The results of late discoveries have pointed out other irregularities in the needle, which, with those above alluded to, are sufficient to weaken our confidence in any surveying instrument founded on a principle so unsteady.

As we are upon this subject, it may not be out of place to mention, that from numerous observations recently made, the variation commences two or three hours before noon, having previously returned to the position it had on the preceding day, and having been quiescent during the night. From this it appears that angles taken early in the morning are more to be depended on than those taken at a more advanced part of the day. It may be of importance to know, that the greatest error resulting from diurnal variation would be 31.067 links, and the least 16.212 links in a distance of one mile, which, upon a map laid scale, would cause great derangement. facts clearly shew how necessary large divided instruments are, when used for very distant objects, as in trigonometrical surveys; and also why the needle cannot be employed on such surveys.

down to a large The preceding and accurately

Some surveyors use the sextant in taking angles, which, except in the absence of a better instrument, should never be used; as the angles observed by it are always incorrect, except when the objects observed happen to be on the same horizontal plane, which is a case that seldom occurs in practice. By it, no doubt, the angular distance subtended at the place of the observer may be measured, but not the horizontal angular distance; therefore the former should be reduced to the latter. The process required to accomplish this is

too tedious and troublesome to be of any use in the ordinary practice of surveying.

To reduce angles taken with the sextant to true horizontal angles, we must know the angles of elevation and depression, which, when very small, are not appreciable by reflecting instruments. Here we must have recourse to calculation to accomplish what falls without the reach of such an instrument by legitimate When the objects are on the same horizontal plane, their angular distances, as taken with the sextant, are the true horizontal angles; but when the objects are on a plane oblique to the horizon, a correction must be made in the observed angle, so as to reduce it to the true horizontal angle.

means.

Let SP and OB be two objects elevated above the surface of the earth; D the place of the observer,

Z his zenith, and S and O two points on the surface of the sphere. The observed angle will be PDB, and the required angle will be SDO measured by the arc SO. In the spherical triangle PZP; PZ, BZ, D and PB are given, to find the angle PZB, the measure of which is the arc SO angle SDO.

sin. PZB/sin. † (PB+ PZ—BZ) sin. † (PB+ZB—ZP) sin PZ, sin BZ

B

But PB is the measure of the observed > PDB; PZ is the complement of the arc SP, which is the measure of the PDS; BZ is the complement of the

arc OB, which is the measure of the angle BDO ; and the difference between any two arcs PZ and BZ, is equal to the difference between their complements SP and OB. Therefore

sin SDO=√/sin † (PDB+PDS-BDO) sin † (PDB+BDO−PDS) cos PDS, cos PDO

If the angle PDB be measured by a sextant, and the vertical angles BDO, PDS, be measured by a theodolite, the angle taken by the former must be reduced to that taken by the latter, as in the following example.

From a station at D, in the

D

P

B

horizontal plane DSO, I took the angle PDS, subtended by the tops of two towers 37° 53′ 20′′, and also the angles of elevation BDO = 4° 23′ 55′′, and PDS 4° 17′ 21". The height of the tower BO is 40 yards, and that of PS 30 yards, from which it is required to find the horizontal distance of my station from each of the towers, and their horizontal distance from each other.

By the above formula, the angle SDO 37° 59′, which, by the sextant, was 37° 53′ 20′′. In this example the angle taken by the sextant requires a correction of 5' 40", shewing that the angles taken by the sextant should never be depended on, except when the objects are situated on the same horizontal plane.

The above formula is sufficiently short for the purpose of a single observation; but when several hundred

of such are to be computed, it becomes of importance to save both time and labour by the application of a more expeditious and easy formula. formula. When the angles of elevation or depression are small, the problem admits of a convenient approximation, which Legendre recommends when the angles do not exceed 2° or 3°.

Let PS and BO be represented by H and h, and A be the observed angle BDP, and a the correction: then A+ will be the required angle ODS. Also, let (H+h)=p, and (H-h)=q and r-radius of the A-2. cot. § A.

sphere; then x=22, tang.

In practice, p and q are generally given in seconds, and therefore r should be expressed in seconds also.

The peculiar construction of the modern theodolite renders it in every respect well adapted for taking horizontal and vertical angles correctly. This instrument, therefore, ought to be the only one used in extensive surveys, or in surveys requiring more than ordinary

accuracy.

To exemplify what has been said, we here give the calculations of a few of the triangles which proceed from the base line. In ordinary surveys, it will be sufficient to measure the angles to one minute by one reading. In the triangles which we here calculate, the angles are given to seconds, being the mean of at least two readings on different parts of the horizontal limb of the theodolite.

The few triangles, whose calculations we here subjoin, will be sufficient to shew how to find the sides and areas of the rest. A greater number of triangles being calculated, would tend rather to tire than instruct

the reader. For this reason the field-book has been omitted, while we exhibit the triangulation and map of the entire survey, with a view to shew how to conduct. large operations.

Having two angles and one side of a triangle, the other sides are found by the following calculation. In the triangle ABC (see the triangulation,) given the angle ACB=42° 19′ 28′′