Appendix. PROBLEM X. To inscribe a circle in a regular polygon. Bisect any two sides of the polygon by the perpendiculars GO, FO, and with their point of intersection O, as a V centre, and OG as a radius describe the circumference of a circle-this circle will touch all the sides of the polygon. H For, draw OA. Then in the two right angled triangles OAG and OAF, the side AO is common, and AG is equal to AF, since each is half of one of the equal sides of the polygon: hence, OG is equal to OF(Bk. I.Th. xix). In the same manner it may be shown that OH, OK and OL are all equal to each other hence, a circle described with the centre O and radius OF will be inscribed in the polygon. C.r. Hence, also the lines OA, ON &c., drawn to the angles of the polygon are equal. APPENDIX OF THE REGULAR POLYGONS. 1. In a regular polygon the angles are all equal to each other (Def. 3). If then, the sum of the inward angles of a regular polygon be divided by the number of angles, the quo tion: will be the value of one of the angles. But the sum of the inward angles is equal to twice as many right angles, wanting four, as the polygon has sides, and we shall find the value in degrees by simply placing 90° for the right angle. Appendix. 2. Thus, for the sum of all the angles of an equilateral triangle, we have 6 × 90°-4 × 90°-540° - 360° — 180° = and for each angle 180°360° : Hence, cach angle of an equilateral triangle, is equal to 60 degrees. 3. For the sum of all the angles of a square, we have 8 × 90°-4 × 90°-720°-360° = 360°, and for each of the angles 360°÷4 90° 4. For the sum of all the angles of a regular pentagɔn, wc have 10 × 90°-4 × 90°-900o.- 360°=540°, and for each angle 540°-5=108°. 5. For the sum of all the angles of a regular hexagon, we have 12 × 90° - 4 × 90° — 1080° — 360°—720°, and of each angle = 720° 6 120°. 6. For the sum of the angles of a regular heptagon, we have 14 × 90°-4 × 90° 1260° - 360° 900°: and for one of the angles = 900°÷7128° 34'+. = 7. For the sum of the angles of a regular octagon, we have 16 x 90°-4 x 90°-1440-360°-1080°: and for each angle 1080°8135° Regular Polygons. 8. Since the sum of the angles about any point is equal tc four right angles (Bk. I. Th. ii. Cor. 3), it may be observed that there are only three kinds of regular polygons, which can be arranged around any point, as C, so as exactly to fill up the space. These are, GEOMETRY. BOOK V. OF PLANES AND THEIR ANGLES. DEFINITIONS. 1. A straight line is perpendicular to a plane, when it is per pendicular to every straight line of the plane which it meets. The point at which the perpendicular meets the plane, is called the foot of the perpendicular. 2. If a straight line is perpendicular to a plane, the plane is also said to be perpendicular to the line. 3. A line is parallel to a plane when it will not meet that plane, to whatever distance both may be produced. Conversely, the plane is then parallel to the line. 4. Two planes are parallel to each other, when they will not meet, to whatever distance both are produced. 5. If two planes are not parallel, they intersect each other in a line that is common to both planes: such line is called their common intersection. 6. The space included between two planes is called a diedral angle: the planes are the faces of the angle, and their intersection the edge. A diedral angle is measured by two lines, one in cach plane, and both perpendicular to the common intersection at the same point. This angle may be acute, obtuse, or a right angle. When it is a right angle, the planes are said to be perpendicular to each other. Of Planes. Let AB be a plane coinciding with the lane of the paper, and ECF a Now, if from any point of the common E H B then will the angle DCE measure the inclination between the two planes. It should be remembered that the line EC is directly over the line CD. 7. A polyedral angle is the angular space included between several planes meeting at the same point. Thus, the polyedral angle S is formed by the meeting of the planes ASB, BSC, CSD, DSA. 8. The angle formed by three planes is called a triedral angle. A B THEOREM I. Two straight lines which intersect each other, lie in the same plane, and determine its position. Let AB and AC be two straight lines which intersect each other at A. Through AB conceive a plane to be passed, and let this plane be turned around AB until it embraces the point C: the plane will then contain the two B lines AB, AC, and if it be turned either way it will depart from the point C, and consequently from the line AC. Hence, |