2. A farmer would mix 14 bushels of wheat, at $1,20 per bushel, with rye at 72cts., barley at 48cts., and oats at 36cts.: how much must be taken of each sort to make the mixture worth 64 cents per bushel ? Ans. 14bu. of wheat; 8bu. of rye; 4bu. of barley; and 28bu. of oats. 3. There is a mixture made of wheat at 4s per bushel, rye at 3s, barley at 2s, with 12 bushels of oats at 18d per bushel : how much is taken of each sort when the mixture is worth 3s 6d ? Ans. 96bu. of wheat; 12bu. of rye; 12bu. of barley; and 12 of oats. 4. A distiller would mix 40gal. of French brandy at 12s per gallon, with English at 7s and spirits at 4s per gallon : what quantity must be taken of each sort, that the mixture may be afforded at 8s per gallon? Ans. 40 gal. French ; 32 English ; and 32 of Spirits. CASE III. 209. When the quantity of the compound is given as well as the price. RULE. II. Then say, as the sum of the proportional quantities, is to the given quantity, so is each proportional quantity, to the part to be taken of each. Ex. 1. A grocer has four sorts of sugar worth 12d, 10d, 6d and 4d per pound : he would make a mixture of 144/6. worth 8d per lb. : what quantity must be taken of each sort? 4 4 12 : 144 ::4:48 6 2 12 : 144 :: 2:24 8 10 2 12 : 144 :: 2:24 12 4 12 : 144 ::4:48 Sum of the proportional parts. 12 Ans. 4816. at 4d; 2416. at 6d; 2476. at 10d, and 4816. at 12d. PROOF BY ALLEGATION MEDIAL. 4816. at 4 d 1920 2416. at 6 d 144d 2416. at 10d 2400 4816. at 12d 576d 144 144)1152(82 The average cost is 8d as above. 2. A grocer having four sorts of tea worth 5s, 6s, es and 9s per pound, wishes a mixture of 871b. worth 7s per pound : how much must be taken of each sort? Ans. 14416. at 5s ; 2916. at 6s; 2916. at 8s and 14 lb. at 9s. 3. A Vintner has four sorts of wine, viz. white wine at 4s per gallon, Flemish at 6s per gallon, Malaga at &s per gallon, and Canary at 10s per gallon : he would make a mixture of 60 gallons to be worth 5s per gallon : what quantity must be taken of each ? Ans. 45gal. of white wine; 5gal. of Flemish; 5gal. of Malaga; and 5gal. of Canary. 4. A silversmith has four sorts of gold, viz. of 24 carats fine, of 22 carats fine, of 20 carats fine, and of 15 carats fine: he would make a mixture of 42oz. of 17 carats fine : how much must be taken of each sort? Ans. 4 of 24; 4 of 22; 4 of 20; and 30 of 16 carats fine. QUESTIONS. $ 205. What is Allegation Medial ? How do you find the price of the mixture ? 8206. What is Allegation Alternate ? How do you prove Allegation Alternate ? g 207. How do you find the proportional parts when the price only is given ? $ 208. What is the rule when a given quantity of one of the simples is to be taken ? g 209. What is the rule when the quantity of the com. pound, as well as the price, is given ? INVOLUTION. $ 210. If a number be multiplied by itself, the product is called the second power, or square of that number. Thus 4X4=16: the number 16 is the 20 power or square of 4. If a number be multiplied by itself, and the product arising be again multiplied by the number, the second product is called the 3d power, or cube of the number. Thus 3X3X3=27: the number 27 is the 3d power, or cube of 3. The term power designates the product arising from multiplying a number by itself a certain number of times, and the number multiplied is called the root. Thus, in the first example above, 4 is the root, and 16 the square or 2d power of 4. In the 2d example, 3 is the root, and 27 the 3d power or cube of 3. The first power of a number is the number itself. $ 211. Involution teaches the method of finding the powers of numbers. The number which designates the power to which the root is to be raised, is called the index or expo nent of the power. It is generally written on the right, and a little above the root. Thus, 4o expressed the second power of 4, or that 4 is to be multiplied by itself once : hence, 42 =4*4=16. For the same reason 33 denotes that 3 is to be raised to the 3d power, or cubed : hence 3'=3x3x3=27 : we may therefore write, 4=4 the 1st power of 4. 4=44=16 the 2d power of 4. 43=4x4x4=64 the 3d power of 4. 4°=4x4x4x4=256 the 4th power of 4. 4 =4x4x4x4x4=1024 the 5th power of 4. &c. &c. &c. $ 212. Hence, to raise a number to any power, we have the following RULE. Multiply the number continually by itself as many times less 1 as there are units in the exponent : the last product will be the power sought. Ex. 1. What is the 3d power of 125? 125 x 125 x 125=1953125. Ans. 2. What is the cube of 7 ? Ans. 343. 3. What is the square of 60 ? Ans. 3600. 4. What is the 4th power of 5 ? Ans. 625. 5. What is the 5th power of 9 ? Ans. 59049. 6. What is the cube of 1 ? Ans. 1. 7. What is the square of f? Ans. + 8. What is the cube of ,1 ? Ans. ,001 9. What is the cube of f? Ans. Y 10. What is the square of ,01 ? Ans. ,0001. 11. What is the square of 2,04? Ans. 4,1616. 12. What is the 5th power of 10 ? Ans. 100000. 13. What is the cube of 27 ? Ans. 117t QUESTIONS. $ 210. If a number be multiplied by itself once, what is the product called ? If it be multiplied by itself twice, what is the product called ? What does the term power mean? What is the root ? $ 211. What is Involution ? What is the number called which designates the power ? Where is it written? $ 212. How do you raise a number to any power ? $213. We have seen $ 211, that Involution teaches how to find the power en the root is given. Evolution is the reverse of Involution : it teaches how to find the root when the power is known. The root is that number which being multiplied by itself a certain number of times, will produce the given nower **^, The square root of a number is that num |