and by the edge of the index draw a dry or obscure line; then measure the distance CD, and lay that distance down on the line CD. Then turn the index about the point C till the mark E be seen through the sights, by which draw a line, and measure the distance to E, laying it on the line from C. to E. In like manner, determine the positions of CA and CB, by turning the sights successively to A and B; and lay the lengths of those lines down. Then connect the points with the boundaries of the field, by draw ing the black lines CD, DE, EA, AB, BC.

2. From a station within the field.

When all the other parts cannot be seen from one angle, choose some place within, or even without, if more convenient, whence the other parts can be seen. Plant the table at O, then fix it with the needle north, and mark the point

E

A

on it. Apply the index successively to, turning it round with the sights to each angle A, B, C, D, E, drawing dry lines to them by the edge of the index, then measuring the distances A, OB, &c. and laying them down upon those lines. Lastly, draw the boundaries AB, BC, CD, DE, EA.

3. By going round the field.

When on account of a wood, or water, or some other obstruction, you cannot measure lines across the field; begin at any point A, and measure round it, either within or without, and draw the directions of all the sides, thus. Plant the table at A, turn it with the needle to the north, or flower-de-luce, fix it and mark the point A. Apply

the

the index to A, turning it till you can see the point E, and there draw a line; and then the point B, and there draw a line; then measure these lines, and lay them down from A to E and B. Next move the table to B, lay the index along the line AB, and turn the table about till you can see the mark A, and screw fast the table; in which position also, the needle will again point to the flower-deluce, as it will indeed at every station, when the table is in the right position. Here turn the index about B till through the sights, you see the mark C; there draw a line, measure BC, and lay the distance on that line after you have set down the table at C. Turn it then again to its proper position, and in like manner find the next line CD. And so on, quite round by E to A, Then the proof of the work will be the joining at A. work be all right, the last direction EA, on the ground, will pass exactly through the point A, on the paper; and the measured distance will also reach exactly to A. If these do not nearly or quite coincide, there is some error, and the work must be examined,

For if the

PROBLEM VIII.

To survey a field with the Theodolite, &c.

1. From one point or station.

When all the angles can be seen from one point, as the angle C (first figure to last Prob.); place the instru ment at C, and turn it about till, through the fixed sights, you see the mark B, and there fix it. Then turn the moveable index about fill the mark A is seen through the sights, and note the degrees cut on the instrument. Next turn the index successively to E and D, noting the degrees VOL. II.

E &

cut...

cut off at each; and all the angles BCA, BCE, BCD, are found. Lastly, measure the lines CB, CA, CE, CD; and enter the measures in a field book, or rather against the corresponding parts of a rough figure, drawn by guess to resemble the field.

2. From a point within or without.

Plant the instrument at O, (last figure) and turn it about till the fixed sights point to any object, as A; and there screw it fast. Then turn the moveable index round till the sights point successively to the other points E, D, C, B, noting the degrees cut off at each of them; and all the angles round the point are found. Lastly, measure the distances OA, OB, OC, OD, OE, noting them down as before, and the work is done.

3. By going round the field.

By measuring round, either within or without the field, proceed thus:

Having set up marks at B, C, &c. near the corners as usual, plant the instrument at any point, A, and turn it till the fixed index be in the direction AB, and there screw it fast; then turn the moveable index to the direction AF, and the degrees cut

B

D

off will be the angle A. Measure the line AB, and plant the instrument at B, and there in the same manner observe the angle B. Measure BC, and observe the angle C. Measure the distance CD, and take the angle D. Measure DE, and take the angle E. Measure EF, and take the angle F. And lastly, measure the distance FA.

To:

To prove the work; add all the inward angles A, B, C, &c. together, and when the work is right, their sum will be equal to twice as many right angles, as the figure has sides, wanting four right angles. And when there is an angle, as F, that bends inward, and you measure the external angle, which is less than two right angles, subtract it from four right angles, or 360 degrees, to find the internal angle greater than a semicircle or 180 degrees.

Otherwise.

Instead of observing the internal angles, you may take the external angles, formed without the figure by producing the sides beyond the angular points. And in this case, when the work is right, their sum will be equal to 360 degrees. But when one of them, as F, runs inward, subtract it from the sum of the rest, to leave 360 degrees.

PROBLEM IX.

To survey a field with crooked hedges, &c.

With any of the instruments measure the lengths and positions of imaginary lines, running as near the sides of the field as you can; and, as you pass along these lines, measure the offsets, in the manner before taught; and you will have the plan, on the paper in using the plane table, by drawing the crooked hedges through the ends of the offsets; but in surveying with the theodolite, or other instrument, set down the measures properly in the field

book,

book, and plan them after returning from the field, by laying down all the lines and angles.

So in surveying the piece ABCDE, set up marks a, b, c, d, dividing it into as few sides as may be. Then begin at any station a, and measure the lines ab, bc, cd, da, and take their positions, or the angles a, b, c, d; and going along the lines, measure all the offsets, as m, n, o, p, &c., along every stationary line.

And this is done either within the field, or without, as may be most convenient. When there are obstructions within, as wood, water, hills, &c. then measure without, as in the following figure.