have

solid AK: solid AZ:: AE: AX.

Multiplying these two proportions in order and omitting in the result the common multiplier solid AK, we obtain

solid AG solid AZ:: ABCD × AE: AMNO × AX.

In the place of the bases ABCD, AMNO, we can substitute AB × AD, AO × AM, which will give

:

solid AG solid AZ :: AB× AD × AE : AO × AM × AX. Therefore any two rectangular parallelopipeds are to each other as the products of their bases by their altitudes, or as the products of their three dimensions.

· 405. Scholium. Hence we may take for the measure of a rectangular parallelopiped the product of its base by its altitude, or the product of its three dimensions. It is on this principle that we estimate all other solids.

In order to understand this measure it is necessary to recollect that by the product of two or several lines is meant the product of the numbers which represent these lines, and these numbers depend upon the linear unit, which may be taken at pleasure; the product therefore of the three dimensions of a parallelopiped is a number which of itself has no meaning, and which would be different according as one or another linear unit is used. But if, in like manner, the three dimensions of another parallelopiped are multiplied together, by estimating them according to the same linear unit, the two products would be to each other as the two parallelopipeds and would give an idea of their relative magnitude.

The magnitude of a solid, its volume, or its extension, constitutes what is called its solidity; and the word solidity is employed particularly to denote the measure of a solid; thus we say that the solidity of a rectangular parallelopiped is equal to the product of its base by its altitude, or the product of its three dimensions.

The three dimensions of a cube being equal to each other, if the side is 1, the solidity will be 1 × 1 × 1, or 1; if the side is

† Content is often employed by English writers to denote both solid and superficial measures. The word solidity, though most commonly used, is exceptionable, as it is likely to suggest to the mind of the student the idea of resistance. The term volume has been adopted by some as preferable to solidity.

2, the solidity will be 2 x 2 x 2, or 8; if the side is 3, the solidity will be 3 x 3 x 3, or 27, and so on; thus, the sides of cubes being as the numbers 1, 2, 3, &c., the cubes themselves, or their solidities, are as the numbers 1, 8, 27, &c. Hence the origin of what in arithmetic is called the cube of a number; it is the product arising from three factors, which are each equal to this number.

If it were proposed to make a cube double of a given cube, it would be necessary that the side of the cube sought should be to the side of the given cube as the cube root of 2 is to 1. Now it is easy to find, by a geometrical construction, the square root of 2; but we cannot, in this way, find the cube root of this number, at least by the simple operations of elementary geometry, which consist in employing only straight lines, two points of which are known, and circles whose centres and radii are determined.

On account of this difficulty, the problem of the duplication of the cube was celebrated among the ancient geometers, as also that of the trisection of an angle, which is nearly of the same character. But the solutions, of which problems of this kind are susceptible, have long been known; and, although less simple than the constructions of elementary geometry, they are not less exact or less rigorous.

THEOREM.

406. The solidity of a parallelopiped, and in general of any prism whatever, is equal to the product of its base by its altitude.

Demonstration. 1. A parallelopiped of whatever kind is equivalent to a rectangular parallelopiped having the same altitude and an equivalent base (401). But the solidity of this last is equal to the product of its base by its altitude (405); therefore the solidity of the first is also equal to the product of its base by its altitude.

2. Every triangular prism is half of a parallelopiped, so constructed as to have the same altitude and a base twice as great (397). Now the solidity of this last is equal to the product of its base by its altitude (405); therefore the solidity of the triangular prism is equal to the product of its base, half of that of the parallelopiped, by its altitude.

3. A prism of whatever kind' may be divided into as many triangular prisms of the same altitude, as there are triangles in the polygon taken for a base. But the solidity of each triangular prism is equal to the product of its base by its altitude; and,' since the altitude is the same in each, it follows that the sum of all the partial prisms is equal to the sum of all the triangles, taken for bases, multiplied by the common altitude. Therefore the solidity of a prism of whatever kind is equal to the product of its base by its altitude.

407. Corollary. If we compare two prisms, which have the same altitude, the products of the bases by the altitudes will be as the bases; therefore two prisms of the same altitude are to each other as their bases; for a similar reason, two prisms of the samo base are to each other as their altitudes.

LEMMA.

408. If a pyramid S-ABCDE (fig. 214) is cut by a plane a b d, Fig, 214, parallel to the base,

1. The sides SA, SB, SC,..... and the altitude SO, will be divided proportionally in a, b, c,..... and o;

2. The section abcde will be a polygon similar to the base ABCDE.

Demonstration. The planes ABC, abc, being parallel, their intersections AB, ab, by a third plane SAB, will be parallel (340); consequently the triangles SAB, S a b, are similar, and SA: Sa:: SB: Sb;

and so on; therefore the sides SA, SB, SC, &c., are cut proportionally at a, b, c, &c. The altitude SO is cut in the same proportion at the point 0; for BO and bo are parallel (340), and consequently

SO: So:: SB: Sb (196).

