When rivers or roads have many abrupt and deep bends, the deter mination of their forms by means of offsets from the station-lines may become impracticable ; and then a separate survey of such details must be made by means of the compass, the eircumferentor, or the theodolite. (Tlisonoters.) The same means must be employed for the survey of a sea-coast, when the operation is to be performed ou land; and it may often be advantageous to determine in like manner the forms of the hedges, walls, &c., in the interior of the tract which is to be surveyed. Ground covered with wood must be surveyed by going quite round it ; poles being set up at remarkable bends on the contour, the distances between them are measured with the chain, and the bearings of the several lines from the meridian are observed with the theodolite.
In order to explain the process of surveying with the theodolite by the method which Is commonly called that of the " back-angle," and which is now almost constantly adopted, let it be required to determine the outline n K c L II A, which may represent the contour of a wood, or of which part may coincide with the course of a road or river. The instrument may be set up at Dr, which may be supposed to be the first station: and let the line n a at each of the stations II, K, o, &c., repre sent the position of the needle or of the magnetic meridian at the station : also let the instrument be adjusted so that the zero point of the horizontal limb may be under the point n (the north point of the needle), or the zero of the degrees in the compass-box may be in coincidence with ti ; and let it be the second station. Turn the upper horizontal plate with the telescope till the object-glass of the latter is directed to K, and make the intersection of the wires appear to coincide with the object at that station ; then the index of the vernier will be at some graduation on the lower horizontal plate, as at x, and the angle all K is that which is observed ; suppose it to be reckoning from the north towards the east, which angle is usually represented by N. 54° E. (N.B. Previously to directing the object glass to K, it might have been directed to any other visible objects, F or D, whose positions it might be required to determine by means of their bearings from the meridian line.] Let the theodolite be now removed to K, a staff being planted in the ground at II : turn the whole instrument round on its vertical axis (the index of the vernier remaining at the graduation N. 54° E.) till the objeetglass of the telescope is directed to u, and the inter. section of the wires appears to coincide with the staff there. Then, if the former angle were correctly taken, and no movement of the hori zontal plates on one another have taken place, the south point s of the needle will lie over the zero of the graduations on the lower plate, or will coincide with the zero of the degrees in the compass-box ; And this circumstance will be a roof of the accuracy of the work, all the meridian lines v s, v 8, being supposed to be parallel to one another.
Now turn the upper horizontal plate with the telescope, till the object-glass of the latter Is directexl to c, and the intersection of the wires appears to coincide with the object there : the telescope in moving from the position K 11 to the position K c having passed over and beyond 8; and the index of the vernier being supposed to be at y, the number of the graduation, these being read from a in the direction any, will be greater than 180: let it be 256 (or 180°+76°); in that ease the observed angle is N. E., and it expresses the bearing of the line PC o from the meridian n K a, or from the meridian n n a. If the telescope in moving from K n should be directed to an object at E, then, the index of the vernier being supposed to be at 3, the number of the graduation will be less than 180° : let it be 110 (or 180°-70°); in that case the observed angle is N. 70° W., and it expresses the bear of the line K E from the meridian n a or n Let the theodolite be removed to c, a staff being left at K, and turn the whole instrument, the index of the vernier remaining at N. 76° E., till the object-glass of the telescope is directed to K, and the inter section of the wires appears to coincide with the staff there; then the point n of the needle should lie over the zero of the graduations. Now turn the sipper horizontal plate till the object-glass of the telescope is directed to t, and the intersection of the wires appears to coincide with the object there ; then the telescope, in turning from the position C K to c t, passing over and beyond a, the number of the graduation coin cident with the index of the vernier will (reckoning from zero at n) be less than 180 ; let it be 133° : in that case the observed angle is N. 133° E., or S. 47° E., and It expresses the bearing of the line c from the meridian n ea or n n a. If the telescope should not pass beyond a, and should be in the position c P, for example, the number of the graduation, reckoned from n, will be greater than 180; let it be 208, or 180° + 26° : in that case the observed angle is S. 26° W., or N. 206° E., or N. 154. W., and it expresses the bearing of the line from n o a or nits. In this manner the process of the survey is con tinued to the end of the road, or till, having passed completely round the wood, the instrument returns to H, from whence it set out.