1. A Quadrant is of a circumference, or 90°; a Sextant

is of a circumference, or 60°.

2. The length of a degree of longitude on the earth's surface at the Equator is 69.16 miles.

3. In astronomical calculations 30° are called a Sign, and there are therefore 12 signs in a circle.

EXERCISES.

1. How many minutes are there in 5°? 6°?

2. In 35 degrees how many seconds are there? In 27°? In 21° 12′ 18"?

3. How many seconds are there in 34° 12′ 43′′?

4. In 468560 seconds how many minutes, etc., are there? 5. In 384500 seconds how many minutes, etc., are there? 6. How many seconds are there in Cir.? In ? In †? 7. How many minutes are there in 2 quadrants? In 2 sextants?

COUNTING.

281. The following denominations are used in counting some classes of articles:

Two things are often called a pair, six things a set, and twenty things a score; as a pair of birds, a set of spoons, a score of years.

STATIONERS' TABLE.

282. The denominations used in the paper trade are:

The terms folio, quarto, octavo, applied to books, indicate the number of leaves into which a sheet of paper is folded. Thus, when a sheet of paper is folded into 2, 4, 8, 12, 16, 18, or 24 leaves, the forms are called respectively, folio, 4to, or quarto, 8vo, or octavo, 12mo, 16mo, 18mo, and 24mo.

EXERCISES.

1. How many eggs are there in 5 dozen? 7 doz.? 10 doz.? 2. How many crayons are there in 2 gross? 3 gr. 5 doz.? 3. How many things are there in a great gross?

4. What will be the cost of 3 dozen brushes at $.45 each? 5. A man lived 3 score and 10 years. What was his age? 6. What will 3 reams of paper sell for at $.15 per quire?

REDUCTION OF DENOMINATE FRACTIONS. 283. The principles, processes and analyses are essentially the same as those of denominate integers.

CASE I.

284. To reduce denominate fractions to equivalent numbers of lower denominations.

EXERCISES.

1. How many hours are there in 1 day? In day? In of a day? In 3 of a day? In § of a day?

2. How many ounces are there in

In lb.? In lb.?

pound avoirdupois ?

3. How many pints are there in of a peck? In pk.? 4. How many pecks and quarts are there in of a bushel? 5. How many pounds and ounces are there in § cwt.? 6. How many inches are there in of a foot? ft.? ft.?

7. Change of a rd. to units of lower denominations.

16. Express

12.

of a peck.

13.

of a day.

14. g of a sq.

rd.

15.

of a cu. yd.

of a gallon as a fraction of a gill.

ANALYSIS. Since in 1 gallon there are 32 gills, in of a gallon there are of 32 gi., orgi. Hence gal. of a gill.

32 145

17. Express of a bushel as a fraction of a pint.

18. Express 21izʊ of a mile as a fraction of a foot.

3 21120

19. Express 94 of a pound as a fraction of a scruple.

1944

20. Express .006 of a bushel as a decimal of a pint. 21. Express in lower denominations .685 of a pound Troy.

PROCESS.

.685
12

8.220 oz.
20

4.400 pwt.

24

9.600 gr.

ANALYSIS. Since in 1 pound there are 12 ounces, in .685 of a pound there are .685 of 12 ounces, or 8.220 ounces.

Since there are 20 pennyweights in 1 ounce, in .220 of an ounce there are .220 of 20 pennyweights, or 4.400 pennyweights.

Since in 1 pennyweight there are 24 grains, in .400 of a pennyweight there are .400 of 24 grains, or 9.600 grains.

Therefore .685 lb. is equal to 8 oz. 4 pwt. 9.6 gr.

285. To change denominate fractions to equivalent fractions of higher denominations.

EXERCISES.

1. What part of a pound Troy is 1 ounce? Is oz.? Is oz.?

2. What part of a ton is 1 pound? Is lb.? Is lb.? 3. What part of a mile is 1 rod? Is rd.? Is rd.? 4. What part of a league is 1 mile? Is 1 rd.? Is rd.? 5. What part of an hour is of a minute? Is of a min.? 6. What part of a week is 1⁄2 of a day? of a day? 7. What part of a bushel is of a pint?

8. Reduce of an inch to the fraction of a yard. 9. Change of a second to the fraction of an hour. 10. Express .375 of a week as a fraction of a year. 11. Express .35 of a pound as a fraction of a ton. 12. Express of a cubic inch as a fraction of a cubic foot.

13. Change of a square yard to a fraction of an acre. 14. Reduce of a pint to a fraction of a barrel.

CASE III.

286. To express one denominate number as a fraction of another.

1. What part of a foot are 3 in. ? 6 in.? 9 in.?

2. What part of an hour are 30 min.? 15 min.? 45 min.?

3. What part of a gallon is 1 pint? 2 pints? 1 quart? 4. What part of a gallon are 2 quarts? 2 qt. 1 pt.? 3 qt. 1 pt.?

5. What part of 3 ft. 6 in. are 2 ft. 3 in.?

ANALYSIS-Since 3 ft. 6 in. =

=2 of 42 in.

= = 42 in., and 2 ft. 3 in. 27 in., 27 in.

6. What part of 3 yd. 2 ft. are 2 yd. 2 ft.?

7. What part of 5 gal. 3 qt. 1 pt. are 2 gal. 1 qt. 1 pt.? 8. What part of 2 pounds Troy are 3 oz. 10 pwt.?

9. What part of 3 pecks are 2 qt. 1 pt.?

10. What part of 3 barrels are 13 gal. 3 qt. 2 pt. 2 gi.? 11. Express 15s. 7d. in the decimal of a pound sterling: