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and he is minded to make a Composition of 60 gallous, worth 9s. per gallon-I demand how much of each sort he must have? Ans. 45 gallons of Canary, and 5 gals. of each other sort.

3. A Brewer hath 3 sorts of ale, to wit, at 10d. at 8d. and at od. per gallon--and he would have a composition of 30 gallons, worth 7d. per gallon- I demand how much of each sort he must have ?

Gals. d.p.gal.
5 at 10

5 at 8

20 at 6. Ans.

30

4. A Goldsmith hath several sorts of Gold, viz. some of 24 carrats fine, some at 22 carrats, and some of 18 carrats fine; and he would have compounded of these sorts the quantity of 60 oz. 20 carrats fine-I demand how much of each sort he must take ?

Oz,
12 at 24 Carrats fine.
12 at 22 Carrats fine.

36 at 18 Carrats fine. Answ.

60

5. A Goldsmith hath Gold of 3 sorts, viz. of 22 carratsa of 21 carrats, and of 20 carrats fine, and he would mix with these so much Alloy, as that the quantity of 21 oz. may bear 18 carrats fine, I demand how much of each sort he must take, and how much alloy? Ans. 6 oz. of each sort of Gold, and 3 oz. of Alloy.

6. A Druggist had 3 sorts of Drugs, one was worth 4s per lb. another 5s, and another 8s. and out of these he made two parcels, one was 21lb. at 6s. per Ib. and the other 35 lb at 7s. per lb. how much of every sort did he take for each parcel ? Ib. sp.lb.

lb s.p.lb. 6 at 4

5 at 4 6 at 5

5 at 5 Ans. 9 at 8

25 at 8

21 at 6s p. Ib. 35 at 7s p. Ib.

2.

OF POSITION.
WHAT is Position ; or Negative Arithmetic ?
A. It discovers the Truth by supposed Numbers.
Q. How many kinds of Position are there?
A. Two; Single and Double.

OF SINGLE PUSITION.
Q. What is Single Position ?
Ă. It discovers the truth by only one supposed number.
Q. How is that supposed number used ?

A. By working with it, as if it was the true number, in the same proportion as the question directs; and if the result be eitker too much or too little, the true number may be found out by the following Rule, viz.

As the Result of the Position,
Is to the Position :
So is the given Number,

To the number required.
Q. How do you prove Position ?

Å. Position, both Single and Double, is proved by adding the several sums required, or the several parts of the sum required, together; and if that sum agrees with the given sum, it is right.

EXAMPLES.

1. Two men, A and B, having found a 'Bag of Money, disputed who should have it; A said the half, third and fourth of the money made 1301. and if B could tell how much was in it, he should have it all, otherwise he should have nothing? I demand how much was in the bag? Ans. 1201.

2. A, B and C, determined to buy together a certain quantity of timber, worth 361. agree that B shall pay more than A, and C 1 more than B-I demand how much cach man must pay? Ans. A. 91. B 121. C 151.

3. A person having about bim a certain number of Crowns, said if the half, third and fourth of them were added together, they would make 65 Crowns--1 demand low mary he had ? Ans. 60 Crowus.

4. C lent D a sum of money, to be paid at 4 payments; when three of them were made, and C came to demand the fourth, D would give him no more, except he would tell him how much was paid already: C said the first payment was a fourth, the second a oftl, and the third a sixth ofilie sum first lent, and altogether made 741. I demand the sum lent? Ans. 1201..

5. One man carrying a bag of money in his hand, another asked him how much was in it; he answered, he could not tells but the third, fourth, and fifth of it made 941. How much was in the bag ? Answer 110l.

6. I have delivered to a Banker a certain sum of money, to receive of him after the rate of 6l. per cent. per annum ; and at the end of ten years he paid me 5001 for Principal and Interest together; I demand the sum de. livered to him at first? Answer 3121 10s.

OF DOUBLE POSITION.. Q. What is Double Position ?

A. It is that which discovers the true number soughts by making use of two supposed numbers.

Q. How are those supposed numbers used?

A. 1. By working with them as if they were the true numbers in the same proportion as the question directs.

2. The result or Errors must be placed against Pos.Er. their Positions, or supposed Numbers thus, 40 28 3. Multiply them Cross-wise.

36 19 4. If the Errors are alike, i. e. both greater, or both less than the given number, take their difference for a divisor, and the difference of the products for a dividend.

5. If the Errors are unlike, take their sum for a divisor, and the sum of the products for a divideud; the quotient thence arising will be the answer.

EXAMPLES. :

1. B, C and D, would divide 1001 between them, so as that C may have 3l more than B, and D 4/ more than C: I demand how much each man must have ? Answ. B 301. C331. D 371.

