EXAMPLES.

16. Remitted from London to Amsterdam a bill of £754. 10. sterling, how many pounds Flemish is the sum, the exchange at 33. 6d. Flemish per pound sterling? Ans. £1263.15.9. Flem. 17. A merchant at Rotterdam remits £1263. 15.9. Flemish to be paid in London, how much sterling money must he draw for, the exchange being at 33s. 6d. Flemish per pound sterling? Ans. £754. 10.0 18. If I pay in London £852. 12. 6. sterling how many guilders must I draw for at Amsterdam, exchange at 34 schel. Agroats Flemish per pound sterling? Ans, 8792 guild. 13 stiv. 144 pennings. 19. What must I draw for at London, if I pay at Amsterdam 8792 guild. 13 stiv. 14) pennings, exchange at 34 schel. 44 groats per pound sterling? Ans. £852. 12.6.

To convert Bank Money into current, and the contrary. NOTE. The Bank Money is worth more than the Current. The difference between one and the other is called agio, and is generally from 3 to 6 per cent. in favour of the Bank.

To change Bank into Current Money.

RULE. AS 100 guilders Bank is to 100 with the agio added :: so is the Bank given: to the current required.

To change Current Money into Bank.

RULE. AS 100 with the agio added: is to 100 Bank: so is the current money given: to the Bank required.

20. Change 794 guilders, 15 stivers, current money into Bank florius, agio 43 per cent. Ans. 761 guilders, 8 stivers, 11117 pens. 21. Change 761 guilders, 9 stivers Bank, into Current Money, agio 4 per cent. Ans. 794 guilders, 15 stivers, 4% pennings.

VI. IRELAND.

22. A gentleman remits to Ireland £575. 15. sterling, what will he receive there, the exchange being at 10 per cent?

Ans. £633...6...6.

23. What must be paid in London for a remittance of £633.....6...6. Irish, exchange at 10 per cent? Ans. £575.15.

COMPARISON OF WEIGHTS AND MEASURES.

EXAMPLES.

1 If 50 Dutch pence be worth 65 French pence, how many Dutch pence are equal to 350 French pence?

Ans. 269. 2. If 12 yards at London make 8 ells at Paris, how many ells at Paris will make 64 yards at London ?

Ans. 42

3. If 30lb. at London make 281b. at Amsterdam, how many b. at London will be equal to 350b. at Amsterdam? Ans. 375. 4. If 956. Flemish make 100%. English, how many lb. English are equal to 275lb. Flemish ?

Ans. 289.

IS when the coin, weight, or measures of several countries are compared in the same question: or it is linking together a væ riety of proportions.

When it is required to find how many of the first sort of coin, weight, or measure, mentioned in the question, are equal to a given quantity of the last.

RULE. Place the numbers alternately, beginning at the left hand, and let the last number stand on the left hand: then multiply the first row continually for a dividend, and the second for a divisor.

PROOF. By as many single Rules of three as the question requires.

EXAMPLES.

1. If zuid. at London make 23lb. at Antwerp, and 1551b. at Antwerp make 18016. at Leghorn, how many ib. at London are equal to 726. at Leghorn?

2. If 12b. at London make 106. at Amsterdam, 100lb. at Amsterdam 120b. at Thoulouse, how many b. at London is equal to 40b. at Thoulouse?

Ans. 40lb. 3. If 140 braces at Venice are equal to 156 braces at Leghorn, and 7 braces at Leghorn equal to 4 ells English, how many braces at Venice are equal to 16 ells English? Ans. 250

80

-288

4. If 406. at London make 3616. at Amsterdam, and 906 at Amsterdam make 116b. at Dantzic, how many ih, at London is equal to 130b. at Dantzic? Ans. 112 4176 When it is required to find how many of the last sort of coin, weight, or measure, mentioned in the question, is equal to the quantity of the first.

RULE. Place the numbers alternately, beginning at the left hand, and let the last number stand on the right hand; then multiply the first row for a divisor, and the second for a dividend.

EXAMPLES.

5. If 12b. at London make 10. at Amsterdam, 100b. at Amsterdam, 120lb. at Thoulouse, how many lb. at Thoulouse are equal to 40%. at London? Ans, 40lb. If 40lb. at London make 26, at Amsterdam, and 90. at Amsterdam 1166. at Dantzic, how many th. at Dantzic are equal to 122b, at London? Ans. 1411. Progression consists of two parts, Arithmetical and Geome

trical.

I when the rank of numbers increase or decrease regularly by the continual adding or subtracting of the equal numbers: As 1, 2, 3, 4, 5, 6, are in Arithmetical Progression by the continual increasing or adding of one; 11, 9, 7, 5, 3, 1. by the continual decreasing or subtracting of two.

Note. When any even number of terms differ by Arithmetical Progression, the sum of the two extremes will be equal to the two middle numbers, or any two means equally distant from the extremes: as 2, 4, 6, 8, 10, 12, where 6+8, the two middle numbers ure 12+2=14, the two extremes 10+4, the two means 14. When the number of terms are odd, the double of the middle term will be equal to the two extremes; or of any two means equally distant from the middle term; as 1, 2, 3, 4, 5, where the double of 3=5+1=2+4=6.

