754. The Altitude of a Prism is any sect perpendicular to both bases.

755. The Altitude of a Pyramid is the perpendicular from its vertex to the plane of its base.

756. A Right Prism is one whose lateral edges are perpendicular to its bases.

757. Prisms not right are oblique.

758. A Parallelopiped is a prism whose bases are parallelograms.

759. A Quader is a parallelopiped whose six faces are rectangles.

760. A Cube is a quader whose six faces are squares.

THEOREM I

761. All the summits of any polyhedron may be joined by one closed line breaking only in them, and lying wholly on the surface.

A

For, starting from one face, ABC. neighboring polygon.

B

each side belongs also to a

Therefore, to join A and B, we may omit AB, and use the remainder of the perimeter of the neighboring polygon a. In the same way, to join B and C, we may omit BC, and use the remainder of the perimeter of the neighboring polygon b, unless the polygons a and b have in common an edge from B. In such a case, draw from B in b the diagonal nearest the edge common to a and b; take this diagonal and the perimeter of b beyond it around to C, as continuing the broken line; .and proceed in the same way from C around the neighboring polygon c.

When this procedure has taken in all summits in faces having an edge in common with ABC .. we may, by proceeding from the closed broken line so obtained, in the same way take in the summits on the next series of contiguous faces, etc.

So continue until the single closed broken line goes once, and only once, through every summit.

THEOREM II.

762. Cutting by diagonals the faces not triangles into triangles, the whole surface of any polyhedron contains four less triangles than double its number of summits.

T= 2(S2).

For, joining all the summits by a single closed broken line, this cuts the surface into two bent polygons, each of which contains S angles, where S is the number of summits.

2 tri

763. COROLLARY. The sum of all the angles in the faces of any polyhedron is as many perigons as the polyhedron has summits, less two.

764. REMARK. Theorem II. is called Descartes' Theorem, and is really the fundamental theorem on polyhedrons, though this place has long been held by Theorem III., called Euler's Theorem, which follows from it with remarkable elegance.

THEOREM III.

765. The number of faces and summits in any polyhedron, taken together, exceeds by two the number of its edges.

CASE I. If all the faces are triangles. Then, by 762,

But also

F= 2(S2).

2E = 3F,

for each edge belongs to two faces, and so we get a triangle for every

By adding, we have 2E 2F + 2(S2); that is,

F + S = E + 2.

CASE II. If not all the faces are triangles.

Since to any pyramidal summit go as many faces as edges, we may replace any polygonal face by a pyramidal summit without changing the

equality or inequality relation of F + S to E+ 2; for such replacement only adds the same number to F as to E, and changes one face to a summit. But, after all polygonal faces have been so replaced, F + S E+2, by Case I. Therefore always the relation was

equality.

THEOREM IV.

766. Quaders having congruent bases are to each other as their altitudes.

HYPOTHESIS. Let a and a be the altitudes of two quaders, e and Q', having congruent bases B.

CONCLUSION. Q: Q :: a : a'.

PROOF. Of a take any multiple, ma; then the quader on base B with altitude ma is mQ.

In the same way, take equimultiples na', nQ.

According as mQ is greater than, equal to, or less than, nQ, we have ma greater than, equal to, or less than na'; therefore, by definition,

Q : & :: a: a'.

EXERCISES. 110. In no polyhedron can triangles and threefaced summits both be absent; together are present at least eight. Not all the faces nor all the summits have more than five sides.

III. There is no seven-edged polyhedron.