Case 3. To find the Value of any Decimal Fraction, in Money, Weight, Measure, &c. RULE. Multiply the decimal by the number of parts of the next inferior denomination, cutting off so many places for decimals to the right hand as your given decimal consists of; and those on the left will be integers; then multiply the remaining decimals by the next inferior denomination, and cut off for decimals as before; thus proceed till you have brought it into the least parts of the integer. EXAMPLES. (25) What is the value of ,878125 of a pound sterling? (31) What is the value of ,0396 of a lb. avoirdupois ? (38) What is the value of,125 of a chaldron of coals? LII. EXTRACTION of the SQUARE ROOT. EXTRACTING the Square Root is finding out such a Number, as, being multiplied into itself, the Product will be equal to the given Number. "As the Square Root of 81 is 9, consequently 9 x9=81 the given Number, as in the following Table. Roots 1|2|3| 4 | 5 | 6 | 7 | 8 | 9 Squares 149 | 16 | 25 364964 81 Case 1. To extract the Square Root of any Number, observe the following RULE. 1. Point the given number or resolvend into periods of two figures, beginning at the unit's place. 2. Find by the Table the greatest square number that is contained in the first period towards the left hand, placing the square number under the first period, and the root thereof in the quotient, as in Division; subtract that square out of the said period, and to the remainder bring down the next period for a dividend. 3. Double the quotient or root, and place it for a divisor, seek how often the divisor is contained in the dividend (reserving always the unit's place), and put the answer in the quotient, and also on the right hand of the divisor. Then multiply the divisor by the last figure put in the quotient, as in common Division, subtract the product from the dividend, and to the remainder bring down the next period, which proIceed with as before. A Note 1. But if it happen that the given resolvend is not a perfect square, cube, &c. then something will remain after extraction has been made throughout all the points. When this is the case, you must annex ciphers, according as the proposed power requires, viz. by pairs or twos in the square; threes in the cube, &c.; and the operation' continued as before. 2. If the given resolvend consist of a whole number and decimals together, make the number of decimals even by adding ciphers to them. EXAMPLES. (1) It is required to extract the square root of 74770609. (2) What is the square root of 60516? (3) What is the square root of 7658? (4) What is the square root of 39342864 ? (5) What is the square root of 8209667940,529 ? (6) What is the square root of,000729 ? (8) is the square root of 2? the square root of 2,2710957? (9) What is the square root of 36,00000625 ? LIII. To extract the SQUARE ROOT of VULGAR FRACTIONS. RULE. Reduce the fraction or fractional parts to its lowest terms, and if it be a mixed number, to an improper fraction; then extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator. Case 3. To extract the square roots of vulgar fractions, when they are surds, i. e. a number where a root can never be exactly found. RULE. Reduce the fraction or fractional part to its lowest terms, then reduce it to a decimal, and annex that decimal to the whole number (if any), and extract the square root therefrom. EXAMPLES. (5) What is the square root of 83 ? LIV. The USE of the SQUARE ROOT. Case 1. To find a mean proportional between any two given numbers. RULE. Multiply the two given numbers together, and extract the square root of the product, which root will be the mean proportional sought. EXAMPLES. (1) What is the mean proportional between 4 and 9? Case 2. To find the side of a square equal in area to any given superficies. RULE. Extract the square root of the given superficies, which root will be the side of the square sought. EXAMPLES. (3) If the area of a given circle be 4276,5, I demand the side of a square, whose superficial content shall be equal thereto. (4) Suppose I have an elliptical or irregular fish-pond, containing in surface 9 acres, 2 roods, 15 perches, and would have a square one of the same content; I desire you will tell how many yards each side must be." (5) If the content of a given circle be 160, what is the side of a square equal thereto ? Case 3. Having the area of a circle, to find the diameter. RULE. As 355: 452 :: or, as 1 : 1,273239 :: so is the area : to the square of the diameter; or multiply the square root of the area by 1,12837, and the product will be the answer. (See Problem VI. in Mensuration.) |