real value; and it involves no other principles than those called out in the mental exercises under the tables.

From these exercises we perceive that quantities may be reduced from a higher to a lower denomination, by multiplication.

Thus, 2 yards are 6 feet: these, reduced to inches, give 72. Conversely, lower denominations may be brought to higher, by division, as pounds to quarters, hundreds weight, or tons. In all these cases, the quantity remains unchanged, although it is expressed in a different denomination.

When higher denominations are to be reduced to lower, the explanation gives the following,

RULE. Multiply the successive denominations, commencing with the highest given, by the number that will make it one of the next lower, adding, in their proper places, the several inferior denominations expressed in the given number.

When lower denominations are to be reduced to higher, the explanation gives the following

RULE. Divide the given denomination by the number that will make one of the next higher; and the quotient thence arising by as many as make one of the denomination next above that, and so on to the required one.

The former of these rules is reduction descending, the latter ascending. Being reverse operations, they prove each other.

As the principles of the operation are the same under each table of weight, measure, or motion, we shall give the examples promiscuously, to exercise quickness of thought.

EXAMPLES

1. Reduce 17 lbs. 11 oz. 15 dwts., to pennyweights.*

* A practical man would at once perceive that this is 5 pennyweights less

than 18 pounds: therefore, 18×12X20-5 the result.

2. In 25 lbs. 9 oz. O dwts. 10 gr., how many grains?

3. Reduce 74616 farthings to pounds.

4)74616

12) 18654 pence.

20)1554 shillings and 6 pence.

77 and 14 shillings over.

Ans. 148330.

Ans. £77, 14s. 6d. 74616 farthings.

6. Reduce 3060288 cubic inches to tons of round timber.

7. Reduce 43 gallons 3 quarts 1 pint, to pints.

43

4

175

2

Ans. 351 pints.

(ART. 27.) 8. Reduce 35 tons to drams.

When there are no inferior denominations, as quarters, pounds, &c., to add in as the operation advances, we may arrange all our multipliers before the eye, and take the product of several of them mentally, to shorten the operation.

In the present example we multiply 35 tons

9. Reduce 23 bushels 3 pecks 5 quarts 1 pint, to pints.

N. B. This is 5 pints less than 24 bushels; therefore, if we reduce 24 bushels to pints, deducting 5, we have the answer.

10. Reduce 11 acres 3 roods 15 rods, to rods.

11. Reduce 24 square rods to square feet.

128
256

1531 Ans.

Ans. 1895.

Ans. 6534.

12. Reduce 10 pipes 1 hogshead and 2 quarts of wine, to pints.

13. Reduce 3 hogsheads to gills.

Ans. 10588.

Ans. 6048.

14. In 1746880 ounces, how many tons.

Ans. 48 tons, and 14 cwt. 3 qrs. 8 lbs. over.

15. In 17645 grains, apothecaries' weight, how many pounds? Ans. 3 lbs., and 6 dr. O scru. 5 grains over.

16. In 168474 feet, how many miles?

Ans. 31 miles, and 7 fur. 58 yds. over. 17. In 2419200 seconds, how many days? Ans. 28. 18. In 547325 farthings, how many pence, shillings, and pounds? Ans. £579 2s. 6d. 1 qr.

19. In 9173 nails, how many yards? Ans. 573 yds. 1 qr. 1 n. 20. How many barleycorns will reach round the earth, supposing it, according to the best calculations, to be 24877 miles?

Ans. 4728620160. 21. How many seconds are in a solar year, or 365 days 5 hours 48 minutes 58 seconds? Ans. 31556938. 22. In a lunar month, or 29 days 12 hours 44 minutes 3 seconds, how many seconds? Ans. 2551443. 23. In an ingot of silver weighing 6 pounds 3 ounces 10 grains, how many grains?

Ans. 36010.

24. In 5 hundreds 2 quarters 16 pounds, how many pounds?

25. In 4 miles 6 furlongs 22 rods, how many rods? 26. In 11456 geographic miles, how many degrees?

27. In 164736 inches, how many miles?

Ans. 632. Ans. 1542.

Ans. 190 deg. 56 min.

Ans. 2 miles 4 fur. 176 yds.

28. In 14764 nails of cloth, how many yards?

Ans. 922 yds. 3 qr.

29. In 2746 quarters, how many ells English?

30. Reduce 2 tuns to gills.

Ans. 549 ells 1 qr.

Ans. 16128.
Ans. 262.

31. Reduce 32 gallons 3 quarts to pints.
32. Reduce 2 hogsheads 27 gallons 3 quarts, to quarts.

Ans. 615.

33. Reduce 3 tuns 1 hogshead 15 gallons 1 quart, to pints.

Ans. 6674. Ans. 185 gal. 2 qts.

34. Reduce 1484 pints to gallons. 35. Reduce 167000 gills to barrels of 32 gallons each. Ans. 163 barrels 2 gals. 3 qts. 36. Reduce 6272640 square inches to acres. Ans. 1. 37. A grocer purchased 16 hundred weight 3 quarters 20 pounds of sugar; how many pounds are there in the whole? Ans. 1896. 38. A tobacconist bought 116 hundred weight 3 quarters of tobacco; how many pounds were in the whole? Ans. 13076.

39. A silversmith has 7 pounds 10 ounces of silver; how many grains are in the whole? Ans. 45120. 40. An apothecary has several sorts of drugs, weighing together 47 pounds 6 ounces 4 drams; how many scruples and grains are in the whole? Ans. 13692 scru., 273840 grs.

41. A merchant wishes to ship 300 bushels of flaxseed in casks containing 7 bushels 2 pecks each; what number of casks are required? Ans. 40.

42. A certain barn is 64 by 36 feet, and 20 feet high; another barn is 32 by 24 feet, and 16 feet high: how much larger is the former than the latter? (See solid measure.) Ans. 3 times larger.

43. How many times will a wheel 16 feet ence, turn round in running 42 miles?

inches in circumferAns. 13440 times.

44. There are two places on the equator, one in longitude 40 deg. east of the meridian of Greenwich, the other 30 deg. west; what is the number of geographical miles between them? Ans. 4200.

45. A ship, during 3 days of storm, had changed her longitude 273 geographical miles; how many degrees and minutes of a degree did she change? Ans. 4 deg. 33 min.

46. How many quires and reams of paper are in 500520 sheets? Ans. 20855 quires, 1042 reams 15 quires.

47. A printer calls for 4 reams 10 quires and 10 sheets of paper to print a book; how many sheets does he call for? Ans. 2170.

48. How many solid inches are in a solid 7 inches long, 7 inches wide, and 7 inches high?

Ans. 343.

49. How many solid inches are in a solid 2 feet 3 inches long, 6 inches wide, and 6 inches high? Ans. 27X36. 50. How many solid inches are in a solid 3 feet 2 inches long by 2 feet 2 inches wide and 1 foot 8 inches high?

Ans. 38×26×20=19760. 51. How many yards of carpeting, 1 yard wide, will be required to carpet a room 18 feet wide and 20 feet long? Ans. 40.

COMPOUND ADDITION.

(ART. 28.) COMPOUND ADDITION is simple addition and reduction combined; but we shall be more clearly comprehended through the medium of an example.

What is the sum of £19 15s. 8d., £27 18s. 6d., £43 9s. 4d., and £7 6s. 6d.

As unlike things cannot be added together, (Art. 5.) we must write pounds under pounds, shillings under shillings, and pence under pence, thus: