point whatever in space and join it to all the vertices of the polyedron. The bases of the different pyramids thus formed will be the faces of the polyedron, and the altitudes of the pyramids the perpendiculars let fall from the point taken upon the planes of these faces. The volume of the polyedron will be the arithmetical or the algebraic sum of the volumes of the pyramids, according as their common vertex (the point taken) is within or without the polyedron.

Sometimes the division of the polyedron into pyramids is effected by taking one of the vertices, and from that drawing diagonals to all the other vertices not adjacent to this.

SCHOLIUM 2. When a point can be found in the interior of a polyedron, equidistant from all its faces, the pyramids composing the polyedron, which have this point for a common vertex, will have as a common altitude the perpendicular let fall from this point on any one of the faces, and the volume of the polyedron will have for its measure the third of the product of its surface by this perpendicular.

As any polygonal pyramid may be divided into triangular pyramids (tetraedrons), it is evident that any polyedron may be divided also into tetraedrons.

PROPOSITION XXVI.

THEOREM.

If a polygonal pyramid, S-ABCDE, and a triangular pyramid, T-FGH, having equivalent bases lying in the same plane and the same altitude, be cut by a plane, abd, parallel to the plane of the bases, the frustums, ABD-abd, FGH-fgh, thus cut off, will be equivalent.

T

For the plane abd, produced, forms in the triangular pyramid a section, fgh, situated at the same height above the common plane of the bases; and therefore, since the base ABCDE is equivalent to FGH, the section abcde will be equivalent to the section fgh (Prop. XXII., Cor.).

a

mids S-ABCDE, T-FGH are equivalent for the same reason; hence, the frustums ABD-abd, FGH-fgh, which remain after taking the small pyramids from the wholes respectively, are equivalent.

PROPOSITION XXVII.

THEOREM.

Every frustum of a pyramid is equal to the sum of three pyramids, having for their common altitude the altitude of the frustum, and for bases the lower base of the frustum, the upper base, and a mean proportional between the two bases.

D

F

E

From the preceding theorem it follows that if the proposition can be proved in the single case of the frustum of a triangular pyramid, it will be true of any other frustum. Let ABC-DEF be the frustum of a triangular pyramid. The planes AEC, DEC divide it into three triangular pyramids, E-ABC, E-DCF, E-DCA. The first, E-ABC, has for its base the lower base, ABC, of the frustum; its altitude is likewise that of the frustum, since its vertex E lies in the plane of the upper base, EDF. If we take the point C for its vertex, the second pyramid, E-DCF, has for its base DEF, the upper base of the frustum; its altitude is also the altitude of the frustum, since its vertex, C, lies in the lower base, ABC.

B

Thus we know two of the pyramids which compose the frustum. It remains to consider the third pyramid, E-DCA. To measure this,

we compare it with the second, E-DCF. These two pyramids having the same altitude (considered with reference to the common vertex, E), are to each other as their bases, CDA, CDF.

But

CDA: CDF :: AC: DF,

since the triangles have the same altitude. Hence

E-DCA E-DCF:: AC: DF.

But since the bases of the frustum, ABC and DEF, are similar,

AC: DF:

Therefore

or

VABC: DEF (Book III., Prop. XXVII.).

Now

I

E-DCF DEF ×

of the altitude of frustum.

3

Therefore, the third pyramid, E-DCA, is equivalent to a pyramid having for its base a mean proportional between the two bases of the frustum, and for its altitude the altitude of the frustum.

Hence, the frustum of a pyramid is equivalent to three pyramids whose common altitude is the altitude of the frustum, and whose bases are respectively the lower and upper bases of the frustum, and a mean proportional between these two bases.

SCHOLIUM. Let V be the volume of a frustum of a pyramid, B and B' its bases, and H its altitude. The volumes of the three pyramids

In two similar tetraedrons, S-ABC, T-DEF, the homologous faces are similar, and the homologous triedrals are equal.

Conversely, Two tetraedrons are similar, 1st, when their homologous faces are similar; 2d, when their homologous triedrals are equal.

First. When the homologous faces are similar, the homologous edges are proportional, and therefore the tetraedrons similar.

Secondly. When the homologous triedrals are equal, the plane angles which form them are respectively equal, and therefore the homologous faces have the angles of the one equal to the angles of the other, and are therefore similar. Hence, the tetraedrons are similar.

COR. 1. Two similar tetraedrons have their six homologous diedrals equal each to each, and conversely.

COR. 2. Every section, abcde, parallel to the base of a pyramid, S-ABCDE, determines another pyramid, S-abcde, similar to the first.

For the planes SEC, SEB, divide the two pyramids into tetraedrons, S-ABE and S-abe, S-BCE and S-bce, S-CDE and S-cde, similar each to each, because their faces are similar. These tetraedrons are also similarly situated. Hence, by definition, the two pyramids are similar.

E

B

1. The homologous faces are similar, each to each, and their inclinations are the same.

2. The homologous solid angles are equal.

First.-The two polyedrons being composed of the same number of tetraedrons, similar each to each and similarly situated, their surfaces are also composed of the same number of triangles, similar each to each and similarly grouped. Moreover, the inclination of two adjacent triangles of the first surface is equal to the inclination of the two homologous triangles of the second; for these inclinations are either the homologous diedrals of two similar tetraedrons, or they are the sums of a like number of homologous diedrals: whence it results that two similar polyedrons are contained by the same number of faces, similar each to each, and equally inclined to each other.

Secondly. The homologous solid angles are equal; for all their plane angles and diedrals are equal each to each and similarly grouped.

COR.-The edges, the diagonals, and in general all the homologous lines of two similar polyedrons, are proportional.

PROPOSITION XXXI.

THEOREM.

Similar tetraedrons are to each other as the cubes of their homologous edges.

We can always place the two tetraedrons so that they shall have a triedral, S, in common (Prop. XXIX, Cor. 2.). Then, since the bases, ABC, DEF, are similar, we have,

And since the angles SAB and SDE are equal, as also SBC and SEF, the plane DEF is parallel to the plane ABC. Therefore,

SH: Sh: SA : SD:: AB : DE,

or,

SH: Sh: AB : DE.

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