H F G M Let the parallelopipedons AG, AL, have the same base BD, then will they be to each other as their altitudes AE, AI. First, suppose the altitudes AE, AI, to be E to each other as two whole numbers, as 15 is to 8, for example. Divide AE into 15 equal parts; whereof AI will contain 8; and through x, y, z, &c. the points of division, draw planes parallel to the base. These planes will cut the solid AG into 15 partial parallelopipedons, all equal to each other, because they have z equal bases and equal altitudes-equal bases, since every section MIKL, made parallel to A the base ABCD of a prism, is equal to that base (Prop. II.), equal altitudes, 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. B K D Again, if the ratio of AE to AI cannot be exactly 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 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 O and I. Let P be the parallelopipedon, whose base is ABCD, and altitude Am; since the altitudes AE, Am, are to each other as the two whole numbers, we shall have sol. AG: P:: AE: Am. But by hypothesis, we have therefore, sol. AG sol. AL:: AE: AO; 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: hence the fourth term of this proportion sol. AG sol. 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 XII. THEOREM. Two rectangular parallelopipedons, having the same altitude, are to each other as their bases. 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, : sol. AQ sol. AK :: AD : AM. 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 :: AB× AD: AO× AM. But AB × AD represents the base ABCD; and AO X AM represents the base AMNO; hence two rectangular parallelopipedons of the same altitude 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 to say, as the products of their three dimensions. sol. AK: sol. AZ:: AE : AX. Multiplying together the corresponding terms of these proportions, and omitting in the result the common multiplier sol. ÁK; we shall have : sol. AG sol. AZ :: ABCD × AE: AMNO × AX. Instead of the bases ABCD and AMNO, put AB× AD and AO× AM it will give sol. AG : sol. AZ :: AB× AD× AE: AO × AM × AX. Hence any two rectangular parallelopipedons are to each other, &c. 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 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 (Book IV. Prop. IV. Sch.). 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 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, forms 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 solidities 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 a 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 XIV. 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 is equivalent to a rectangular parallelopipedon, having the same altitude and an equivalent base (Prop. X.). Now the solidity of the latter is equal to its base multiplied by its height; hence the solidity of the former is, in like manner, equal to the product of its base by its altitude. In the second place, any triangular prism is half of the parallelopipedon so constructed as to have the same altitude and a double base (Prop. VII.). But the solidity of the latter is equal to its base multiplied by its altitude; hence that of a triangular prism is also equal to the product of its base, which is half that of the parallelopipedon, multiplied into its altitude. In the third place, any prism may be divided into as many triangular prisms of the same altitude, as there are triangles capable of being formed 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 altitude is the same for all, it follows that the sum of all the partial prisms must be equal to the sum of all the partial 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. Comparing two prisms, which have the same altitude, the products of their bases by their altitudes will be as the bases simply; hence two prisms of the same altitude are to each other as their bases. For a like reason, two prisms of the same base are to each other as their altitudes. And when neither their bases nor their altitudes are equal, their solidities will be to each other as the products of their bases and altitudes. PROPOSITION XV. THEOREM. Two triangular pyramids, having equivalent bases and equal altitudes, are equivalent, or equal in solidity. Let S-ABC, S-abc, be those two pyramids ; let their equivalent bases ABC, abc, be situated in the same plane, and let AT be their common altitude. If they are not equivalent, let S-abc |