of the quantities, or differences thus determined, is to the given quantity, so is each ingredient, found by linking, to the required quantity of each. EXAMPLES. 1. How many gallons of water at os. per gallon must be mixed with wine worth 3s. per gallon, so as to fill a vessel of 100 gallons, and that a gallon may be afforded at 2s. 6d. ? Ans. 83 gallons of wine, and 16 of water. 2. A grocer has currants at 4d. 6d. 9d. and 11d. per Ib. and he would make a mixture of 240lb. so that it may be afforded The rates of the simples are 92 and 52, and of the compound 64; therefore And the sum of these is 12+28=40, which should have been but 10; whence, by the rule, 40: 10 :: 12: 3lb. of copper} 40: 10: 28: 7lb. of T the answer. afforded at 8d. per pound; how much of each sort must he take? Ans. 72lb. at 4d. 24 at 6d. 48 at 9d. and 96 at itd. 3. How much gold of 15, of 17, of 18 and of 22 carats fine, must be mixed together to form a composition of 40 ounces of 20 carats fine? Ans. 5oz. of 15, of 17 and of 18, and 25 of 22. When one of the ingredients is limited to a certain quantity = take the difference between each price and the mean rate as before; then, As the difference of that simple, whose quantity is given, is to the rest of the differences severally, so is the quantity given to the several quantities required. EXAMPLES. 1. How much wine at 5s. at 5s. 6d. and 6s. the gallon must be mixed with 3 gallons at 4s. per gallon, so that the mixture may be worth 5s. 4d. per gallon? 3 : 6 10 : 20 :: ΙΟ : 20 :: 3 : 6 Ans. 3 gallons at 5s. 6 at 5s. 6d. and 6 at 6s. 2. A * In the very same manner questions may be wrought, when several of the ingredients are limited to certain quantities, by finding first for one limit and then for another. The two last rules can want no demonstration, as they evidently result from the first, the reason of which has been already explained. 2. A grocer would mix teas at 12s. Ios. and 6s. with 20lb. at 4s. per pound; how much of each sort must he take to make the composition worth Ss. per lb ? Ans. 20lb. at 45. 10 at 6s. 10 at 10s. and 20 at 12s. 3. How much gold of 15, of 17 and of 22 carats fine, must be mixed with 5oz. of 18 carats fine, so that the composition may be 20 carats fine? Ans. 5oz. of 15 carats fine, 5 of 17, and 25 of 22. INVOLUTION. A POWER is a number produced by multiplying any given number continually by itself a certain number of times. Any number is itself called the first power; if it be multiplied by itself, the product is called the second power, or the square; if this be multiplied by the first power again, the product is called the third power, or the cube; and if this be multiplied by the first power again, the product is called the fourth power, or biquadrate; and so on; that is, the power is denominated from the number, which exceeds the multiplications by 1. Thus, 3 is the first power of 3. 3X39 is the second power of 3. 3X3X327 is the third power of 3. 3X3X3X3=81 is the fourth &c. power of &c. 3. And in this manner is calculated the following table of powers. TABLE TABLE of the first twelve POWERS of the 9 DIGITS, 1oth Pow. 11024 59049 1048576 9765625 60466176 282475249 1073741824 3486784401| NOTE I. The number, which exceeds the multiplica tions by I, is called the index, or exponent, of the power; so the index of the first power is 1, that of the second power is 2, and that of the third is 3, &c. NOTE 2. Powers are commonly denoted by writing their indices above the first power: so the second power of 3 may be denoted thus 3, the third power thus 33, the fourth power thus 3*, &c. and the sixth power of 503 thus 503. Involution is the finding of powers; to do which we have evidently the following RULE. Multiply the given number, or first power, continually by itself, till the number of multiplications be less than the index of the power to be found, and the last product will be the power required.* NOTE. Whence, because fractions are multiplied by taking the products of their numerators and of their de nominators, they will be involved by raising each of their terms to the power required. And if a mixed number be proposed, *NOTE. The raising of powers will be sometimes shortened by working according to this observation, viz. whatever two or more powers are multiplied together, their product is the power, whose index is the sum of the indices of the factors; or if a power be multiplied by itself, the product will be the power, whose index is double of that, which is multiplied: so if I would find the sixth power, I might multiply the given number twice by itself for the third power, then the third power into itself would give the sixth power; or if I would find the seventh power, I might first find the third and fourth, and their product would be the seventh; or lastly, if I would find the eighth power, I might first find the second, then the second into itself would be the fourth, and this into itself would be the eighth. |