number of faces of the polyedral angle; each side of the polygon will be the measure of one of the plane angles formed by the edges of the polyedral angle; and each angle of the polygon will be equal to the diedral angle contained between two consecutive faces of the regular polyedron.

To find the required diedral angle, therefore, it only remains to deduce a formula for finding one angle of a regular spherical polygon when the sides are given.

E

D

C

Let ABCDE represent a regular spherical polygon, and let P be the pole of a small circle passing through its vertices. Suppose P to be connected with each of the vertices by arcs of great circles; there will thus be formed as many equal isosceles triangles as the polygon has sides, the vertical angle in each being equal to 360° divided by the number of sides. Through P draw the arc of

a great circle, PQ, perpendicular to AB: then will AQ be equal to BQ, and the angle APQ to the angle QPB (B. IX., P. XI., C.). If we denote the number of sides of the spherical polygon by n', the angle APQ will be equal to

In the right-angled spherical triangle AQP, we know the base AQ, and the vertical angle APQ; hence, by Napier's rules for circular parts, we have

denoting the side AB of the polygon by s', and the angle PAQ, which is half the angle EAB of the polygon, by A, we have

n'

To find the volume of a regular polyedron.

126. If planes be passed through the centre of the polyedron and each of the edges, they will divide the polyedron into as many equal right pyramids as the polyedron has faces. The common vertex of these pyramids will be at the centre of the polyedron, their bases will be the faces of the polyedron, and their lateral faces will bisect the diedral angles of the polyedron. of each pyramid will be equal to the product of its base and one third of its altitude, and this product multiplied

The volume

by the number of faces, will be the volume of the polyedron.

It only remains to deduce a formula for finding the altitude of the several pyramids, i. e., the distance from the centre to one face of the polyedron.

Conceive a perpendicular OC OC to be drawn from O, the centre of the polyedron, to one face; the foot of this perpendicular will be the centre of the face. From C, the foot of this perpendicular, draw a perpendicular to one side of the

D

face in which it lies, and connect the point D with the centre of the polyedron. There will thus be formed a right-angled triangle, OCD, whose base, CD, is the apothem of the face, whose angle ODC is half the angle CDL contained between two consecutive faces of the polyedron, and whose altitude OC is the required altitude of the pyramid, or, in other words, the radius of the inscribed sphere. This will be true for any one of the regular polyedrons-the hexaedron is taken here for simplicity of illustration.

Denote the line CD by p, the angle ODC by A, and the perpendicular OC by R. p may be found by the formula, given in Art. 101, for finding the apothem of a regular polygon; A may be found from the formula for sin ¿Ā, given in Art. 125; then, in the right-angled triangle OCD, we have, formula (3), Art. 37,

R p tan A.

Compute the area of one of the faces of the given polyedron and multiply it by R, as determined by the formula just given, and multiply the result thus obtained by the number of faces of the polyedron; the final product will be the volume of the given regular polyedron.

The volumes of all the regular polyedrons have been computed on the supposition that their edges are each equal to 1, and the results are given in the following

From the principles demonstrated in Book VII., we may write the following

RULE.-To find the volume of any regular polyedron, multiply the cube of its edge by the corresponding tabular volume; the product will be the volume required.

15?

Examples.

1. What is the volume of a tetraedron, whose edge is Ans. 397.75.

is 12?

What is the volume of a hexaedron, whose edge
Ans. 1728.

3. What is the volume of an octaedron, whose edge is 20? Ans. 3771.236. 4. What is the volume of a dodecaedron, whose edge is 25? Ans. 119736.2328.

5. What is the volume of an icosaedron, whose edge Ans. 17453.56.

is 20?

REMARKS. In the following table, in the nine right-hand columns of each page, where the first or leading figures change from 9's to 0's, points or dots are introduced instead 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.