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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, 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

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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 AB and DE Cross one another at right angles, and thus the periphery is divided into four equal parts, as BD, D., AE, 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

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 deg ces, or the fourth part of the circle, which is 360 degrees.

21. The versed sine of an are is that part of the diameter that lies between the rght sine and the circumference: thus LBs the ersed sine of the arc HB. fig. 8.

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26. The sine, tangent, or sear plement of any arc, is called tee gent, or co-secant of the arc itsek z 2 sine, DI the tangent, and C 1 arc DH: or they are the co-secant of the arc HB. fig..

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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 thus if multiplied by D, be equal to B multiplied by C, then A is said to be to Bas C is to D.

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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, &c. part.

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.

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