of beginning and ending fixed by the intersection of two lines cut into the head of the bolt.

The measurement may be made with steel tape or rods. If a tape is used, it should be carefully drawn out each time to its standard length, and should be compared with a standard both before and after measurement, and correction made for its variation, if any, from standard. The mean of several measurements should be taken for the correct measurement. If the measurement has been made on an incline, instead of on a level, the measured distance should be reduced to the horizontal distance by multiplying the inclined distance measured by the cosine of the angle of inclination.

247. For a description of the base-line apparatus used in the U. S. Coast and Geodetic Survey, see Reports of that Survey for 1854, 1857, and 1880.

248. The alignment of the measuring tape, or rods, both vertically and horizontally, or in the line of the slope if the measurement be not horizontal, is of the greatest importance, since there is no compensation of errors, a faulty alignment always resulting in a measured length greater than the true length.

249. Having carefully measured the base, it is then necessary to reduce the measurement to the sea level.

L

h

But his so small as compared with R that we may, without sensible error, make R+h= R; whence we have,

250. The trigonometrical stations are marked by signals, which may be constructed in a great variety of ways, depending upon the locality of the stations and the lengths of the sides of the triangles.

Sometimes a signal has to be raised above the level of the adjacent country, in which case it is constructed of timbers, and upon the apex is placed a vertical staff bearing a flag. The exact trigonometrical point is determined by a plumb-line suspended from the apex of the signal.

A temporary signal may be constructed with three or four pieces of scantling framed and braced, as shown in the annexed figure, with a short pole projecting from the apex. The plumb determines the point B, which is the exact trigonometrical point over which the theodolite is to be placed. Where the sides of the triangles are not very great, a pole, planted vertically and surmounted by a flag, will answer as a signal.

FIG. 120.

In order to distinguish the different signals, the flags which they bear should be different from each other. They may be formed by arranging stripes of white and red, according to some prearranged plan, and the flags of the different stations should be entered in a book. For the purpose of future reference, the trigonometrical point at each station, as B, should be indi

cated by a permanent mark. If the point falls upon a rock, a hole may be drilled to show the locality; or if not, a mark-stone may be sunk under the point, deep enough to be beyond the reach of accident. A record of the monument should be preserved, together with its reference to some of the permanent objects in the neighborhood.

251. A Heliotrope is necessary in long sights, and is always of great service in short sights in directing the observer to the station-mark. It consists essentially of a small mirror, so directed by an assistant as to throw a beam of sunlight into the telescope of the distant observer. Let a silvered glass, about 3 inches square, be mounted on a board in a manner similar to the telescope mount of a transit so as to have a motion about a horizontal axis and at the same time about a vertical axis; in front of this, at a distance of two or three feet, mount a board with a hole in it, across which hole threads are to be stretched at right angles to each other, and adjust this hole over the station by a plumb-line. At the centre of the back of the mirror scrape away the silver, making a small sight-hole; if now an assistant, sighting through the hole in the mirror, moves it so that the cross threads come in line with the distant station, it will be easy to keep the beam upon the observer by properly inclining and revolving the mirror. As the light reflected from so large a mirror would be too intense to observe with the telescope, it is necessary to make the cross-thread hole in the board quite small, not more than inch diameter for distances not exceeding 5 miles, about one inch diameter for distances of ten miles, and

so on.

A small pocket mirror will be found very useful as a means of telegraphing instructions, by combinations of flashes according to a system previously agreed upon. It can be directed to the observing station (or to the observed station) with sufficient

accuracy for signalling, by setting a vertical staff in line with the distant station and causing flashes to travel up and down the staff.

252. The extent of the survey, and the standard of accuracy to which the results are required to conform, must determine the size and perfection of the instrument to be employed in the measurement of angles. The angles of the primary triangles of the United States Coast and Geodetic Survey are measured with theodolites, whose horizontal circles are 24 or 30 inches in diameter; and to eliminate, as much as possible, every source of error, great numbers of observations are made at each station, the readings being made on different points of the arc by different verniers. Usually from 40 to 60 observations are made for each angle-one measurement, with the telescope direct, and one with it reverted, constituting a complete observation. With these precautions, it has been found that the error in a primary triangle (where the sum of its three angles has been compared with 180°), has fallen much within 3 seconds. The error of 3 seconds has been adopted as the highest admissible limit of error in such triangles.

253. The theodolite does not differ essentially in the principles of its construction and use from the transit, which has already been described. It is fitted with many appliances, for accuracy in the observation and the reading of angles, which it is unnecessary to describe here. Fig. 121 is a representation of the 8 to 12 inch theodolite used in the U. S. Coast and Geodetic Survey, taken from the Report of that Survey for 1880.