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meter (cm.3); 1000 cubic centimeters=1 cubic decimeter (dm. 3); 1000 cubic decimeters=1 cubic meter (Mo.).

NOTE.—The higher denominations are not generally used. I indicate the cubic measures with an exponent, instead of writing cu. before the denominations.

MEASURES OF CAPACITY. 422. The Liter is the unit of capacity. It equals a cubic decimeter; that is, a cubic vessel whose edge is onetenth of a meter. It is used for measuring liquids and dry substances. The liter is a cylinder, and holds 2.1135 pints wine measure, or 1.816 pints dry measure.

TABLE.—10 milliliters (ml.)=1 centiliter (cl.); 10 centiliters=1 deciliter (di.); 10 deciliters=1 liter (L.); 10 liters =l decaliter (DL.); 10 decaliters=1 hectoliter (HL.); 10 hectoliters=1 kiloliter (KL.); 10 kiloliters=l myrialiter (ML.).

Notes.-1. The liter is principally used in measuring liquids, and the hectoliter in measuring grains, etc.

2. The liter equals nearly 118 liquid quarts, or % of a dry quart, or nearly ste of a bushel measure.

3. The hectoliter is about 23 bushels or < of a barrel. 4 liters are a little more than a gallon ; 35 liters, very nearly a bushel.

MEASURES OF WEIGHT. 423. The Gram is the unit of weight. It is the weight of a cubic centimeter of distilled water at the temperature of melting ice. The gram equals 15.432 Troy grains.

TABLE.–10 milligrams (mg.)=1 centigram (eg.); 10 centigrams=1 decigram (dg.); 10 decigrams=1 gram (G.); 10 grams=1 decagram (DG.); 10 decagrams=1 hectogram (HG.); 10 hectograms=1kilogram (KG., or K.); 10 kilograms=1 myriagram (MG.).

Notes.-1. The gram is used in weighing letters, in mixing and compounding medicines, and in weighing all very light articles. The five-cent coin adopted 1866 weighs 5 grams.

2. The kilogram is the ordinary unit of weight, and is generally abbreviated into kilo. It equals about 21 pounds avoirdupois. Meats, sugar, etc., are bought and sold by the kilogram.

3. In weighing heavy articles, two other weights, the quintal (100 kilograms) and the tonneau (1000 kilograms) are used. The tonneau is between our short ton and long ton.

4. The avoirdupois ounce is about 28 grams; the pound is a little less than } a kilo. The U. S. post offices receive 15 grams, though a little overweight, as equivalent to an ounce avoirdupois.

5. Some of the old weights and measures are still used in France; 1 livre=} a kilogram ; 1 marc=} a livre; 1 once=} a marc; 1 gros =

an once ; 1 grain = 12 gros; 1 toise=2 métres ; 1 pied or foot=1 métre; 1 inch iz pied or foot; 1 aune=1} métres; 1 boisseau or bushel = 121 litres ; 1 litron 1.074 Paris pints. When these are employed, the word usuel is annexed to them, signifying customary.

424. Units of the common system may readily be changed to those of the Metric System by the following

TABLE.

1 Inch=2.54 Centimeters.

1 Cu. Inch=16.39 Cu. Centim. 1 Foot= 30.48 Centimeters. 1 Cu. Foot=28320 Cu, Centim. 1 Yard=.9144 Meter.

1 Cu. Yard=.7646 Cu. Meters. 1 Rod=5.029 Meters.

1 Cord 3.625 Steres. 1 Mile=1.6093 Kilometers. 1 Fl. Ouncer 2.958 Centiliters. 1 Sq. Inch=6.4528 Sq. Centimeters. 1 Gallon=3.786 Liters. 1 Sq. Foot=929 Sq. Centimeters. 1 Bushel=.3524 Hectoliters. 1 Sq. Yard=.8361 Sq. Meters. 1 Troy Gr.= 64.8 Milligrams. 1 Sq. Rod=25.29 Centiares. 1 Troy lb.=.373 Kilo. 1 Acre=40.47 Ares.

