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two bases of the frustum: therefore the convex surface of

the frustum is equal to

(ABCDE+abcde) × Hf.

Hence, The convex surface of a frustum, &c.

PROPOSITION XVIII. THEOREM.

The solidity of a pyramid SABCDE, is measured by the product of its base by a third of its altitude.

For, extending the planes SEB, SEC through the diagonals EB, EC, the polygonal pyramid SABCDE will be divided into several triangular pyramids all having the same altitude SO. But each of these pyramids is measured by multiplying its base ABE, BCE, or CDE, by the third part of its altitude SO; (15. 7. Cor. ;) hence the sum of these triangular pyramids, or the polygonal

E

S

C

pyramid SABCDE will be measured by the sum of the triangles ABE, BCE, CDE, or the polygon ABCDE, multiplied by SO. Hence,

The solidity of every pyramid is equal to the product of its base into a third of its altitude.

Cor. 1. Every pyramid is the third part of the prism which has the same base and the same altitude.

Cor. 2. Two pyramids having the same altitude are to each other as their bases.

Scholium. 1. Let B represent the base of a pyramid; H its altitude; the solidity of the pyramid will be BxH, or H× B, or BH.

Scholium. 2. The solidity of any polyedral body may be

computed, by dividing the body into pyramids; and this division may be accomplished in various ways. One of the simplest is to make all the planes of division pass through the vertex of one solid angle; in that case, there will be formed as many partial pyramids as the polyedron has faces, minus those faces which form the solid angle whence the planes of division proceed.

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If a pyramid be cut by a plane parallel to its base, the frustum that remains when the small pyramid is taken away, is equal to the sum of three pyramids having for their common altitude the altitude of the frustum, and whose bases are the lower base of the frustum, the upper one, and a mean proportional between the two bases.

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two bases may be regarded as situated in the same plane; in which case, the plane abcde, if produced, will form in the triangular pyramid a section fgh, situated at the same altitude above the common plane of the bases; and therefore the section fgh will be to the section abd as the base FGH is to the base ABD, (13. 7. Cor. 1,) and since the bases are equivalent, the sections will be so likewise. Hence the pyramids Sabcde, Tfgh are equivalent, for their altitude is the same and their bases are equivalent. The whole pyramids

SABCDE, TFGH are equivalent for the same reason; therefore the frustums ABDdab, FGHhfg, are equivalent: consequently if the proposition can be proved in the single case of the frustum of a triangular pyramid, it will be true of every other.

Let FGHhfg be the frustum of a triangular pyramid, having parallel bases through the three points F, g, H, extend the plane FgH; it will cut off from the frustum the triangular pyramid gFGH. This pyramid has for its base the lower base FGH of the frustum; its altitude likewise is that of the frustum, because the vertex g

lies in the plane of the upper base fgh.

This pyramid being cut off, there will remain the quadrangular pyramid gfhHF, whose vertex is g, and base fhHF. Extend the plane fgH through the three points f, g, H; it will divide the quadrangular pyramid into two triangular pyramids gFfH, gfhH. The latter has for its base the upper base gfh of the frustum; and for its altitude, the altitude of the frustum, because its vertex H lies in the lower base. Thus, we already know two of the three pyramids which compose the frustum.

It remains to examine the third gFƒH. Now, if gK be drawn parallel to ƒF, and if we conceive a new pyramid fFKH, having K for its vertex and FfH for its base, these two pyramids will have the same base FfH; they will also have the same altitude, because their vertices g and K lie in one line gK, parallel to Ff, and consequently parallel to the plane of the base: hence these pyramids are equivalent. But the pyramid ƒFKH may be regarded as having its ver

tex in f, and thus its altitude will be the same as that of the frustum as to its base FKH, we are now to show that this is a mean proportional between the bases FGH and fgh. Now in the triangles FHK, fgh, the angle F=f, and the side FK=fg; hence we have, (24. 4,)

FHK fgh: FH×FK: fhxfg: Dividing analogous terms by the equals FK, fg, (13. 3. Cor.,) we have,

FHK fgh: FH : fh.

Also, (24. 4,) FGH: FHK:: FHXFG: FHFK or fg Hence, (13. 3. Cor.,)

FGH FHK: : FG : fg.

And since the triangles FGH, fgh, are similar, we have, (Def. 3. 4,)

FG fg FH: fh:

Hence, (11. 3,) FGH FHK: FHK fgh:

:

Consequently FHK is a mean proportional between the two bases FGH, fgh. (Def. 13. 3.) Hence,

The frustum of a triangular pyramid, with parallel bases, is equivalent to three pyramids whose common altitude is that of the frustum, and whose bases are the lower base of the frustum, the upper base, and a mean proportional between the two bases.

Scholium. If H represents the altitude of a frustum of a pyramid B and b its parallel bases; then Bb will be a

mean proportional between them. (7. 3. Cor.) The solidity of the frustum will be,

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Hence,

Two similar pyramids are to each other as the cubes of their homologous sides.

C

For two pyramids being similar, the smaller may be placed within the greater, so that the solid angle S shall be common to both. In that position, the bases ABCDE, abcde, will be parallel; because, since the homologous faces are similar, the angle Sab is equal to SAB, and Sbc to SBC; hence the plane ABC is parallel to the plane abc. (13. 6.) Now let SO be the perpendicular drawn from the vertex S to the plane ABC, and o the point where this perpendicular meets the plane abc: from what has already been shown, (13, 7,) we shall have

A

SO: So :: SA: Sa :: AB: ab;

consequently, SO: So: AB: ab.

::

E

C

B

But the bases ABCDE, abcde being similar figures, we have, (27. 4,)

ABCDE: abcde: : AB2 : ab1.

Multiplying the corresponding terms of these two proportions, there results the proportion

ABCDESO : abcde x}{So : : AB3 : ab3.

Now ABCDESO is the solidity of the pyramid SABCDE, and abcdexSo is that of the pyramid Sabcde. (18. 7.) Hence, Two similar pyramids are, &c.

Scholium. Let P and p represent the solidities of two similar polyedrons; A and a two homologous sides or diagonals of these polyedrons: then we shall have,

P:p:: A': a3.

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