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3. When the chord is greatest it becomes a diameter, and then the segments are equal; and each segment is a semicircle.
19. A sector of a circle is a part thereof less than a semicircle, which is contained between two radii and an arc: thus the space contained between the two radii CH, CB, and the arc HB is a sector. fig. 8.
20. The right sine of an arc, is a perpendicular line let fall from one end thereof, to a diameter drawn to the other end : thus HL is the right sine of the arc HB.
The sines on the same diameter increase till they come to the centre, and so become the radius; hence it is plain that the radius CD is the greatest possible sine, and thence is called the whole sine.
Since the whole sine CD (fig. 8.) must be perpendicular to the diameter (by def. 20.) therefore producing DC to E,the two diameters ÁBand DE cross one another at right angles, and thus the periphery is divided into four equal parts, as BD, DA, AĚ, and EB ; (by def. 10.) and so BD becomes a quadrant or the fourth part of the periphery: therefore the radius DC is always the sine of a quadrant, or of the fourth part of the circle BD.
Sines are said to be of as many degrees as the arc contains parts of 360: so the radius being the sine of a quadrant becomes the sine of 90 degrees, or the fourth part of the circle, which is 360 degrees.
2]. 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 arc HB. fig. 8.
24. What an arc wants of a quadrant is called the complement thereof: 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 are 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, DI 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, HE 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 if 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; 1st, 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.
34. Or isosceles, having two equal sides, as B.
35. Or scalene, having the three sides unequal, as C. fig. 12.
36. In respect to its angles, it is either rightangled, having one right angle, as D. fig. 13.
37. Or obtuse angled, having one obtuse angle, as E. fig. 14.
38. Or acute angled, having all the angles acute, as F. fig. 15.
39. Acute and obtuse angled triangles are in general called oblique angled triangles, in all which any side may be called the base, and the other two the sides.
40. The perpendicular height of a triangle is a line 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 A 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.
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 : thus if A multiplied by D, be equal to B multiplied by C, then A is said to leto Bas C is to D.