which is made up of the triangles ABC and A'BC is equivalent to ABC + AC'B'.

Equations (1), (2), and (3) give by addition,

(4). BACA' + ABCB' + ACBC' 2 ABC + ACB + AB'C' + ACB' + ABC'

(5). But, BACA' + ABCB' + ACBC' — 2 A·T+ 2B T2C T...

(229);

(6). and ACB + AB'C' + ACB' + ABC' 4 T, the surface of the hemisphere BCB'A... (226).

(7). And 2 ABC = 2 S, from our hypothesis.

Now, substituting in (4) values found in (5), (6), and (7); then,

(8). 2 A T + 2 B · T + 2 C · T = 2 S + 4 T ;

(9). or, 2 S 2A T+2B T+2C T-4T;

or, S = (A + B + C − 2) T.

Q. E. D.

232. SCHOLIUM. By dividing a spherical polygon into a number of spherical triangles we may readily deduce from our theorem that, —

The area of a spherical polygon is equal to the product of its spherical excess by a tri-rectangular triangle.

233. GENERAL SCHOLIUM. In the foregoing demonstrations it has been shown that certain similar solids (see prisms, pyramids, cylinders, cones, and spheres) are to each other as the cubes of corresponding edges or dimensions, and that the surfaces of these similar solids are to each other as the squares of corresponding edges or dimen

sions. By the methods already used we may show that, any two similar solids are to each other as the cubes of their corresponding dimensions; and that, the surfaces of any two similar solids are to each other as the squares of their corresponding dimensions.

EXERCISES.

234. LEMMA. Any point in the plane which bisects a diedral angle is equally distant from the faces of the angle.

From any point P in the bisecting plane BE, draw perpendiculars PN and PM to the faces of the diedral angle. Through PM and PN pass a plane intersecting the faces of the angle and the bisecting plane in the lines FN, FM, and PF. Show that the plane angles PFN and PFM are the measures of corresponding diedral angles, and hence equal. From which find the triangles PNF and PMF to be equal, and hence PM equals to PN.

235. THEOREM. A sphere may be inscribed in a triangular pyramid.

B

Bisect the diedral angle between the base BCD and each face, by planes; then see (234).

236. THEOREM. The surface of a sphere is to the surface of a circumscribed right cylinder as two is to three.

237. THEOREM. The volume of a sphere is to the volume of a circumscribed right cylinder as two is to three.

238. THEOREM. Tangents drawn to a sphere from a common external point are equal.

SECTION V.

REGULAR POLYEDRONS.

DEFINITIONS.

239. A polyedron has already been defined as a solid bounded by plane surfaces, the bounding planes being called faces.

240. Polyedrons are named from the number of their faces; one of four faces is called a tetraedron, one of six faces a hexaedron, etc.

241. A regular polyedron is a polyedron of which the faces are equal regular polygons, and each polyedral angle is bounded by the same number of faces.

PROPOSITION XLIII.

242. THEOREM. There cannot be more than five regular polyedrons.

Each face is a regular polygon (241). And the value of one angle of a regular polygon is,

But since the sum of the face angles at any vertex of a polyedral angle is less than 360°. . . (45); and since there are at least three face angles meeting at any vertex, the hexagon cannot be the face of any regular polyedron; for

120° x 3 is not less than 360°. In like manner the heptagon, octagon, etc., are excluded as faces.

There remains then to be considered only the triangle, the square, and the pentagon. With respect to these, The polyedral angle may be formed by the meeting at a point of,

(a) 3 triangles, since 60° × 3 = 180°, or by
(b) 4 triangles, since 60° x 4 = 240°, or by
(c) 5 triangles, since 60° x 5 = 300°, or by
(d) 3 squares, since 90° × 3 = 270°, or by
(e) 3 pentagons, since 108° × 3 = 324°.

All other cases are excluded by the limiting value of a polyedral angle, i.e., 360°.

Thus then there cannot be more than,

three regular polyedrons of triangular faces; see (a), (b), (c),

one regular polyedron of square faces; see (d),

one regular polyedron of pentagonal faces; see (e). Five in all.

Q. E. D.

243. SCHOLIUM. The five regular polyedrons are named from the number of their faces, the tetraedron, the hexaedron, the octaedron, the dodecaedron, and the icosaedron.

244. The five regular polyedrons (243) may be constructed out of cardboard in a very simple manner.

TETRAEDRON.

HEXAEDRON.

OCTAEDRON.