Undecagon. Dodecagon. O 44. A circle is a plane figure, bounded by a curve line, called the circumference, which is every where equidistant from a certain point within, called the centre. NOTE. The circumference itself is often called a circle. 45. The radius of a circle is a right line, drawn from the centre to the circumference. 46. The diameter of a circle is a right line, drawn through the centre, and terminating in the circumference on both sides. 47. An arc of a circle is the circumference. D any part of 48. A chord is a right line, joining the extremities of an arc. 49. A segment is any part of a circle, bounded by an arc and its chord. 50. A semicircle is half the circle, or a segment cut off by a diam eter. 51. A sector is any part of a circle, bounded by an arc, and two radii, drawn to its extremities. 52. A quadrant, or quarter of a circle, is a sector, having a quarter of the circumference for its arc, and its two radii are perpendicular to each other. 48. The height, or altitude, of a figure is a perpendicular, let fall from an angle, or its vertex, to the opposite side, called the base. 54. In a right-angled triangle, the side opposite to the right angle is called the hypotenuse; and the other two the legs, or sides, or sometimes the base and perpendicular. 56. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; and each degree into 60 minutes, each minute into 60 seconds, and so on. Hence a semicircle contains 180 degrees, and a quadrant 90 degrees. 57. The measure of a rightlined angle is an arc of any circle, contained between the two lines, which form that angle, the angular point being the centre; and it is estimated by the number of degrees, contained in that arc. Hence a right angle is an angle of 90 degrees. 3 58. Identical figures are such, as have all the sides and all the angles of one respectively equal to all the sides and all the angles of the other, each to each; so that, if one figure were applied to, or laid upon, the other, all the sides of it would exactly fall upon and cover all the sides of the other; the two becoming coincident. 59. An angle in a segment is that, which is contained by two lines, drawn from any point in the arc of the segment to the extremities of the arc. 60. A right-lined figure is inscribed in a circle, or the circle circumscribes it, when all the angular points of the figure are in the circumference of the circle. 61. A right-lined figure circumscribes a circle, or the circle is inscribed in it, when all the sides of the figure touch the circumference of the circle. 62. One right-lined figure is inscribed in another, or the latter circumscribes the former, when all the angular points of the former are placed in the sides of the latter. 63. Similar figures are those, that have all the angles of one equal to all the angles of the other, each to each, and the sides about the equal angles proportional. 64. The perimeter of a figure is the sum of all its sides, taken together. 65. A proposition is something, which is either proposed to be done, or to be demonstrated, and is either a problem or a theorem. 66. A problem is something proposed to be done. 67. A theorem is something proposed to be demonstrated. 68. A lemma is something, which is premised or previously demonstrated, in order to render what follows more easy. 69. A corollary is a consequent truth, gained immediately from some preceding truth, or demonstration. 70. A scholium is a remark, or observation, made upon something preceding it. PROBLEMS. PROBLEM. I: To divide a given line A B into two equal parts. From the centres A and B, with any radius greater than half A B, describe arcs, cutting each other in m and n. Draw the line m C n, and it will cut the given line into two equal parts in the middle point C.* PROBLEM II. To divide a given angle A B C into two equal parts. From the centre B, with any radius, describe the arc A C. From A and C, with one and the same radius, describe arcs, intersecting in m. Draw the line B m, and it will bisect the angle, as required.t Suppose right lines to be drawn from A to m, m to B, B to n, and n to A; then Am B n is a parrallelogram, and its diagorals A B, mn, mutually bisect each other. bisected in C. Therefore A B is + Suppose right lines to be drawn from A to m and from C to |