THE TRIGONOMETER. SECTION I. DESCRIPTION OF THE PARTS, COMBINATION, AND GENERAL METHODS OF USING THE INSTRUMENT. 1. The Distance of a line, is its rectilinear length.* 2. A Meridiant is a line that runs directly North or South, in the plane of the visible horizon, or on the map of a field. 3. A Parallel of Latitude is a line running directly East or West through any point, at right angles to the Meridian which passes through the same point. 4. A Circle is a plane figure, bounded by one line called the circumference, from which all straight lines drawn to a certain point within the figure are equal to each other; and this point is called the center of the circle. Thus ABE, Fig. 1., is a circle; of which F is the center, ABCDE the circumference; and each of the straight lines AF, BF, CF, DF, is called a radius of the circle. 5. A Diameter of a circle is a straight line passing through its center, and terminated both ways by the circumference; as * When confined to Land Surveying, the distance of a line is its horizontal length. † All Meridians passing through any survey of ordinary extent may be considered straight, parallel lines. Strictly, Meridians are vertical circles, cutting the horizon at its North and South points at right angles. Hence no two of them are any where exactly parallel. So also strictly, a parallel of latitude is a circle passing through the earth at right angles to its axis. It is called a parallel, because it is parallel to the plane of the Equator. In ordinary field operations in which the compass is used, and also in mapping or plotting, the direction of the needle, for the sake of convenience, is for the time assumed as the meridian; the true meridian and the variation of the needle always being appended when the map is finished. AC, Fig. 1. It divides the circle into two equal parts called 6. An Arc is any portion of the circumference; as BC, or CD, Fig. 1. 7. A Quadrant is the fourth part of a circle; as AFB, Fig. 1. 8. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds, &c. 9. The Measure of an angle is the arc intercepted between the two lines that form the angle; and the angular point is the center of the circle. Thus the angle ABC, Fig. 2, is measured by the arc DE, and contains the same number of degrees that the arc does. D FIG. 3. H G E A Hence, the measure of a right angle is 90 degrees; for DG, Fig. 3, which measures the right angle DCG, is a fourth part of the circumference, or 90 degrees. Degrees are usually indicated by a small circle, thus (°); minutes thus ('); and seconds thus ("); 20° 15′ 25′′ therefore denotes that the arc or angle contains 20 degrees, 15 minutes, and 25 seconds. E 10. The Complement of an arc or angle, is the difference between the arc or angle and 90°; and the supplement of an arc or angle, is what it wants of 180°. Thus BG, Fig. 3, is the complement of the arc AB; and BCG is the complement of the angle ACB. So also the arc BD is the supplement of the arc AB; and BCD is the supplement of the angle ACB. 11. The sum of the three angles of a triangle is equal to two right angles, or 180°. Hence, if the sum of any two angles of a triangle be subtracted from 180°, the remainder will be the third angle. And if one of the angles be subtracted from 180°, the remainder will be the sum of the other two angles. Hence, also, if the triangle be right angled, and one of the acute angles be subtracted from 90°, the remainder will be the other acute angle. FIG. 4. N 12. The Bearing or Course of a line, is the angle which it makes with a meridian passing through one of its terminal points. It is reckoned from the North or South points of the horizon towards the East or West points. Thus, supposing NS, Fig. 4, a meridian, the angle NAB, is the bearing or course of the line AB; and if it contains 35°, it is read North 35° West; or N. 35° W. 13. The Reverse Bearing of a line, is the bearing taken from the other end of the line. Thus, c the reverse bearing of the line AB, Fig. 4, is BA, or the angle S'BA; that is, South 35° East, or S. 35° E. S' S A Sometimes the term Course signifies the Bearing and Distance of a line taken collectively. 14. Latitude is the distance measured on a given meridian, between two parallels of latitude. It is called also Northing or Southing; or, Difference of Latitude. Thus AC, Fig. 4, is the Northing of the course AB; and BC' is the Southing of BA. 15. Longitude or Meridian distance is the distance measured on a parallel of latitude, of any point from a given meridian. Thus, in Fig. 4, the longitude of the course AB; that is of the point B, is BC. 16. Double Longitude is the sum of two adjacent meridian distances; being always either the base of a right angled triangle, or the sum of the parallel sides of a trapezoid. Thus, the double longitude of the right angled triangle ABb, Fig. 25, Art. 54, is Bb; and that of the trapezoid bBCc, is Bb+Cc. |