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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 20d, 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? Answ. 3 caracts of 17, 1 caract of 18, 3 caracts of 22 and 4 of 24 fine.

1.2. A vintner would make a mixture of Malaga worth 7s. 6d. gallon, with Canary at 6s. 9d. 4 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 ds. P gallon?

Answ. 12 of Malaga; 18 of sherry; 21 Canary; 9 white wine, or otherwise.

THE

SECT II.

ALLIGATION PARTIAL.

HE particular rates of the ingredients proposed to be 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. bushel, that the whole mixture may bear 4s. 4d. per bushel?

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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 quantity required.

14. How much alloy must be put to bullion of 103oz. fine, to bring it to 7 oz. fine?

Answ. 54oz.

15. How much water must be mixed with 63 gallons of brandy, at 5s 5d. the gallon, to reduce it to 4s. 6d. galion? Answ. 125 gallons. 16. How much brass of 144. the ib must I melt with 50

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b. and pewter of 1040. of copper worth 16d, the

that the whole may stand me in Is. the bi Answ. 200. at 10 d. and 50 at 14d.

17. How much gold of 21 and 23 caracts fine, must be mixt with 30oz. of 20 caracts fiße, to bring it to 22 caracts fine? Answ. 30 of 21 and 90′ of 23.

18. With 60 gallons of brandy at 6s. P gallon, I mix brandy of 5s. 4d. P gallon, and some water; then I find it stood me in 3s. 6d. gallon: I demaud how much brandy, and how much water I took?

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 30oz. of 20 caracts fine, to bring it to 18 caracts fine? Answ. 16oz. alloy, 30oz.

of 21, and 30oz. of 23 caracts fine.

How is the above answer proved to be true?

T

SECT. IV.

ALLIGATION TOTAL.

HE particular rates of all the ingredients proposed to be mixed; and the sum of all their quantities, with the mean rate of that sum being given; to find the parti. cular 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 quantities 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

mixture?

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22. A goldsmith hath 2 sorts of silver bullion, the one of 10 oz. and the othor of 5 oz fine, and has a mind to mix a tb. of it, so that it shall be 8 oz. fine; I demand how much of each be must take? Answ. 44 of 5 oz. fine,

and 7 of 10 oz. fine.

23. A grocer has sugar of 12d. the b. and of 6 d. the b. and has a mind to mix a Cwt. of it, so that he may sell it at 8d. the . I demand how inuch of each sort he must take? Answ. 1621b. of 61⁄2d. and 61. at 12d.

24. A refiner has silver of 114oz. and of 7 oz. fine, and has a mind to make a piece of work requiring 351. of 93oz. fine: How much must he take of each ?

Answ. 21tb of 111⁄2 oz. fiue, and 1313b. of 7 oz. fine. 25. Brandy of 3s. 6d. and 5s. 9d. the gallon is to be mixed, so that a hhd. of 63 gallons may be sold for 121. 12s. I demand how many gallons must be taken of each?

Answ. 14 gallons of 5s. 9d. and 49 gallons of 33. 6d. 26. A vintner has 3 sorts of wine, viz. of 24d. 22d. and 18. the gallon: now he has a mind to mix a cask of

60

60 gallons, so that he may sell it at 20d. the gallon: how much must he take of each? Answ. 12 at 24d. 12 at

22d. and 36 at 18d.

27. A goldsmith has 3 sorts of silver, viz. of 11, 8 and 50%. fine, and has a mind to make a piece of work that shall weigh 10. of 9oz. fine: how much of each must he take? Answ. 5lb. of 11oz. fine, 23lb. of 8oz. fine, and 23lb. of 50%. fine.

28. A cask of 58 gallons is filled with liquor of 7, 8, and 10d. the gallon, and then it stands in 94d. the gallon; I would know how many gallons of each sort was taken ? Answ. 40. gallons of 10d. 87 gallons of 8d. and 8% of 7d.

BOOK IV.

EXTRACTIONS, PROGRESSIONS, &c.

CHAP. I.

OF THE SQUARE ROOT.

Fa Number be multiplied into itself any number of

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 4X4 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.

Roots

A TABLE

Of the Squares of the single Figures.

| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 Squares | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81

This Table being committed to memory, we are to shew

how to extract the square root of any number.

QUES

QUESTIONS.

Quest. 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
Doubled 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.

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§ 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 multiplica.

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