To find the Solidity of a Parabolic Conoid or Paraboloid. RULE.-Multiply the square of the diameter of its base by 3927, and that product by the height; the last product will be the solidity. NOTE. A paraboloid is half its circunscribed cylinder; its Ans. 2261-952. To find the Solidity of the Frustum of a Paraboloid. RULE.-Multiply the sum of the squares of the diameters of the two ends by the height, and that product by 3927, for the solidity. In a parabolic frustum whose height is 12, the diameters of the ends are 20 and 10, what is the solidity? Ans. 2357.4. The diameters of the frustum of a paraboloid are 15 and 12, and the height 8; what is the solidity? Ans. 1159.8408. The diameters of the frustum of a paraboloid are 10 and 5, the length 12; required the solidity. Ans. 589.05. ON THE RATIOS OF SIMILAR FIGURES TO EACH OTHER. Figures, whether superficial or solid, are said to be similar when, though differing in size, their relative proportions remain the same. Equilateral triangles, for instance, however they may vary in size, must have always their three sides and three angles equal; they are, therefore, always similar;-so also squares, the regular polygons, and circles, which always maintain their proportions unchanged. In solids-spheres, cubes, and the other four regular solids, are also invariable in their proportions, and consequently always similar. With other figures, they are only similar when all their parts as to lines or superficies are proportionally increased or diminished, and the angles remain unaltered. A rectangle, for instance, whose sides were 2 and 4, would be similar to a rectangle whose sides were 4 and 8, but not to a rhomboid whose sides were 4 and 8, because the latter would differ in the angles. Similar surfaces bear to each other the same proportions as the squares of their like dimensions. In a square this fact is very palpable: thus, a square whose side is 2, has a surface of 4, which is the square of its side; a square whose side is 3, has a surface of 9, or the square of its side;-therefore, those squares whose sides are 2 and 3, are to each other as 4 to 9, or as the squares. The same would be true if the entire perimeter or boundary of each were taken; that of the first would be 8, of the second 12, the squares of which are 64 and 144, and which are to each other as 4 to 9;-so, also, if the diagonals of each were taken as the dimension. Circles are as the squares of their diameters and circumferences, and the regular polygons as the squares of either of their sides, or of the radii of the circles in which they are inscribed. With solids, the proportion is as the cubes of their like dimensions. Thus, a cube whose side is 2 has a solidity of 8, and a cube whose side is 3 has a solidity of 27. Their proportions are, therefore, as 8 to 27. So also of spheres or other similar solids, as may be proved by comparing the proportions of their actual solidity with the proportion found in cubing any simple dimension of either. If a solid be doubled or otherwise increased in one dimension only say its length, its total bulk is only increased to that amount; if it be increased superficially in the same ratio, that is in the dimensions of length and breadth, the total increase will be as the squares of the dimensions, and if the same amount of increase be made in the depth also, the proportion will be as the cubes. This will appear quite evident by a reference to plate II., fig. 13, in which a and b are to each other as their lengths; a and c as the squares of their lengths; and a and d as the cubes of their lengths;-and this proportion holds good with any rate of increase, however great or minute. GAUGING. OF THE IMPERIAL GALLON, AND THE MODE OF MENSURATION ADAPTED TO IT FOR FIGURES OF VARIOUS KINDS. In the calculations hitherto made, our unit of measurement has been founded on the Imperial Yard and its derivatives. In gauging liquids, however, a different unit is employed—not founded on the yard, but on an original standard peculiar to itself; it is called the Imperial Gallon. The standard for the Imperial Gallon is the space occupied by 10 lbs. avoirdupoise (or 70,000 grains) of distilled water, taken at the temperature of 62° Fahr., with the barometer at 30 inches. The space thus filled, expressed in cubic inches, is 277-274, a cubic foot of liquid being a trifle over 6 gallons, or nearly 61 gallons (6.23). This number, 277-274, is called the Square Divisor for Imperial Gallons, because it is used for the purpose of dividing the number of cubic inches in any vessel, to obtain its contents in Imperial Gallons, as every time the quantity it represents is found in the vessel, so often is there an Imperial Gallon.* If we suppose a square vessel to be formed, with an area of 277 274 square inches, and of the depth of 1 inch, it is obvious * Before the Imperial Gallon was fixed by law, different measures were used for ale and wine. Ale gallons were 282 cubic inches, and wine gallons 231. To reduce old wine gallons to new, divide by 1-2, or set 5 on the slide at the back of the head rod to 6 on the top of the rule, and under the number of wine gallons will be the Imperial Gallons on the top line of the slide. Multiply old ale gallons by 60, and divide by 59, for Imperial Gallons. that such vessel would contain the same number of cubic as of superficial inches, or just one gallon; and the side of the square thus formed would be the square root of its area, or 16·65 inches; this is called the Square Gauge Point, or unit of calculation, in gauging square vessels. In a square prism, the side of whose base is 16.65 inches, every inch of its depth will represent one gallon, and the total inches in that dimension will stand for the number of gallons it contains. If, however, the sides of the base of the prism were longer than this, say 33.3 inches, or double the former, the area of the base would be increased four times. Each inch of depth would then represent 4 gallons, and the total content be increased in the same proportion. It is therefore evident that, in all square vessels, their area in gallons will be as the square of 16.65 is to the squares of the sides of such vessels; or, taking 16.65 as unity, the square of the side of any vessel taken as a multiple of such unit, will be its area in gallons. Most vessels for holding liquids have a circular form, and the square divisor and gauge point would be of little or no use in gauging such figures, except by the tedious process of reducing their contents to cubic inches, and dividing by 277.274. In order to avoid this, a circular divisor and a circular gauge point have been obtained, on the same principle as the foregoing. A circle, whose area would be 277.274 inches, and which would, therefore, at the depth of 1 inch, contain a gallon, would have a diameter of 18.789 inches,-or, as usually taken, 18.79; this, therefore, is the circular gauge point, while the circular divisor is the square of such diameter, or the area of a square described about such a circle, which is 353-036, generally taken 353.04. Now, in gauging a cylindrical vessel, if we were to use the square divisor, it would be necessary to square the diameter in inches, and multiply by the length in inches, and then multiply by 7854 to correct the excess over the true content, before dividing by 277.274. But the use of the circular divisor obviates this, as its excess over the true number of inches in a allon is in the proportion of 1 to 7854, and the result obtained |