T66. CAPACITIES OF SUPERFICIES AND SOLIDS. Of all plane figures, the circle is the most capacious; that is, a given perimeter will enclose the greatest area in a circular form. ६ Of quadrilateral figures, the square is the most capacious: and of regular polygons, having the same perimeter, that is the most capacious which has the greatest number of sides. Of all solids, the sphere is the most capacious; that is, it comprehends the greatest solidity under a given surface. Of all quadrilateral prisms, the square prism is the most capacious; and of all prisms, the cylinder is the most capacious. EXAMPLES. 1. The standard bushel measure is 18.5 inches in diameter, and 8 inches deep; and its capacity is 2150.42 cubic inches. Suppose that such a measure in seasoning becomes warped, and its form is changed to an ellipse, whose diameters are 19.5 and 17.5 inches; required its capacity. Ans. 2144.137 cubic inches. 2. A pewter pint ale measure contains 35.25 cubic inches; what will be its capacity if it be beaten into a square form? Ans. 28.9 cubic inches. 3. The areas of a circle and a square are each 4 inches; required the circumference of the circle and the perimeter of the square. Ans. 7.08983; and 8 inches. 67. THE BALANCE, OR SCALES. The balance is a straight inflexible rod or beam, turning about a fixed point in the middle. If the point of suspension be not exactly in the middle of the beam, the balance is imperfect. The balance serves to compare the weights of bodies, and to determine whether two or more bodies differing in bulk have the same or equal quantities of matter. A perfect bal ance requires the point of sus D E pension of the beam to be in a right line with the points of suspension of the scales, and that there be as little friction as possible about the point on which the beam turns. Such a balance will be in equilibrio when the scales are empty, and likewise when the scales are loaded with equal weights. To discover a false balance :-Weigh the same body in each scale, and if the weights differ, the balance is false. To find the true weight of a body by means of a false bal ance: Weigh the body, first in one scale, and then in the other; and the square root of the product of the weights will be the true weight of the body. EXAMPLE.-If a body is found to weigh 11 pounds in one scale and 12 pounds in the other; what is its true weight? Ans. 11.489 pounds. The weights 1, 3, 9, 27, 81, 243, and 3 times 243, and so on, will weigh any number of pounds from 1 to the sum of the weights. T68. SPECIFIC GRAVITY OF BODIES. The specific gravity of a body is the ratio of its weight to the weight of the same volume of some other body assumed as a standard. Pure water being the standard commonly as sumed, the specific gravity of a solid body may be regarded as the weight of a given volume of the body compared with that of an equal volume of water; and it shows how much heavier or lighter it is than the same volume of water. Thus, if the weight of a cubic foot of any body is 10 times the weight of a cubic foot of water, its specific gravity is 10; or if the weight of a cubic foot of any body be 5 times the weight of a cubic foot of water, its specific gravity is 5, &c. To find the specific gravity of a body heavier than water: Weigh the body in the air, and then in water, and divide its weight in the air by the difference between its weight in the air and in water, and the quotient will be its specific gravity. EXAMPLE. What is the specific gravity of a body which weighs 3 pounds in the air and 1 pound in water? Ans. 3, or 1.500. To find the specific gravity of a body lighter than water:To the lighter body attach a heavier one that will cause it to sink, having previously ascertained the weight of the heavier body and its Loss in water. Then weigh the compound mass in water, and from its Loss, when weighed in water, subtract the LOSS of the heavier body when weighed by itself, and the remainder, or difference, is the Loss of the lighter body; by which Loss divide its absolute weight, and the quotient will be its specific gravity. EXAMPLE. What is the specific gravity of a block of wood, which weighs 15 pounds in the air, if, when attached to a piece of copper, which weighs 18 pounds in the air and 16 in water, the compound mass weighs in water only 6 pounds? Ans., or 0.600. To find the specific gravity of a fluid : Find the loss of weight of one and the same body in water, and likewise in the fluid whose specific gravity is to be ascertained; then divide the loss in the given fluid by the loss in water, and the quotient will be the specific gravity. EXAMPLE. Suppose a piece of iron, when weighed in pure water, loses 8 pounds, and when weighed in beer, loses 9 pounds, what is the specific gravity of the beer? Ans., or 1.125. The following table of specific gravities shows the absolute weight of each body in avoirdupois ounces. A cubic foot of distilled water weighs (at the temperature of 60°) 621⁄2 pounds, or 1000 ounces avoirdupois; and we call its specific gravity 1, or 1.000; and the specific gravity of brick is 2, or 2.000. A cubic foot of brick will consequently weigh 2000 ounces avoirdupois. |