C and A are found, by Problem v of the last Chapter; and then the angle B, by Problem I of this Chapter. FORMULA. Ι ba+c: a-c: d. CD=b+d. AD=b-d. CD a radius: sec C. AD c radius: sec A. d being the difference of the segments of the base. EXAMPLE. The three sides of a triangle are 28, 40, and 16: what are the angles? NOTE. The rule proceeds on the supposition that the two sides are unequal. If they are equal, the first part of the rule is evidently not required, since, in that case, the segments are each half the base. EXERCISE 1. The three sides are 735, 595, and 350 feet what are the three angles? Ans. 28° 4′+, 98° 48', and 53° 8' 2. What are the three angles, supposing the three sides respectively 0.532, 0.259, and 0.371 miles? Ans. 39° 33'+, 26° 24′ —, and 114° 3′-. 3. The base of a triangle measures 64 ft. 8 in., and the two sides respectively 45 ft. 7 in., and 36 ft. 3 in. : what are the angles? Ans. 32° 57'+, 43° 12′+, and 103° 51′—. CHAPTER IX. OF THE APPLICATIONS OF PLANE The principal applications of Trigonometry are to the measurement of Heights and Distances by means of angular observations, to Navigation, to Astronomy, and to Mechanics and the other departments of Natural Philosophy. Trigonometry is much employed in Land-surveying, when conducted on an extensive scale, such as the National Survey of the British Isles, but the processes employed are the same as those for the determination of Heights and Distances. When a survey is spread over so large a portion of the Earth's surface that it cannot be regarded as a plane, the aid of Spherical Trigonometry then becomes essential. Questions relating to that class of operations do not belong to the subject of this volume. Many works on Trigonometry describe various simple modes of ascertaining unknown heights and distances entirely independent of angular measurements, and include questions relating to these methods. Such questions belong more properly to Mensuration of Lines. In the course of the following Exercises, allusion will frequently be made to instruments employed for the mea surement of angles of observation,—that is, for determining the angular elevation or depression of any distant object above or below the horizon, or the angular distance of any two such objects from each other, the point of observation being taken as the centre or angular point. For the sake of perspicuity it is proper to explain here briefly the nature of each of these instruments and its appropriate uses. The first and simplest of these is the old-fashioned Quadrant represented in the an nexed diagram. It is in the form of a sector of a circle, is commonly made of wood, and has its arc divided into ninety degrees commencing at the extremity A. A plumb line is suspended from the centre by a thread, and on the side opposite A are placed two sight vanes, E and E': these are two pieces of brass with a small hole in each, or other E A mark, the said holes or other marks being in a line parallel to the adjacent radius. In taking angles of elevation we look from E to E'; but, in taking angles of depression, we look from E' to E. The instrument is either held in the hand by the person taking the observation, while the degrees are read by an assistant, or is placed on a support, and, when properly directed (with the object in a line with the sights), is fixed in that position till the angle is read. An improved form of the quadrant has a telescope, T, instead of the sight vanes, and a spirit level, L, to serve the same purpose as the plumb line, in a different manner. The degrees are numbered from B both ways, the instrument being limited to angles of not more than 45 degrees, whether of It is elevation or of depression. placed on a stand on which is a fixed index pointing out the number of B degrees, while the quadrant turns on the fixed centre, O. The spirit level is attached either to the quadrant or to the fixed support. In the latter case it remains level during the operation: in the former it is level only when the index is at zero. In adjusting the instrument previously to the observation, the index is fixed at zero, the telescope is turned towards the object, and the stem is brought as nearly as possible to an upright position by means of the feet on which it stands; and, when that is done, a screw acting on the stem brings the spirit level to a horizontal position and completes the adjustment in the direction of the telescope, while that in the cross direction, being considered of comparatively little importance, is determined by the eye, and attained by the use of the feet of the instrument only. The exact point to which the telescope is directed is determined by two hairs or other fine lines, crossing the sight at right angles to each other, -one being horizontal, the other vertical. The quadrant in either of the preceding forms is used only for vertical angles, that is, angles in a vertical plane. But even for that purpose it is an imperfect instrument, and is not employed in nice operations. In these it is superseded by the following: The Theodolite consists of a vertical semicircle, S, connected with a horizontal circle, C, both graduated at the circumference. The graduated horizontal circle remains fixed during an observation, while another concentric circular plate, I, turns upon it, having an index on its margin: but the vertical arc moves, while its index remains fixed. While the vertical semicircle and the horizontal index-circle turn freely, each on its own axis, the axes themselves are firmly connected with each other. Two spirit levels, L', L', placed on the circle, at right angles to each other, deter T mine its accurate position in a horizontal plane, and, at the same time, the position of the semicircle in a vertical plane; the instrument being brought nearly into that position by shifting the feet on which it stands, and then adjusted exactly by the screws A, A. A telescope, T (having its sight crossed by two hairs, as before described), is fixed to the vertical semicircle parallel to its diameter, and, consequently, turning with it, so that the angular elevation or depression of the telescope is determined by the number of degrees passed over by the index at the circumference; the instrument, previously to any observation, being so adjusted (by means of a third level L), that, when the telescope is directed to an object in the horizon, the index of the vertical semicircle points to zero.* Again, when the telescope turns from one object to another, both in the horizon, it carries the index-circle round with it, the graduated circle showing the number of degrees of revolution. But when the telescope turns from one object to another, neither in the same vertical nor in the same horizontal plane with each other and with the place of observation, both the vertical semicircle and the horizontal indexcircle turn on their axes; but, in that case, neither of them shows the angular distance between the two observed objects. The former merely shows the angular elevation or depression of each object above or below the horizon, while the latter shows the difference of their bearings or horizontal directions, that is, the angular distance of the seats of the two objects on the horizontal plane of the point of observation. Thus, E being the point of observation, let p and q be two other points in the same horizontal plane with it, and let Pp and Qq be two perpendiculars to that plane. If the telescope at E turn from the object P to Q, the horizontal circle turns exactly the same number of degrees as if the telescope had been directed first to p and then to q; for the telescope would evidently pass E from P to p, or from Q to q, without any movement of the horizontal circle. Therefore the horizontal circle measures the angle pEq, not the angle PEQ, although the telescope turns from P to Q. The latter angle, when * Or if not exactly, the error must be added or subtracted at each observation. |