A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

Parallelopipeds and Tetrahedra. 131
EXERCISES.-XXXIII.
1. Find the values of E, F, S for each of the regular polyhedra.
2. Prove that the centres of the faces of the polyhedra P4, P6, P, P2, P, 20
are respectively the summits of polyhedra P4, P8, P6, P20, P12.
3. Find the ratios between the volumes of a tetrahedron or cube and the
volume of the solid, whose summits are the centres of its faces.
4. Prove that the inradius of P4 = thee times its circumradius.
5. In the same' case, the radius of the sphere touching its six edges is
a mean proportional between the inradius and circumradius.
6. Prove that the ratio of the inradius to circumradius is the same in
P6 and Ps, and also in P12 and 2Po.
7. In any convex polyhedron (regular or irregular), prove that the number
of faces having an odd number of sides is even, and that the number of solid
angles having an odd number of edges is uneven.
8. In every convex polyhedron, the number of triangular faces increased
by the number of trihedral angles is equal to or greater than eight.
9. Every convex polyhedron must have either triangular, or quadrangular,
or pentagonal faces, and trihedral, or tetrahedral, or pentrahedral angles.*
SECTION II.-PARALLELOPIPEDS AND TETRAHEDRA.
123. To find the volume of a parallelopiped in terms of three
conterminous edges and their inclinations.
Let OA, OB, OC be the three edges, and let their lengths
be a, b, c, respectively; and let the angles BOC, COA, A OB
be denoted by a, /3, y. Draw AD perpendicular to the plane
B O C, and describe a sphere, with O as centre, meeting the lines
OA, OB, OC, OD in the points a, b, c, d, respectively. The
* The most important recent works which treat of the polyhedra are
ALLMAN'S "Greek Geometry from THALES to EUCLID," and "Lectures on
the Icosahedron," by PROFESSOR KLEIN, Gottingen. This is a very remarkable work, showing the great importance of the polyhedra in the higher
departments of modern Analysis.
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