6. What is the biquadrate root of 2,998,219,536 ? 7. What is the sixth root of 1,178,420,166,015,625? 8. What is the eighth root of 722,204,136,308,736 ? 9. What is the ninth root of 387,420,489 ? 10. What is the twelfth root of 282,429,536,481?

GENERAL RULE FOR EXTRACTING ALL ROOTS.

A. 234.

A. 325.
A. 72.

A. 9.

A. 9.

11. Point off, from the unit's place, the periods, as the required root directs; that is, for the fourth root point off periods of four figures each; for the fifth root, periods of five figures, &c.

12. Find by trial the greatest root in the left hand period, and subtract its power from the said period.

13. To the remainder bring down the next figure in the next period, for a dividend.

14. Involve the root to the power next inferior to that which is given, and multiply the result by the index of the given power for a divisor.

15. Divide the dividend by the divisor, and consider the quotient the next figure of the root.

16. Involve the whole root to the given power, and subtract it from as many left hand periods as the root has places of figures.

17. To the remainder bring down the next period for a new dividend, to which find a new divisor as before, and so on till the periods are all brought down.

18. What is the sursolid or 5th root of 701,583,371,424?

25

70 15 8 3 3 7 14 24 (234
32 subtrahend.

2 × 5 divisor 8 0 ) 3 8 1 dividend.

5

23 = 6 4 3 6 3 4 3 subtrahend.

23 × 5 = 13 9 9 205) 5 7 9 4 9 0 7 dividend.

2345

=

70158337 1 4 2 4 subtrahend.

19. Observe that only one figure is brought down to form the dividend, and that the subtrahend is in each instance taken directly from the periods in the top line.

20. What is the fifth root of 1,934,917,632 ?

21. What is the seventh root of 10,030,613,004,288?

A. 72.

A. 72.

22. What is the tenth root of 3,486,784,401? The better method is to extract the 5th root of the square root.

A. 9.

23. If the amount of $100 for 8 years at compound interest be $159.38480745308416, what is the amount for the first year, and what is the rate per cent.? A. $106; 6 per cent.

Q. What is the fourth or biquadrate root of 256?-of 10,000? In the rule which is applicable to all powers, what is the direction for pointing off? 11. What is the rule for obtaining the dividend? 12, 13. What, for finding the divisor? 14. What, for finding the second figure in the root? 15. Describe the rest of the process? 16, 17

ALLIGATION.

CII. 1. ALLIGATION is the method of mixing several simples of different qualities, so that the compound or composition may be of a mean or middle quality.

2. When the quantities and prices of the several things or simples are given, to find the mean price or mixture compounded of them, the process is called

3. A farmer mixed together two bushels of rye, worth 50 cents a bushel, 4 bushels of corn, worth 60 cents a bushel, and 4 bushels of oats, worth 30 cents a bushel; what is a bushel of this mixture worth?

5. Divide the whole cost by the whole number of bushels, &c.; the quotient will be the mean price or cost of the mixture.

6. A grocer mixed 10cwt. of sugar at $10 per cwt., 4 cwt. at $4 per cwt., and 8cwt. at 7 per cwt.; what is 1cwt. of this mixture worth ?-what is 5cwt. worth?

A. lcwt. is worth $8, and 5cwt. is worth $40. 7. A composition was made of 5lb. of tea at $1 per lb., 9lb. at $1.80 per lb., and 17lb. at $11 per lb.; what is a pound of it worth? A. $1.5467+.

8. If 20 bushels of wheat, at $1.35 per bushel, be mixed with 15 bushels of rye, at 85 cents per bushel, what will a bushel of this mixture be worth? A. $1.135+. 9. If 4lb. of gold, of 23 carats fine, be melted with 2lb. 17 carats fine, what will be the fineness of this mixture? A. 21 carats.

CIII. 1. ALLIGATION ALTERNATE is the process of finding the proportional quantity of each simple, from having the mean price or rate, and the mean prices or rates of the several simples given; consequently, it is the reverse of ALLIGATION MEDIAL, and may be proved by it.

2. A farmer has oats worth 25 cents a bushel, which he wishes to mix with corn worth 50 cents per bushel, so that the mixture may be worth 30 cents per bushel, what proportion or quantities of each must he take?

CII. Q. What is Alligation? 1. Alligation Medial? 2. Rule? 5.
CIII. Q. What is Alligation Alternate? 1.

3. In this example, it is plain, that if the price of the corn had been 35 cents, that is, had it exceeded the price of the mixture (30 cents) just as much as it 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, 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 differences, these differences will represent the same proportion.

