By the mensuration and protraction of lines and angles we determine the lengths, heights, depths, or distances, of bodies and objects. And this branch is commonly called Heights and Distances, or Altimetry and Longimetry. Accessible lines are measured by applying to them some certain measure, as an inch, a foot, &c. a numbet of times ; but inaccessible lines must be measured by taking angles, or by some similar method, drawn from the principles of Geometry and Trigonometry. When instruments are used for taking the quantities of the angles in degrees, the lines are then calculated by Trigonometry. In the other methods the lines are calcu. lated VOL. II. X lated from the principle of similar triangles, without any regard to the quantities of the angles. Angles of elevation, or depression, are usually taken either with a theodolite, or with a quadrant, divided into degrees and minutes, and furnished with a plummet sus, pended from the centre, and two sights fixed perpendicalarly upon one of the radii. PROBLEM I. To take an angle of altitude and depression with the quadrant. Let A be any object, as the top of a tower, hill, or other eminence ; or the sun, moon, or a star ; and let it be required to find the measure of the angle ABC, which a line, drawn from the object, makes with the horia zontal line BC. Fix the centre of the quadrant in the angular point, and move it round there as a centre, till with one eye at D, the other being shut, you perceive the object A through the two sights E, F ; then will the arc GH of the quad. tant, cut off by the plumb line BH, be the measure of the angle ABC required. The angle ABC of depression of any object A is taken in the same manner, except that here the eye is applied to the centre, and the measure of the angle is the arc GH. The observations with the quadrant, necessary to determine the heights and distances of objects, will be sufficiently apparent from the manner, in which the following examples are proposed ; and the solutions may casily be given by any one, who understands Plane Trigonometry The construction of the figures to the following exam. ples is omitted ; but'is to be performed as in the problems of Trigonometry PROBLEM By the Sliding Rule. Set 40 on C to 46-4 on D; then against 24 on D stands 1095 32 on D stands 19'0 57 on D stands 62'0 Sum 91.5 ale gal. Set 40 on C to 42o on D ; then against 24 on D stands 1360 32 on D stands 232 57. on D stands 75'0 Sum 111'2 wine gal. 2. What is the content of a cask, whose length is 20, the bung diaineter being 16, the head diameter 12, and the diameter in the middle between them 143? Ans. 11'4479 ale gal. 13-9010 wine gal. PROBLEM VI. of an To find the content of any cask from three dimensions only. RULE. Add into one sum 39 times the square of the bung diameter, 25 times the square of the head diameter, and 26 times the product of the two diameters ; then multiply the sum by the length, and the product by :00034 for ale gallons, or by 2003, or '0000370, for wine gallons. EXAMPLES. EXAMPLES. |