But, in the right-angled triangle ABH, we have, REMARK 1.—When the polygon in question is the equi lateral triangle, the diagonal becomes a side, and conse quently, half the diagonal becomes half a side of the triangle. 360° REMARK 2. The perpendicular BH = AB sin 2n 25. To determine the angle included between two adja. cent faces of either of the regular polyedrons, let us suppose a plane to be passed perpendicular to the axis of a polyedral angle, and through the vertices of the polyedral angles which lie adjacent. This plane will intersect the convex surface of the polyedron in a regular polygon; the number of sides of this polygon will be equal to the number of planes which meet at the vertex of either of the polyedral angles, and each side will be a diagonal of one of the equal faces of the polyedron. Let D be the vertex of a polyedral angle, CD the intersection of two adjacent faces, and ABC the section made in the convex surface of the polyedron by a plane perpendicular to the axis through D. C A F B Through AB let a plane be drawn perpendicular to CD, produced, if necessary, and suppose AE, BE, to be the lines in which this plane intersects the adjacent faces. Then will AEB be the angle included between the adjacent faces, and FEB will be half that angle which we will represent by A. Then, if we represent by n the number of faces which meet at the vertex of the solid angle, and by m the number of sides of each face, we shall have, from what has already been shown, BF BC cos· 360° 360° and EB BC sin " 2m = sin FEB = sin A, to the radius of unity; EB This formula gives, for the diedral angle formed by any two adjacent faces of the Having thus found the diedral angle included between the adjacent faces, we can easily calculate the perpendicu lar let fall from the centre of the polyedron on one of its faces, when the faces themselves are known. The following table shows the solidities and surfaces of the regular polyedrons, when the edges are equal to 1. A TABLE OF. REGULAR POLYEDRONS WHOSE EDGES ARE 1. 26. To find the solidity of a regular polyedron. 1. Multiply the surface by one-third of the perpendicular let fall from the centre on one of the faces, and the product will be the solidity. Or, 2. Multiply the cube of one of the edges by the solidity of a similar polyedron, whose edge is 1. The first rule results from the division of the polyedron into as many equal pyramids as it has faces, having their common vertex at the centre of the polyedron. The second is proved by considering that two regular polyedrons having the same number of faces may be divided into an equal number of similar pyramids, and that the sum of the pyramids which make up one of the polyedrons will De to the sum of the pyramids which make up the other polyedron, as a pyramid of the first sum to a pyramid of the second (B. II., P. 10); that is, as the cubes of their homologous edges (B. VII., P. 20); that is, as the cubes of the edges of the polyedron. Ex. 1. What is the solidity of a tetraedron whose edge is 15? Ans. 397.75. 2. What is the solidity of a hexaedron whose edge is 12? Ans. 1728. 3. What is the solidity of a octaedron whose edge is 20? Ans. 3771.236. 4. What is the solidity of a dodecaedron whose edge Ans. 119736.2328. is 25? 20? 5. What is the solidity of an icosaedron whose edge is Ans. 17453.56. REMARK. In the following table, in the nine right hand columns of each page, where the first or leading figures change from 9's to O's, points or dots are introduced in stead of the O's, to catch the eye, and to indicate that from thence the two figures of the Logarithm to be taken from the second column, stand in the next line below. 145 161368 1667 1967 2266 2564 2863 3161 3466 3758 4055 299 146 4353 4650 4947 5244 5541 5838 151 8977 9264 9552 6430 6726 7022 297 147 7317 7613 7908 8203 8497 8792 90S6 9330 | 9674 9968 295 148 170262 0555 0848 1141 1434 1726 2019 2311 2603 2895 293 149 3186 3478 3769 4060 | 4351 4641 4932 5222 5512 5802 150 176091 6381 6670 6959 7248 8113 8401 8689 9839 126 6134 291 7536 7825 289 413 699 985 1272 1558 287 152181844 2129 2415 2700 2985 3270 3555 3839 153 154 4691 4975 5259 5542 5825 4123 4407 285 6674 6956 7239 283 8366 8647 8928 9209 9490 9771 0051 281 1171 1451 1730 2010 2289 3125 3403 3681 157 5899 6176 6453 158 8657 8932 9206 159 201397 1670 1943 2216 2488 2567 2846 279 3959 4237 6729 7005 7832 4514 4792 5069 5346 5623 278 7281 7556 8107 8382 276 029 303 2761 3033 577 €850 1124 274 3305 3577 3848 272 |