4. These are 20 and 5, that is, 20 bushels of oats to 5 of corn, which are the quantities or proportions required. In determining those differences, it will be found convenient to write them down in the following manner:

30

cts. bushels.
$.25- -20
$.50

Ans.

5. It will be recollected, that the difference between 50 and 30 is 20; that is, 20 bushels of oats, which must

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; again, the difference between 25 and 30=5, that is, 5 bushels of corn opposite the 50, the price of the corn.

6. The answer, then, is 20 bushels of oats to 5 bushels of corn, or in that proportion.

7. By this mode of operation, it will be perceived that there is precisely as much gained by one quantity as there is lost by another, and therefore the gain or loss on the whole is equal.

8. 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.

9. 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 two quantities of two 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. 10. Proof.-We will now ascertain the price of the mixture by the last rule, thus:

20 bushels of oats at 25 cents per bushel=$5.00

11. Having reduced the several prices to the same denomination, Q. Why does not the operation affect the total value of the commodity? 7, 8. Why is not the result confined to one answer? 9. Rule? 11, 12, 13.

connect by a line each 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.

12. Place the difference between the mean rate and that of each of the simples opposite the price with which they are connected.

13. Then, if only one difference stands against any price, it expresses the quantity of that price; but if there be several, their sum will express the quantity.

14. A merchant has several sorts of tea, some at 10s., some at 11s., some at 13s. and some at 24s. per lb.; what proportions of each must be taken to make a composition worth 12s. per lb.?

15. How much wine, at 5s. per gallon and 3s. per gallon, must be mixed together, that the compound may be worth 4s. per gallon? A. 1 gallon of each.

16. 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 bushel? A. 14bu. at 42c.; 3 at 60; 4 at 67; 22 at 78.

17. A grocer would mix different quantities of sugar, viz.-one at 20, one at 23, and one at 26 cents per lb.; what quantity of each sort must be taken to make a mixture worth 22 cents per lb.?

A. 5lb. at 20 cents; 2 at 23; 2 at 26. 18. 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, 1, 1, 4.

19. We have seen that we can take 3 times, 4 times,,, or any proportion of each quantity, to form a mixture.

20. 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 following

RULE.

21. As the proportional quantity of that piece whose quantity is given is to each proportional quantity, so is the given quantity to the quantities or proportions of the compound required.

22. A grocer wishes to mix one gallon of brandy, worth 15s. per gallon, with rum worth 8s., so that the mixture may be worth 10s. per gallon; how much rum must be taken?

23. 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 brandy. A. 2 gallons.

24. 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

8:10: 10:12

8:32:10:40

bushels of rye.

25. How much water must be mixed with 100 gallons of rum, worth 90cts. per gallon, to reduce it to 75cts. per gallon. A. 20gal. 26. A grocer mixes teas at $1.20, $1, and 60 cents, with 20lb. at 40c. per lb.; how much of each sort must he take to make the composition worth 80c. per lb. A. 20 at $1.20, 10 at $1, 10 at 60c.

27. A grocer has currants at 4 cents, 6 cents, 9 cents, and 11 cents per lb.; and he wishes to make a mixture of 240lb., worth 8 cents per lb.; how many currants of each kind must he take? In this example we can find the proportional quantities by linking, as before; then it is plain that their 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.

RULE.

28. 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.

29. A grocer, having sugars at 8c., 12c., and 16c. per lb., wishes to make a composition of 120lb., worth 13c. per lb.; what quantity of each must be taken? A. 30lb. at 8, 30lb. at 12, 60lb. at 16.

30. How much water, at 0 per gal., must be mixed with wine, at 80c. per gal., so as to fill a vessel of 90gal., which may be offered at 50c per gal.? A. 56 gallons of wine, and 339 gallons of water. 31. How much gold, of 15, 17, 18, and 22 carats fine, must bẻ mixed together, to form a composition of 40 ounces of 20 carats fine? A. 5oz. of 15, of 17, of 18, and 25oz. of 22.

ARITHMETICAL PROGRESSION. CIV. 1. ARIthmetical Progression, or SERIES, is any rank of numbers more than two, that increase by a constant addition, or decrease by a constant subtraction, of the same number.

2. THE COMMON DIFFERENCE is the number added or subtracted as above.

3. AN ASCENDING SERIES is one formed by a continual addition of the common difference, as 2, 4, 6, 8, 10, &c.

CIV. Q. What is Arithmetical Progression? 1. What the Common Differ ence? 2. An Ascending Series? 3. A Descending Series? 4. Give an ex ample of each. 3, 4. What are the terms? 5.