ches at the bung diameter, is ordered to make another cask of the saine shape, but to hold just twice as much; what will be the bung diameter and length of the new cask ? 40x40x40x2=128000 then 128000=50,3+ length. 32X32X32X2=65536 and 3/65536=40,3+bung diam. A General Rule for Extracting the Roots of all Powers. RULE. 1. Prepare the given number for extraction, by point ing off from the unit's place, as the required root directs 2. Find the first figure of the root by trial, and subtract its power from the left hand period of the given number. 3. To the remainder bring down the first figure in the next period, and call it the dividend. 4. Involve the rost to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor. 5. Find' how many times the divisor may be liad in the dividend, 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 num ber 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 trom which it is to be taken, the last quotient figure must be taken less, &c. IXAMPLES. 1. Required the cube root of 155706,744 by the above general method. 5X5X3=75 first divisor. 514x514x514=135796744 third subtrahend. 3. Required the sursolid, or fifth root of 6436343. 6436545)23 root. 2x2x2x2x5=80)323 dividend. 23x23x23x23x23=6436343 subtrahend. Note: The roots of most powers may be found by the square and cube roots only; therefore, when any even power is given, the easiest method will be (especially in a very high power) to extract the square root of it, which reduces it to half the given power, then the square root of 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 square root of that reduces it to a sixth power: and the square root of a sixth power to a cube. IXAMPLES. 3. What is the biquadrate, or 4th root of 19987175576? Ans. 376. 4. Extract the square, cubed, or 6th root of 12230590 404. Ans. 48. 5. Extract the square, biquadrate, or 8th rooi of 72138 95789338336. Ans. 96 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 : : 80 is any part of the composition : to its mean price. EXAMPLES. 1. A farmer mixed 15 bushels of rye, at 64 cents a bushel, 18 bushels of Indian corn, at 55 cts. a bushel, and 21 bushels of oats, at 23 cts. a bushel ; I demand what a bushel of this mixture is worth? bu. cts. Scts. Wu. $ cts. tu. 15 at 6459,60 As 54 : 23,38 : :1 1 cis. 54)25,58(,47 Answer. 54 25,39 2. If 20 bushels of wheat at 1 dol. 35 cts. per bushel, be mixed with 10 bushc's of rye at 90 cents per bushei, what will a bushel of this mixture bc worth: Ars. S1, 20cts. 3. A Tobacconist mixud 36!!). of Tobacco, at 1s. 6 per lb. 12 lb. at 2s. a pound, with 12 lb. at 1s. 10d. per Ib.; what is the price of a pound of this mixture ? Ans. ls. 8d. 4. A Grocer mixed 2 C. of sugar, at 56s. per C. and i C. at 433. per C. and 2 C. at 50s. per C. together; I demand the price of 5 cwt. of this mixture ? Aris. 7 13s. 5. A Wine merchant mixes 15 gallons of wine at 4s. 2d. per gallon, with 24 gallons at 6s. &. and 20 gallons, at 6s. Sd.; what is a galon of this composition worth? Ans. 55. 10d. 27%91S. 6. A grocer hath screral sorts of sugar, viz. one soit at 8 dois. per cwt. another sort at 9 dols. per cwt. a thiru sort at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt. and he would mix an equal quantity of each together; I demand the price of 33 cwt. of this mixture ? Ans. $34 12cts. im. 7. A Goldsmith melted together 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 oz. fine, and 15 lb. of 6 oz. pray what is the quality, or fineness of this composition Ans. 6oz. 15pwt. Sgorr. fine. 8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats fine, and i lb. of alloy be melted together; what is the quality, or fineness of this mass ? Ins. 13 carats fine fine; ALLIGATION ALTERNATE, is the method of finding what quantity of each of the ingredients, whose rates are given, will compose a misture of a given rate; so that it is the reverse of alligation medial, and may be proved by it. CASE. I. When the mean rate of the whole mixture, and the rates of all the ingredients are given without any limited quantity. RULE. 1. Place the several rates, or prices of the simples, be. ing reduced to one denomination, in a column under each other, and the mean price in the like name, at the left hand. 2. Connect, or link, the price of each simple or ingredient, which is less than that of the mean rate, with one or any number of those, which are greater than the mean rate, and each greater rate, or price with one, or any number of the less. 3. Place the difference, between the mean price for mixture rate) and that of each of the simples, opposite mintos with which they are connected. Answer. Ib. ; 4. Then, if only one difference stands against any rate, it will be the quantity belonging to that rate, but if there be several, their sum will be the quantity. EXAMPLES. 1. A merclic at has spices, some at 9d. per lb. some at 1s. some at 23. and some at 2s. 6il. per lb. how much of ei ch sort must he mix, that he may sell the mixture at Is 8d. per pound? d. d. lb. a 16. 9 10 at 97 97 4 d. 12 4 12 ( Gives the d. | 12 10 201 24 8 24 | Answer, or 20 24 11 30.11 30 SO 8 2. A grocer would mix the following quantities of sugar; viz. at 10 cents, 13 cents, and 16 cts. per what quantity of each sort must be taken to make a mixture worth 12 cents per pound ? ns. 5lb. at 10cts. 2lb. at 13cts. and alb. at 16 cts. per 3. A grocer has two sorts of tea, viz. at 9s. and at 15s. per Ib. how must he mix them so as to afford the coinposition for 12s. per lb. ? Ans. He must mix an equal quantity of each sort. 4. A goldsmith would mix gold of 17 carats fine, with some of 19, 21, and 24 carats tine, so that the compound inay be 22 carats fine; what quantity of each must ire take. Ans. 2 of each of the first three sorts, and 9 of the last. 5. It is required to mix several sorts of rum, viz. at 5s. 75. and 9s. per gallon, with water at O per gallon together, so that the mixture may be worth 6s. per gallon ; how much of each sort must the mixture consist of ? Ans. I gal.of Rum at 5s. 1 do. at 7s. 6 do at 9s. and 3 gals. water. Or, s gals. rum at 5s. 6 do. at 7s. 1 do. at 9s. and 1 gal. water, 6. A grocer hath several sorts of sugar, viz.. one sort at 12 cts. per Ib. another at 11 cts. a third at 9 cts. and a fourth at 8 cts. per lb. ; I demand how much of each sort must he inis together, that the whole quantity may ba afiorded at 10 Centsyer pound? ? |