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28. 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.
29. The sine of the supplement of an arc, is the same with the sine of the arc itself; for drawing them according to def. 22, there results the self-same line; thus HL is the sine of the arc HB, or of its supplement ADH. fig. 8.
30. 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 legs contain a greater or less number of degrees of the whole circle.
31. 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.
32. Parallel lines are such as are equi-distant from each other, as AB, CD. fig. 9.
33. 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 mixt figure.
34. The most simple rectilineal figure is a triangle, being composed of three right lines, and is considered in a double capacity; first, with respect to its sides; and second, to its angles.
35. In respect to its sides, it is either equilateral, having the three sides equal, as A. fig. 10;
36. Or isosceles, having two equal sides, as B. fig. 11.
37. Or scalene, having the three sides unequal, as C. fig. 12.
38. In respect to its angles, it is either rightangled, having one right angle, as D. fig. 19.
39. Or obtuse angled, having one obtuse angle, as E. fig. 14.
40. Or acute angled, having all the angles acute, as F. fig. 15.
41. Acute and obtuse angled triangles are in general called oblique angled triangles, in all which
be called the base, and the other two the sides.
42. 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.
43. Any figure of four sides is called a quadrilateral figure.
44. Quadrilateral figures whose opposite sides are parallel, are called parallelograms : thus, ABCD is a parallelogram. Fig. 3. 17. and AB fig. 18 and 19.
45. A parallelogram whose sides are all equal and angles right, is called a square, as ABCD. fig. 17
46. A parallelogram whose opposite sides are equal and angles right, is called a rectangle or an oblong, as ABCD. fig. 3.
47. A rhombus is a parallelogram of equal sides, and has its angles oblique, as A. fig. 18. and is an
48. 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.
49. A quadrilateral figure that is not a parallelogram, is called a trapezium. Plate 7. fig. 3.
50. Figures which consists 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.
51. 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 be to B as C is to D.
POSTULATES OR PETITIONS.
1. That a right line may be drawn from any one given point to another.
2. That a right line may be produced or continued at pleasure.
3. That from any centre and with any radius, the circumference of a circle may be described.
4. It is also required that the equality of lines and angles to others given, be granted as possible : that
it is possible for one right line to be perpendicular to another, at a given point or distance : and that every magnitude has its half, third, fourth,
Note, Though these postulates are not always quoted, the reader will easily perceive where, and in what sense they are to be understood.
AXIOMS, OR SELF-EVIDENT TRUTHS.
1. Things that are equal to one and the same thing, are equal to each other.
2. Every whole is greater than its part.
3. Every whole is equal to all its parts taken together.
4 If to equal things, equal things be added, the whole will be equal.
5. If from equal things, equal things be deducted the remainders will be equal.
6. If to or from unequal things equal things be added or taken, the sums or remainders will be unequal.
7. All right angles are equal to one another.
8. If two right lines not parallel, be produced towards their nearest distance, they will intersect each other.
9. Things which mutually agree with each other, are equal.