28. In 6 bales, each 12 pieces, and each piece 18 yards, how many ells Flemish? Ans. 1728.

29. In 4 ingots of silver, each weighing 2 lb. 6 oz. 11 pwt. how many grains? Ans. 58656.

30. How many hours, minutes, and seconds in one year? Ans. 8766 hr. 525960 min. and 31557600 sec.

31. In 1597 quarts, how many bushels, &c.? 3 pk. and 5 qt.

Ans. 49 bu.

QUESTIONS.-What are compound numbers? How do they increase? What are included under this head? Let the 14 tables of Compound Numbers be made familiar, before the scholar proceeds with Reduction. What is Reduction? How are high denominations brought into low denominations? And how are low denominations brought into high? What is the rule when high denominations are brought into low? And what is it, when low denominations are brought into high? What should the scholar notice before commencing to reduce any quantity? In circular measure, how is the circle regarded? What are the divisions of the circle? To what is this measure applied? How many days and hours does the year contain? To what do the 6 hours amount in 4 years? How many days does every fourth year contain? What is this fourth year called? How may it be found? In dividing the given year by four, what does the figure that remains (if any) show? What is the more usual division of the year? How many days are contained in each of the 12 months? When quantities are to be exchanged, what is the rule for the common mode of operation ? What is the rule for canceling? What is Note 1st? What is Note 2d ?

COMPOUND RULES.

The scholar will recollect, that in simple numbers, the denominations increase in value in the constant ratio of 10. ↑

The peculiarity of the numbers in the preceding rule, and in the four next following, is, that they have no uniform ratio of increase, common to all denominations; but each denomination has its own peculiar ratio. These ratios are all represented in the tables of Compound Numbers.

In simple numbers, 10 units make 1 ten; 10 tens make 1 hundred; 10 hundred make 1 thousand, &c. In operations with these numbers, we therefore carry for 10.

In the table of English money, 4 farthings make 1 penny; 12 pence 1 shilling, and 20 shillings 1 pound. For the same

reason, therefore, that we carry for 10 in simple numbers, we carry for 4, 12, and 20, in operations with pounds; shillings, pence, and farthings; that is, from farthings to pence, we carry for 4, because 4 farthings make 1 penny; from pence to shillings, by 12, for a similar reason; and from shillings to pounds, for 20. The same general principle and reasoning may be applied to the other compound tables.

There is one peculiarity noticeable in writing compound numbers. In simple numbers, we always know that any figure sustains a ten-fold relation to the figures next it; that is, the one on the left of it is of 10 times more value; and the one on the right, of 10 times less value than they would be in its own place. Hence, all that is here necessary, is that the figures preserve their proper order. In compound numbers, each denomination is known only by its own appropriate mark. There is, therefore, an obvious necessity for each denomination to be separately written.

ION

COMPOUND ADDITION.

Compound Addition is an operation by which several numbers of different denominations, as pounds, shillings, pence, &c. are united together. The rule to be observed in writing down these numbers is, to place those of the same name under each other.

Let it be required to add together 3 £. 15 s. 9 d. 3 qr.; 5 £. 6 s. 8d. 2 qr.; 8 £. 13 s. 11 d. 3 qr.; and 10 £, 12 s. 8 d. 2 qr. The following is a convenient mode of writing them:

The amount of the right hand column is 10 farthings=2 d. and 2 qr. Like simple numbers, the 2 qr. are set down; and the 2 d. added to the column of pence, the amount of which 38 d. 3 s. and 2 d. Setting down the 2 d. and carrying the 3 s. to the column of shillings, we 2 £. and 9 s. Lastly, setting down

make this column=49 s.

and carrying as before, we find the amount of the column of pounds to be 28, which we write at the foot of the column. We therefore find the amount of the four given numbers to be 28 £. 9 s. 2 d. 2 qr.

From the preceding example, the scholar will see the appropriateness of the following rule:

RULE. Write the numbers so that each denomination shall

occupy a separate column. Then, commencing with the lowest denomination, add each column by itself.

Notice at the addition of each column, to how many of the denomination next above, the amount obtained is equal, and how many remain. Write down those that remain, and carry the other number to the next column. Proceed thus through all the denominations.

