(Art. 82.) Log. tan. + B-C=log. AC—AB+log. tan. 90°—÷A —log. AC+AB; then ÷ B+C++B-C=ang. B} by Art. 69. +B+C+B-C-ang. whence the three angles A, B, and C, are known *. EXAMPLES.-1. Given the side AB=12, AC=11, and BC= 10, to find the angles A, B, and C. By construction. 1. Draw the straight line BC=10, taken from any convenient scale of equal parts, from B as a centre with the radius 12 describe an arc, and from C with the radius 11 cross the above arc in A, (both the latter distances being taken from the same scale with BC,) and join AB, AC. 2. Measure the angles by means of the scale of chords, or protractor, and they will be nearly as follows; viz. A=514, B 59°, and C=69°. By calculation. First, let AD be perpendicular to BC; see the last figure but one. BD+DC=BC=10 Then log. sin. BAD=log. BD+10-log. AB=0.7888751 +10-1.0791812=9.7096939; ·. ang. BAD=39° 50′, and ·.· its complement B=59° 10′. In like manner, log. sin. CAD=log. CD+10-log. CA= 0.5854607+10-1.0413927-9.5440680. ang. CAD=20° 29'; the complement of which is 69° 31' the angle C. Also the ang. BAC-BAD+CAD=30 50'+20° 29′= 51. 19'. • On having found the angle A, the remaining angles B and C may be found (perhaps more conveniently) by Art. 67. thus BC: CA:: sin A: sin B= CA.Sin A BC known. .. ; B is known; whence also C=180-4+ B; . C is likewise The solution without a perpendicular; see the last figure. Natural cos A .6250000 angle A=51° 19′; ·. C+B=180°-51° 19′ AB+AC=0+10.3184222-1.3617278-8.9566944 ·.· C-B 2 1. Extend from 10 to 23 on the line of numbers; this extent will reach, on the same line, from 1 to 2, the difference of the segments of the base. 2. Extend from 12 to 6.15 on the numbers; this extent will reach on the sines from 90° (radius) to 30° 50′=BAD, the complement of which is 59° 10' ang. B. 3. Extend from 11 to 3.85 on the numbers; that extent will reach from 90° to 20° on the sines, the complement of which is 69C, Second method. 1. Extend from 264 (=2BA.AC) to 165 (=BA)2+AC2-BC2) on the numbers; that extent will reach from 90° to 38 on the sines, the complement of which is 514-angle A. 2. Extend on the numbers from 23 to 1; that extent will reach from 64° to 45°; and back again to 54 on the tangents, for half the difference of the angles B and C. Ex. 2. Given the three sides, viz. AB=100, AC=40, and BC=70.25; to find the three angles? Ans. ang. A=33° 35′, ang. B=18° 22′, ang. C=128° 3'. 3. Given AB=368.95, 4C=472, and BC=700, to find the angles? Ans. ang. A=112° 6′, ang. B=38° 40′, ang. C 29° 14'. THE APPLICATION OF PLANE TRIGONOMETRY TO THE FINDING OF THE HEIGHTS AND DISTANCES OF INACCESSIBLE OBJECTS. The uses to which Plane Trigonometry may be applied are so various and extensive, that merely to point them out would require a very large volume; and to understand them, the student must be well acquainted with Geography, Astronomy, and the numerous branches of Natural Philosophy, of which this science forms a necessary part. At present we shall confine ourselves to one of its immediate and obvious applications, namely, that of determining the heights and distances of inaccessible objects. The following instruments are used in this branch of mensuration, namely, a quadrant, a theodolite, a mariner's compass, a perambulator, Gunter's chain, measuring tapes, a measuring rod, station staves, and arrows; the description and uses of which are as follow: 84. THE QUADRANT 'is an instrument for measuring angles in a vertical position; that is, to determine the angular altitude f Besides the common surveying quadrant, of which that described above is the simplest form, there are various other kinds, as the astronomical quadrant, the sinical quadrant, the herodictical quadrant, Davis's, Gunter's, Hadley's, Cole's, Collins's, Adams's, and some others. Quadrants may be had at any price from one to twelve guineas. The height of an object may be taken in two senses, viz. 1. its perpendicular distance (in fathoms, yards, feet, &c.) from the ground; 2. its angular height, or the number of degrees contained in the angle at the eye of the observer, which the perpendicular height subtends; the former we have, for distinction, denominated height, the latter altitude. of any proposed object. 0 ABC is a quadrant, to the centre C of which the weight W is freely suspended, by means of the string CW; ss are two sights, through which the eye of an observer at A sees the object O. The arc AB of the quadrant is divided into degrees, which are subdivided into halves, quarters, or single minutes. In using this instrument, the observer turns it about the centre C, until the ob ject O is visible through the sights ss; and as he turns it, the line CW, revolving freely about the centre C, moves along the circumference AB; when he sees the object 0 through the sights, the arc BW will be the measure of its angular altitude, that is, of the angle OAD. Draw OD perpendicular, and AD parallel to the plane of the horizon; then because the angles at E and D are right angles and the angle A common, the triangles CAE, OAD are equiangular (32. 1.), . the angle ACE=AQD; but DOA+ DAO (a right angle) ACB, from these equals take the equals DOA=ECA, and the remainder DA0=ECB. And since the arc BW is the measure of the angle ECB (Part 8. Art. 237.) it is likewise the measure of DAO, or of the angular altitude of the object O above the plane of the horizon. 85. THE THEODOLITE, in its simplest form, consists of a brass * Some of the best theodolites are adapted to measuring vertical as well as horizontal angles to a single minute; being fitted with vertical arch, level, telescopic sights, and rack-work motions. The prices of theodolites are from two to forty guineas. The circumferentor is an instrument for measuring horizontal angles, chiefly used in wood lands, and its price is from two to five circle of about a foot in diameter, having its circumference divided into 360 degrees, and these subdivided into halves, quarters, or minutes; the index sCs turns about the centre C, and has fixed on it two sights s s; there are likewise fixed on the circumference two-sights nn; this circle is fixed in a hori zontal position on three legs of a convenient height for making observations. The theodolite is used for measuring the angular distances of objects situated on the plane of the horizon; thus, Let A and B be two objects, place the instrument in such a position that one of them, as A, may be seen through the fixed sights n and n by an eye at F. Turn the index s s about the centre C, until the other object B appears B through the sights ss to an eye situated at E; then will the angle ACB, which is measured by the arc nr, be the angular distance of the given objects and B. 86. THE MARINER'S COMPASS is an instrument used for finding the position or bearings of objects with respect to the meridian, and for determining the course of a ship: what principally requires explanation is the card; it is a round piece of stiff pasteboard, having its circumference divided into thirty-two guineas. The semicircle is a much simpler and cheaper instrument than the theodolite, and serves very well for measuring angles on the plane of the horizon where very great accuracy is not required. The invention of the mariner's compass is usually ascribed to Flavio Gioia, an Italian, A. D. 1302; but it is stated by some authors that the Chinese had a knowledge of it as early as the year 1120 before Christ. The price of this useful instrument is from half-a-crown to twelve guineas. |