2. Since a b is parallel to AB, bc to BC, cd to CD, &c., the angle abc ABC, the angle b c d = BCD, and so on. Moreover, on account of the similar triangles SAB, Sab,

=

AB: ab:: SB: Sb;

SB: Sb:: BC: bc;

and, on account of the similar triangles SBC, Sbc,

BC: bc:: CD: cd,

in like manner,
and so on. Therefore the polygons ABCDE, abcde, have their
angles equal, each to each, and their homologous sides propor-
tional; that is, they are similar.

409. Corollary. Let S-ABCDE, S-XYZ, be two pyramids that have a common vertex, and whose altitudes are the same, or whose bases are situated in the same plane; if these pyramids be cut by a plane parallel to their bases, the sections a b c d e, xyz, thus formed, will be to each other as the bases ABCDE, XYZ. For, the polygons ABCDE, abc de, being similar, their surfaces are as the squares of their homologous sides AB, ab; but AB: ab:: SA: Sa,

whence

SA: Sa:: SX: Sx,

ABCDE: abcde:: XYZ: xyz;

therefore the sections abcde, xyz, are to each other as their bases ABCDE, XYZ.

Fig. 215.

LEMMA.

410. Let S-ABC (fig. 215), be a triangular pyramid, of which S is the vertex and ABC the base; if the sides SA, SB, SC, AB, AC, BC, be bisected at the points D, E, F, G, H, I, and through these points the straight lines DE, EF, DF, EG, FH, EI, GI, GH, be drawn; we say that the pyramid S-ABC may be considered as composed of two prisms AGH-FDE, EGI-CFH, equivalent to each other, and two equal pyramids S-DEF, E-GBI.

Demonstration. It follows from the construction, that ED is parallel to BA, and GE to AS (199); hence the figure ADEG is a parallelogram. For the same reason, the figure ADFH is also a parallelogram; consequently the straight lines AD, GE, HF, are equal and parallel; therefore the solid AGH-FDE is a prism (346).

It may be shown, in like manner, that the two figures EFCI, CIGH, are parallelograms, and that thus the straight lines EF, IC, GH, are equal and parallel; therefore the solid EGI-CFH is

also a prism. Now we say that these two triangular prisms are equivalent to each other.

Indeed, if upon the edges GI, GE, GH, the parallelopiped GX be formed, the triangular prism EGI-CFH will be half of this parallelopiped (397); on the other hand, the prism AGH-FDE is also equal to half of the parallelopiped GX (406), since they have the same altitude, and the triangle AGH, the base of the prism, is half of the parallelogram GICH (168), the base of the parallelopiped. Therefore the two prisms EGI-CFH,AGH-FDE, are equivalent to each other.

These two prisms being taken from the pyramid S-ABC there will remain only the two pyramids S-DEF, E-GBI; now we say that these two pyramids are equal to each other.

Indeed, since the following sides are equal, namely, BE=SE, BG = AG= DE, EG=AD=SD, the triangle BEG is equal to the triangle ESD (43). For a similar reason, the triangle BEI is equal to the triangle ESF; moreover the mutual inclination of the two planes BEG, BEI, is the same as that of the two planes ESD, ESF, since BEG, ESD, are in the same plane, and BEI, ESF, are also in the same plane. If then, in order to apply the one pyramid to the other, we place the triangle EBG upon its equal EDS, the plane BEI must fall upon the plane ESF; and, since the triangles are equal and similarly disposed, the point I will fall upon F, and the two pyramids will coincide throughout (384).

Therefore the entire pyramid S-ABC is composed of two triangular prisms AGF, GIF, equivalent to each other, and two equal pyramids S-DEF, E-GBI.

411. Corollary 1. From the vertex S let fall upon the plane ABC the perpendicular SO, and let P be the point, where this perpendicular meets the plane DEF, parallel to ABC; since SD = 1 SA, we have SP = ¦ SO (408), and the triangle DEF = 1 triangle ABC (218); consequently the solidity of the prism AGH-FDE=ABC SO;

and the solidity of the two prisms AGH-FDE, EGI-CFH, taken together, is equal 1 ABC × SO. These two prisms are less than the pyramid S-ABC, since they are contained in it; therefore the solidity of a triangular pyramid is greater than the fourth part of the product of its base by its altitude.