Nore.—The foregoing example gives a perfect root; and if, when all the periods are exhausted, there happens to be a remainder, you may annexperiods of ciphers, and continue the operation as far as you think it necessary. Rule.-1. Find by trial, a cube near to the given number, and call it the supposed cube. 2. Then, as twice the supposed cube, added to the given number, is to twice the given number added to the supposed cube, so is the root of the supposed cube, to the true root, or an approximation to it. 3. By taking the cube of the root thus found, for the supposed cube, and repeating the operation, the root will be had to a greater degree of exactness. ExAMPLEs. 1. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube; then—1,3× 1,3× 1,3–2,197—supposed cube. Then, 2,197. 2,000 given number. As 6,394 : 6,197 : : 1,3 - 1,2500 root. which is true to the last place of decimals; but might by repeating the operation be brought to greator to actress. 2. What is the cube root of 584; **** - --Lutton, of Extraction of Roots. 3. Required the cube root of 7290011011 Ans. 900,000+ QUESTIONs. Showing the use of the Cube Root. 1. The statute bushel contains 2150,425 cubic or solid inches. I demand the side of a cubic box, which shall contain that quantity? V250,425–12,907 inch. Ans. Note—The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides or diameters. 2. If a bullet 3-inches diameter weigh 4 lb. what will a bullet of the same metal weigh, whose diamo-er is 6 in ches? 3x3x3–27 tox6x6–216. As 27: 4 in. : : 216. olb. Ans. 3. If a solid globe of silver, of 3-inche. diameter, b. worth 150 dollars; what is the value of another globe ol silver, whose diameter is six inches? 3 & 8 × 3–27 6-6 x 6=216, As 27 : 150 :: 216 81.200. Ans. The side of a cube being given, to find the side of tha cube which shall be double, triple, &c. in quantity to the given cube. Rule.-Cube your given side, and multiply by the given propor tion between the given and required cube, and the cube root of th product will be the side sought. ---------4. If a cube of silver, whose side is two inches, be worth 20 dollars; I demand the side of a cube of like silver whose value shall be 8 times as much? 2x2x2–8, and 8×8–64 vo-4 inches. Ans. 5. There is a cubical vessel, whose side is 4 feet; I de mand the side of another cubical vessel, which shall con tain 4 times as much 4×4×4=64, and 64x4–256/355-6349+ ft. Ans. 6. A cooper having a cask to inches long, and 32 in ches at the bung diameter, is ordered to make another cask of the same shape, but to hold just twice as much; wha will be the bung diameter and length of the new cask? 10x40x40x2-12-000 then J 128000–50,34-length. #232:32:2-65536 and wood-4034 oung diam. -A General Rule for extracting the Roots of all Powers. RULE. 1. Prepare the given number for extraction, by pointing off from the unit's place, as the required root directs. 2. Find the first figure of the root by trial, and subtract is power from the left hand period of the given number. 3. To the remainder bring down the first figure in the *ext period, and call it the dividend. 4. Involve the root to the next inferior power to that *hich is given, and multiply it by the number denoting the oven power, for a divisor. 5. Find how many times the divisor may be had in the lividend, and the quotient will be another figure of the root. 6. Involve the whole root to the given power, and subtract it (always) from as many-periods of the given number as you have found figures in the root. 7. Bring down the first figure of the next period to the remainder for a new dividend, to which find a new divisor as before, and in like manner proceed till the whole be finished. Note—When the number to be subtracted is greater than those periods from which it is to be taken, the last ruotient figure must be taken less, &c. Ex-MPLEs. 1. Required the cube root of 135796,744 by the above general method. wo Evolution, to Extraction of Roc rs Nore.—The roots of most powers may be found by th: square and cube roots only; therefore, when any ever power is given, the easiest method will be (especially in t very high power) to extract the square root of it, which re duces it to half the given-power, then the square root o that power reduces it to half the same power; and so on till you come to a square or a cube. For example: suppose a 12th power be given; the squart root of that reduces it to a 6th power: and the square roo of a 6th power to a cube. Ex-MPLEs. 3. What is the biquadrate, or 4th root of 1998.71733761 Ans. 376. 4. Extract the square, cubed, or 6th root of 12230590 464. Ans. 48. 5. Extract the square guadrate, or 8th root of 72138 95.789838.336. Arts 96. ALLIGATION. 177 ALLIGATION, is the method of mixing several simples of different qualities, so that the composition may be of a mean or middle quality: It consists of two kinds, viz. Alligation Medial, and Alligation Alternate. ALLIGATION MEDIAL, Is when the quantities and prices of several things are given, to find the mean price of the mixture composed of those materials. RULE. As the whole composition : is to the whole value: : so is any part of the composition : to its mean price. Exo-M'LES. 1. A farmer mixed 15 bushels of rye, at 64 cents a bushe, is bushels of Indian corn, at 55 cts, a bushel, and 21 bushels of oats, ato-ets, a bushel; I demand what a bushel of this mixture is worth? bu, cts sets. bu. $ cts, bu15 at 64–9,60. As 54 - 25,38 : : 1 is 55–9,90 I 21 28–5, SS cts. - - 54)25,38(,47 Ans. 54 25,38 2. If 20 bushels of wheat at I dol. 35 cts. per bushel be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of this mixture be worth 2 - Ans. $1,20 cits. 3. A tobacconist mixed 36 lb. of tobacco, at 1s. 6d. per Ib. 12 lb. at 2s. a pound, with 12 lb. at 1s. 10d. per lb.; what is the price of a pound of this mixture? Ans, 1s. 8d. 4. A grocer mixed 2 C. of sugar at 56s. per C. and 1 C. at 43s. per C. and 2 C. at 50s, per C. together; I demand the price of 3 cwt. of this mixture? Ans. E7 13s. 5. A wine merchant mixes 15 gallous of wine at 4s. od. per gallon, with 24 gallous at 6s. 8d. and 20 gallons at 6s. 8d.: what is a gallon of this composition worth? 4ns. 5s 10d. 24% ars, |