Fig. 30.—Method of plotting a three-point survey, n, w, s, the three fixed points. A the point to be found. B, C, centres of struck circles.

Though the ordinary methods of survey need not be stated here, the box-sextant is so seldom seen that some account should be given of its use. The objection to its use on short distances, that parallax between the direct and reflected ray causes errors, can be avoided by overlapping the images about ¾ inch, the usual amount of the parallax. The main use of the sextant is for three-point survey. Over broken ground where many isolated points have to be fixed, within a few inches on a few hundred feet, there is no method so quick and useful as the nautical three-point method, when improved by rigid plotting. At any three points which shall be visible from the whole of the ground, and within its general plane, three signals are placed, best lettered by the quarter of the horizon nearest to each, say n, s, w. The three points must be so placed that the one circle passing through them all shall not pass through points needed in the survey; otherwise they may be in any position, though best as a triangle of about equal sides. The three angles and one side are to be measured, thus defining the whole triangle. Then at any point to be fixed, A, the two angles between n to s and w to s are measured with the sextant, and these suffice to fix the position. For plotting ([Fig. 30]), lay down the triangle of the three fixed points, say to scale 1/100th (the triangle with shaded corners n, s, w), and the perpendiculars to each side of it; this is most accurately done by a large protractor with vernier, setting out the radii and perpendiculars of the triangle from its centre. Then tabulate the half of each base × cotan. angles observed on that base, e.g.

logs.n.n.logs.n.n.logs.n.n.
½ bases n·27314s ·36621w ·29223n
x cotan.
angles at
places 1, 2, 3		{ 1·432232·705·262721·831
{ 2·566713·687·482143·035
{ 3·419952·630·677094·754

Here the log. half base n to s is ·27314; this added to log. cotan. of angle subtended by n-s from station 1 is log. ·43223, giving a value 2·705 inches. From station 1 the angle s-w was observed; and from stations 2 and 3 the angle w-n was observed. All this calculation can be rapidly done in this form, placing the sheet upon the log. book, with the written log. half base next below the printed log. cotan. angle, and writing down the sum of the two against the number of the station. Then on the plan, plot these (½ base × cotan.) on the perpendiculars of their respective bases as at B and C, marking the station number to each. Then with compasses sweep an arc from one centre B, with radius Bs equal to the distance from the centre to its two points of the triangle. The same from the other centre C that has the same number of station. The intersection of the arcs is the point A of that station on the plan.

Of course the prolonged perpendiculars (broken lines) are used as often as the direct perpendiculars; the aspect of the angle from the station, whether n-s or s-n showing on which half of the perpendicular we should lay off the centre. For angles over 90° the complement of the angle should be used in calculation, the centre then laid off on the wrong half of the perpendicular, and the arc swept across the right half. This mode of plotting gives the fullest accuracy, such as is never possible with the use of station-pointers, or trial and error devices which are used in nautical survey. A field of 40 stations can be easily calculated in an hour, and plotted in a couple of hours more. If it is needful to work any point with pure calculation instead of plotting, it can be accurately done by the principle that the line joining the two centres of arcs, B and C, forms with their common point s an equal and opposite triangle to that which they form with the survey point A. It will be seen on looking at the diagram that w-s, the angle by which B is plotted, is equal to the angle w-s from A; and similarly the angle of the half base n-s from C, is equal to n-s from A. Hence the points n, s, w subtend from A, the observed angles, and A is the point from which they must have been observed.

For levelling, the handiest instrument is a short rigid pendulum, with mirror attached, to hang truly vertical. The reflection of the eye back to itself is then a truly horizontal line, and can be sighted on to any distance. The pendulum is best made about 5 inches long, with tetrahedral net of suspension thread, to avoid twisting, passing through two eyes on the mirror and two eyes on the holder, and a covering tube to shield it from wind. With this, readings can easily be taken to an inch on 100 feet, and this is sufficient accuracy for most archaeological work.

CHAPTER VI
COPYING

Paper squeezes.