although in the instance of a pyramid on a square base, the content may be found at one operation, as follows: on D. on C. (15.197 on D. 46.37 on C. [20.95 W.G. Sq. G.P. 16.79 to 10 against 22 are 17.16 A. G. 2.25 M. B. PROBLEM VIII. To determine the Content of a Sphere. RULE. By the Pen. Multiply the cube of the diameter in inches, by the decimal 5236, and the product will be the content in cubic inches: then divide by the number of cubic inches in the proposed integer. By the Sliding Rule. Set the circular gage-point, for the proposed integer, on D, to two-thirds of the diameter on C, and against the entire diameter on D, will be the content on C. EXAMPLE. Let ABCD be a globe or sphere 30 inches in diameter; the content is required in wine gallons, ale gallons, and malt bushels. Multiply by 30 diameter Product 900 square of the diameter Multiply by 30 diameter Product 27000 cube of the diameter Multiply by 5236 constant decimal 36652 10472 Product 14137.2 content in cubic inches. Wherefore it will be, W.G. divisor 231) 14137.2 (61.20 wine gallons. A. G. divisor 282) 14137-2 (50-13 ale gallons. M. B. divisor 2150) 14137-2 ( 6.57 malt bushels. BY THE SLIDING RULE. on D. on C. on D. on C. 61.20 W.G. C. G. P. 18.95 to 20 against 30 are 50·13 A. G. 52.32 6.57 M.B. The Rule by the pen is founded on this property, that globes are to one another as the cubes of their diameters, and that when the diameter is 1 the content is ⚫5236. The reason of setting two-thirds of the diameter on the line C, to the circular gage-point on D, is, that every globe, is equal to two-thirds of its circumscribing cylinder; or to a cylinder two-thirds the altitude of the globe, on a base, having for diameter the diameter of the globe. All this being well understood, it will be easy to construct a set of divisors and gage-points for spheres, as follows: Divisor Dividend Quotient Square roots 5236) 231.0000 (441·18 (21-0 wine gallons. •5236) 282-0000 (538.58 (23.2 ale gallons. •5236) 2150-4200 (4107-0 (64.1 malt bushels. Divisor Dividend Sph. Divisors Spherical Gage-points. In like manner may spherical divisors, and spherical gage-points, be obtained for any other integer whatever. USE OF THE SPHERICAL DIVISORS AND SPHERICAL GAGE-POINTS. To determine the Content of a Globe by the Pen. RULE. Divide the cube of the diameter by the spherical divisor for the proposed integer, and the quotient will be the content. To determine the Content of a Globe by the Sliding Rule. RULE. Set the spherical gage-point, for the proposed integer on D, to the diameter of the globe on C, and against the diameter on D, will be the content on C. EXAMPLE. The content of a globe, 30 inches in diameter, is required in wine gallons, ale gallons, and malt bushels. SOLUTION BY THE Pen. 30 diameter Multiply by 30 diameter Product 900 square of the diameter Multiply by 30 diameter Product 27000 cube of the diameter Wherefore it will be, W.G. sph. divisor 441·18) 27000 (61.19 wine gallons. A. G. sph. divisor 538.58) 27000 (50·13 ale gallons. M. B. sph. divisor 4107) 27000 (6.57 malt bushels. BY THE SLIDING RULE. on D. on C. (21.0) on D. on C. (61.2 W.G. Sph. G. P. 23-2 to 30 against 30 are 50-1 A. G. Multiply the transverse diameter by the square of the conjugate, and divide the product by the spherical divisor for the proposed integer. By the Sliding Rule. Set the spherical gage-point, for the proposed integer, on D, to the transverse diameter on C, and against the conjugate on D, will be the content on C. EXAMPLE. Let MNOP be a spheroid, whereof the longer axis, MO, is 42 inches, and the shorter axis, NP, 31 inches; |