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 1×1×1=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 O 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. S 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 be the smaller and suppose Aa to be the altitude of a prism, which having ABC for its base, is equal to their difference. Divide the altitude AT into equal parts Ax, xy, yz, &c. each less than Aa, and let k be one of those parts; through the points of division pass planes parallel to the plane of the bases; the corresponding sections formed by these planes in the two pyramids will be respectively equivalent, namely DEF to def, GHI to ghi, &c. (Prop. III. Cor. 2.). This being granted, upon the triangles ABC, DEF, GHI, &c. taken as bases, construct exterior prisms having for edges the parts AD, DG, GK, &c. of the edge SA; in like manner, on bases def, ghi, klm, &c. in the second pyramid, construct interior prisms, having for edges the corresponding parts of Sa. It is plain that the sum of all the exterior prisms of the pyramid S-ABC will be greater than this pyramid; and also that the sum of all the interior prisms of the pyramid S-abc will be less than this pyramid. Hence the difference, between the sum of all the exterior prisms and the sum of all the interior ones, must be greater than the difference between the two pyramids themselves. Now, beginning with the bases ABC, abc, the second exterior prism DEF-G is equivalent to the first interior prism def-a, because they have the same altitude k, and their bases DEF, def, are equivalent; for like reasons, the third exterior prism GHI-K and the second interior prism ghi-d are equivalent; the fourth exterior and the third interior; and so on, to the last in each series. Hence all the exterior prisms of the pyramid S-ABC, excepting the first prism ABC-D, have equivalent corresponding ones in the interior prisms of the pyramid S-abc : hence the prism ABC-D, is the difference between the sum of all the exterior prisms of the pyramid S-ABC, and the sum of the interior prisms of the pyramid S-abc. But the difference between these two sets of prisms has already been proved to be greater than that of the two pyramids; which latter difference we supposed to be equal to the prism a-ABC: hence the prism ABC-D, must be greater than the prism a-ABC. But in reality it is less; for they have the same base ABC, and the altitude Ax of the first is less than Aa the altitude of the second. Hence the supposed inequality between the two pyramids cannot exist; hence the two pyramids S-ABC, S-abc, having equal altitudes and equivalent bases, are themselves equivalent. PROPOSITION XVI. THEOREM. Every triangular pyramid is a third part of the triangular prism having the same base and the same altitude. Let F-ABC be a triangular pyramid, ABC-DEF a triangular prism of the same base and the same altitude; the pyramid will be equal to a third of the prism. B Cut off the pyramid F-ABC from the prism, by the plane FAC; there will remain the solid F-ACDE, which may be considered as a quadrangular pyramid, whose vertex is F, and whose base is the parallelogram ACDE. Draw the diagonal CE; and pass the plane FCE, which will cut the quadrangular pyramid into two triangular ones F-ACE, F-CDE. These two triangular pyramids have for their common altitude the perpendicular let fall from F on the plane ACDE; they have equal bases, the triangles ACE, CDE being halves of the same parallelogram; hence the two pyramids F-ACE, F-CDE, are equivalent (Prop. XV.). But the pyramid F-CDE and the pyramid F-ABC have equal bases ABC, DEF; they have also the same altitude, namely, the distance between the parallel planes ABC, DEF; hence the two pyramids are equivalent. Now the pyramid F-CDE has already been proved equivalent to F-ACE; hence the three pyramids F-ABC, F-CDE, F-ACE, which compose the prism ABC-DEF are all equivalent. Hence the pyramid F-ABC is the third part of the prism ABC-DEF, which has the same base and the same altitude. A F PROPOSITION XVII. THEOREM. C Cor. The solidity of a triangular pyramid is equal to a third part of the product of its base by its altitude. The solidity of every pyramid is equal to the base multiplied by a third of the altitude. |