2. A man lying at the point of death, said he had in a certain coffer 100l which he bequeathed to 3 of his friends after this manner: the first must have a certain portion, the second must-have twice as much as the first, wanting 8!. and the third must have three times as much as the first, wanting 151. I demand how much each man must have ? Answer, the first 201, ios, second 331. third 461. 10s.

.3. B, C and D built an House which cost 1001. of which B paid a certain sum, Cpaid 101. more than B, and D paid as much as B and C-1 demand each man's share in that charge? Ans. B 201. C 301. D 501.

4. Three persons discoursed together concerning their ages ; s.ys B, I am 20 years of age; says C, I am as old as B, and liali ] ; and says D, I am as old as you both; I

demand the age of each person ? Answer, B was 20, C 60, D 80 years of age.

5. A man lying at the point of death, left to his 3 sons all his estate in money, viz to F half wanting 501. to Gone third, and 10 H the rest, which was to less than the share of G-1 demand the sum left, and each man's part ? Ans. the sum left was 360l. whereof F had 1301. G 1201. H 1101

6. A certain man having drove his Swine to the market, viz. Hogs, Sows and Pigs, received for them all 501. being paid for every Hog 18s. for every Sow 16s. for every Pig 2s. There were as many Hogs as Sows, and for every Sow there were three Pigs~ demand how many there were of each sort? Answ. 25 Hogs, 25 Sows, 75 Pigs.

7. A surly old fellow being demanded the ages of his four children, answered, you may go and look: but if you must needs know, my first son was born just 1 year after I was married to his mother, who, after his birth, lived 5 years, and then died in ehild bed, with

my
second son;

four years after that I married again, and within two years had my third and fourth sons at a birth: the sum of whose two ages is now cqual to that of the eldest-I demand their several ages ? Answer, the first son was 22 years old, the second 17, the third 11, and the fourth 11 years old.

OF COMPARATIVE ARITHMETIC. Q. WHAT is Comparative Arithmetic?

J. It is such as answers Questions by Numbers, having Relation one to another.

Q. Wherein does this relation consist?
Ă. It consists either in Quantity or Quality.
Q. What is the Relation of Numbers in Quantity?
A. It is the Respect that one Number has to another.
Q. How many are the Numbers propounded ?

A. They are always two, the Antecedent and the Consequent.

Q. In what does relation of numbers in quantity consist?

Å. It consists in the Difference, or else in the Rate or Reason that is found between the Terms propounded.

Note: The difference of any two Numbers is the Remainder? but the Rate or Reason is the Quotient of the Antecedent divided by the Consequent.

Q. What is relation of numbers in quality or progression ?

. Progression or Proportion is the respect that the Reason of Numbers have one to another.

Q. How many must the terms be?

A. Three, or more, but never less, because less than 5: will not admit of a comparison of reason or differences.

OF PROGRESSION. Q. How

many kinds of Progression are there? Å. Two--Arithmetical and Geometrical.

OF ARITHMETICAL PROGRESSION. Q. What is Arithmetical Progression ?

X. Arithmeticai Progression is when several Numbers have equal differences as 1, 2, 3, 4, differ by 1, or 2, 4, 6, 8 differ by 2.

Note 1. If any number of terms differ by Arithmetical Progression, the sum of the two extremes will be equal to the sum of any two means equally distant from the extremes. As in 2, 4, 6, 8,: where 2x 8 are=4X6=10, and so of any larger number of terms.

2. If the number of terms be odd, the middlemost supplies the place of two terms, as in 1, 2, 3; where 1x3 are=2x2=-4.

Case 1.
Q. What do you observe in this first case ?

À. When the two Extremes, and the number of terms in any series of numbers in arithmetical progression are given, and the sum of all the terms is required, then multiply the sum of the two extremes by half the nuinber of terms: Or,

Multiply half the sum of the extremes by the whole number of terms, the product is the total of all the terms.

EXAMPLES.

1 How many Strokes does the Hammer of a Clock strike in 12 hours? Answer 78.

2 A merchant hath sold 100 yds. of superfine cloth, viz. the 1st yard for Is. the 2d 2s. the 3d for 3s, &c. Idemand how much he received for the said cloth? Ans. 252/ 10s.

3 Bought 19 yards of Shalloon, and gave id for the first yard, 3d for the second, 5d for the third, &c. increasing 2d every yard -1 demand what I gave for the 19 yards? Ans. 11 10s 1d.

4 A Mercer sold 20 yards of silk, at 3d for the first yard, 6d for the 2d, 9d for the 3d, &e. increasing 3d every yard—I demand what he sold the 20 yds. for? Ans. 21 128 6d.

5 A Butcher bought 100 head of Cattle, viz. Oxen, and gave

for the first Ox i Crown, for the second Ox 2 Crowns, for the third Ox 3 Crowns, &e.--I demand wbat the Cattle cost him? Ans. 12621 tos.

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