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10. A wine merchant has four sorts of wine, viz. of 20d. 16d. 12d. and 7d, the quart; how much of each sort must he take to sell a quart for 14d. Answ. 6 quarts of 7d. 2 quarts of 12d. 2 quarts of 16d. and 7 quarts of 206. or otherwise. 11. A goldsmith has gold of 17, 18, 22 and 24 caracts fine ; how much must he take of each to make it 21 caracts fine P Answ. 3 caracts of 17, 1 caract of 18, 3 caracts of 22 and 4 of 24 fine. 12 A vintner would make a mixture of Malaga worth 7s. 6d. p. gallon, with Canary at 6s. 9d. #' gallon; sherry at 5s. #' gallon, and white wine at 4s. 3d. #' gallon : what quantity of each must he take, that the mixture may be sold for Gs. P. gallon? Answ. 12 of Malaga; 18 of sherry ; 21 Canary; 9 white wine, or otherwise.

SECT. III.
ALLIGATION PARTIAL.

HE particular rates of the ingredients proposed to be T mixed, the mean rate of the whole mixture, and any one of the quantities to be mixed being given, to find how much of every one of the other ingredients is requisite to compose the mixture.

13. How much wheat at 5s. the bushel must be mixed with 12 bushels of rye, at 3s. 6d. 45° bushel, that the whole mixture may bear 4s. 4d. per bushel :

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First, I find a quantity of bu. wheat, which being mixed 60, 10 of wheat, will bear the price proposed *} viz. 10 bushes of wheat 42 8 of rye. and 8 of rye: but the given 8–12–10 quantity of rye being 12 IO bushels, I must find a quan- tity of wheat so proportion- 8)12O ed to 10 bushels as 12 to 8 — viz. 8 : 12:: 10:15, whence Answ. 15 bushels. appears the reason of the or 8–10–12

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Rule.

Set down all the particulars and find their differences, Then say, As the difference standing against the price of which the quantity is given, is to the said given quantity, so is each other difference to the quantify required. ... 14. How much alloy must be put to bullion of 1030t. fine, to bring it to 7 oz. fine Answ. 53.8% oz. 15. How much water must be mixed with 63 gallons of brandy, at 5s 5d. the gallon, to reduce it to 4s. 6d. p’ gali, n ° Answ. 12; gallons. 16. How much brass of 14d. §2 ft. and pewter of 104d. the ib must I melt with 59th of copper worth 16d. the fö, so that the whole may stand me in is, the sh; Answ. 2008. at 10:d. and 50 at 14d.

17. How much gold of 21 and 23 caracts fine, must be mixt with 30.oz. of 20 caracts fille, to bring it to 22 caracts fine 2 Answ. 30 of 21 and 90 of 23. 18. With 60 gallons of brandy at 6s. f* gallon, I mix brandy of 5s. 4d. p. gallon, and some water; then I find it stood me in 3.s. 6d. p. gallon : I demaud how much brandy, and how much water I took 2 * Answ. 60 at 5s. 4d. and 74% of water. 19. How much Malaga of 7s. 5d. the gallon, and sherry of 5s. 2d, the gallon, and white wine at 4s. 2d. the gallon, must be mixed with 20 gallons of Canary at 6s. 8d. the gallon, so that one gallon of the mixture may stand in 6s. the gallon Answ. 44 gallons at 7s. 5d., 16 gallons at 5s. 2d. and 34 gallons at 4s.2d. 20. How much alloy, and how much gold of 21 and 23 caracts fine, must be put to 3002, of 20 caracts fine, to bring it to 18 caracts fine * Answ. 1630z. alloy, 30oz. of 21, and 3002. of 23 caracts fine. How is the above answer proved to be true

SEC T. IV.
ALLIGATION TOTAL.

HE particular rates of all the ingredients proposed to

be mixed; and the sum of all their quantities, with the nean rate of that sum being given; to find the particular quantities of the mixture.

Rule.

Rule.

Set down all the particular rates with the mean rate, as before, and find, their differences, and add together all the differences into one sum ; then say, As the sum of all the differences: is to the sum of all the quantitles given : So is every particular difference: to its particular quantity.

21. Let it be required to mix wheat at 5s. the bushel, with rye at 3s.6d, the bushel, so as that the whole quantity may be 27 bushels, to be sold for 4s. 4d. a bushel: What quantity of each must be taken to make up the

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I* a Number be multiplied into itself any number of Times, the product is called a Power of that Number, and the Number multiplied in respect of the Product is called its Root, particularly

If a Number be multiplied into itself the Product is a Square Number, viz. the Square (or second power) of the number multiplied, which number is likewise the Square Root of the Product. As 4×4=16, So 16 is the Square of 4, and 4 the Square Root of 16.

Having the Root given to find the Square thereof, is only to find the product of the given number multiplied by itself, and thus we construct.

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QUESTIONS.

2uest. What is a square number? A. That which is produced from the multiplication of any number into itself, which number is called the root with respect to its square. 2. Repeat the squares of the single figures. A. The square of 1 is 1, of 2 is 4, &c. 2. How must I extract the square root ” A. By the following rule. First to prepare the square, this do, Point off the figures two by two: Beneath the last the square next less Put; and its root i' th' quotient place: From the last period take the square, Then the next lower period there To the remainder must be brought; Be this a dividend: The quote ..I)oubled must the divisor be To all but units place; then see How oft the greater holds the less, That figure must the quote express, And the divisor units too, Then as in plain Division do. Thus every period one by one We manage and the work is done. 2. How is the work proved A. By multiplying the root into itself, and adding the remainder, if any.

Examples

1. What is the square root of 256* Answ. 16.
2. -— 57.6 °
3. Find the square root of 234256
4. What is the square root of 45 1584

5. - 23007636 2

6. 15132 l 2
7. —--—— 14 122564 *
8. ———- 298.5984
9. —-——— 15437,041 *
10. ——-—— 2990667969
I l. -—— 572199960721 *

§ 2. To extract the Square Root of Fractions. A Fractional power may be considered either as an immediate power, i.e. the immediate product of the multiplication

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