69. A plane perpendicular to a horizontal plane, is called a vertical plane. 70. The lines of a horizontal plane, as well as all lines which are parallel to it, are named horizontal lines. 71. Lines which are perpendicular to a horizontal plane, are called vertical lines, and lines which are inclined to it, oblique lines. 72. The horizontal distance between two points, is the horizontal line intercepted between the two vertical lines passing through those points. 73. A horizontal angle is one whose sides are horizontal ; its plane also is horizontal. A horizontal angle may also be defined, the angle included between two vertical planes passing through the angular point, and the two objects which subtend the angle. 74. A vertical angle is one whose plane is vertical. 75. An angle of elevation, is a vertical angle having one of its sides horizontal, and the inclined side above the horizontal side. 76. An angle of depression, is a vertical angle having one of its sides horizontal, and the inclined side under the horizontal side. 77. An oblique angle, is one whose plane is oblique to the horizontal plane. 78. All lines, which can be the object of measurement, must belong to one of the three classes above named: that is, they are either horizontal, vertical, or oblique. The angles also are distributed into three classes; horizontal angles, vertical angles, and oblique angles: the class of vertical angles being subdivided into angles of elevation, and angles of depression. CHAPTER II. OF THE MEASUREMENT AND CALCULATION OF LINES AND ANGLES. 79. It has been shown (47), that at least one side and two of the other parts of a plane triangle, must be given or known, before the other parts can be found by computation. When, therefore, it is proposed to ascertain distances by trigonometrical calculations, the first steps necessary are, to measure certain lines on the ground, and also to measure as many angles as are required to render at least three parts of every triangle known; and then, by the aid of trigonometry, the other sides and angles may be calculated. 80. Our attention, then, is directed first, to the measurement of lines; secondly, to the measurement of angles; and thirdly, to the calculations for the unknown or required parts. 81. Any tape, rod, or chain, by means of which we can ascertain equal parts, may be used as a measure, and the unit of measure may be a foot, a yard, a rod, a mile, or any other ascertained distance. The measure in general use is a chain of 4 rods, or 66 feet, in length, called Gunter's chain, from the name of the inventor. This chain is made up of 100 links; every tenth, from either end, is marked by a small brass plate attached to it, and notched, to designate its number from the end. The division of the chain into one hundred equal parts, is a very convenient one, since the divisions, or links, are decimals of the whole chain, and in the calculations may be treated as such. The length of the chain being 4 poles, or 66 feet, is equal to 792 inches, which being divided by 100, gives 7.92 inches for the length of each link. A mile being equal to 320 rods, 80 chains=1 mile, 40 chains a mile, and 20 chains=4 Besides the chain, there are wanted for measuring, 10 marking pins, and two staves about 6 feet in length, having a spike in the lower end, to aid in holding them firmly, and a horizontal strip of iron passing through them, to prevent the chain from slipping off; these staves are to be passed through the rings, at the ends of the chain. TO MEASURE A HORIZONTAL LINE. 82. Being provided with a chain, the staves, and the marking pins, let two signal staves be planted, one where the measurement is to begin, and the other where it is to terminate; or, perhaps, some distant object, in the direction of the line to be measured, may serve as a sufficient guide, and render the second staff unnecessary. Let the ten marking pins, and one end of the chain be taken by the person that is to go forward, who is called the leader, and let him plant the staff as nearly as possible in the direction of the stations: then, taking the staff in his right hand, let him stand off at arm's length, so that the person at the other end of the chain can align it exactly; when the alignment is made, let the chain be stretched, and a marking pin placed: and so on till the whole line is measured. Great care must be taken to keep the chain horizontal; and if the acclivity or declivity of the ground be too great to admit of measuring a whole chain at a time, a part of a chain only must be measured: the sum of all the horizontal lines so measured, is evidently the horizontal distance between the stations. 