2. If 17.25 lb. of candles cost 12.75s. what will 5.75 lb. cost? As 17.25 5.75 :: 12.75; 4.25=4 3 Ans. As 17.25 12.75 :; 5.75; 4.25=4 3 Ans. 3. What will 28.5 quarts of wine cost, if 9.758. be given for 3.25 quarts? Ans. £.4 5s. 6d. 4. If cwt. of tobacco cost £.615s.; what will be the price of 3 cwt. 2 qr. 7 lb? Ans. £.48 1s. 101d. 5. If £.3 5s. be given for 1 cwt. of tallow; what will be the price of 5 cwt. 3 qr. 14 lb. at the same rate? 6. What is the duty of 2431 27s. 6d. is paid for 3 barrels? Ans. £.19 1s. 101d. barrels of ale, when Ans. £.111 98. 91d. 7. If the duty of 3 barrels of table beer, amounts to 5s. 6d. ; what will be the duty of 18 barrels ? Ans. £.1 14s. 41d. 8. If the duty of of a barrel of ale comes to 6s. 101d.; what will be the duty of 161 barrels ? Ans. £.7 8s. 111d. 9. What is the duty of 7 of a 1000 of polished bricks, when the duty of of a 1000 amounts to 9s. 74d.? 24 Ans. 11s. 2 d. 10. If the duty of of a 1000 of common bricks comes to 31d.; what will be the duty of 8 of a 1000? Ans. 58. 81d. 11. What is the duty of of a 1000 of common bricks, when the duty of of a 1000 is 13d.? Ans. 4s. 8d. 12. If the duty of 5 pounds of hard soap comes to 111d.; what is the duty of 42364 pounds? Ans. £.397 3s. 3d. C SQUARE ROOT. To extract the Square Root, is to find such a number as being multiplied by itself will produce the given number. Thus the Square Root of 25, is 5; because 5 squared, or multiplied by 5=25, the proposed number. RULE. Divide the given number into periods of two figures each, by placing a point above every second figure; beginning at units place, and proceeding to the left in integers, and to the right in decimals. Find the greatest square root contained in the first period on the left-hand, and place it on the right-hand of the given number, after the manner of a quotient figure in division. Subtract the square of this root from the said period; and to the remainder bring down the two figures of the next period, for a dividend. Double the root above mentioned, for a divisor; find how often it is contained in the dividend, exclusive of the right-hand figure; and place the result both in the quotient and the divisor. Multiply the divisor thus augmented, by the last quotient figure; subtract the product from the dividend; and bring down the next period to the remainder, for a new dividend. Repeat the same process for each period, and the quotient thus obtained will be the root required. A Table of Squares and Roots. Squares 1. 4. 16. 25. 36. 49. 64. 81. Roots 1. 2. 4. 5. 6. 7. 8. 9. Note 1. The root will consist of as many integers and decimals as there are periods of each in the given number. 2. When the decimals in the given number do not consist of an even number of figures, they must be made even, by placing a cypher on the right; and when all the periods are brought down, the operation may be continued at pleasure, by annexing two cyphers to each remainder. 3. If the square root of a vulgar fraction be required, reduce it to a decimal, and then extract the root. 4. The best method of doubling the root, to form new divisors, is to add the last quotient figure to the last divisor. 5. The method of proof is to multiply the root into itself, add the remainder, if any, to the product, and the sum will be equal to the given number, if the work be right. EXAMPLES. 1. What is the square root of 642.1156 ? 642.1156(25.34 Ans. 4 45)242 225 503)1711 5064) 20256 20256 Proof. 25.34 25.34 101.36 7602 12670 5068 642.1156 2. What is the square root of 1048576? 3. Required the square root of 983. 4. What is the square root of 8104.2634? 5. Find the square root of 744.326. 7. What is the square root of q? 8. Extract the square root of $118. Ans. 1024. Ans. 31.352. Ans. 90.023. Ans. .00423. Ans. .802. Ans. .8367. 9. What is the square root of 18 of 3 of 13 of 14? Ans. .7099. SQUARE ROOT, APPLIED IN SOLVING USEFUL MATHEMATICAL PROBLEM I. To find a mean proportional between two given numbers. RULE. Multiply the two given numbers together, and the square root of their product will be the mean proportional sought. EXAMPLES. 1. What is the mean proportional between 20 and 45? 2. There are three numbers in geometrical progression ; the first is 48, and the third 243, what is the middle number? Ans. 108. 3. In a pair of false scales, a body weighed 36 lb. in one scale, but only 16 lb. in the other; required its true weight; and the ratio of the lengths of the two arms of the balance on each side of the point of suspension. Ans. The true weight of the body is 24 lb. and the arms of the balance are to each other as 16 to 24, or as 24 to 36, or as 2 to 3. PROBLEM II. To find the side of a square equal in area to any given superficies. RULE. Extract the square root of the number expressing the superficies of the given figure; and it will be the side of a square equal in area. Note. The definitions of Geometrical Figures, and the methods of of finding their areas may be seen in Parts III. and IV. EXAMPLES. 1. The area of a rectangular floor is 2025 square feet; what will be the side of a square floor of equal area? Ans. 45 feet. 2. The area of the base of a triangular cistern is 324 square feet; required the side of a square cistern that shall be equal in area. Ans. 18 feet. 3. A maltster has a rectangular kiln whose area is 144 square feet; what must be the side of a square one, which he intends to build, so that the latter may dry four times as much malt at a time as the former ? PROBLEM III. Ans. 24 feet. To find the diameter of a circle, when the area is given. RULE. Divide the area by .7854, and the square root of the quotient will be the diameter required. EXAMPLES. 1. The area of a circle is 8824.7 square inches; what is the diameter ? Ans. 106 inches, nearly. 2. The area of the head of a cask is 254.5 square inches; required the head diameter, Ans. 18 inches. PROBLEM IV, To find the hypothenuse of a right-angled triangle, when the base and perpendicular are given, RULE. square of the per To the square of the base add the pendicular, and the square root of the sum will be the hypothenuse. |