on. 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 Hence a semicircle contains 180 degrees, and a quadrant go 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 numa ber of degrecs, contained in that arc. Hence a right angle is an angle of go degrees. 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 ex« tremities 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 62. One righi-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 thië 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 cither a problem for 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. 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 mcii, and it will cut the given line into two equal parts in the middle point C. A B 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, in- A tersecting in m. Draw the line B m, and it will bisect the angle, as required. NOTE. By this operation the arc AC is bisected ; and in a similar manner may any given arc of a circle be bisected. PROBLEM PROBLEM III. To divide a right angle A B C into three equal parts, From the centre B, with any radius, describe the arc AC. From the centre A, with the same radius, A cross the arc AC in n; and with the centre C, and the same radius, cut the arc AC in m. Then through the points m and n draw B m and B n, and they will trisect B the angle, as required. To draw a Line parallel to a given Line A B. CASE 1. When the parallel Line is to be at a given Disa tance C. CASE 2.-When the parallel Line is to pass through a given Point C. r From any point m, in the line AB, with the radius mC, D describe the arc Cn. From the centre C, with the same A radius, describe the arexmr. Take the arc Cn in the compasses, and apply it from m to r. Through C and r draw DE, the parallel required. Beb NOTE Note. In practice, parallel lines are more easily drawn with a Parallel Rule. PROBLEM V. To erect a Perpendicidar from a giveri Poini A in a given Line BC. CASE 1.-When the Point is near the middle of the Line. On eacn side of the point A, take any two equal distances Ani, An. From the centres m and ng radius grcater than Am or An, describe two arcs intersecting in r. Through A and r draw the line Ar, and it will be the perpendicular required. A m ns. CASE 2.-When the Point is near the end of the Line. With the centre A, and any radius, describe the arc From the point m, with the same radius, turn the compasses twice over on the arc, at n and s. Again, with the centres n and's, C* describe arcs intersecting in r. Then draw Ar, and it will be the perpendicular required. B m A |