83. We come now to the measurement of angles, and for this purpose several instruments are used. The one, however, which affords the most accurate results, and which indeed can alone be relied on for nice work or extensive operations, is called a theodolite. This instrument only will be described at present, others will be subsequently explained. OF THE THEODOLITE. 84. The theodolite is an instrument used to measure horizontal and vertical angles. It is usually placed on a tripod ABC (Pl. II. Fig. 1), which enters by means of a screw the body of the instrument. Through the horizontal plate DE, four small hollow cylinders are inserted with female screws in their interior, which receive four screws with milled heads, that work against a second horizontal plate, FG. The upper side of the plate DE terminates in a curved surface, which encompasses a ball, that is nearly a semisphere, with the plane of its base horizontal. This ball, which is hollow, is firmly connected with the smaller base of a hollow conic frustum, that passes through the curved part of the plate DE, and screws firmly into the curved part of the second horizontal plate FG. A hollow conic spindle passes through the middle of the ball, and the hollow frustum with which it is connected. To this spindle, a third horizontal and circular plate HI, called the limb of the instrument, is permanently attached. Within this spindle, and concentric with it, there is a second spindle, called the inner, or solid spindle. To this latter, is united a thin circular plate, called the vernier plate, which rests on the limb of the instrument, and supports the upper frame work. The two spindles terminate at the base of the spherical ball, where a small male screw enters the inner one, and presses a washer against the other, and the base of the ball. On the upper surface of the plate FG, rests a clamp which goes round the outer spindle, and being compressed by the clamp screw K, is made fast to it. This clamp is thus connected with the plate FG. A small cylinder a, is fastened to the plate FG : through this cylinder a thumb screw L passes, and works into a small cylinder b, connected with the clamp. The cylinders b and a admit of a motion round their axis, to relieve the screw L of the pressure which would otherwise be occasioned by working it. Directly above the clamp, is the lower telescope MN. This telescope is connected with a hollow cylinder, which is worked freely round the outer spindle, by the thumb screw P, having a pinion acting with a concealed cog-wheel, that is permanently fastened to the limb of the instrument. By means of a clamp screw Q, the telescope is made fast to the limb, when it will have a common motion with the limb and outer spindle. The circular edge of the limb is chamfered, and is generally made of silver, and on this circle the graduation for horizontal angles is made. In the instrument described, the circle is cut into degrees and half degrees; the degrees are numbered from 0 to 360. On the circular edge of the vernier plate, is a small space of silver called a vernier; this space is divided into 30 equal parts, and numbered from the line marked 0 to the left. There are two levels attached to the vernier plate, at right angles to each other, by small adjusting screws; one of them is seen in the figure. The vernier plate, turning with the inner spindle, is moved round by the thumb screw T, which has a pinion working with a concealed cog-wheel, that is part of the limb of the instrument. The clamp screw S, fastens the vernier plate to the limb. In some theodolites, there is a tangent screw, similar to the screw L, by means of which, the smaller motions of the vernier plate are regulated. There is a compass on the vernier plate that is concentric with it, the use of which is explained under the head, Compass. The frame work which supports the horizontal axis of the vertical semicircle UV, and the upper telescope with its attached level, rests on the upper plane of the vernier plate, to which it is made fast by three adjusting screws, placed at the angular points of an equilateral triangle. The vertical semicircle UV, is called the vertical limb; its motions are governed by the thumb screw Z, which has a pinion, that works with the teeth of the vertical plate. On the face of the vertical limb, opposite the thumb screw Z, the circle is divided into degrees and half degrees: the degrees are numbered both ways from the line marked 0. There is a small plate resting against the graduated face of the vertical limb, called the vernier; it is divided into 30 equal parts, and the middle line is designated by 0. On the other face of the vertical limb, are two ranges of divisions, commencing at the 0 point, and extending each way 45°. The one shows the vertical distance of any object to which the upper telescope is directed, above or below the place of the instrument in 100th parts of the horizontal distance: the other, the difference between the |