168. PROP. The area of the triangle being given in square feet, it is required to find from it the spherical excess in seconds.

Let a be the area of the triangle to radr, the earth's radius; then,

SECTION IX.

ON REGULAR POLYHEDRONS, &c.

169. PROP. If F = number of regular faces,

(1.) Since every edge is made by two sides, therefore the whole number of sides in the polyhedron* is equal to 2 E; but this number is also equal to n F,

(2.) Take any point within the polyhedron, and from it draw lines to all the angular points; then, if a spherical sur

* For the definitions of the regular polyhedrons, &c. the student is referred to Euc. XI. DEF.

=

face (rad 1) be supposed to be described about this point, cutting the lines just drawn, and the points of intersection be joined so as to form as many spherical polygons as there are faces in the polyhedron; the area of one of these polygons

=

П

180° angles of polygon (n-2) 180°};

hence, the area of all these polygons, or

π

180- {Fx angles of polygon-F (n−2) 180°} =4 π,

which is the surface of the sphere rad = 1. Now, it has been shown, that nF = 2 E, and all the spherical angles that can be formed about a point on the surface of a sphere 360°, therefore Fx angles of polygon = S. 360°;

π

hence, 180°

{S. 360°-(2 E-2 F) 180°} = 4π,

or SE + F = 2,

.. S+FE+ 2.

=

It is manifest this equation is also true when the polyhedron is not regular.

170. COR. EF-S- 2.

171. PROP. The sum of all the plane angles, which form the solid angles of a polyhedron

For all the interior angles of one of the faces of (n) sides= (n -2) 180°, therefore the sum of all the plane angles of all the plane faces = F (n − 2) 180°,

172. PROP. If n = number of sides in a plane face,

m = number of plane angles containing a solid angle;

hence the sum of all the plane angles, which form one solid

180°, and, therefore, the sum of all the

plane angles which form all the solid angles = m.

n 2

n

(1.)

n

=

F=

and F =

2

2 (m + n)

2 m (n-2)

2 (m + n) — m n2

4m

2 (m + n)

,

mn'

And, since E+2S+ F; there results,

174. PROP.

There can be only five regular polyhedrons, viz. the tetrahedron, the hexahedron, the octahedron, the dodecahedron, and the icosahedron.

Since S, F, and E, are by the supposition positive integers, their values as found in (art. 170.) must also be positive intem+ n gers; hence 2 (m+n) must be greater than mn, or must be greater than

n

2་

m

(1.) Now the least values of m and n are 3 and 3, in which

case

m + n

m

n

is greater than; and the values of S, F, and E

are 4, 4, and 6 respectively: hence the solid is a regular tetrahedron, or pyramid.

m+ n
m

n

(2.) Let m = 3, and n = 4: then is greater than ;

2

and S=8, F=6, and E-12; thence the solid, in this case, is a hexahedron, which is the same as the cube.

(3.) Let m = 4, and n = 3: then

=

and S6, F=8, and E 12. Or, the solid has eight plane triangular faces, for which reason it is called an octahedron.