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NOTE.—The carat of weight must be carefully distinguished from the assay carat. The former is an absolute weight; the latter is used to denote the proportion of pure gold in a mass, and is a twenty-fourth part of the mass. Thus gold 18 carats fine has 18 parts goid and 6 parts alloy.

EXAMPLES FOR PRACTICE. 1, Change 7 lb. 3 oz. 6 pwt. 3 gr. to Apothecaries' Weight.

Ans. 7 lb. 33. 23. 13. 7 gr. 2. Change 14 lb. 6 oz. 12 pwt. 6 gr. to Avoirdupois Weight.

Ans. 11 lb. 15.5749 oz. 3. Change 4 cwt. 72 lb, 8 oz. to Troy Weight

Ans. 574 lb. 2 oz. 12 pwt. 12 gr. 4. Change 6 cwt. 20 lb. 12 oz. U. S. to English weight.

Ans. 5 cwt. 2 qr. 4 lb. 12 oz. 5. Which is heavier, and how much, a pound of gold or a pound of iron ?

Ans. The latter, 1240 gr. 6. Which is heavier, and how much, an ounce of silver or an ounce of lead ?

Ans. The former, 424 gr. 7. If Commodore Nutt weighs 23 lb. Avoirdupois, how much would he weigh by Troy Weight? Ans. 27137 lb.

8. I shipped 125 tons of iron from England; how much did it weigh by United States weight? Ans. 140 tons.

9. What is the weight of 24 gold eagles and 72 silver dollars ?

Ans. 6 lb. 2 oz. 15 pwt. 12 gr. 10. What is the weight of 10 sovereigns, 5 shillings, and 8 pence?

Ans. 7 oz. 9 pwt. 13.09 gr. 11. What is the weight of $437.985 in English sovereigns; also in shillings?.

Ans.
ime si lb. 11 oz. 2 pwt. 6.66 gr. in sovereigns.

*727 lb. 3 oz. 5 pwt. 6 gr. in shillings. 12. What is the weight of the gold and also of the alloy in 720 sovereigns ?

(Gold, 14 lb. 1 oz. 10 pwt. .84 gr. Ans. Ans. 3

Alloy, 1 lb. 3 oz. 8 pwt. 4.44 gr. 13. If I owe $1,000,000, and pay it in gold, what will be its weight Av. ?

Ans. 36855 lb. 14. An apothecary bought 14 lb. 12 oz. of opium by Avoirdupois weight, at 62} cts. an ounce, and retailed it at 5 cts. a scruple; how much did he gain? Ans $110.621.

MEASURES OF LENGTH. 370. Measures of Length are used in measuring length, breadth, height, distance, etc.

371. A Line is that which has length without breadth or thickness. It is estimated by ascertaining how many times it contains a unit of measure. 372. An Angle is the opening between

D two lines which diverge from a common point. Thus ACD and DCB are angles.

B 373. The Vertex of an angle is the point from which the two lines diverge; thus, C is the vertex of the angle BCD.

374. A Right Angle is formed by one line perpendicular to another; as, ABC or CBD. One line is perpendicular to another when it makes the two adjacent angles. A equal.

LONG MEASURE. 375. Long Measure is used for the general purposes of measuring length and distances.

TABLE.

12 inches (in.) . .= 1 foot 3 feet . . . .= 1 yard . . yd. 54 yards, or 161 feet .= l rod . . rd. 320 rods . . .=l mile . . mi. 3 miles . . . = 1 league .. lea. 69.16 miles . . . = 1 degree of latitude, deg. or o,

or of longitude at the equator. mi. rd. yd.

ft.

in.
1 = 320 = 1760 = 5280 = 63360

1 = 52 = 161 = 198 SCALE.- Ascending, 12, 3, 53, 320, 69.16; descending, 69.16, 320, 5), 3, 12.

I. TERMS.—The units of length are nearly all derived from the different parts of the human body and from other objects. The ancient yard of England was the length of the arm of King Henry I. The term inch is from uncia, a twelfth; foot is from the human foot; yard was a rod or

shoot; rod is from a measuring stick or rod; furlong, now obsolete, is from fur, furrow, and lang, long, the length of a furrow; mile is from mille passuum, 1000 paces ; span is the space measured from the end of the thumb to the end of the little finger extended ; cubit, from the elbow to the end of the middle finger; fathom, the length of the two arms extended.

II. UNIT.-The standard unit of length is the yard, from which all other measures of length, and also those of capacity, weight, etc., are derived. It is identical with the Imperial yard of Great Britain, which, under William IV., was declared to be fixed by dividing a pendulum, which vibrates seconds in a vacuum, at the level of the sea, at 62° Fah., in the latitude of London, into 391393 equal parts, and taking 360000 of these parts for the yard. Subsequent scientific experiments have proved that such a standard is impracticable.- See Brooks's Philosophy of Arithmetic.

III. THE MILE.— The geographic or nautical mile is equal to 1 minute of one of the great circles of the earth; hence it equals ob of sto of the circumference of the earth, which equals about 1.15 statute miles. The English mile is the same as that of the United States. The German short mile equals 6857 yd., or about 3% statute miles; the German long mile equals 10125 yd., or about 5 statute miles; the Prussian mi 8237 yards, or about 47 statute miles. 3 statute miles make a land league ; 3 nautical miles a nautical league.

