Place live homologous faces, ABEDE

and A'B'C'L'E'

518

GEOMETRY. BOOK VII.

PROPOSITION XXII. THEOREM.

586. Two similar polyhedrons may be decomposed into the same number of tetrahedrons similar, each to each, and similarly placed.

S

Let ABCDE-OPQRS and A'B'C' D' E'-O' P' Q' R' S' be two similar polyhedrons of which P and P' are homologous vertices.

We are to prove that A B C D E-OPQRS and A'B'C' D' E'O'P'Q' R'S' can be decomposed into the same number of tetrahedrons similar and similarly placed.

Place these polyhedrons in the same plane, having any two homologous edges, as A B and A'B' and lying in the same direction.

On any two corresponding faces not adjacent to P and P', as ABCDE and A'B'C' D' E', from two homologous vertices, as E and E', draw diagonals dividing these faces into ▲, similar and similarly placed.

From the homologous vertices P, P' of the polyhedrons draw straight lines to the vertices of these A.

Repeat this construction for each of the faces not adjacent to P, P'.

Then the polyhedrons will be divided into the same number of tetrahedrons;

that is, into as many tetrahedrons as there are ▲ in these faces.

Now, any two corresponding tetrahedrons, as P-A BE and

P-A' B' E', are similar;

for the faces EA B and PAB are similar respectively to the faces E'A' B' and P' A' B',

(being similarly situated ▲ of similar polygons).

In the APBE and P' B'E'

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PB is to P' B', and B E to B' E',

(since they make equal

respectively with the || lines A B and A'B') ;

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.. LPBE=/ P'B' E',

(two not in the same plane having their sides || and lying in the same

and

direction are equal);

Also, in the APAE and P'A' E'

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§ 284

PE

PB

PA

AB

A E

§ 278

PE P'B' P'A' A'B'. A' E

(being homologous sides of similar ▲ ).

.. face PA E is similar to face P'A' E'.

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Moreover, since any two corresponding trihedral of these

tetrahedrons are formed by three planes which are equal, each

to each, and similarly situated, they are equal.

§ 492 $584

.. P-A BE and P'-A' B' E' are similar. In like manner we may show that any other two tetrahedrons similarly situated are similar.

That is, the two similar polyhedrons have the same number of tetrahedrons similar each to each, and similarly situated.

Q. E. D.

587. COROLLARY. Any two homologous lines in two similar polyhedrons have the same ratio as any two homologous edges.

PROPOSITION XXIII. THEOREM.

588. Similar tetrahedrons are to each other as the cubes

Let S-BCD and S'-B' C'D' be two similar tetrahedrons having for bases the similar faces BCD and B'C' D', and for altitudes SO and S' O'.

Apply the tetrahedron S'-B' C' D' to the tetrahedron S-BCD, so that the polyhedral S shall coincide with S.

Then the base B'C' D' will be I to the face BCD,
since their planes make equal & with the face SBO),

and the LSO, 1 to B C D, will also be to B'C' D'.
SO' will be the altitude of the tetrahedron S-B' C' D'.

Now

S-B C D BCDX SO BCD SO
X
S-B'C' D' B'C' D'X SO' B'C' D' SO

=

=

(any two tetrahedrons are to each other as the products of their bases and

altitudes).

(in two similar polyhedrons any two homologous lines are in the same ratio as uny two homologous edges).

589. COROLLARY 1. Two similar polyhedrons are to each other as the cubes of any two homologous edges.

For, two similar polyhedrons may be decomposed into tetrahedrons similar, each to each, and similarly placed, of which any two homologous edges have the same ratio as any two homologous edges of the polyhedrons. And, since any pair of the similar tetrahedrons are to each other as the cubes of any two homologous edges, the entire polyhedrons are to each other as the cubes of any two homologous edges.

§ 266

590. COR. 2. Similar prisms or pyramids are to each other as the cubes of their altitudes; and similar polyhedrons are to each other as the cubes of any two homologous lines.

Ex. 1. The portion of a tetrahedron cut off by a plane parallel to any face is a tetrahedron similar to the given tetrahedron.

Ex. 2. Two tetrahedrons, having a dihedral angle of one equal to a dihedral angle of the other, and the faces including these angles respectively similar, and similarly placed, are similar.

Ex. 3. Given two similar polyhedrons, whose volumes are 125 feet and 12.5 feet respectively; find the ratio of two homologous edges.

ON REGULAR POLYHEDRONS.

591. DEF. A Regular polyhedron is a polyhedron all of whose faces are equal regular polygons, and all of whose polyhe dral angles are equal.

The regular polyhedrons are the tetrahedron, octahedron and icosahedron, all of whose faces are equal equilateral triangles; the hexahedron, or cube, whose faces are squares; the dodecahedron, whose faces are regular pentagons.

Only these five regular polyhedrons are possible, for a polyhedral angle must have at least three face angles, and must have the sum of its face angles less than four right angles, (§ 488). Hence :

I. If the faces be equilateral triangles, polyhedral angles may be formed of them in groups of 3, 4, or 5 only, as in the tetrahedron, octahedron and icosahedron. Since each angle of an equilateral triangle is two-thirds of a right angle, the sum of six such angles is four right angles, and therefore greater than a convex polyhedral angle.

II. If the faces be squares, polyhedral angles may be formed of them in groups of three only, as in the regular hexahedron, or cube; since four such angles would be four right angles.

III. If the faces be regular pentagons, polyhedral angles may be formed of them in groups of three only, as in the regular dodecahedron; since four such angles would be greater than four right angles.

IV. We can proceed no farther; for a group of three angles of regular hexagons would equal four right angles, and of regular heptagons, etc., would be greater than four right angles.