Required the distance of the object O from the station C, and

its height above the plane of the base BC.

The angles of the triangles OAC, OAB, OBC, are

These three triangles form the sides of the pyramid whose vertex is O, and base ACB: we have therefore to find its height OD, and the point D where the perpendicular OD meets the plane of the base.

The sides of the triangle ABC

BC = 370

AC 369

AB = 3305 give the angle ACB = 53° 8′ (228).

Let OP, OR, meet AC, BC produced, at right angles in P and R; and suppose OD is the perpendicular on the plane of the base, and join PD, CD, RD. Then OCP = 39° 54′ (the supplement of OCA); and OCR 30° 50' (the supplement of OCB);

Then, 39° 54' cosine 9.884889
CO log. 2603957

CP 308 2 log. 2'488346

30° 50′ cosine 9.933822 CO log. 2 603957 CR 345 log. 2.537779

Now in the quadrilateral CRDP (in the plane of the base ABC) we have the sides CP, CR, and their included angle = 53° 8', whence (226) we get the angle CRP = 57° =CDP (because the angles CPD, CRD being right ones, a circle will circumscribe the quadrilateral), therefore CP and all the angles of the right angled triangle CPD are given; whence the distance CD 367-5 yards; from this side and the hypotenuse CO, the perpendicular OD will be found 162-3 yards.

If the triangle ABC is on level ground, CD is the horizontal distance of the object O from the station C, and OD its height.

OF TERRESTRIAL REFRACTION.

235. As the apparent or observed elevations of objects are always greater than the true, it may not be improper to give a short explanation of Refraction.

Let E be the place of an observer's eye, EH the horizontal line, and O an object, suppose on the summit of a distant hill.

E

T

H

Then if the rays of light proceeded from the object O to the eye at E in a straight line, the object would appear in its true place at O, and OEH would be the elevation (considering EO as a right line); but the rays in passing through the atmosphere are continually attracted or bent downwards from a rectilineal direction, by which means the object is seen in the direction ET, which is supposed to be a tangent to the curve at E, and therefore the apparent or observed elevation is the angle TEH; and the angle TEO, or rather the angle comprehended by TE and a right line from O to E, will be the refraction.

This Refraction which is called the terrestrial, to distinguish it from that which affects the altitudes of the heavenly bodies, is not constant at the same elevation and distance, but is found to vary with the changes in the atmosphere, as heat, a different density, moist vapours, &c. &c. At the distance of 8 or 10 I miles it is sometimes no more than about 30 seconds, but in particular states of the air we find it amount to upwards of 2 minutes.

236. It is a difficult operation to determine the exact quantity of refraction at any particular time. The following

OF REFRACTION.

method however, has been successfully practised in the Trigonometrical Survey carried on by order of the Board of Ord

nance.

Let A and B be two stations, SS the intercepted or corresponding arc of the earth's circumference, C the centre of the earth; AG, BH, the horizontal lines at A and B drawn to meet CG, CH.

An instrument being at each of the stations A and B, the reciprocal observations are made at the same instant

R

of time, which is determined by means of signals or watches previously regulated for that purpose; that is, the observer at A takes the depression (for example) of B while the other person at B observes the depression of A.

If a and b represent the apparent places of the objects A and B, the angle AB is the refraction at A, and aBA that at B; therefore, half the sum of those angles will be the refraction, if we suppose it equal at each station.

In the quadrilateral AOBC the angles at A and B are right ones, therefore the sum of the other two angles at O and C are equal to two right angles, and consequently the angles OAB, OBA are together equal to the angle C or are SS, therefore if the sum of the two depressions or angles HBa + GAb is taken from the sum of the angles HBA + GAB or the angle C, the remainder is the sum of both refractions or angles aBA + bAB; therefore half the difference between the sum of the two depressions and the contained arc SS (or angle C) is the refraction.

TT 2

If one of the objects (B) instead of being depressed, is ele vated, suppose to the point R; then the sum of the angles dAB + dBA will be greater than the sum OAB + OBA (or angle C) by the angle of elevation RAG; but if from the sum dAB + dBA we take the depression HBA, there will remain dAB+ aBA the sum of the two refractions; therefore, if the depression be subtracted from the sum of the contained arc and elevation, half the remainder is the refraction in this case.

It is almost unnecessary to remark that the distance between the places of observation A and B should be known sufficiently near to give the contained arc SS true to a very few seconds of a degree. The refraction however, is generally too minute to be of consequence in the operations with a common Theodolite, which are usually confined to moderate distances.

OF SURVEYING.

237. SURVEYING is the Art of laying down the true positions of the principle features, and exhibiting an exact representation of the boundary of a country, or any part thereof, on a plane or paper, so that the dimensions, &c. may be readily measured by means of a scale of miles, yards, chains, &c. &c. When fields or other inclosures, and Gentlemen's estates are surveyed, not only a correct delineation of the boundaries is required, but the superficial content in acres, &c. must be computed. This is called Land Surveying, or Land Measuring.

238. To lay down or make a Map or Plan of any consider able extent of Country, a series of connected triangles should be carried in all directions to its boundaries from a long and well measured base as the foundation: For that purpose the most conspicuous points, as the summits of hills, roofs of

church-towers, &c. &c. must be chosen for stations; and all remarkable objects in view should be intersected at every place. where the instrument for taking the angles is set up. When a high pointed spire, or the like, upon which the instrument cannot be conveniently placed, presents itself as a proper situation for carrying on the triangles, it should always be intersected from several stations in order to compare, or correct the connecting distances by a computation from independent triangles.

239. It will be adviseable to observe every angle of the prin cipal triangles if the situations permit; then, as the sum of the three angles of each triangle ought to be very nearly equal to two right ones, the deviations will in some measure, enable us to judge of the accuracy of the work.

240. The sides of the principal triangles should be calcu lated. But objects situated within those triangles may be laid down by means of a protractor: these objects however, should if possible, be intersected from three stations.

241. The principal triangles and interior objects laid down on a large scale, suppose 5 or 6 inches to the mile, will be a sufficient ground work for Military sketches which are usually drawn by eye without any actual measurement. The method of adapting a scale to the Plan; and enlarging or diminishing it to any particular size is given in Art. 167.

212. But the most difficult and tedious operation connected with a Survey, is that of measuring a base-line accurately. We shall therefore recommend a perusal of the Account of the Trigonometrical Survey (236) to those who may engage in an undertaking of this kind when great exactness is required. A base for common surveys may be measured with a 20 feet deal-rod: for this purpose a rope not less than 100 yards should be stretched very tight along the ground; the rod must then be applied to the rope, and its extremity may be marked with a small pin stuck in the rope to preserve the distance while the rod is removed.