d') [b + ¦ (d + d') ] — ¦ d × } d — ¦ d' × } d' = } [(d+d') b + 3dd']

parallel sides de, cb, multiplied by the altitude af. Notes of the nature of the soil are kept in the field book of the compass, and the amount of digging and wheeling or carting that can be done in a day in different kinds of earth, having been ascertained by experiment, and the price of day labor being known, the data for determining the expense of the work are all known. To the above must be added the accidental expenses of culverts, blasting rocks, construction of tunnels, &c., which are all subjected to the same general rules, and are functions of the price of materials, mechanic labor, and experiments as to relations of time and amount of performance.

TO COMPUTE THE CONTENTS OF FIELDS.

1. Compute the contents of the figures, whether triangles or trapeziums, &c., by the proper rules for the several figures. If the linear measures be in links, the result is acres, after cutting off five figures on the right for decimals. Then bring these decimals to roods and perches, by multiplying first by 4, and then by 40.

2. In small and separate pieces, it is usual to cast up their contents from the measures of the lines taken in surveying them, without making a correct plan of them.

3. In pieces bounded by very crooked and winding hedges, measured by offsets, all the parts between the offsets are most accurately measured separately as small trapezoids.

4. But in larger pieces, and whole estates, consisting of many fields, it is the common practice to make a rough plan of the whole, and from it compute the contents quite independent of the measures of the lines and angles that were taken in surveying. For, then, new lines are drawn in the fields in the plan, so as to divide them into trapeziums and triangles, the bases and perpendiculars of which are measured on the plan by means of the scale from which it was drawn, and so multiplied together for the contents. In this way the work is very expeditiously done, and sufficiently correct; for such dimensions are taken as afford the most easy method of calculation; and, among a number of parts thus taken and applied to a scale, it is likely that some of the parts will be taken a small matter too little, and others too great; so that they will, upon the whole, in all probability, very nearly balance one another. After all the fields and particular parts are thus computed separately, and added all together into one sum, calculate the whole estate independently of the fields, by dividing it into large and arbitrary triangles and trapeziums, and add these also together. Then if this sum be equal to the former, or nearly so, the

work is right; but if the sums have any considerable difference, it is wrong, and they must be examined and recomputed, till they nearly

agree.

5. But the chief secret in computing consists in finding the contents of pieces bounded by curved or very irregular lines, or in reducing such crooked sides of fields or boundaries to straight lines, that shall inclose the same or equal area with those crooked sides, and so obtain the area of the curved figure by means of the right-lined one, which will commonly be a trapezium. Now, this reducing the crooked sides to straight ones, is very easily and accurately performed in this manner:-Apply the straight edge of a thin, clear piece of lantern-horn to the crooked line which is to be reduced, in such a manner that the small parts cut off from the crooked figure by it, may be equal to those which are taken in; which equality of the parts included and excluded you will presently be able to judge of very nicely by a little practice; then with a pencil or point of a tracer, draw a line by the straight edge of the horn. Do the same by the other side of the field or figure. So shall you have a straight-sided figure equal to the curved one, the content of which, being computed as before directed, will be the content of the curved figure proposed.

Or, instead of the straight edge of the horn, a horse-hair may be applied across the crooked sides in the same manner; and the easiest way of using the hair is to string a small slender bow with it, either of wire, or cane, or whalebone, or such like slender or elastic matter; for, the bow keeping it always stretched, it can be easily and neatly applied with one hand, while the other is at liberty to make two marks by the side of it, to draw the straight line by.

EXAMPLE.

Let it be required to find the contents of the irregular figure below, to a scale of 4 chains to an inch.

Draw the four dotted straight lines AB, BC, CD, DA, cutting off equal quantities on both sides of them, which they do as near as the eye can judge; so is the crooked figure reduced to an equivalent right-lined one of four sides, ABCD. Then draw the diagonal BD, which, by applying a proper scale to it, measures 1256. Also the perpendicular, or nearest distance, from a to this diagonal measures 456; and the distance of c from it is 428.

Then, half the sum of 456 and 428, multiplied by the diagonal 1256, gives 555,152 square links, or 5 acres, 2 roods, 8 perches, the content of the trapezium, or of the irregular crooked piece.

TO FIND THE CONTENT OF A FIELD WITHOUT PLOTTING.

Take the bearings and lengths of the sides of the field, and enter them in a field book as course and distance, and take out the difference of latitude and departure corresponding to each, and enter them in two double columns, marked N. s. and E. w. as at Art. 98. To obtain these, if the bearings are given in degrees, recourse may be had to a table of difference of latitude and departure for every degree and minute of the quadrant, such as is found in Bowditch's Navigator, or instead of this, the difference of latitude and departure may be calculated for each course and distance.* Another double column must be added, entitled double meridian distances. The meridian distance of any line is the distance of its middle point from an assumed meridian, which should be taken through some corner of the field. The double meridian distance, corresponding to the first course adjoining the assumed meridian, will be equal to the departure of that course. Double the meridian distance of any other course will be equal to the double meridian distance of the preceding course, plus the departure of the preceding course, plus its own departure.†

In applying this rule, distances to the right should be considered +, those to the left The double meridian distances east of the meridian

* The sum of the numbers in the column marked N. ought to equal that of the numbers in the column marked s. If such be not the case, the difference between the two sums should be half of it subtracted from the numbers in the column having the greater sum, being distributed among them in proportion to their magnitude; the other half should be added in the same way to the numbers in the column producing the less sum. For in going round a field and returning to the same point, the distance gone north must be equal to that gone south. The same remark applies to the columns marked E. and w. New columns will then be derived which may be called corrected diff. of lat. and departure.

†This may be seen by making and inspecting a diagram.