IV. DEGREES.-A degree of longitude at any point is to of the circle passing through the latitude of that point, and as these circles diminish as we pass from the equator, the degrees of longitude will diminish. Thus, at the equator, the length of a degree of longitude is about 691 statute miles ; at 25° of latitude, 62,7 miles ; at 40° of latitude, 53 miles ; at 42°, 511 miles ; at 49°, 454 miles; at 60°, 317 miles, etc. A degree of latitude also varies, being 68.72 miles at the equator; from 68.9 to 69.25 miles in middle latitude; and from 69.30 to 69.34 miles in the polar regions.

V. OTHER MEASURES.— The following denominations are frequently used : in clock-making, 6 points=1 line, and 12 lines=1 inch ; in measuring the foot, 3 barleycorns or sizes=1 inch ; in measuring the height of horses, 4 inches =1 hand, the measure being taken directly over the foreshoulder; 1 span=9 inches; 1 common cubit = 18 inches, and 1 sacred cubit = 21.888 inches; 1 pace=3.3 feet; a knot is equal to a nautical mile. Formerly we had 40 rods equal 1 furlong and 3 furlongs one mile, but these are now seldom used.

SURVEYORS' LINEAR MEASURE. 376. Surveyors' Linear Measure is used by surveyors and engineers in measuring the dimensions of land, distances, etc.

TABLE. 7.92 inches (in.) . . = 1 link . . li. 100 links . . . = 1 chain . . ch. 80 chains . . . = l mile . . mi. mi.

li. 1 = 80 = 8000

63360 1 = 100

792 SCALE.--Ascending, 7.92, 100, 80; descending, 80, 100, 7.92. I. NAME.- Gunter's chain is named after the reputed inventor, Edmund Gunter, an English mathematician, born 1581.

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172

II. UNIT.-The unit is a chain called Gunter's Chain, which consists of 100 links, and is 4 rods, 66 feet, or 792 inches long.

III. The denomination rods is seldom used by surveyors, distances being represented in chains and links. Since each link is to of a chain, the number of links is generally expressed as a decimal; thus, 5 chains and 47 links are written 5.47 chains. Engineers generally use a chain 100 feet

long.

MARINERS' AND CLOTH MEASURES. 377. Mariners' Measure is used by seamen in measuring distances, the depth of the sea, etc. Cloth Measure is used for measuring cloth, ribbons, etc. MARINERS' MEASURE.

CLOTH MEASURE. 6 feet = 1 fathom. 1 yard = 36 inches. 120 fathoms = 1 cable length. 1 yard = 18 inches. 880 fathoms = 1 mile. 4 yard = 9 inches.

į yard = 41 inches. I. The foot and yard of these two measures are the linear foot and yard. The nail in Cloth Measure is obsolete. At the custom-houses, the yard is divided into tenths, hundredths, etc.

II. In the old table of Cloth Measure there were given 3 gr. =1 Ell Flemish ; 5 qr. =1 Ell English ; 6 qr.=1 Ell French; 4 qr. 1} in. =1 Ell Scotch.

EXAMPLES FOR PRACTICE. 1. Reduce 120 ch. 25 li., to miles. Ans. I mi. 161 rd. 2. Reduce 575 stat. miles to geog. miles. Ans. 500. 3. Reduce 12 cable lengths 60 fathoms, to chains.

Ans. 1364 ch. 4. Required the distance round the earth.

Ans. 24897.6 mi. 5. If a horse is 16. hands high, what is its height in feet and inches?

Ans. 5 ft. 6 in. 6. A ship was sailing in 121 fathoms of water; how deep was the water ?

Ans. 75 ft. 7. If a vessel sails 14 knots an hour, how niany statute miles will it sail in 12 hours? Ans. 193 miles 64 rd.

8. The soldier's common step is 28 inches, his double quick step 32 inches; how many of each must be take in marching a mile?

Ans. 22626 ; 1980. 9. Two towns in Germany are 26 “long miles” from each other; what is the distance in “short miles” and in English miles ? Ans. 382684 short miles; 149191 Eng. miles.

MEASURES OF SURFACE. 378. A Surface is that which has length and breadth without thickness.

379. A Square is a plane surface which has four equal sides and four right angles, as in the margin.

380. A Rectangle is a surface which has four sides and four right angles, as in the margin. A slate, a door, the sides of the room, etc., are examples of rectangles.

381. All Surfaces are measured by ascertaining the number of times they contain a small square regarded as the unit of measure.

Thus, in the surface in the margin there are three rows of squares, each row containing 4 squares; hence there are 3 times 4 or 12 squares in all; and since these make up the entire surface, the measure of the surface, called its area, is 12 square units.

SURFACE OR SQUARE MEASURE. 382. Surface or Square Measure is used in measuring surfaces, as land, boards, amount of painting, papering, plastering, paving, etc.

TABLE.

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144 square inches ( sq. in.) = 1 square foot, sq. ft. 9 square feet

1 square yard, sq. yd. 304 square yards, or ) 2724 square feet

1 perch or sq.rod, P. 160 perches . . . = 1 acre, . . A 640 acres . . . = 1 square mile, sq. mi

A. P. sq. yd. sq. ft. sq. in.
1 = 160 = 4840 = 43560 = 6272640
1 = 301 = 2721 = 39204
1 =

9 = 1296 SCALE.- Ascending, 144, 9, 301, 160, 640; descending, 640, 160, 301, 9, 144.

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