XXI. SOLID MEASURE. 230. A Table of Solid Measure might be given as extensive as the Table of Square Measure of Art. 126; but it will be sufficient to observe that 1728 cubic inches make 1 cubic foot, 231. The connection which subsists between the systems of measures and of weights must be noticed. The grain is thus determined: a cubic inch of pure water weighs 252-458 grains. A pound Avoirdupois contains 7000 grains. 7000 A cubic foot of pure water weighs 1728 × 252.458 grains, 16 x 1728 × 252.458 that is, Avoirdupois ounces: it will be found that this number to three decimal places is 997·137. Thus it is usually sufficient in practice to take 1000 Avoirdupois ounces as the weight of a cubic foot of pure water. A gallon is a measure which will hold 10 Avoirdupois lbs. of pure water, that is 70000 grains. Hence the num ber of cubic inches in a gallon is 70000 252.458 it will be found that this number to three decimal places is 277.274. Thus it is usually sufficient in practice to take 277 as the number of cubic inches in a gallon. XXII. RECTANGULAR PARALLELEPIPED. 232. Suppose we have a rectangular parallelepi ped, which is 4 inches long, 3 inches broad, and 2 inches high. Let the rectangular parallelepiped be cut by planes, an inch apart, parallel to the faces; it is thus divided into 24 equal solids, each of which is a cube, being an inch long, an inch broad, and an inch high such a cube is called a cubic inch. The rectangular parallelepiped then contains 24 cubic inches; this fact is also expressed thus: the volume of the rectangular parallelepiped is 24 cubic inches. The word content, or the word solidity may be used instead of the word volume. The number 24 is the product of the numbers 4, 3, and 2, which denote respectively the length, the breadth, and the height of the rectangular parallelepiped. 233. If a rectangular parallelepiped be 8 inches long, 7 inches broad, and 5 inches high, we can shew in the same manner that its volume is 8 times 7 times 5 cubic inches, that is, 280 cubic inches. Similarly, if a rectangular parallelepiped be 15 inches long, 12 inches broad, and 10 inches high, its volume is 15 times 12 times 10 cubic inches, that is, 1800 cubic inches. And so on. 234. In the same manner, if a rectangular parallelepiped be 4 feet long, 3 feet broad, and 2 feet high, its volume is 24 cubic feet; that is, the rectangular parallelepiped might be divided into 24 equal solids, each being a foot long, a foot broad, and a foot high. If a rectangular T. M. 9 parallelepiped be 4 yards long, 3 yards broad, and 2 yards high, its volume is 24 cubic yards. And so on. 235. The beginner will observe, that the way in which volumes are measured is another case of the general principle explained in Art. 131. We fix on some volume as a standard, and we compare other volumes with this standard. The most convenient standard is found to be the volume of a cube; it may be a cubic inch, or a cubic foot, or any other cube. 236. In order then to find the volume of a rectangular parallelepiped we must express the length, the breadth, and the height in terms of the same denomination; and the product of the numbers which denote the length, the breadth, and the height, will denote the volume. If the length, the breadth, and the height, are all expressed in inches, the volume will be expressed in cubic inches; if the length, the breadth, and the height, are all expressed in feet, the volume will be expressed in cubic feet; and so on. 237. In the example given in Art. 232, we find that the volume is equal to 4 × 3 × 2 cubic inches. Now suppose we take for the base of the rectangular parallelepiped the rectangle which is 4 inches by 3; then the height is 2 inches, and the area of the base is 12 square inches. Thus the number denoting the volume is equal to the product of the numbers denoting the area of the base and the height. If we take for the base the rectangle which is 4 inches by 2; then the height is 3 inches: and, as before, the number denoting the volume is equal to the product of the numbers denoting the area of the base and the height. Or we may take for the base the rectangle which is 3 inches by 2; then the height is 4 inches: and, as before, the number denoting the volume is equal to the product of the numbers denoting the area of the base and the height. 238. The student will now be able to understand the way in which we estimate the volumes of solids, and to use correctly the rules which will be given: the rules will be stated with brevity, but this will present no difficulty to those who have read the foregoing explanations. 239. To find the volume of a rectangular parallelepiped. RULE. Multiply together the length, the breadth, and the height, and the product will be the volume. Or. Multiply the area of the base by the height, and the product will be the volume. 240. Examples. (1) The length of a rectangular parallelepiped is 2 feet 6 inches, the breadth is 1 foot 8 inches, and the height is 9 inches. 2 feet 6 inches = 30 inches, 1 foot 8 inches=20 inches, 30 x 20 x 95400. Thus the volume is 5400 cubic inches. (2) The area of the base of a rectangular parallelepiped is 15 square feet, and the height is 3 feet 9 inches. 3 feet 9 inches=3.75 feet. 15 x 3'75 56°25. Thus the volume is 56′25 cubic feet. 241. If we know the volume of a rectangular parallelepiped, and also the area of its base, we can find the height by dividing the number which expresses the volume by the number which expresses the area of the base; and similarly if we know the volume and the height we can find the area of the base. Of course we must be careful to use corresponding denominations for the volumes and the known area or height: see Art. 132. 242. Examples. (1) The volume of a rectangular parallelepiped is 576 cubic inches, and the area of the base is half a square foot find the height. Half a square foot=72 square 576 inches; = = 8. 72 Thus the height is 8 inches. (2) The volume of a rectangular parallelepiped is 8 cubic feet, and the height is 1 foot 4 inches: find the area of the base. 8 8 3 1 foot 4 inches = 1}} feet; =- X-= 6. 1층 4 Thus the area of the base is 6 square feet. 243. A cube is a rectangular parallelepiped having its length, breadth, and height equal; hence to find the volume of a cube we multiply the number which denotes the length by itself, and multiply the product by the number again. Thus we see the reason for using the term cube of a number to denote the result obtained by multiplying a number by itself, and the product by the number again. 244. The statements made in Art. 230 as to the connection between cubic inches, cubic feet, and a cubic yard, will be easily understood by the aid of the explanations of the present Chapter. Take, for example, the first statement that 1728 cubic inches make 1 cubic foot: a cubic foot is a cube 12 inches long, 12 inches broad, and 12 inches high; and therefore by the method of Art. 232 we see that a cubic foot contains 12 × 12 × 12 cubic inches, that is, 1728 cubic inches. 245. We will now solve some exercises. (1) Find how many bricks will be required to build a wall 25 yards long, 15 feet high, and I foot 10 inches thick; a brick being 9 inches long, 4 wide, and 3 deep. The number of cubic inches in the wall is 9 the number of cubic inches in a brick is 9 x ×3; divide the former number by the latter, and the quotient is 30000, which is therefore the number of bricks required. |