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21. The versed sine of an arc is that part of the diameter that lies between the right sine and the circumference: thus LB is the versed sine of the arc HB. fig. 8.
22. The tangent of an arc is a right line touching the periphery, being perpendicular to the end of the diameter, and is terminated by a line drawn from the centre through the other end : thus BK” is the tangent of the arc HB. fig. 8.
23. And the line which terminates the tangent, that is, CK, is called the secant of the are HB. fig. 8.
24. What an arc wants of a quadrant is called the complement thereot: Thus DH is the complement of the arc HB. fig. 8.
25. And what an arc wants of a semicircle is called the supplement thereof: thus AH is the supplement of the arc HB. fig. 8.
26. The sine, tangent, or secant of the complement of any arc, is called the co-sine, co tangent, or co-secant of the arc itself: thus FH is the sine, Dl the tangent, and CI the secant of the arc DH: or they are the co-sine, co-tangent, or co-secant of the arc HB. fig. 8.
27. The sine of the supplement of an arc, is the same with the sine of the arc itself ; for drawing them according to def. 20, there results the self-same line; thus HL is the sine of the arc HB, or of its supplement ADH. fig. 8.
28. The measure of a right-lined angle, is the arc of a circle swept from the angular point, and
contained between the two lines that form the angle : thus the angle HCB (fig. 8.) is measured by the arc HB, and is said to contain so many degrees as the arc HB does ; so if the arc HB is 60 degrees, the angle HCB is an angle of 60 degrees.
Hence angles are greater or less according as the arc described about the angular point, and terminated by the two sides, contains a greater or less number of degrees of the whole circle.
29. The sine, tangent, and secant of an arc, is also the sine, tangent, and secant of an angle whose measure the arc is : thus because the arc HB is the measure of the angle HCB, and since HL is the sine, BK the tangent, and CK the secant, BL the versed sine, HF the co-sine, DI the co-tangent, and CI the co-secant, &c. of the arc BH; then HL is called the sine, BK the tangent, CK the secant, &c. of the angle HCB, whose measure is the arc HB. fig. 8.
30. Parallel lines are such as are equi-distant from each other, as AB, CD. fig. 9.
31. A figure is a space bounded by a line or lines. If the lines be right it is called a rectilineal figure, if curved it is called a curvilineal figure ; but if they be partly right and partly curved lines, it is called a mixed figure.
32. The most simple rectilineal figure is a triangle, being composed of three right lines, and is considered in a double capacity ; lst, with respect to its sides ; and 2d, to its angles.
33. In respect to its sides it is either equilateral, having the three sides equal, as A. fig. 10.
DI the the B.
40. The perpendicular height of a triangle is aline drawn from the vertex to the base perpendicularly: thus if the triangle ABC, be proposed, and BC be made its base, then if from the vertex 4 the perpendicular AD be drawn to BC, the line AD will be the height of the triangle ABC, standing on BC as its base. Fig. 16.
Hence all triangles between the same parallels have the same height, since all the perpendiculars are equal from the nature of parallels.
41. Any figure of four sides is called a quadrilateral figure.
42, Quadrilateral figures, whose opposite sides are parallel, are called parallelograms : thus
ABCD is a parallelogram. Fig. 3. 17, and AB fig. 18 and 19.
43. A parallelogram whose sides are all equal and angles right, is called a square, as ABCD. fig. 17.
44. A parallelogram whose opposite sides are equal and angles right, is called a rectangle, or an oblong, as ABCD. fig. 3.
45. A rhombus is a parallelogram of equal sides, and has its angles oblique, as A fig. 18. and is an inclined square.
46. A rhomboides is a parallelogram whose opposite sides are equal and angles oblique; as B. fig. 19. and may be conceived as an inclined rectangle.
47. Any quadrilateral figure that is not a parallelogram, is called a trapezium. Plate 7. fig. 3.
48. Figures which consist of more than four sides are called polygons; if the sides are all equal to each other, they are called regular polygons. They sometimes are named from the number of their sides, as a five-sided figure is called a pentagon, one of six sides a hexagon, &c, but if their sides are not equal to each other, then they are called irregular polygons, as an irregular pentagon, hexagon, &c.
49. Four quantities are said to be in proportion when the product of the extremes is equal to that of the means : tlius if A multiplied by D, be equal to B multiplied by C, then Ä is said to be to Bas C is to D.