falls short, he must have taken equal quantities of each sort; but, since the difference between the price of the corn and the mixture price is 4 times as much as the difference between the price of the oats and the mixture price, consequently, 4 times as much oats as corn must be taken, that is, 4 to 1, or 4 bushels of oats to 1 of corn But since we determine this proportion by the dif ferences, hence these differences will represent the same proportion. These are 20 and 5, that is, 20 bushels of oats to 5 of corn, which are the quantities or proportions required. In determining these differences, it will be found convenient to write thien down in the following manner: OPERATION. cts. bushels. 30 S$,25$,50 20 5 Ans. It will be recollected, that the difference be tween 50 and 30 is 20, that is, 20 bushels of oats, which must, of course, stand at the right of the 25, the price of the oats, or, in other words, opposite the price that is connected or linked with the 50; likewise the difference between 25 and 30= 5, that is, 5 bushels of corn, opposite the 50, (the price of the corn.) The answer, then, is 20 bushels of oats to 5 bushels of corn, or in that proportion. By this mode of operation, it will be perceived, that there is precisely us much gained by one quantity as there is lost by another, and, therefore, the gain or loss on the whole is equal. The same will be true of any two ingredients mixed together in the same way. In like manner the proportional quantities of any number of simples may be determined; for, if a less be linked with a greater than the mean price, there will be an equal balance of loss and gain between every two, consequently an equal balance on the whole. It is obvious, that this principle of operation will allow a great variety of answers, for, having found one answer, we may find as many more as we please, by only multiplying or dividing each of the quantities found by 2, or 3, or 4, &c.; for, if 2 quantities of 2 simples make a balance of loss and gain, as it respects the mean price, so will also the double or treble, the, or part, or any other ratio of these quantities, and so on to any extent whatever. PROOF. We will now ascertain the correctness of the foregoing operation w the last rule, thus: 20 bushels of oats, at 25 cents per bushel, =$5,00 25 25)7,50(30 0 Ans. 30 cts., the price of the mixture. Hence we derive the following RULE. 1. Reduce the several prices to the same denomination. I. Connect, by a line, cach price that is less than the mean rate, with one or more that is greater, and each price greater than the mean rate with one or more that is less. III. Place the difference between the mean rate and that of each of the simples opposite the price with which they are con nected. IV. Then, if only one difference stands against any price, it expresses the quantity of that price; but if there be several, their sum will exvress the quantity. 2. A merchant has several sorts of tea, some at 10 s., some at 11 s., some at 13 s., and some at 24 s. per lb.; what proportions of each must be taken to make a composition worth 12 s. per lb. ? 3. How much wine, at 5s. per gallon, and 3 s. per gallon, must be mixec together, that the compound may be worth 4 s. per gallon? A. An equal quantity of each sort 4 How much corn, at 42 cents, 60 cents, 67 cents, and 78 cents, per bushel, must be mixed together, that the compound may be worth 64 cents per bushe!! A. 14 bushels at 42 cents, 3 bushels at 60 cents, 4 bushels at 67 cents, and Dushels at 78 cents. 5. A grocer would mix different quantities of sugar; viz. one at 20, one a 23, and one at 26 cents per lb.; what quantity of each sort must be taken t make a mixture worth 22 cents per lb. ? A. 5 at 20 cents, 2 at 23 cents, and 2 at 26 cents 6. A jeweller wishes to procure gold of 20 carats fine, from gold of 16, 19, 21, and 24 carats fine; what quantity of each must he take? A. 4 at 16,1 at 19, 1 at 21, and 4 at 24, We have seen that we can take 3 times, 4 times,,, or any proportion of each quantity, to form a mixture. Hence, when the quantity of one simple is given, to find the proportional quantities of any compound whatever, after having found the proportional quantities by the last rule, we have the follow ing RULE. As the PROPORTIONAL QUANTITY of that price whose quantity is given is to EACH PROPORTIONAL QUANTITY:: so is the GIVEN QUANTITY: to the QUANTITIES or PROPORTIONS of the compound required. 