37. A trapezium is a quadrilateral, which has not both pair of its opposite sides parallel.

38. A trapezoid has only one pair of opposite sides parallel.

39. A diagonal is a right line, joining any two opposite angles of a quadrilateral.

40. Plane figures, having more than four sides, are, in general, called polygons; and they receive other particular names, according to the number of their sides or angles.

41. A pentagon is a polygon of five sides; a hexagon has six sides; a heptagon, seven; an octagon, eight; a nonagon, nine a decagon, ten; an undecagon, eleven; and a dodecagon, twelve.

42. A regular polygon has all its sides and all its angles equal. If they be not both equal, the polygon is ir regular.

43. An equilateral triangle is also a regular figure of three sides, and the square is one of four; the former be ing also called a trigon, and the latter a tetragon.

Pentagon.

Hexagon.

Heptagon.

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 cir cle, contained between the two lines, which form that angle, the angular point being the centre ; and it is estimated by the num◄ ber of degrees, contained in that

arc. Hence a right angle is an angle of 90 degrees.

58. Identical figures are such, as have all the sides and all the angles of one respectively equal to all the angles of the other, each to each;

all the sides and

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 eircle, 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 cirele, 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 anether, 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 immediate ly from some preceding truth, or demonstration.

70. A scholium is a remark, or observation, made upon something preceding it.

PROBLEMS.

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 AB, describe arcs, cutting each other in m and n. Draw the line mcn, and it will cut the given line into two equal parts in the middle point C.

PROBLEM II.

To divide a given Angle ABC into two equal parts.

From the centre B, with any radius, describe the arc AC. From A and C, with one and the same radius, describe arcs, in- A tersecting in m. Draw the line Bm, and it will bisect the angle, as required."

B

m

B

NOTE. By this operation the arc AC is bisected; and in a similar manner may any given arc of a circle be bisected.

PROBLEM