Sun 59′ 8′′ 20"; now supposing them both to start from the same point in the heavens, how far will the Moon have gained on the Sun in 24 hours? Ans. 12° 11' 26" 40". 23. A farmer raised 136bu. of wheat; if he sells 49bu. 2pk. 7qt. 1pt., how much has he remaining?

Ans. 86bu. Ipk. Oqt. 1pt. 24. If from a stick of round timber, containing 2T. 18ft. 1410in., there be taken 38ft. 1720in., how much will be left? Ans. 1T. 19ft. 1418in. 25. If from 1b. of ipecacuanha there be taken at one time 43 23 13gr., and at another, 33 13 29 14gr., how much will be left? Ans. 43 33 29 13gr. 26. A brewer has in one cellar 18bbl. 3gal. 2qt. of beer, and in another, 13bbl. 1p.; what is the whole quantity, and how much more is in one cellar than the other?

Ans. 31bbl. 3gal. 2qt. 1pt. 5bbl. 3gal. 1qt. 1pt.

66

27. If from $100.00 there be paid at one time $17.28,5, at another time $ 10.00,5, and at another $37.15, how much will remath? Ans. $35.56.

SECTION XII.

REDUCTION.

THE object of Reduction is to change the denomination of numbers without altering their value. It consists of two parts, Descending and Ascending. The former is performed by Multiplication, the latter by Division.

Reduction Descending teaches to bring numbers of a higher denomination to a lower; as, to bring pounds into shillings, or tons into hundred weights.

Reduction Ascending teaches to bring numbers of a lower denomination into a higher; as, to bring farthings into pence, or shillings into pounds.

REDUCTION DESCENDING.

EXAMPLE.

1. In 48£. 12s. 7d. 2 qr., how many farthings?

£.

48

20

972

12

11671

4

46686

S. d. qr. 12 7 2

In this example, we multiply the 48£. by 20, because it takes 20 shillings to make a pound; and to this product we add the 12s. in the question. Then we multiply by 12, because it takes 12 pence to make one shilling; and to the product we add the 7 pence in the question. We then multiply by 4, the number of farthings in a penny, and to the product we add the 2 farthings, and the work is done.

From the above example and illustration, we deduce the following

RULE.

Multiply the highest denomination given by the number required of the next lower denomination to make one of the denomination next above it, and add to the product thus obtained the corresponding denomination of the multiplicand. Proceed in this way, till the reduction is brought to the denomination required by the question.

NOTE 1. To multiply by, we divide the multiplicand by 2, and to multiply by, we divide by 4.

NOTE. 2. The answers to Reduction Descending will be found in the questions of Reduction Ascending.

2. In 127£. 15s. 8d. how many farthings?

3. In 28£. 19s. 11d. 3 qr. how many farthings?

4. In 378£. how many pence?

5. How many grains in 28lb. 11oz. 12dwt. 15gr. troy? 6. In 17lb. 12dwt. troy, how many pennyweights ?

7. If a silver tankard weigh 3lb. 11oz., how many grains will it be?

8. How many scruples in 23lb. apothecaries' weight?

9. If a load of hay weigh 3T. 16cwt. 2qr. 18lb., how many ounces will it be?

10. Required the number of drachms in a hogshead of sugar, weighing 2T. 17cwt. 3qr. 16lb. 15oz. 13dr.

11. In 57yd. how many nails?

12. In 83947E.E. 4qr. how many nails? 13. In 2263E.F. 2qr. how many quarters ?

14. How many feet in 79 miles?

15. How many inches in 396 furlongs ?

16. How many inches from Haverhill to Boston, the distance being 30 miles ?

17. How many barleycorns will it take to reach round the world?

18. In 403m. 7fur. 35rd. 2yd. Oft. Oin. 1bar. how many barleycorns?

19. In 413Le. 2m. 2fur. 38rd. lyd. Oft. 7in. how many inches?

20. In 144m. 1fur. 8rd. lyd. 1ft. how many feet?

21. How many inches in 105lyd. 2ft. 5in.?

22. In 3576fur. 12rd. 3yd. how many yards?

23. How many square feet in 25 acres?

24. How many square rods in 365 square miles?

25. The surface of the earth contains 196563942 square miles. What would it be in square inches?

26. Required the number of feet in 10A. 3R. 38p. 6yd. 5ft.

72in.

