The terms folio, quarto, octavo, duodecimo, etc., indicate the number of leaves into which a sheet of paper is folded.

75 Words make 1 Folio or sheet of Common Law.

160. An Aliquot Part of a number is such a part as will exactly divide that number; thus, 3, 5, 7 are aliquot parts of 15.

An aliquot part may be a whole or mixed number, while a factor must be a whole numbe:.

REDUCTION.

167. Reduction is the process of changing the denomination of a number, without altering its value.

168. Reduction Descending is changing a number of one denomination to another denomination of less unit value, and is performed by multiplication; thus, $1 = 10 dimes 100 cents=1000 mills; 1 yd.=3 feet 36 in.

1. Reduce 6 gal. 2 qt. 1 pt. to pints.

OPERATION.

6 gal. 2 qt. 1 pt.

4

26 qt. 2

Ans. 53 pt.

ANALYSIS. Since in 1 gal. there are 4 qt., in 6 gal. there are 4 qt. × 6 =24 qt., and the 2 qt. in the given number added, makes 26 qt. in 6 gal. 2 qt. Since in 1 qt. there are 2 qt., in 26 qt. there are 2 pt. × 26=52 pt., and the 1 pt. in the given number added, make 53 pints in the

given compound number. As either factor may be used as a multiplicand (61), we may consider the numbers in the descending scale as multipliers.

RULE. I. Multiply the highest denomination of the given number by that number of the scale which will reduce it to the next lower denomination, and add to the product the given number, if any, of that lower denomination.

II. Proceed in the same manner with the result obtained in each lower denomination, until the reduction is brought to the denomination required.

EXAMPLES FOR PRACTICE.

2. In 8 lb. 10 oz. how many ounces?

Ans. 138 oz.

3. In £12 6s. 9d. how many pence?

Ans. 2961d.

4. In 4 yd. 1 ft. 10 in. how many inches?

5. In 3 mi. 226 rd. how many rods?

6. In 18s. 8d. 3 far. how many farthings?

Ans. 899 far. 7. Reduce 3 lb. 9 oz. 12 pwt. to pennyweights. 8. In 2 hhd. 15 gal. 2 qt. how many pints?

9. Reduce 4 da. 5 hr. to minutes. Ans. 6060 min. 10. Reduce 10 bu. 1 pk. 6 qt. to pints. Ans. 668 pt. 11. Reduce 14 A. 140 sq. rd. to square rods.

12. Reduce 4 cd. 3 cd. ft. 9 cu. ft. to cubic inches.

13. Reduce 4 yr. 7 mo. to hours.

Ans. 39600 hr.

14. Change 2 T. 11 cwt. to pounds.

15. Change 9 lb. 9 oz. 10 pwt. to grains.

Ans. 5100 lb.

16. Change 5 lb. 63 43 2 10 gr. to grains.

17. Change 3 mi. 240 rd. to feet.

18. In 40 chains how many links?

Ans. 19800 ft.

Ans. 4000 1.

Ans. 1113840 sq. in.

19. In 28 sq. rd. 12 sq. yd. 4 sq. ft. how many square inches? 20. In 16 A. 4 sq. ch. 8 P. 80 sq. 1. how many square links? Ans. 1645080 sq. 1. 21. In 12 tons of round timber how many cubic inches?

Ans. 2224 pt.

22. In 8 bbl. 26 gal. how many pints?

28. Change 2 reams 10 quires to sheets.

29. In 40 score how many single things? Ans. 800.

30. In 14 great gross how many dozens?

31. In 30° 20' 24" how many seconds?

32. In the 3 Autumn months how many hours?

33. In the three Summer months how many minutes? 34. In 75 cords how many cubic feet?

169. Reduction Ascending is changing a number of one denomination to another of greater unit value, and is performed by Division; thus, 1000 mills=100 cents=$1. 1. Reduce 53 pints to gallons.

OPERATION.

2) 53

4) 26 qt. +1 pt.

6 gal. +2 qt.

Ans. 6 gal. 2 qt. 1 pt.

ANALYSIS. Dividing the given number of pints by 2, because there are as many quarts as pints, we obtain 26 quarts plus a remainder of 1 pt. Next divide 26 quarts by 4, because there are as many gallons as quarts, and

we obtain 6 gallons and a remainder of 2 qt. This last quotient, with the several remainders annexed, forms the answer.

2. Reduce 4902 inches to rods.

OPERATION.

12) 4902

161) 408 ft.+6 in. 2

33) 816

24 rd.+24=12 ft. Ans. 24 rd. 12 ft. 6 in.

ANALYSIS. Divide successively by the numbers in the ascending scale in the same manner as in the preceding example. But in dividing the 408 fr. by 161, first reduce 408 ft. to halves by multiplying by 2, and we have 816 halves; and reducing 16 to halves, we have 33 halves. Then dividing

816 by 33 we obtain 24 rd. plus a remainder of 24 halves to 12 ft., which, with the preceding remainder annexed to the last quotient, gives the answer.

RULE. I. Divide the given number by that number of the scale which will reduce it to the next higher denomination.

II. Divide the quotient by the next higher number in the scale; and so proceed to the highest denomination required. The last quotient, with the several remainders annexed in a reversed order, will be the answer.