1. In the surveys of woods, lakes, harbours, &c., by ranging a system of lines round them, and connecting them by taking the angles at their several points of junction. 2. In the surveys of winding roads, rivers, &c., especially where they are obstructed by woodlands, buildings, &c., on both sides, by taking the angles, as in the first case. 3. In the surveys of the streets, of parts or the whole of cities, and large towns. 4. In determining the positions and distances of several stations, in an extensive engineering or other survey, by taking angles to them from two or more stations, the distances of which are known. 5. In railway surveys, &c., by connecting the main line with angles, as in the first and second cases, by determining the inaccessible parts that may occur, as in the fourth case ; in ranging a long line over high summits. This case frequently embracing all the four preceding ones. PROBLEM I. TO SURVEY WOODS, LAKES, HARBOURS, &C., BY THE THEODOLITE. 1. Required the plan and area of the wood represented in the following figure, the field notes, adapted to the theodolite, being given. Fix station flags round the wood, so that the lines compassing it may be as near it as possible, for the convenience of taking the offsets ; the stations at the same time being made on proper ground for fixing the theodolite. Let ABCDE be the stations, and the field book as below. THE METHOD OF TAKING THE ANGLES, &c. It will be seen, from the field notes, that the line AB is first measured, as a base for the plan. The first angle ABC is then taken, which is found to be 81° 29', and shews the direction of the second line B C. In taking this angle, the theodolite is fixed directly over O B; the two zeros of the horizontal plates being then clamped together, the object glass of the telescope is directed to the flag or other mark at O A, and the whole instrument clamped ; the upper plate is now unclamped, and the telescope directed to the flag at o C, when the angle ABC is found to be 81° 29', as shewn in the field notes. In a similar manner, the angles at stations C and D are respectively found to be 11]° 39' and 46° 51', the three lines BC, CD, DE each bending respectively to the left of the line preceding it. The angle at o E is found to be 241° 38', which being greater than 180°, that is, greater than the semicircle pq, shews that the line E A turns to the right. Finally, the angle at o A is found to be 58° 23', shewing that the line A B turns to the left of EA. Thus the magnitude of the angle shews whether the new line inclines to the right or the left of the old one, the new line turning to the left of the old one, when the angle is less than 180°, and to the right when greater, the zero of the instrument being always directed to the commencement of the old line : therefore, the bearing of all the lines, except the first, may be omitted in the field notes; the remainder of which, being simi. lar to those already giren, need no further explanation. PLANNING AND PROVING THE WORK. Draw the base line A B in the given direction, indefinitely, and lay off the given length 2302 links thereon. Place the centre of the protractor at o B, with its straight side close against A B, and prick off 81° 29' from its end towards A ; then, through o B and the protractor-mark, draw BC, making it the given length 2678. Lay down similarly the two follow ing lines with their angles C and D. The angle at O E is 241° 38', therefore E A must turn to the right, and the angle to be laid off at E is 360° — 241° 38' = 118° 22'; or, if a circular protractor be used, the whole angle may be laid off at once, and E A being drawn, must reach to © A, where the survey began, or so very near to it that the error may be con sidered immaterial; but if it do not reach to O A, by a considerable distance, there has been an error either in taking the angles or measuring the lines. But since the sum of all the interior angles of a polygon is equal to twice as many right angles as the figure has sides, lessened by four right angles, and since the given figure has five sides, the sum of all its five interior angles will be 5 x 2 . 4: = 6 right angles = 90° x 6 = 540°. This will be found to result by adding all the angles of the figure, as below. Angle at B = 81° 29' C= 111 39 46 51 E = 241 38 58 23 D = A = Proof, as respects the angles 540° 0 The above result shews that the angles have been accurately taken; if, therefore, the work do not close, that is, if E A does not reach to 0 A, the error is in the measuring of some of the lines, or in making a wrong entry in the field notes, which may now be readily detected, if required. When the work is planned the content will be found to be 37a. 3r. 4p. It will be seen that the above survey might have been ef. fected by the method given in Prob. VIII., Chap. III.; but it is here assumed that the lines compassing the wood cannot be prolonged for the purpose of measuring tie lines, on account of obstructions, as is often the case. Lakes, Meres, and large Ponds are surveyed and planned in the same manner as the wood just given. 2. The following figure represents a gulf or inlet of the sea, the survey of which is required to adapt it for the purposes of a harbour for ships. The coast, here shewn, is the boundary of high water; the survey begins at O A, station-flags being fixed at B, C, D, E, F, and G, and angles taken at С B A to F and G. The line BA, being first prolonged backwards to high water mark is then measured to B, and the angle A B C taken. Similarly all the succeeding lines are measured, and the angles D taken; also at F and G the F angles are taken to O A, and the line FG prolonged to high E water mark, all the offsets being taken as the work proceeds. The figure may now be laid down, precisely as in the last example, the magnitude of the angles shewing the directions of the lines, and the lines A F, A G, which could not be measured on account of the great width of the entrance of the harbour, proving the work by means of the angles, taken at A, and F, and G. PROBLEM II. THE SURVEYS OF ROADS AND RIVERS, 1. The following figure represents a river, the survey and plan of which is required, for the purpose of improving its navigation, making locks, &c. E P D a a Station flags being set up, at or near the principal windings of the river, as at A, B, C, D, E, the line A B is measured, and the offsets taken to the nearest shore of the river, its widths, if very great, being determined by Prob. IV., Chap. III.; or by throwing a leaden ball across the river, with a slender cord attached thereto at one end, and holding the other end in the hand, and then drawing it back, and measuring the length of cord, required to reach the opposite shore. This last method is impracticable where the width of the river is very great. A flag is left at o P in A B, where the sight, in the general direction of the river, is unobstructed for a considerable distance. The measuring of A B being now finished, the angle at B is taken, which, being less than 180°, shews that B C turns to the left. On measuring to C the angle there is found to be greater than 180°, shewing tha, CD turns to the right; and thus the work proceeds to O E, where an angle is taken to the nowdistant flag at P. This last angle will prove the accuracy o: the work when laid down. If the width of the river be very great and unequal, a similar system of lines must be used on the opposite shore; otherwise, a correct map cannot be obtained, the two systems of lines being connected by finding occasional widths of the river, as already stated; thus the lips, B C may be prolonged across the river to connect another system of lines, if required. 2. If a road be represented by the winding figure, in the last example, it may be surveyed precisely in the same manner, excepting that it would be more convenient to have the system of lines ABCDE upon the road, instead of the side of it, that the offsets may be readily taken to the right and left of the several lines, recollecting to leave a flag, or some other prominent object, in or near the first line, as at OP, in the last figure, to which an angle may be taken, after the survey has proceeded a considerable distance, to prove the work. The method of planning either a river or road will be sufficiently clear, from the first example in Prob. I. The map of a river or road being thus obtained, if the area be required it may be readily found by the methods previously given, or, by taking widths at the end of every chain, estimated along the middle of the river or road, and taking a mean of the widths, which, multiplied by the length, will give the area sufficiently correct for most cases, a |