27. In 2R. Op. 24yd. 3ft. how many inches?

28. In 1A. 3R. 34p. 27yd. 4ft. 54in. how many inches?

29. In 17 cords of wood, how many inches ?

30. In 19 tons of round timber, how many inches?

31. How many cubic feet of wood in 128 cords?

32. In 4899hhd. 4gal. 3qt. how many quarts?

33. In 1224 tuns lp. 1hhd. 19gal. 1qt. Opt. 1gi. how many gills? 34. How many pints in 790p. Ohhd. 58gal. Oqt. 1pt. ?

35. In 460 butts 1hhd. 31gal. of beer, how many gallons? 36. In 36hhd. 26gal. 3qt. 1pt. how many pints?

37. In 16 tons of round timber, how many inches ?

38. How many seconds from the deluge, it being 2348 years B. C., to the year 1836 ?

39. How many days did the last war continue, it having commenced June 18, 1812, and ended Feb. 17, 1815?

40. How many pecks in 676 chaldrons?

41. In 657 cents, how many mills? 42. In 3165 dimes, how many mills? 43. In 63 dollars, how many cents? 44. In 27 eagles, how many mills?

From the preceding illustration and example, we deduce the following

RULE.

Divide the lowest denomination given by the number which it takes of that denomination to make one of the next higher; so proceed, until it is brought to the denomination required. Any remainders occurring in the successive divisions will be of the same denominations with the dividends to which they respectively belong.

NOTE 1. To divide by 5, we multiply the multiplicand by 2, and divide the product by 11; to divide by 161⁄2, we multiply by 2 and divide by 33; and to divide by 2721, we multiply by 4 and divide by 1089.

NOTE 2.-The answers to Reduction Ascending are the questions in Reduction Descending.

NOTE 3.- When we divide by 11 or 33, and there is a remainder, we divide that remainder by 2 to get the true remainder; and, when we divide by 1089, we must divide the remainder by 4.

2. In 122672 farthings, how many pounds?
3. In 27839 farthings, how many pounds?
4. In 90720 pence, how many pounds?
5. In 166863 grains, how many pounds troy?
6. How many pounds in 4092 pennyweights ?
7. How many pounds troy in 22560 grains?
8. In 6624 scruples, how many pounds?
9. In 137376 ounces, how many tons ?
10. In 1660157 drachms, how many tons?
11. How many yards in 912 nails?

12. Required the ells English in 1678956 nails.
13. Required the ells Flemish in 6791 quarters.
14. Required the miles in 417120 feet.

15. Required the furlongs in 3136320 inches.
16. Required the miles in 1900800 inches.

17. How many degrees in 4755801600 barleycorns?
18. How many miles in 76789567 barleycorns?
19. How many leagues in 78653167 inches?
20. Required the miles in 761116 feet.
21. Required the yards in 37865 inches.
22. Required the furlongs in 786789 yards.

23. How many acres in 1089000 square feet?

24. How many square miles in 37376000 square rods? 25. How many square miles in 789,103,900,894,003,200 square inches?

26. In 478675 square feet, how many acres?
27. In 3167856 square inches, how many roods?
28. How many acres in 12345678 square inches?
29. How many cords in 3760128 cubic inches?

30. How many tons of round timber in 1313280 cubic inches ?

31. How many cords in 16384 cubic feet?

32. How many hogsheads of wine in 1234567 quarts?
33. How many tuns of wine in 9877001 gills?
34. In 796785 pints of wine, how many pipes?
35. How many butts of beer in 49765 gallons?
36. In 15767 pints, how many hogsheads of beer?
37. In 1105920 inches, how many tons of round timber
38. In 132005440800 seconds, how many years?

39. In 974 days, how many years and calendar months?
40. How many chaldrons in 97344 pecks?
41. How many cents in 6570 mills?
42. How many dimes in 316500 mills?
43. How many dollars in 6300 cents?
44. How many eagles in 270000 mills?

COMPOUND REDUCTION.

1. In 57£. 15s. how many dollars? Ans. $192.50cts. 2. In 67£. 14s. 9d. how many crowns at 6s. 7d. each ?

Ans. 205cr. 5s. 2d.

3. How many pounds and shillings in 678 dollars ?

4. How many ells English in 761 yards?

Ans. 203£. 8s.

Ans. 608E. E. 4qr.
Ans. 45yd. 3qr.

5. How many yards in 61 ells Flemish ? 6. Ho many bottles, that contain 3 pints each, will it take to hold & hogshead of wine? Ans. 168.

7. How many steps, 2ft. 8in. each, will a man take in walking from Bradford to Newburyport, the distance being 15 miles? Ans. 29700.

8. How many spoons, each weighing 2oz. 12dwt., can be made from 5lb. 2oz. 8dwt. of silver?

Ans. 24. 9. How many times will the wheel of a coach revolve, whose circumference is 14ft. 9in. in passing from Boston to Washington, the distance being 436 miles ? Ans. 156073,32.

10. I have a field of corn, consisting of 123 rows, and each row contains 78 hills, and each hill has 4 ears of corn; now if it take 8 ears of corn to make a quart, how many bushels does the field contain? Ans. 149bu. 3pk. 5qt. Opt. 11. If it take 5yd. 2qr. 3na. to make a suit of clothes, how many suits can be made from 182 yards? Ans. 32.

12. A goldsmith wishes to make a number of rings, each