1. A grocer wishes to mix gallon of brandy, worth 15 s. per gallon, with rum, worth 8 s., so that the mixture may be worth 10 s. per gallon; how much rum must be taken? By the last rule, the differences are 5 to 2; that is, the proportions are 2 of brandy to 5 of rum; hence he must take 2 gallons of rum for every gallon of A. 2 gallons brandy. 2. A person wishes to mix 10 bushels of wheat, at 70 cents per bushel, with rye at 48 cents, corn at 36 cents, and barley at 30 cents per bushel, so that a bushel of this mixture may be worth 38 cents; what quantity of each must be taken? We find by the last rule, that the proportions are 8, 2, 10, and 32. Then, as 8: 2:: 10: 2 bushels of ryc. 8 10 10 12 bushels of corn. } Ans. 3. How much water must be inixed with 100 gallons of rum, worth 90 cents per gallon, to reduce it to 75 cents per gallon? A. 20 gallons. 4. A grocer mixes teas at $1,20, $1, and 60 cents, with 20 lbs. at 40 cents per lb.; how much of each sort must he take to make the composition worth 80 cents per lb.? A. 20 at $1,20, 10 at $1, and 10 at 60 cents. 5. A grocer has currants at 4 cents, 6 cents, 9 cenus, and 11 cents per lb.; and he wishes to make a mixture of 240 lbs., worth 8 cents per Ib.; how many currants of cach kind must he take? In this example we can find theropor tional quantities by linking, as before; then it is plain that the sum will be In the same proportion to any part of their sum, as the whole compound is to any part of the compound, which exactly accords with the principle of Fellowship. Hence we have the following RULE. As the sum of the PROPORTIONAL QUANTITIES found by linking as before is to EACH PROPORTIONAL QUANTITY : : so is the WHOLE QUANTITY or compound required to the REQUIRED QUANTITY of each. We will now apply this rule in performing the last question. 1. A grocer, having sugars at 8 cents, 12 cents, and if cents per pound wishes to make a composition of 120 lbs., worth 13 cents per ponud, withon gain or less; what quantity of each must be taken? A. 30 lbs. at 8, 30 lbs at 12, ano ba 16. 2. How much water, at 0 per gallon, must be mixed with wine, at 80 cents per gallon, so as to fill a vessel of 90 gallons, which may be offered at 50 cents per gallon? A. 56 gallons of wine, and 338 gallons of water. 3 How much gold, of 15, 17, 18, and 22 carats fine, must be mixed together, to form a composition of 40 ounces of 20 carats fine? A. 5 oz. of 15, of 17, of 18, and 25 oz. of 22. ↑ LXXXIV. make? INVOLUTION. Q. How much does 2, multiplied into itself, or by 2, Q. How much does 2, multiplied into itself, or by 2, and that product by 2, make? Q When a number is multiplied into itself once or more, in this manner, what is the process called? A. Involution, or the Raising of Powers. Q. What is the number, before it is multiplied into itself, called? first power, or root. A. The Q. What are the several products called? A. Powers. Q. In multiplying 6 by 6, that is, 6 into itself, making 36, we use 6 twice what, then, is 36 called? A. The second power, or square of 6. Q. What is the 2d power or square of 8? 10? 12? A. 64, 100, 144. Q. In multiplying 3 by 3, making 9, and the 9 also by 3, making 27, we use the three 3 times; what, then, is the 27 called? A. The 3d power, or cube of 3. Q. What is the 3d power of 2? 3? 4? A. 8, 27, 64. Q. What is the figure, or number, called which denotes the power, as, 3d Bower, 2d power, &c.? A. The index or exponent. Q When it is required, for instance, to find the 3d power of 3, what is the index, and what is the power? A. 3 is the index, 27 the power. Q. This index is sometimes written over the number to be multiplied, thus, ; what, then, is the power denoted by 21? A. 2×2×2×2=16. Q. When a figure has a small one at the right of it, thus, 65, what does it sean? A. The 5th power of 6, or that 5 must be raised to the 5th power. 