RULE 2. Multiply the cube of the diameter by 5236, and the product is the solid content. EXAMPLE 1. Required the convex surface of a sphere, whose diameter AB 251 inches. 25·52 × 3·1416 = 2042·8254 square inches, ÷ 144 = 14·1862 square or superficial feet. EXAMPLE 2. Required the solid content of a sphere whose diameter A B 25 inches. B A 25′5a × ·5236 = 8682·00795 cubic inches; ÷ 1728 = 5·0243 cubic feet. PROBLEM IX. To find the convex surface and solid content of the segment of a sphere. RULE 1. Multiply the height of the segment by the whole circumference of the sphere, and the product is the curved surface. RULE 2. Add the square of the height to three times the square of the radius of the base; multiply that sum by the height, and by 5236, and the product is the solid content. EXAMPLE 1. The diameter A B of the sphere B ABCD = 20 inches; what is the convex surface of that segment of it whose height E D = 8 inches? D F E A C 3·1416 × 20 × 8 = 502·656 square inches; ÷ 144 = 3·49 superficial feet. EXAMPLE 2. The base FG of the segment FDG = 18 inches, and perpendicular ED = 8; what is the solid content ? 82 = 64, and 92 × 3 = 243; then 243+64 × 8 × ·5236 = 1285·9616 cubic inches, ÷ 1728 — 7441 cubic feet. EXAMPLE 3. Suppose ABCD to be a sugar-pan, and that the diameter of the 'mouth A B is 4 feet, the depth DC being 25 inches, how many imperial gallons will it contain? 252 = 625, and 24a × 3 = 1728; D D Ꭺ PROBLEM X. To find the solidity of a spheroid. RULE. Multiply the square of the revolving axis by the fixed axís, and by 5236, and the product will be the solidity. B D A. EXAMPLE 1. Required the solid content of the prolate spheroid ABCD, whose fixed axis AC is 50, and revolving axis B D 30. 302 x 50 x 5236 23562, the solidity. EXAMPLE 2. What is the solid content of an oblate spheroid, the fixed axis being 30, and revolving axis 50? 502 x 30 x 5236 39270, the solid content. PROBLEM XI. To find the solidity of the segment of a spheroid when the base is circular or parallel to the revolving axis. RULE. From triple the fixed axis take double the height of the segment; multiply the difference by the square of the height, and by 5236; then say, as the square of the fixed axis is to the square of the revolving axis, so is the former product to the solidity. EXAMPLE 1. Required the solid content of the segment AB C, whose height Br is 10; the revolving axis E F being 40, and fixed axis B D 25. B EXAMPLE 2. What is the solid content of the segment of a spheroid whose height = 20 inches, the revolving axis being 25, and fixed axis 50? 50 × 3 20 × 2 = 110, and 110 × 20a × 5236 = 23038·4; then, as 502: 252 :: 23038′4 : 5759.6 inches, the solid content. PROBLEM XII. To find the convex surface and solid content of a cylindric ring. RULE 1. Multiply the thickness of the ring added to the inner diameter by the thickness and by 9·8698, and the product will be the convex surface. RULE 2. To the thickness of the ring add the inner diameter; multiply that sum by the square of the thickness and by 2·4674, and the product will be the solid content. EXAMPLE 1. The thickness of a cylindric ring AC or DB2 inches, and inner diameter= 18, required the convex superficies. 18 + 2 × 2 × 9·8698 = 394-792 square inches, and ÷ 1442741 superficial feet nearly. EXAMPLE 2. Required the solid content of the ring as above. 18+ 2 × 2a × 2·4674 197·392 cubic inches, and ÷ 1728 = ·114 cubic feet. Note.-A cubic foot is equal to 1728 cubic inches, or 2200 cylindrical inches, Also, the cubic foot being considered unity, or 1, ... A cylinder 1 foot in diameter and 1 foot in length.. Decimal Approximations, FOR FACILITATING CALCULATIONS IN MENSURATION. *7854 *5236 *2619 INSTRUMENTAL ARITHMETIC; OR, UTILITY OF THE SLIDE RULE. The slide rule is an instrument by which the greater portion of operations in arithmetic and mensuration may be advantageously performed, provided the lines of division and gauge points be made properly correct, and their several values familiarly understood. The lines of division are distinguished by the letters A B CD, A B and C being each divided alike, and containing what is termed a double radius, or double series of logarithmic numbers, each series being supposed to be divided into 1000 equal parts, and distributed along the radius in the following manner: From 1 to 2 contains 301 of those parts, being the log. of 2. 3 477 3. 4. 5. 6. 7. 8. 9. 1000 being the whole number. The line D, on the improved rules, consists of only a single radius; and although of larger radius, the logarithmic series is the same, and disposed of along the line in a similar proportion, forming exactly a line of square roots to the numbers on the lines B C. Numeration. Numeration teaches us to estimate or properly value the numbers and divisions on the rule in an arithmetical form. Their values are all entirely governed by the value set upon the first figure, and, being decimally reckoned, advance tenfold from the commencement to the termination of each radius: thus, suppose 1 at the joint be one, the 1 in the middle of the rule is ten, and 1 at the end one hundred. Again, suppose 1 at the joint ten, 1 in the middle is 100, and 1 or 10 at the end is 1000, &c., the intermediate divisions on which complete the whole system of its notation. To Multiply Numbers by the Rule. Set 1 on B opposite to the multiplier on A; and against the number to be multiplied on B is the product on A. Multiply 6 by 4. Set 1 on B to 4 on A: and against 6 on B is 24 on A. The slide thus set, against 25 100, &c., &c. To divide Numbers upon the Rule. Set the divisor on B to 1 on A, and against the number to be divided on B is the quotient on A. Divide 63 by 3. Set 3 on B to 1 on A, and against 63 on B is 21 on A. Proportion, or Rule of Three Direct. Rule. Set the first term on B to the second on A, and against the third upon B is the fourth upon A 1. If 4 yards of cloth cost 38 shillings, what will 30 yards cost at the same rate? Set 4 on B to 38 on A, and against 30 on B is 285 shillings on A. 2. Suppose I pay 31s. 6d. for 3 cwt. of iron, at what rate is that per ton? 1 ton 20 cwt. Set 3 upon B to 31'5 upon A, and against 20 upon B is 210 upon A. Rule of Three Inverse. Rule. Invert the slide, and the operation is the same as direct proportion. 1. I know that six men are capable of performing a certain given portion of work in eight days, but I want the same performed in three: how many men must there be employed? Set 6 upon C to 8 upon A, and against 3 upon C is 16 upon A. 2. The lever of a safety valve is 20 inches in length, and 5 inches between the fixed end and centre of the valve: what weight must there be placed on the end of the lever to equipoise a force or pressure of 40 lbs. tending to raise the valve? Set 5 upon C to 40 upon A, and against 20 on C is 10 on A. 3. If 8 yards of cloth, 1 yards in width, be a sufficient quantity, how much will be required of that which is only ths in width, to effect the same purpose? Set 1:5 on C to 8'75 on A, and against 8'75 upon C is 15 yards upon A. Square and Cube Roots of Numbers. On the engineer's rule, when the lines C and D are equal at both ends, C is a table of squares, and D a table of roots, as Squares, 1 4 9 16 25 36 49 64 81 on C. To find the geometrical mean proportion between two numbers. Set one of the numbers upon C to the same number upon D, and against the other number upon C is the mean number or side of an equal square upon D. Required the mean proportion between 20 and 45. Set 20 upon C to 20 upon D, and against 45 upon C is 30 on D. |