CHAPTER XI. APPLICATION OF TRIGONOMETRY TO MEASURING HEIGHTS AND DISTANCES. 1. T is by means of the relations between the sides and angles of triangles that the operations of surveying are conducted. To go into the details of this branch of the subject with any degree of completeness would alone require as large a volume as has been devoted to this treatise. It is however the expedients to which it is necessary to have recourse in actual practice, which constitute a large portion of a work specially devoted to surveying. The trigonometrical principles of the calculations involve but little difficulty, and we shall be able by the solution of a few general problems to illustrate them sufficiently for the purposes of this work. 2. In making any measurement the first requisite is a fixed standard length very accurately measured, to which we can refer other lengths as a unit. On paper, and for small distances, an accurately marked ruler or some equivalent instrument might be employed. When however the measurements are on a large scale, a straight line is very accurately measured, and by means of the principles of Trigonometry, we are able to express the other distances we require in terms of this length. This line or base, as it is technically termed, answers the same purpose as a ruler or measuring rod, except that it is stationary, and the other distances to be measured are calculated by referring them to the base, by means of trigonometrical calculations. There are various instruments which enable the surveyor to measure the angles which any given point subtends at two other points. He must also be furnished with a set of logarithmic and trigonometrical tables. 3. The angle of elevation of a point anywhere situated above the eye of the observer, is the angle which the straight line joining his eye and the point makes with the horizon. The angle of depression of a point situated below the eye of the observer, is the angle which the line joining his eye and the point makes with the horizon. 4. To find the distance of an inaccessible object upon a horizontal plane. Let P be the distant object which owing to some obstacle cannot be reached. Let A be the place of the observer. Measure a base AB, the length of which call a. Measure the angles PAB and PBA, which suppose to be found to be a and ẞ respectively. Then we have .. log AP=log a+ log sin ẞ-log sin (a+B), and log BP log a+ log sin a-log sin (a +ß). Thus AP and BP are known from the tables. 5. This problem may also be solved without any angular measurement, which in the absence of instruments may be very convenient. The method is due to Colonel Everest, who surveyed India. It is also given in Mr Galton's Art of Travel, page 287 (3rd edition). Having measured the base AB as before, measure any length Ad in AB and an equal one Ae along AP. Then measure de. Similarly measure Bƒ and Bg each equal to Ad or Ae, and then measure fg. Since eAd is an isosceles triangle, a perpendicular from A upon de will bisect de. Thus a and B are known by the aid of the tables, and the problem is now completed as before. Mr Galton has drawn up a very ingenious table in which, by referring to a value of de given in one column and the value of fg given in another, the magnitudes of the angles A and B and the distances AP, BP are given. In the table the length of the lines Ad, Ae, Bf, Bg is supposed to be one-tenth of the base AB. 6. To measure the distance of the summit P of a hill from a point A, and the height of the hill above the horizontal plane in which A lies. B P A Measure a base AB of length a say. Observe the angles BAP (a), ABP (B), and the angle of elevation PAC (y). Then, from the triangle PAB, we have |