1 = 12 = 360 = 21600 = 1296000.

SCALE-ascending, 60, 60, 30, 12; descending, 12, 30, 60, 60.

NOTES.-1. Minutes of the earth's circumference are called geographic or nautical

miles.

2. The denomination, sign, is confined exclusively to Astronomy.

3. A degree has no fixed linear extent. When applied to any circle it is always 80 part of the circumference. But, strictly speaking, it is not any part of a circle.

4. 90° make a quadrant or rightangle; 60o make a sextant or of a circle.

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

A sheet folded in 2 leaves is called a folio.

DENOMINATE NUMBERS.

383. A Denominate Number is a number composed of one or more units of any denomination, and may be simple or compound.

384. A Simple Denominate Number is a number composed of but one denomination, as, 7 miles; 6 hours; 5 tons.

385. A Compound Denominate Number is a number composed of two or more denominations of the same nature; as, 10 lb. 6 oz.; 6 yr. 3 mo.

386. A Denominate Fraction is a concrete fraction whose integral unit is one of a denomination of some compound number. Thus, of a day is a denominate fraction, the integral unit being one day; so are of a bushel, of a mile, etc., denominate fractions.

387. Most of the tables in Denominate Numbers, except those in the Metric System, are based on varying scales.

NOTE. The only other tables that are not based on a varying scale are the table of counting 12 things make 1 dozen, 12 dozen 1 gross, 12 gross 1 great gross, etc., based on the duodecimal scale; the table of United States Currency, and some of the tables of Foreign Currency, based on the decimal scale.

388. Reduction is the process of changing a number from one denomination to another without altering its value. Reduction is of two kinds, Descending and Ascending.

389. Reduction Descending is changing a number of one denomination to a lower denomination or one of less unit value; thus, $1 = 10 dimes = 100 cents = 1000 mills.

390. Reduction Ascending is changing a number of one denomination to a higher denomination or one of greater unit value; thus, 1000 mills = 100 cents = 10 dimes $1.

REDUCTION DESCENDING.

Examples.

391. To reduce a compound denominate number to lower denominations.

1. Reduce 3 mi. 57 rd. 2 yd. 1 ft. 8 in. to inches.

OPERATION.

3 mi. 57 rd. 2 yd. 1 ft. 8 in.

320

1017 rd.

51 508 5085

55951 yd.

=

=

SOLUTION. Since in 1 mile there are 320 rd., in 3 miles there are 3 x 320 rd. = 960 rd., and the 57 rd. in the given number added, 1017 rd. in 3 mi. 57 rd. Since in 1 rd. there are 5 yd., in 1017 rd. there are 1017 × 51 yd. = 5593 yd., which with the 2 yd. in the given number 5595 yd. in 3 mi. 57 rd. 2 yd. Since in 1 yd. there are 3 ft., in 5595 yd. there are 55951 × 3 ft. = 16786} ft., which with the 1 ft. in the given number = 167871⁄2 ft. in 3 mi. 57 rd. 2 yd. 1 ft. And since in 1 ft. there are 12 in., in 167871 ft. there are 16787 × 12 in. = 201450 in., which with the 8 in. in the given number=201458 in. in the given compound number. On examining the operation, 201458 in. Ans. We find that we have successively multiplied by the numbers in the descending scale of linear measure from miles to inches, inclusive. As either factor may be used as a multiplicand, we may consider the numbers in the descending scale as multipliers.

3 16787 ft. 12

RULE.-I. Multiply the highest denomination of the given compound 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 results obtained in each lower denomination, until the reduction is brought to the denomination required.

392. All the numbers in the metric system are based on a uniform scale of 10, 100, or 1000. Hence we have the following principle for the reduction of metric denominations.

PRINCIPLE. A denominate number in the metric system may be reduced to the next higher or lower denomination by the removal of the decimal point one place to the left or right in

measures of length, capacity, and weight; two places to the left or right in measures of surface, and three places to the left or right in measures of volume, thus:

NOTE. When numbers in the metric system are reduced to like denominations, they may be added, subtracted, multiplied, or divided like any other whole numbers or decimals.

2. How many grains are there in 16 lb. 10 oz. 18 pwt.

5 gr.?

3. How many farthings are there in £133 6s. 8d.?

4. Change 100 mi. to inches.

5. How many rods of fence will inclose a farm 11⁄2 miles square?

6. The gray limestone of Central New York weighs 175 lb. to the cubic foot. What is the weight of a block 8 ft. long and 1 yd. square?

7. What will be the cost of 1 hogshead of molasses at $.28 per gallon.?

8. Express 3.565% in centigrams.

9. A man wishes to ship 1548 bu. 1 pk. of potatoes in barrels containing 2 bu. 3 pk. each. How many barrels must he obtain?

10. A grocer bought 10 bushels of chestnuts at $3.75 a bushel, and retailed them at $.06 a pint. What was his whole gain?

11. Reduce 90° 17' 40" to seconds.

12. How many days were there in the 18th century?

13. At 64 cts. each, what will be the cost of a great gross of writing books?

14. Reduce 6 O. 14 f3 3 f3 45 m to minims.

15. How large an edition of an octavo book can be printed from 4 bales 4 bundles 1 ream 10 quires of paper, allowing 8 sheets to the volume?

16. Suppose your age to be 18 yr. 24 da.

How many

minutes old are you, allowing 4 leap years to have occurred in that time?

17. How many pence are there in 481 sovereigns?

18. Reduce $73 to mills.

19. In £6 10s. 10d. how many dollars United States currency are there?

20. Which is the cheaper and how much, to buy ribbon at $.35 a yard or at $.40 a meter?

21. How many dollars Canada currency are equal to £126 12s.?

22. How many square centimeters are there in a lot which is 50 ft. long and 100 ft. wide?

23. How many steps of 2 ft. 9 in. each will a man take in walking a distance of 95 miles?

24. How many decimeters are there in 5 yards?

ΚΙ

25. Express .0037081 K in hectoliters; in centiliters.

26. A grocer bought 12 barrels of cider at $1 a barrel, and after converting it into vinegar, he retailed it at 6 cents a quart. What was his whole gain?

27. In 75 A. 4 sq. ch. 18 sq. rd. 118 sq. 1. how many square links are there?

28. How many inches high is a horse that measures 16 hands?

29. If a vessel sails 150 leagues in a day, how many statute miles does she sail?

30. If 14 A. are sold from a field containing 50 A., how many square rods will the remainder contain?

31. A man returning from Pike's Peak has 36 lb. 8 oz. of pure gold; what is its value at $1.04 per pwt. ?

32. I paid $360 for 2 tons of cheese, and retailed it for 30 cents a kilogram. What was my whole gain?