tudes, because the altitudes are the equal divisions, Ax, xy, yz, &c. But of those 15 equal parallelopipedons, 8 are contained in AL; hence, the solid AG is to the solid AL as 15 is to 8, or generally, as the altitude AE is to the altitude AI.

Second. If the ratio of AE to AI cannot be expressed exactly in numbers, it may still be shown, that we shall have

solid AG : solid AL :: AE AI

:

For, if this proportion is not correct, suppose we have, sol. AG : sol. AL :: AE AO greater than AI. Divide AE into equal parts, such that each shall be less than OI; there will be at least one point of division m, between 0 and I. Let P denote the parallelopipedon, whose base is ABCD, and altitude Am; since the altitudes AE, Am, are to each other as two whole numbers, we have

sol. AG : P : AE : Am.

But by hypothesis, we have

sol. AG : sol. AL :: AE : AO;

therefore (B. II., P. 4),

sol. AL : P :
:: AO: Am.

But AO is greater than Am; hence, if the proportion is correct, the solid AL must be greater than P. On the contrary, however, it is less therefore, AO cannot be greater than AI. By the same mode of reasoning, it may be shown that the fourth term cannot be less than AI; therefore, it is equal to AI: hence, rectangular parallelopipedons having equal bases, are to each other as their altitudes.

PROPOSITION XII. THEOREM.

Two rectangular parallelopipedons, having equal altitudes, are to each other as their bases.

Let the parallelopipedons AG, AK, have the same altitude AE; then will they be to each other as their bases AC, AN.

altitudes AB, AO: in like

B

manner, the two solids AQ, AK, having the same base AOLE, are to each other as their altitudes AD, AM. Hence, we have

sol. AG: sol. AQ :: AB A0;

:

also, sol. AQ sol. AK :: AD : AM.

4Q :

Multiplying together the corresponding terms of these proportions, and omitting, in the result, the common multiplier sol. AQ; we shall have

sol. AG sol. AK: ABXAD : AOXAM.

But ABX AD represents the area of the base ABCD; and AOXAM represents the area of the base AMNO; hence, two rectangular parallelopipedons having equal altitudes, are to each other as their bases.

PROPOSITION XIII. THEOREM.

Any two rectangular parallelopipedons are to each other as the products of their bases by their altitudes; that is, as the products of their three dimensions.

dons AK, AZ, having the same base NA, are to each other as their altitudes AE, AX; hence, we have,

sol. AK : sol. AZ :: AE : AX.

Multiplying together the corresponding terms of these proportions, and omitting in the result the common multiplier sol. AK; we shall have,

sol. AG : sol. AZ :: Instead of the bases ABCD and AMNO, put ABXAD and AOXAM, and we shall have,

ABCDXAE : AMNOXAX.

sol. AG: sol. AZ :: ABXAD×AE: AO×AM×AX: hence, any two rectangular parallelopipedons are to each other, as the products of their three dimensions.

Scholium 1. The magnitude of a solid, its volume or extent, is called its solidity; and this word is exclusively employed to designate the measure of a solid: thus, we say the solidity of a rectangular parallelopipedon is equal to the product of its base by its altitude, or to the product of its three dimensions.

In order to comprehend the nature of this measurement, it is necessary to consider, that the number of linear units in one dimension of the base multiplied by the number of linear units in the other dimension of the base, will give the number of superficial units in the base of the parallelopipedon (B. IV., P.4, s.) For each unit in height, there are evidently, as many solid units as there are superficial units. in the base. Therefore, the number of superficial units in the base multiplied by the number of linear units in the altitude, gives the number of solid units in the parallelopipedon.

If then, we assume as the unit of measure, the cube whose edge is equal to the linear unit, the solidity will be expressed numerically, by the number of times which the solid contains its unit of measure.

Scholium 2. As the three dimensions of the cube are equal, if the edge is 1, the solidity is 1x1x1=1: if the edge is 2, the solidity is 2x2x2=8; if the edge is 3, the solidity is 3x3x3=27; and so on. Hence, if the edges of a series of cubes are to each other as the numbers 1, 2, 3, &c., the cubes themselves, or their solidities, are as the numbers 1, 8, 27, &c. Hence it is, that in arithmetic, the cube of a number is the name given to a product which results from three equal factors.

If it were proposed to find a cube double of a given cube, we should have, unity to the cube-root of 2, as the edge of the given cube to the edge of the required cube. Now, by a geometrical construction, it is easy to find the square root of 2; but the cube-root of it cannot be found, by the operations of elementary geometry, which are limited to the employment of the straight line and circle.

Owing to the difficulty of the solution, the problem of the duplication of the cube became celebrated among the ancient geometers, as well as that of the trisection of an angle, which is a problem nearly of the same species. The solutions of these problems have, however, long since been discovered; and though less simple than the constructions of elementary geometry, they are not, on that account, less rigorous or less satisfactory.

PROPOSITION XIV. THEOREM.

The solidity of a parallelopipedon, and generally of any prism, is equal to the product of its base by its altitude.

·First. Any parallelopipedon is equivalent to a rectangular parallelopipedon, having an equal altitude and an equivalent base (P. 10). But, the solidity of a rectangular parallelopipedon is equal to its base multiplied by its height; hence, the solidity of any parallelopipedon is equal to the product of its base by its altitude.

Second. Any triangular prism is half a parallelopipedon so constructed as to have an equal altitude and a double base (P. 7). But the solidity of the parallelopipedon is equal to its base multiplied by its altitude; hence, that of the triangular prism is also equal to the product of its base, which is half that of the parallelopipedon, multiplied into its altitude.

Third. Any prism may be divided into as many triangular prisms of the same altitude, as there are triangles formed by drawing diagonals from a common vèrtex in the polygon which constitutes its base. But the solidity of each triangular prism is equal to its base multiplied by its altitude; and since the altitudes are equal, it follows that the sum of all the triangular prisms must be equal to the sum of all the triangles which constitute their bases, multiplied by the common altitude.

Hence, the solidity of any polygonal prism, is equal to the product of its base by its altitude.

Cor. Since any two prisms are to each other as the products of their bases and altitudes, if the altitudes be equal, they will be to each other as their bases simply; hence, two prisms of the same altitude are to each other as their bases. For a like reason, two prisms having equivalent bases are to each other as their altitudes.