Note.-The whole amount of the left hand column must be written down, if it be in the highest denomination.

If it be

not in the highest denomination it should be reduced as far as practicable.

The column of farthings amounts to 6 qr. 1 d. and 2 qr. The column of pence is 35 d. 2 s. 11 d. The column of shillings is 44 s.=2 £. and 4 s. and the column of pounds 69 £. Carrying and setting down agreeably to rule, we obtain the annexed amount.

2 17 4 2

The farthings are 6-1 d. 2qr. The pence are 28=2 s. 4 d. The shillings are 57-2 £. 17 s. which is written agreeably to the above note.

13. Bought a horse for 26 £. 12 s.; a yoke of oxen for 31 £. 17 s. 8d.; a cow for 7 £. 16 s. 9 d.; and paid 15 s. 8 d. for a bridle. How much did they all cost me? Ans. 67 £. 2 s. 1 d.

14. Bought cloth for 29 £. 6 s. 10 d.; ribbon for 3 s. 4 d. 3 qr.; a pair of boots for £1. 6 s. 3 d.; and paid 2 s. 8 d. 2 qs. for mending a pair of shoes. What was my bill for the whole? Ans. 30 £. 19 s. 2 d. 1 qr.

11 s. 6 d.

15. Bought at one time goods to the amount of 175 £. 16 s. 11 d. ; at another, to the amount of 35 £. 19 s. 8 d.; paid for carting 7 £. 8 s. 9 d. 3 qr.; and for insurance 3 £. 9s. 7 d. What was the amount of my bills? Ans. 222 £. 14 s. 11 d. 3 qr. 16. Sold at one time goods to the amount of 35 £. 3 qr.; at another, to the amount of 56 £. 19 s. 7 d. 1 qr.; at a third time, to the amount of 75 £. 1 s. 3 d.; and at a fourth, to the amount of 63 £. 13 s. 4d. 2qr. What was the whole amount of my sales? Ans. 231 £. 5 s. 9 d. 2 qr.

17. Bought a quantity of corn for 113 £. 11 s. 11 d.; of rye, for 32 £. 19 s. 3 d.; of wheat, for 136 £. 16 s. 8 d.; and of oats, for 22 £. 14 s. 9 d. What was the whole amount? Ans. 306 £. 2 s. 7 d.

18. A man sold his farm for 856 £.; his sheep, for 67 £. 17 s.; his swine, for 19£. 19 s. 11 d.; and his grain, for 36 £. 13 s. 2 d. How much money did he receive? Ans. 980 £.

10 s.
1 d.

14

1.

lb.

oz. pwt. gr.

15 6 13

17

42 11

11 3 11 19 4 3 21 17 15

87 2

6

21

TROY WEIGHT.

The column of grains amounts to 69 gr. 2 pwt. 21 gr. The pwt. amount to 46 2 oz. and 6 pwt. The ounces amount to 262 lb. and 2 oz. The pounds amount to 87; the whole of which is to be written down.

The scholar will observe, that as we have left the table of English money, we have no longer to carry by 20, 12, and 4. Our carrying numbers now are 12, 20, and 24.

7. Purchased at one time, 4 lb. 3 oz. 16 pwt. 15 gr. of silver, and at another, 7 lb. 8 oz. 18.pwt. 23 gr.; besides a quantity of jewelry, weighing 5 lbs. 11 oz. and 13 pwt. What was the whole weight? Ans. 18 lb. 8 pwt. 14 gr.

1

8. Add 3 lb. 9 oz. 13 pwt. 19 gr. ; 2 lb. 10 oz. 9 pwt. 17 gr.; 6 lb. 11 oz. 18 pwt. 22 gr. ; 9 oz. 11 pwt. 12 gr.; and 8 lb. 3 oz. 6 pwt. 20 grs. Ans. 22 lbs. 9 oz. 18 gr.

9. Again, add 6 lb, 2 oz. 16 pwt. 14 gr.; 3 lb. 8 pwt. 2 gr.; 12 lb. 4 oz. 15 pwt. 22 gr.; 8 oz. 16 gr.; 5 lb. 13 gr.; and 23 gr. Ans. 27 lb. 4 oz. 2 pwt. 18 gr.