NETWORK in turn. The first station should read about o°, the base plate FIG. 4 being so set, and then the other stations would be observed in order and the round of angles would close again on the first sta tion. If this round were taken with the telescope to the right, the next round would be taken with the telescope to the left. The number of rounds taken will depend on the order of the work, the zero being shifted at each round, -or at each alternate round. For rapid work a good system would be First zero, o'o', telescope right Second zero 225° 1' left. Telescope right is sometimes called "Face Right," and telescope left, "Face Left." The angles having been all booked, the means are taken, and if each angle of a triangle has been observed, it is clear that the sum of the three angles should add up to 180°, plus the spherical excess, an excess which is due to the fact that the earth's surface is not plane, but is curved. Spherical excess varies directly as the area of the triangle, and may be taken to amount to 1.32 seconds for every hundred square miles, and other areas in proportion. It is generally neglected in rough work, and the excess or defect of the sum of the three angles of a triangle is, in such work, dis tributed equally amongst the three angles. Thus, if the sum of the three observed angles amounted to 180° I' 3o", each observed angle would be diminished by 3o" for use in the computations.
Over large areas, and in work which is likely to be extended to great distances, or in which, for some special reason, the geographical positions are required, it is necessary to calculate for each trigonometrical station its latitude and longi tude, starting from some station for which these elements are known. But, in the case of small or isolated surveys, such elabo ration is unnecessary, and we can proceed as follows :—Take any one station as the origin, and the meridian through that station as the initial meridian ; then, having observed an azimuth, or true bearing, at the origin, we shall know the azimuth of each line which radiates from the origin. Then if
is the length of one of these lines, and if a is the angle which it makes with the initial me ridian, then
cos a is the north-south co-ordinate, and
sin a is the east-west co-ordinate of the end of that line. For lines radiating from the end of the line in question, if 13 be the angle that one of them makes with a line parallel to the initial meridian, then if
be the length of this second line, the co-ordinates of the end of it with reference to the beginning will be
cos/3 and sin/3, so that the co-ordinates of any point in the triangulation, with reference to the origin, will be
cosa
cosy ± • • . . , and
sina+12 sin/3+/3 sin-y+ . . .. Each point in the triangulation can now be plotted with reference to the origin and the initial meridian, due regard being paid to the signs of the trigo nometrical functions. With these co-ordinates available it is, of course, easy to compute the distance of any one point in the tri angulation from any other point.
In using this simple method, we have assumed that the earth's surface is plane ; actually it is a spheroidal surface. The errors involved are chiefly in a north-south direction. At sixty miles'
distance from the origin the error amounts to 8,800, and the error increases as the square of the distance, so that at i oo miles the error is ein and if we were content with this error as a maximum, we could survey 200 miles on this system.
Points fixed as abqve described, and plotted on paper at intervals of, say, four or five inches, give a sufficient control for the horizontal work of the detail surveyor; but he will also require a frame-work of heights if he is to contour, or approxi mately contour, the terrain. Heights may be determined in a variety of ways and with several degrees of accuracy; the most accurate method, and at the same time, the slowest and most expensive, is levelling, which will be dealt with later. The next method in order of accuracy is trigonometrical determination by vertical angles, taken with a theodolite, and this will now be bfiefly described. On the same day that the horizontal angles are observed at a trigonometrical station, vertical angles are observed to all the distant marks, and are read on the vertical circle of the theodolite. It has been found by experience that vertical refraction is least during the middle of the day, and vertical angles would, there f ore, be taken between noon and three. It is best to eliminate refraction as far as possible by observing vertical angles at each of the two stations of which the difference of height is required. Then it is easily shown that, if one angle is an elevation,
and the angle at the other station is a depression,
then the dif ference ference of height is ctan ; or if both are depressions,— which may often happen with long rays,—the difference of height is ctan Db —
c being the distance between the two stations. If 2 only one angle has been observed, it will be necessary to correct this for refraction and curvature; the average amount of. this cor rection may be taken as roughly about 4-1- seconds for every thousand feet of horizontal distance between the stations, and for other distances in proportion. Barometers are best used for determining differences of height, and not absolute heights. All heights should, when possible, be based on some determination of mean sea-level, which is now the universally accepted datum.
A "Level" is an optical surveying instrument, which, when in adjustment, has its line of collimation horizontal; that is, the intersection of the cross-wires will cut an object, seen through the telescope, on the same horizontal plane as the optical axis. In using a level there is no need to consider the curvature of the earth, because the distances between the level and the forward and back level-staves are always kept short ; in good work such a distance should not exceed 5o yards. If the distances in question are always kept approximately equal, any error due to faulty collimation will tend to disappear. Excellent modern levels are now made which enable the observer to read the bubble with out moving from the eye end of the telescope, and the instru ment is finally levelled before the reading is taken. Readings are taken on two staves, usually some ien feet long; each staff gradu ated in feet and tenths of feet, and hundredths may be estimated.