1 Av. lb.=.4536 Kilo. 1 Sq. Mile=259 Hectares.

1 Ton=.907 Tonneau.

NUMERATION AND NOTATION. 425. In the Metric System the decimal point is placed between the unit and its divisions, the whole quantity being regarded as an integer and a decimal. Thus, 3 decagrams, 5 grams, 6 decigrams, 8 centigrams, are written 35.368 grams.

NOTE.-The initials of the denomination may be placed either before or after the quantity, though they are most frequently placed after it; thus, 27 grams may be written G27, or 27G.

EXERCISES IN NUMERATION AND NOTATION. 1. Read 48.64 M., 85.87 A., 48.89 M2. 2. Read 854 17 S., 506.347 L., 4007.563 G. 3. Write 12 meters, 3 decimeters, 5 centimeters. 4. Write 8 hectares, 10 ares, 17 centiares. 5. 9 kilograms, 5 hectograms, 4 grams and 1 centigram.

REDUCTION OF THE METRIC SYSTEM TO THE

COMMON SYSTEM.
1. How many pounds Av. in 488.125 grams?

Ans. 1 lb. 1 oz. 95.245 gr. 2. Grams in 24 pounds Troy? Ans. 8958.009 G. 3. Meters in 4 mi. 240 rd ?

Ans. 7644.399 M.

4. Miles in 2000 meters? Ans. 1 mi. 77 rd. 11 ft. 2 in. 5. Acres in 1011.2 ares ? Ans. 24 A. 156.2624 P. 6. Ares in 11 A. 48 P.?

Ans. 457.489 A. 7. Cu. ft. in 429.56 steres ? Ans. 15170.5987 cu. ft. 8. Steres in 32 cu. yd. 16 cu. ft.? Ans. 24.918 S. 9. Gallons in 90.1 liters ?

Ans. 23 gal. 3 qt. 10. Liters in 73 gallons ?

Ans. 276.319 L. 11. Bushels in 130.5 liters ? Ans. 3 bu. 2 pk. 6.49 qt.

PRACTICAL PROBLEMS. 1. What cost 48.625 meters of cloth, if 9.725 meters cost $36.75 ?

Ans. $183.75. 2. What must I pay for 75.25 steres of wood at the rate of $2.65 a stere ?

Ans. $199.41. 3. Bought 15.25 liters of wine in Bordeaux, at 75.5 francs a liter ; what is the cost in U. S. money? Ans. $222.22.

4. How much must be paid for 12.5 grams of jewels, at $6.50 a gram?

Ans. $81.25. 5. What is the cost of 672.25 grams of opium at 621¢ a gram?

Ans. $420.16. 6. Mr. Brown imported for his house 35.429 meters of French carpet, at 19.75 francs a meter, including duty; required the whole cost.

Ans. 699.72 +fr. 7. Mr. Winslow bought a valuable gem in Paris which weighed 245.25 grams, @ 10.25 francs, duty $4.75; how must be sell it a gram to clear $100 ?

Ans. $2.41. 8. An importer bought 428.5 grams of drugs in France, at 12.5 francs a gram, paid 31į cents a gram duty and freight, and sold them for $2.25 a gram ; how much was gained or lost?

Ans. Lost $204.61. 9. I bought 175.25 liters of French brandy at 7.50 francs a liter, paid 15 cents a liter duty and freight, and sold it in New York at $1.65 a liter; how much did I gain?

Ans. $9.20. 10. Jordan, Marsh, & Co. bought 200 meters of silk in Lyons, at 16.25 francs a meter; after paying $2 a yard duty and freight, they sold it in Boston at $6.12 a yard; what was their profit?

Ans. $274.98.

REDUCTION OF COMPOUND NUMBERS. 426. Reduction is the process of changing a number from one denomination to another, without altering its value.

427. There are Two Cases: Reduction Descending and Reduction Ascending.

These two cases have been considered in the examples under the tables, but we will present a few more problems under their proper heads.

