But, if it were required to construct a map on a large, cadastral scale, more fixed points still would be needed, and a tertiary tri angulation would be carried out, depending upon the secondary work. This tertiary triangulation would have its points only a mile or two apart, and its triangular error would not exceed 15 seconds. It is now customary to classify a triangulation according to its accuracy, thus, a triangulation of the 1st order has a tri angular error not exceeding I second; a triangulation of the 2nd order has a triangular error not exceeding 5 seconds; a triangula tion of the third order has a triangular error not exceeding 15 seconds; and one of the 4th order has a triangular error greater than 15 seconds.
But it will easily be imagined that certain areas do not readily lend themselves to triangulating, and in such areas, occasionally, traversing is resorted to for the frame-work. This will be de scribed later. Also, in the rapid exploration of a new country it may be necessary that the frame-work be astronomical. But this is an inaccurate make-shift, and should always be avoided when ever possible.
It should be emphasized that whatever may be the extent of the area to be surveyed, whether it be a backyard or a continent, the area should be dealt with as a whole, and the frame-work should be designed to stiffen the whole area. Anyone who is charged with the execution of a survey should, therefore, carefully consider how he can best design the control or frame-work.
Bases for geodetic work are measured with very great refinement, and the probable error of such a base would be of the order of one in one-million. Such bases in times past have been measured with glass rods, wooden rods, steel bars, and compound bars of brass and steel. But now almost every accurate base, whether for geodetic purposes, or for secondary work, or for very good topographical work, would be measured with metallic (usually invar) wires or tapes supported in catenary, i.e., allowed to hang free in a natural curve, being slung over trestles. Invar is an alloy of steel and nickel, containing 36% of the latter, and this alloy is used because it has the smallest known expansion of any metal or alloy. Invar wires and tapes require, however, careful handling, and testing, and on this account are not freely used for rapid topographical bases. It is possible to get quite good results from the use of ordinary steel (not invar) tapes laid along the surface of the ground. Let us then, take the case of the measurement of a topo graphical base with steel tapes laid along the ground. The first
thing to do is to select the site and the positions of the terminal points. These must be so chosen that well-conditioned triangles can be built up on the base, connecting it with the main points of the triangulation. (See fig. 2.) Then the site must be cleared of obstructions, and small undula tions levelled. Then the base terminals must be marked, and a theodolite set up over one of them and directed to a pole put up over the other. Intermediate marks, in the form of pickets driven in flush with the ground, should be fixed. The actual meas urement is made with a steel tape, kept at a fixed tension with a spring balance, and marks are made on the tops of the pickets, which are at tape-lengths from centre to centre; the tops of the pickets may be conveniently covered with strips of zinc, on these strips the positions of the ends of the tape are marked when meas uring. A convenient length of tape is 30o ft., with a width of inch. A convenient tension is 20 pounds. A base so measured requires corrections for standard, temperature, slope if any, and height above sea. The chief difficulty in this sort of measurement is to ascertain the temperature of the tape.
A base of this kind might be some two or three miles long; but it should be remembered that it is more important to be able to extend the base by well-conditioned triangles than to measure a long base. The base having been measured, it is now necessary to proceed with the execution of the triangulation.
Fig. 2 shows the base extension, and in the case shown, the longest side reached is more than 8 times the length of the base, so that if the base were 2 miles long, the side of the main triangu lation would be more than 16 m. long.
The process of triangulating consists of observing the angles at the apices of the triangles and at the two ends of the base.
The apices and the ends of the base must be marked. The per manence of the marks will depend upon the character of the triangulation. Thus, in first-order work, a mark-stone would be buried deep in the soil, and iso lated from the platform sur rounding it. Vertically over the mark, when the observations were finished, a stone pillar would be built. In such work the signals would be luminous, either lamps or heliographs. But in work of a lower order the marks would usu ally be opaque ; various different patterns have been used, poles, baskets, quadripods and so on. In very rapid work the tops of spires, pagodas, temples, or soli tary trees are observed too ; in some cases it has been found to be useful to clear a hill of all trees but one. In very rough work sometimes the tops of prominent hills are used, without any mark.