The ordinate to the E. or W. distance x sin re

duced bearing.

These quantities may be obtained direct from traverse tables or with the help of tables of sines and cosines as may be preferred.

4th. The final independent co-ordinates for any point, taken with reference to the starting point of the traverse, are obtained by summing up the successive co-ordinates up to that point, treating them as positive or negative in accordance with their signs of direction.

The formula for checking the sum of the inward angles of any closed traverse is (N−2) × 180°, as explained in the collection of formulæ, at p. 48, Chapter I.

By adopting the above method of reduction, which affords a check on itself, the plotting can be afterwards effected by co-ordinates with the minimum of error.

Secondary triangles.-The above main traverse-work is of course the mere skeleton of the survey; and as it frequently happens that these surveys are very narrow, being often confined to a mere road, and that the detail required consists merely in frontages of property and minor detail close to the road; in such a case the detail is surveyed simultaneously with the traverse by the method of distances and offsets. When the limits of deviation are of pronounced width, and the detail within them cannot be very satisfactorily managed by chainings and offsets, aided by compass bearings, recourse must be had to a series of secondary triangles, which may be conveniently arranged by measurement to intermediate points along the main lines of traverse; at these secondary points, angles are taken to lateral points near the intended limits of deviation on each side, and check

angles are also observed at the lateral points. For this purpose the box sextant is useful.

The filling in, or detail.—This is applied either to the main traverse-work or to the secondary triangulation before mentioned, or perhaps to a secondary set of traverse-lines. The amount of such work in traverse surveys is comparatively small, and generally occurs in detached pieces at places where the construction of some special work of art or building is contemplated. It is, however, generally required with much precision, hence the compass-survey work and plane-table-work 'suited to subsequent reduction for small-scale plans is not often fitted for this object; and the more accurate results of chain-survey or good telemeter-survey assisted by gccasional compass bearings are better for these purposes.

Plate VI., with the corresponding detailed field-book, No. IV., is an illustration of applying the method of traversing with the theodolite to an engineering survey "of a portion of railway about one mile in length.

The various examples of field-books actually recorded in the field, and the illustrative field-sketches or reduced plans given in the Plates, are intended to serve for the guidance of those newly entering on the subject.

PART II.

LEVELLING.

Section I. INSTRUMENTS AND APPLIANCES.

LEVELLING is that branch of Geodesy which is limited to operations for determining and measuring the relative heights and depths, or elevations of places, points, and objects on and comparatively near the earth's surface.

Absolute elevation of points or places is estimated in linear units (in England, in English feet) above or below mean sea-level; relative elevation is estimated in the same way with reference to any convenient datum level, such as that of Trinity high-water mark, the zero of the Nilometer, the plinth of the nearest town-hall or church, the rail-level at the middle of the platform of the nearest railway-station, or any arbitrary visible or imaginary point. A bench-mark is a subordinate datum-point, marked on some permanent common object, such as a kerbstone, milestone, plinth, stone step, lower hook of gatepost, etc., whose elevation above some datum level has been carefully determined and is known; bench-marks made on surfaces or objects not liable to wear or to change, are useful for reference and saving the relevelling of long distances.

Any set of relative levels of a series of points may be recorded in either of the following ways.