In Arithmetical Progression five things are to be observed, vix. 1. The first term; better expressed thus, F.

2. The last term,

3. The number of terms,

4. The equal difference......

5. The sum of all the terms....

Any three of which being given the other two may be found. The first, second, and third terms given, to find the fifth.

RULE. Multiply the sum of the two extremes by half the number of terms, or multiply half the sum of the two extremes by the whole number of terms, the product is the total of all the terms: or thus,

I. F. L. N. are given to find S.

N

F+LX-S

2

EXAMPLES.

1. How many strokes does the hammer of a clock strike in 12 hours? 12+1=13, then 13×6=78.

2. A man buys 17 yards of cloth, and gave for the first yard 28. and for the last 10s. what did the 17 yards amount to? Ans. £5.2.0.

3. If 100 eggs were placed in a right line, exactly a yard asunder from one another, and the first a yard from a basket, what length of ground does that man go who gathers up these 100 eggs singly, returning with every egg to the basket to put it in ? Ans. 5 miles, 1300 yards.

The first second and third term given to find the fourth. RULE. From the second subtract the first, the remainder die vided by the third less one, gives the fourth: or thus, II. F. L. N. are given to find D.

L-F

N-1

D

EXAMPLES.

4. A man uad eight sons, the youngest was 4 years old and the eldest 32, they increase in Arithmetical Progression; what was the common difference of their ages? Ans. 4

32—4—28, then 28÷8—1—4 common difference. 5. A man is to travel from London to a certain place in 12 days, and to go but 3 miles the first day, increasing every day by an equal excess, so that the last day's journey may be 58 miles, what is the daily increase, and how many miles distant is that place from London? Ans. 5 daily increase. Therefore, as three miles is the first day's journey, 3+5 8 the second day,

8+5=13 the third day, &c.

The whole distance is 366 miles.

The first, second, and fourth terms given, to find the third. RULE. From the second subtract the first, the remainder divide by the fourth, and to the quotient add 1, gives the third: or thus,

III. F. L. D. are given to find N.

L-F

EXAMPLES.

6. A person travelling into the country, went 3 miles the first day, and increased every day 5 miles, till at last he went 58 miles in one day, how many days did he travel?

58-3-55. then 55-5-11. 11+1=12 the Ans. 7. A man being asked how many sons he had, said that the youngest was 4 years old, and the oldest 32; and that he increased one in his family every 4 years, how many had he?

Ans. 8.

The second, third, and fourth terms given, to find the first. RULE. Multiply the fourth by the third made less by 1, the product subtracted from the second gives the first or thus, IV. L. N. D. are given to find F.

L-DXN-1=F.

EXAMPLES.

8. A man in 10 days went from London to a certain town in the country, every day's journey increasing the former by 4, end the last he went was 46 miles, what was the first?

Ans. 10 miles. 4× 10-136, then 46--36-10 the first day's journey. 9. A man takes out of his pocket at 8 several times, so many aifferent numbers of shillings, every one exceeding the former by 6, the last at 46, what was the first?

The third, fourth, and fifth given to find the first.

Ans. 4.

RULE. Divide the fifth by the third, and from the quotient subtract half the produce of the fourth multiplied by the third less 1 gives the first or thus,

V. Ñ. D. S. are given to find F
S DXN-1

N

=F

EXAMPLE.

10. A man is to receive £360 at 12 several payments, each to exceed the former £4, and is willing to bestow the first payment on any one that can tell him what it is, What will that person have for his pains? Ans. £8.

4x12-1

360÷12-30, then 30

2

8 the first payment.

The first, third, and fourth given to find the second.

RULE. Subtract the fourth from the product of the third, multiplied by the fourth, that remainder added to the first gives the second or thus,

VI. F. N. D. are given to find L.

ND-D+FL.

EXAMPLE.

11. What is the last number of an Arithmetical Progression, beginning at 6, and continuing by the increase of 8 to 20 places?

Ans. 158.

20X8-8-152, then 158+6=158 the last number.

GEOMETRICAL PROGRESSION

S the increasing or decreasing of any rank of numbers by some common ratio; that is, by the continual multiplication or division of some equal number: as 2, 4, 8, 16, increase by the multiplier 2, and 16, 8, 4, 2, decrease by the divisor 2.

NOTE. When any number of terms is continued in geometrical Progression, the product of the two extremes will be equal to any two means, equally distant from the extremes; as 2:4, 8, 16, 32, 61, where 64×2 are 4×82, and 8×16=128.

When the number of terms are odd; the middle term multiplied into itself will be equal to the two extremes, or any two means, equally distant from the mean, ás 2, 4, 8, 16, 32, where 2×32 4X16=8X8 64.

In Geometrical Progression the same five things are to be observed as are in Arithmetical.