235. Let us now suppose that the sides of A B C D, instead of being a whole number of inches, contain some inches and a fraction. For example, let A B be 3½ inches, or (114) ⁷/₂ of an inch, and let A C contain 2½ inches, or ⁹/₄ of an inch. Draw A E twice as long as A B, and A F four times as long as A C, and complete the rectangle A E F G. The rest of the figure needs no description. Then, since A E is twice A B, or twice ⁷/₂ inches, it is 7 inches. And since A F is four times A C, or four times ⁹/₄ inches, it is 9 inches. Therefore, the whole rectangle A E F G contains, by (234), 7 × 9 or 63 square inches. But the rectangle A E F G contains 8 rectangles, all of the same figure as A B C D; and therefore A B C D is one-eighth part of A E F G, and contains ⁶³/₈ square inches. But ⁶³/₈ is made by multiplying ⁹/₄ and ⁷/₂ together (118). From this and the last article it appears, that, whether the sides of a rectangle be a whole or a fractional number of inches, the number of square inches in its surface is the product of the numbers of inches in its sides. The square itself is a rectangle whose sides are all equal, and therefore the number of square inches which a square contains is found by multiplying the number of inches in its side by itself. For example, a square whose side is 13 inches in length contains 13 × 13 or 169 square inches. 236. EXERCISES. What is the content, in square feet and inches, of a room whose sides are 42 ft. 5 inch. and 31 ft. 9 inch.? and supposing the piece from which its carpet is taken to be three quarters of a yard in breadth, what length of it must be cut off?--_Answer_, The content is 1346 square feet 105 square inches, and the length of carpet required is 598 feet 6⁵/₉ inches. The sides of a rectangular field are 253 yards and a quarter of a mile; how many acres does it contain?--_Answer_, 23. What is the difference between 18 _square miles_, and a square of 18 miles long, or 18 _miles square_?--_Answer_, 306 square miles. 237. It is by this rule that the measure in (215) is deduced from that in (214); for it is evident that twelve inches being a foot, the square foot is 12 × 12 or 144 square inches, and so on. In a similar way it may be shewn that the content in cubic inches of a cube, or parallelepiped,[48] may be found by multiplying together the number of inches in those three sides which meet in a point. Thus, a cube of 6 inches contains 6 × 6 × 6, or 216 cubic inches; a chest whose sides are 6, 8, and 5 feet, contains 6 × 8 × 5, or 240 cubic feet. By this rule the measure in (216) was deduced from that in (214). [48] A parallelepiped, or more properly, a _rectangular_ parallelepiped, is a figure of the form of a brick; its sides, however, may be of any length; thus, the figure of a plank has the same name. A cube is a parallelepiped with equal sides, such as is a die. SECTION II. RULE OF THREE.