To extract the cube root of a vulgar fraction. RULE. Reduce the fraction to its lowest terms, then extract the cube root of the numerator and denominator for a new numerator and denominator; but if the fraction be a surd, reduce it to a decimal, and then extract the root from it. To extract the cube root of a mixed number. RULE. Reduce the fractional part to its lowest terms, and then the mixed number to an improper fraction, extract the cube roots of the numerator and denominator for a new numerator and denominator; but if the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root therefrom. 1. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches doth it contain? Ans. 103823. 2. There is a cellar dug that is 12 feet every way, in length, breadth, and depth, how many solid feet of earth were taken out of it? Ans. 1728. To find the side of a cube that shall be equal in solidity to any given solid, as a globe, cylinder, prism, cone, &c. RULE. The cube root of the solid content of any solid body given is the side of the cube of equal solidity. EXAMPLE. If the solid content of a globe is 10648, what is the side of a cube of equal solidity? Ans. 22. The side of the cube being given, to find the side of that cube, that shall be double, treble, &c. in quantity to the given cube. RULE. Cube the side given, and multiply it by 2, 3, &c. the cube root of the product is the side sought. EXAMPLE. There is a cubical vessel, whose side is 12 inches, and it is required to find the side of another vessel that is to contain three times as much? Ans. 17,306. EXTRACTION OF THE BIQUADRATE ROOT. To extract the Biquadrate Root is to find out a number, which being involved four times into itself, will produce the given number. RULE. First extract the square root of the given number, then extract the square root of that square root, and it will give the biquadrate root required. EXAMPLES. 1. What is the biquadrate of 27? Ans. 531441. 2. What is the biquadrate root of 531441? A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. 1. PREPARE the number given for extraction, by pointing off from the unit's place as the root required directs. 2. Find the first figure in the root, by the table of powers, which subtract from the given number. 3. Bring down the first figure in the next point to the remainder, and call it the dividend. 4. Involve the root into the next inferior power to that which is given; multiply it by the given power, and call it the divisor. 5. Find a quotient figure by common division, and annex it to the root; then involve the whole root into the given power, and call that the subtrahend. 6. Subtract that number from as many points of the given power as is brought down, beginning at the lowest place, and to the remainder bring down the first figure of the next point for a new dividend. 7. Find a new divisor, and proceed in all respects as before. EXAMPLES. 1. What is the square root of 141376? 2. What is the cube root of 53157376 ? Ans. 376. Ans. 376. 3. What is the biquadrate root of 19987173376 ? Ans. 376. DUODECIMALS. DUODECIMALS, or Cross Multiplication, is a rule made use of in measuring and computing the dimensions of the several parts of buildings; it is likewise used to find ships' tounage and the contents of bales, cases, &c. Dimensions are taken in feet, inches, and parts. Artificers' work is computed by different measures, viz. Glazing, and masons' flat work, by the foot. Painting, paving, plastering, &c. by the yard. Partitioning, flooring, roofing, tiling, &c. by the square of 100 feet. A perch of masons' work is 24 feet. A square or cubic fathom is 216 feet. The contents of bales, cases, &c. by the ton of 40 cubic feet. RULE FOR MULTIPLYING DUODECIMALLY. 1. Under the multiplicand write the corresponding denominations of the multiplier. 2. Multiply each term in the multiplicand, (beginning at the lowest) by the feet in the multiplier; write each result under each respective term, observing to carry an unit from each lower denomination to its superior. 3. In the same manner, multiply the multiplicand by the inches in the multiplier, and write the result of each term, one place more to the right hand of them, in the multiplicand. 4. Work in the same manner with the other parts in the multiplier, setting the result of each term two places to the right hand of those in the multiplicand, and so on for thirds, fourths, &c. 5. Proceed in the like manner with all the rest of the denominations, and their sum will give the answer required. EXAMPLES. 1. Multiply 4 feet 9 inches by & inches. 4 9 2. Multiply 9 feet 6 inches by 4 feet 9 inches. Ans. 45 feet 1 inch and 6 twelfths. 3. In a load of wood 8 feet 4 inches long, 4 feet 3 inches wide, 3 feet 6 inches high, how many cubic or solid feet? 8. What is the price of a marble slab, whose length is 5 feet 7 inches, and breadth 1 foot 10 inches, at one dollar per foot. Ans. 10 dols. 23 cents. 9. There is a house with three tiers of windows, 3 in a tier, the height of the first tier is 7 feet 10 inches, of the second 6 feet 8 inches, and of the third 5 feet 4 inches, and the breadth of each is 3 feet 11 inches; what will the glazing come to, at 14d. per foot? Ans. 13 118. 104d. 10. If a house measures within the walls 52 feet 8 inches in length, and 30 feet 6 inches in breadth, and the roof be of a true pitch or the rafters & of the breadth of the building, what will it come to, roofing at 10s. 6d. per square? Ans. £12 12s. 11ąd. APPLICATION OF DUODECIMALS. To find how many cubic or solid square feet (in order to ascertain the freight) are contained in cases, bales, &c. that is, how many cubic feet they will take up in a ship. EXAMPLES. 1. Suppose the dimensions of a bale be 7 feet 6 inches, 3 feet 3 inches, and 1 foot 10 inches; what is the solid content? Ans. 44 feet 8 inches. 2. What is the freight of a bale containing 65 feet 9 inches, at 15 dols. per ton of 40 feet, or 37 cts. per foot? Ans. $24,65 cts. 3. A merchant imports from London 6 bales of the following dimensions, viz. 4. What are the solid contents, and how much will the freight amount to, at 20 dollars per ton? The contents are, viz. |