REDUCTION DESCENDING. 428. Reduction Descending is the process of reducing a number to a lower denomination.

1. Reduce £8 6 s. 4 d. to pence.

OPERATION.
£ S. d.
8 6 4
20

SOLUTION.-In 1 pound there are 20 shillings, and in £8 there are 8 times 20 shillings, plus 6 shillings are 166 shillings : in 1 shilling there are 12 pence, and in 166 shillings there are 166 times 12

166 s. pence, plus 4 pence equals 1996 pence. Therefore,

12 etc.

1996 d., Ans. Rule.—I. Multiply the number of the highest denomination given, by the number of units of the next lower denomination which equals one of this higher, and to the product add the number given, if any, of this lower denomination.

II. Multiply this result as before, and proceed in the same manner until we arrive at the required denomination. 2. Reduce 8 lb. 4 oz. 6 pwt. 12 gr. to gr.

Ans. 48156. 3. Reduce 9 lb. 11 3 3 3 2 2 4gr. to gr.

Ans. 57344. 4. Reduce 124 A. 140 P. to sq. yd. Ans. 604395. 5. Reduce 120 cd. 6 cd. ft. to cubic feet. Ans. 15456. 6. Reduce 52 hhd. 24 gal. 3 qt. to pints. Ans. 26406. 7. 6 Circ. 10 S. 16° 20' 20" to seconds. Ans. 8914820. 8. Cong. vij. 0.iv. fz vj. fz iij. to minims. Ans. 463860.

9. A farmer sold 16 A. 132 P. of land at $1.25 a square rod; how much did he receive ?

Ans. $3365. 10. A man bought 6 bu. 3 pk. 5 qt. of berries for $10.25, and sold them at 10 cents a quart; how much did he gain?

Ans. $11.85.

OPERATION.

REDUCTION ASCENDING. 429. Reduction Ascending is the process of reducing a number to a higher denomination.

1. In 246374 grains, how many pounds ?

SOLUTION.—There are 24 gr. in 1 pwt., hence in 246374 gr. there are

gr. as many pwt. as 24 is contained

24)246374 times in 246374, which is 10265 pwt. and 14 gr. remaining: there

20)10265+14 gr. are 20 pwt. in 1 oz., hence in 10265

12)513+5 pwt. pwt. there are as many ounces as 20

42 lb.+9 oz. is contained times in 10265, which Ans. 42 lb. 9 oz. 5 pwt. 14 gr. are 513 oz., and 5 pwt. remaining : there are 12 oz. in 1 pound, and in 513 oz. there are as many pounds as 12 is contained times in 513, which are 42 lb. and 9 oz. remaining. Therefore in 246374 grains there are 42 lb. 9 oz. 5 pwt. 14 gr.

Rule.-1. Divide the given number by the number of units in that denomination which equals one of the next higher.

II. Divide the quotient in the same way, and thus proceed until we arrive at the required denomination.

III. The last quotient and the remainders, if any, will be the result required. 2. 346256 gr. to lb.

Ans. 60 lb. 13 2 3 2 16 gr. 3. 4763254 li. to miles. Ans. 595 mi. 32 ch. 54 li. 4. 764325 cu. in. to cubic yards.

Ans. 16 cu. yd. 10 cu. ft. 549 cu. in. 5.74625 m. to Cong.

Ans. 1 Cong. 10. 11 fz 3 f3 45 in. 6. 25627542 sq. li. to acres.

Ans. 256 A. 2 sq. ch. 7542 sq. li. 7. The side of a square field is 360 ft. long; how many rods of fence will enclose it? Ans. 87 rd. 1 yd. 1 ft. 6 in.

8. A dealer sold 1 ton of fish at $4.00 a quintal; what did it amount to?

Ans. $80.00. 9. A miller sold 2560 lb. of flour at the rate of $9.00 a barrel ; what did it amount to?

Ans. $117.55. 10. Bought 7420 square rods of land at $172 an acre, and sold it for $7000; how much did I lose ? Ans. $976.50.

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