5. How much is 14, or the 4th power of 1? A. 1 8. What is the cube of??? A. A. 4, 8 12 Involve 13 Involve 16. What is the value of 1*? 4. • Exercises for the Slate. 2. What is the cube of 211? A. 9393931. 3. What is the biquadrate or 4th power of 80? A. 40960000. 4 What is the sursolid or the 5th power of 7? A. 16807. 5. Involve,,, cach to the 3d power. 64 59319 6. What is the square of 530. 7. What is the square of 16? A. 272. ? What is the value of 85? A. 32768. 9. What is the value of 104? A. 10000 10. What is the value of 61? A. 1296. 11. What is the cube of 25? A. 15625. EVOLUTION. ! LXXXV. Q. What number, multiplied into itself, will make 16. hat is, what is the first power or root of 16? A. 4. Q. Why? A. Because 4X4 16. Q. What number, multiplied into itself three times, will make 27 ? that is what is the 1st power or root of 273? A. 3. Q. Why? A. Because 3×3 × 3 = 27. Q. What, then, is the method of finding the first powers or roots of 2d jd, &c. powers called? A Evolution, or the Extraction of Roots." In Involution we were required, with the first power or root being given, to find higher powers, as 2d, 3d, &c. powers; but now it seems, that, with the 2, 3d, &c. powers being given, we are required to find the 1st power A. It is or root again; how, then, does Involution differ from Evolution? exactly the opposite of Involution. Q. How, then, may Evolution be defined? A. It is the method of finding the root of any number. 1. What is the square root of 144? A. 12. Q. Why? A. Because 3 X3X 3X3=81. We have seen, that any number may be raised to a perfect power by Involu tion; but there are many numbers of which precise roots cannot bo obtained; as, for instance, the square root of 3 cannot be exactly determined, there being no number, which, by being multiplied into itself, will make 3. By the aid of decimals, however, we can come nearer and nearer; that is, approximate towards the root, to any assigned degree of exactness. Those numbers, whose roots cannot exactly be determined, are called SURD Roors, and those, whose roots can exactly be determined, are called RATIONAL ROOTS. To show that the square root of a number is to be extracted, we prefix this character, V. Other roots are denoted by the same character with the index of the required root placed before it. Thus V9 significs that the square root of 9 is to be extracted; 3/27 signifies that the cube root of 27 is to be extracted; 4/64: the 4th root of 64. When we wish to express the power of several numbers that are connected ogether by these signs, +, -, &c., a vinculum or parenthesis is used, drawn from the top of the sign of the root, and extending to all the parts of it; thus he cube root of 30-3 is expressed thus, 30-3, &c. EXTRACTION OF THE SQUARE ROOT. ↑ LXXXVI. Q. We have seen (T LXXXV.) that the root of any number is its 1st power; also that a square is the 2d power: what, then, is to be done, in order to find the 1st power; that is, to extract the squaro root of any number? A. It is only to find that number, which, being multiplied into itself, will produce the given number. Q. We have seen (T LXXIX.) that the process of finding the contents of a square consists in multiplying the length of one side into itself; when, hen, the contents of a square are given, how can we find the length of each ide; or, to illustrate it by an example, If the contents of a square figure be 9 feet, what must be the length of each side? A. 3 feet. Q. Why? A. Because 3 ft. X3 ft. 9 square feet. Q. What, then, is the difference in contents between a square figure whose sides are each 9 feet in length, and one which contains only 9 square feet? 9X91-972, Ans Q. What is the difference in contents between a square figure containing 3 square feet, and ono whose sides are each 3 feet in length? A. 6 square feet Q. What is the square root of 144? or what is the length of each side of a figure, which contains 144 square feet? A 12 square feet. Q. Why? A. Becauso 12 X 12 144 |