Fig. 6. 16. A chord is a right line drawn from one end of an arc to the other, and is the measure of the arc; as HG is the chord of the arc HIG. Fig. 6. A B D Nore. The chord of an arc of 60 degrees is equal in length to the radius of the circle of which the arc is a part. 17. The segment of a circle is a part of a circle cut off by a chord ; thus the space comprehended between the arc HIG and the chord HG is called a segment. Fig. 6, 19. A sector of a circle is a space contained between two radii and an arc less than a semicircle ; as BCD or ACD. Fig. 6. 20. The sine of an arc is a line drawn Fig. 7. from one end of the arc, perpendicular to the radius or diameter drawn through the other end: or, it is half the chord of double D the arc; thus HL is the sine of the arc HB. Fig. 7. F 21. The sines on the same diameter increase in length till they come to the cen. A tre, and so become the radius, after which they diminish.—Hence, it is plain that the sine of 90 degrees is the greatest possible sine, and is equal to the radius. 22. The versed sine of an arc is that part of the diameter or radius which is between the sine and the circumference ; thus LB is the versed sine of the arc HB. Fig. 7. 23. The tangent of an arc is a right line touching the circumference, and drawn perpendicular to the diameter; and is terminated by a line drawn from the centre through the other end of the arc; thus BK is the tangent of the arc BH. Fig. 7. Note. The tangent of an arc of 45 degrees is equal in length to the radius of the circle of which the arc is a part. through one end of the arc till it meets the tangent; thus CK is the secant of the arc BH. Fig. 7. 25. The complement of an arc is what the arc wants of 90 degrees, or a quadrant : thus HD is the complement of the arc BH. Fig. 7. 26. The supplement of an arc is what the arc wants of 180 degrees, or a semicircle ; thus ADH is the supplement of the arc BH. Fig. 7. [Note. It will be seen, by reference to Fig. 7, that the sine of any arc is the same as that of its supplement. So likewise, the tangent and secant of any arc are used also for its supplement.] 27. The sine, tangent or secant of the complement of any arc is called the co-sine, co-tangent, or co-secant of the arc; thus, FH is the sine, DI the tangent, and CI the secant of the arc DH ; or they are the co-sine, co-tangent, and co-secant of the arc BH. Fig. 7. [The terms sine, tangent and secant, are abreviated thus : sin. tan, and sec. So likewise, co-sine, co-tangent, and co. secant, are written co-sin, co-tan. and co-sec.] 28. The measure of an angle is the arc of a circle contain: ed between the two lines which form the angle, the angular point being the centre; thus, the angle HCB. Fig. 7. is mea. sured by the arc BH: and is said to contain as many degrees as the arc does. NOTE An angle is esteemed greater or less according to the opening of the lines which form it, or as the arc intercepted by those lines contains more or fewer degrees. Hence it may be observed, that the size of an angle does not depend at all upon the length of the including lines ; for all arcs described on the same point, and intercepted by the same right lines, contain exactly the same num. ber of degrees, whether the radius be longer or shorter. 29. The sine, tangent, or secant of an arc is also the sine, tangent, or secant of the angle whose measure the arc is. Fig. 8. B 30. Parallel lines are such as are equal. ly distant from each other; as AB and Fig. 9. 31. A triangle is a figure bounded by three lines ; as ABC. Fig. 9. 32. An equilateral triangle has its three sides equal in length to each other. Fig. 9. Fig. 10. 33. An isoceles triangle has two of its sides equal. Fig. 10. Fig. 11. 34. A scalene triangle has three unequal sides. Fig. 11. Fig. 12. 35. A right angled triangle has one right angle. Fig. 12. Fig. 13. 36. An obtuse angled triangle has one obtuse angle. Fig. 13. 37. An acute angled triangle has all its angles acute. Fig. 9, or 10. 38. Acute and obtuse angled triangles are called oblique angled triangles, or simply oblique triangles ; in which the lower side is generally called the base and the other two, legs. 39. In a right angled triangle the longest side is called the hypothenuse, and the other two, legs, or base and perpendic. ular. Note. The three angles of every triangle being added to gether will amount to 180 degrees; consequently the two acute angles of a right angled triangle amount to 90 degrees, the right angle being also 90. Fig. 14. А 40. The perpendicular height of a trian. gle is a line drawn from one of the angles perpendicular to its opposite side ; thus, the dotted line AD. Fig. 14. is the perpendicu. lar height of the triangle ABC. B с Note. This perpendicular may be drawn from either of the angles; and whether it falls within the triangle, or on one of the lines continued beyond the triangle, is immaterial. Fig. 15. D 41. A square is a figure bounded by four equal sides, and containing four right angles. Fig. 15. Fig. 16. 42. A rectangle* is a figure bounded by four sides, the opposite ones being equal and the angles right. Fig. 16. 43. A rhombus is a figure bounded by four equal sides, but has its angles oblique. Fig. 17. B Fig. 18. A 44. A rhomboid is a figure bounded by four sides, the opposite ones being equal, but the angles oblique. Fig. 18. B * In previous editions, this figure is denominated a parallelogram. The [45. Any four-sided figure, having its opposite sides paral. lel, is called a parallelogram. Figs. 15, 16, 17, and 18.] 46. The perpendicular height of a parallelogram is a line drawn from one of the angles to its opposite side ; thus, the dotted lines AB. Fig. 17. and Fig. 18, represent the perpendicular height of the rhombus and rhomboid. Fig. 19. 47. A trapezoid is a figure bounded by four sides, two of which are parallel though of unequal lengths. Fig. 19. and Fig. 20. Fig. 20. Note. Fig. 19. is sometimes called a right angled trape. zium. Fig. 21. 48. A trapezium is a figure bounded by four unequal sides Fig. 21. B 49. A diagonal is a line drawn between two opposite angles; as the line AB. Fig. 21. A 50. Figures which consist of more than four sides are cal. led polygons ; if the sides are equal to each other they are called regular polygons, and are sometimes named from the number of their sides, as pentagon, or hexagon, a figure of five or six sides, &c.; if the sides are unequal, they are called ir. regular polygons. |