all equal to each other, having equal bases and equa altitudes equal bases, because every section MIKL made parallel to the base ABCD of a prism, is equal to that base, (B. VII, Prop. I, Cor. ;) and equal altitudes, because these altitudes are the same 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.

Again, if the ratio of AE to AI cannot be expressed in numbers, it is to be shown that, notwithstanding, we shall have solid AG solid AL:: AE : AI.

:

For, if this proportion is not correct, suppose we have solid AG solid AL AE AO, greater than AI. Divide AE into equal parts, such that each shall be less than OI; there will vision m between O and I. don, whose base is ABCD altitudes AE, Am are to each other as two whole numbers, we shall have solid AG: P:: AE: Am. But, by hypothesis, we have

be at least one point of diLet P be the parallelopipeand altitude Am. Since the

solid AG solid AL

solid AL P

AE

AO; therefore : : 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; hence the fourth term of this proportion, solid AG : solid AL:: AE: x, cannot possibly be a line greater than AI.

By the same mode of reasoning, it might be shown that the fourth term cannot be less than AI; therefore it is equal to AI. Hence rectangular parallelopipedons having the same base, are to each other as their altitudes.

PROPOSITION IX.

THEOREM. Two rectangular parallelopipedons having the same altitude, are to each other as their bases.

third parallelopipedon AQ, which may be compared with each of the parallelopipedons AG, AK. The two solids AG, AQ having the same base AEHD, are to each other as their altitudes AB, AO. In like manner, the two solids AQ, AK having the same base AOLE, are to each other as their altitudes AD, AM. Hence we have the two proportions sol. AG sol. AQ: : AB: AO;

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

sol. AG sol. AK: AB× AD: AO× AM.

But AB × AD represents the base ABCD, and AO× AM represents the base AMNO: hence two rectangular parallelopipedons of the same altitude are to each other as their bases.

PROPOSITION X.

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

For, having placed the two solids AG, AZ, so that their surfaces have the common angle BAE, produce the interior planes necessary for completing the third parallelopipedon AK, having the same altitude with the parallelopipe

H

E

L

B

don AG. By the last proposition, we shall have

sol. AG sol. AK : : ABCD: AMNO.

N

M

But the two parallelopipedons AK, AZ having the same base AMNO, are to each other as their altitudes AE, AX: hence we have

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

Multiply together the corresponding terms of these proportions, omitting in the result the common multiplier sol. AK: we shall have

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

:

Instead of the bases ABCD and AMNO, put ABX AD and AO× AM: it will give

sol. AG sol. AZ:: ABX ADX AE: AOX AMX AX.

Hence any two rectangular parallelopipedons are to each other, etc.

Scholium. We are consequently authorized to assume, as the measure of a rectangular parallelopipedon, the product of its base by its altitude; in other words, the product of its three dimensions.

In order to comprehend the nature of this measurement, it is necessary to reflect, that by the product of two or more lines is always meant the product of the numbers which represent them; those numbers themselves being determined by their linear unit, which may be assumed at pleasure. Upon this principle, the product of the three dimensions of a parallelopipedon is a number, which signifies nothing of itself, and would be different if a different linear unit had been assumed; but if the three dimensions of another parallelopipedon are valued according to the same linear unit, and multiplied together in the same manner, the two products will be to each other as the solids, and will serve to express their relative magnitude.

The magnitude of a solid, its volume or extent, form what 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.

As the cube has all its three dimensions equal, if the side is 1, the solidity will be 1x1x1=1; if the side is 2, the solidity will be 2×2×2=8; if the side is 3, the solidity will be 3×3×3=27, and so on: hence, if the sides of a series of cubes are to each other as the numbers 1, 2, 3, &c., the cubes themselves (or their so

lidities,) will be as the numbers 1, 8, 27, &c. Hence it is, that in arithmetic, the cube of a number is the name given to the product which results from three factors, each equal to this number.

If it were proposed to find a cube double of a given cube, the side of the required cube would have to be to that of the given one, as the cube root of 2 is to unity. Now, by a geometrical construction, it is easy to find the square root of 2; but the cube root of it cannot be so found, at least not by the simple operations of elementary geometry, which consist in employing nothing but straight lines, two points of which are known, and circles whose centres and radii are determined.

Owing to this difficulty, 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 nearly of the same species. The solutions of which such problems are susceptible, 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 XI.

. THEOREM.

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

For, in the first place, any parallelopipedon, (B. VII, Prop. vII,) is equal to a rectangular parallelopipedon having the same altitude and an equal base. Now the