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137.

ARITHMETICAL TABLES.

39

Weights.*

IV. TROY WEIGHT. 21 grains, grs. make 1 penny weight, put. grs. 24 puts.] 07. lb. 20 penny weights

480 20 12 ounces

Ib
lb. 5760 240 12 1

- 1 ounce,
" I pound,

grs. 20

V. APOTHECARIES' WEIGHT. 20 grains, gr. make ] scruple, sc.

sc. 1 drms, 1 oz. Ib. 3 scru; les 1 dram, dr.

60 3 1 8 dram

1 ounce,
Oz

480 24 8 1 12 ounes

1 pound,

Ib. 5760 280 96 i 121

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All the rest have thirty-one,
Excepting February alone,
Which hath twenty-eight, nay more,

Hath twenty-nine one year in four. The true solar year consists of 365 days, 5 h. 48 m, 57 8. or nearly to 8654 days. A commion year is 365 days, and one year is added in Leap years to make up the loss of l of a day in each of the three preceding years. This method of reckoning was ordered by Julius Cæsar, 40 year before the birth of Christ, and is called the Julian account, or Old Style But as the true year fell 11 m. 3 s. short of 3654 days, the aduition of a day every fourth year was too much by 44 m. 12 s.

This amounted to one day in about 130

years.

To correct this error, Pope Gregory, in 1582, ordered that ten days should be struck out of the calendar, hy call. ing the 5th of October the 15th; and to prevent its recurrence, he ordered that each succeeding century, divisible by 4, as 16 hundred, 20 hundred, ar.d 24 hundred, should be Leap vears, but that the centuries riot dirisible by 4, as 17 hundred, 18 hundred, and 19. hundred, should be common years. This reckoning is called the Gregorian or New Style. The New Style differs now twelve days from the old style.

* The original standard of all our weights was a corn of wheat, taken from the iniddle of the car, and well dried. These were called grains, and 32 of them made one pennyweight. But it was afterwards thought Bufficient to divide this saine pennyweight into 24 equal parts, still calling the parts grains, and these are the basis of the table of Troy weight, by which are weighed gold, silver and jewelry. Apothecaries' weight is the same as 'Troy weight, only baving different divisions between graing and ounces. Apoi hecaries make use of this weight in compounding their medicines, but they buy and sell their drugs by Avoirdupois weight. In buying and selling coarse and drossy articles, it becanie customary to allow a greater weight than that used for small and precious articles, and this custom at length- established the Avoirdupois, or corpmon weight, by which all articles are now weighed, with the foregoing exceptions. Avoir. dupois weight is about che sixth part more than Troy weight, 2 pound of zhe former leing 7000 grains, ana of the latter 5760 grains. 'n buying and selling by the hundred weight, 28 pounds have been called a quarter, and 112 pounds a cut. but this practice of grossing, as it is called, is now pretty generally laid aside, and 25 pounds are considered a quarter and 4 quarters, or 100 pounds, a hundred weight.

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oz.

- 1 quarter,

66

yd.

VI. AVOIRDUPOIS, or COMUN' H'EIGHT. 16 drama make 1 ounce,

dr. 16 oz. Į, Ihs. qrs. cwt. con. 16 ounces • 1 pound,

lb. 256 16 1 28 pounds

7168 qr.

448 28 1 4 quarters '« I hundred, cwt.

4

28672 1792 112 20 hundred " 1 ton, ton.

573440 35840 2240 80 20 1 Measures. *

VII. LONG MEASURE. 3 barley corns make 1 inch, in.llin. 12 ft. lyds. rds.fur.lmi. 1 foot,

36 8 112 inches

1 3 feet, 1 yard,

198 164 51 1 58 yards, or 163 st. 1 rod, or pole, rd. i 7920 660 220 40 40 rods

1 furlong, fur. 63360 5280 1760 320 8 1 8 furlongs

1 mile, mr. 7.92 in. make I link, 3 miles

I league, lea. 25 li. I rod, 69.2 miles

1 degree, deg. 4 rd. or 100 li. 1 chain, ches. 1360 degrees

1 circle of earth.

180 chains 1 mile, mi. * The original standard of English long measure was a barley com taken from the middle of thc ear, and well dried. Three of these in length were called an inch, and then the others as in the table. Long measure is employed for denoting the distance of places, and for measuring any thing where length only is concerned. When measure is applied to surfaces, where length and breadth are both concerned, it is called square measure. A square inch is a square measuring an inch on every side. The table of square measure is made from that of long measure by multiplying the several numbers of the latter into themselves. Thus, 12 inches are a foot in length, a square foot then is a square which measures 1 foot, or 12 inches, on every side, and contains 12X12=144 square inches. 3 feet in

length make a yard; a square yard is a square measuring 3 feet on each side: but such a square contains (see figure) nine (3X3=9) squares measuring a foot on each side, or 9 square feet; and when we say that a surface contains so many square

feet, or square yards, we mean that the surface is equal to such a number of squares measuring a foot or a yard, on each side. When measure is applied to solids which have length, breadth, and thickness, it is called solid or cubic imcasure.

A solid inch is a body, or block, 3ft.

having six sides, each of which is an inch square, and the number of inches in a solid foot is equal to the number of such blocks that would be required to make a pile a foot square and a foot high. Now it would require 144 blocks to cover a square foot one inch high. Hence to raise the pile twelve inches high would require twelve timts 144=1728 blocks or inches. In like manner it would require 9 solid blocks, foot each way, to cover a square yard to the heiglit of one foot, and 3 times 9=27, to raise it three feet, or make one se! yard. This will be obvious from an inspers

3ft.

1872.

ARITHMETICAL TABLLS.

41

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VIII.' CLOTH MEASURE. 24 inches make 1 nail, na. 3 quarters make 1 ell Flemish, E. F. 4 nails

1 quarter, qr. 5 quarters 1 ell English, E. E. 4 quarters 1 yard, yd. 37.2 in.

1 ell Scotch, E.S. IX. SQUARE MEASURE. 144 inches make 1 square foot, ft. in. 1441 ft. 1 yds. /rds fro.lmi. 9 feet

1 sq. yard, yd. 1296 9 1 304 yards, 1 sq. rod, rd. 39204 2724 304 1 2724 feet,

rd. 1568160 10890 1210 40 11 40 rods

1 rood,

ro. 6272640) 43560 4840 160 4 1 4 roods,

acr,

10 sq. chains make 1 acre, acr. 640 acres 1 sq. mile, mi.

6400 chains make 1 sq. mile, mi. X. SOLID, or CUBIC MEASURE. 1728 inches, in. make 1 foot, ft. in. 1728 feet 14 yard cord 27 feet

1 yard, yd. 46656 27
128 feet
« 1 cord, cor.

221184 128

4341

1 40 ft. of round timber, or 50 ft. of hewn timber, make 1 ton, ton.

1 sq. rod,

1 acre,

1 quart,

bar,
hhd.
pipe.

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XI. WWE MEASURE.* 4 gills, gls. make 1 pint, pt. cu.28}/pt. 11 qts.gal. 2 pints

qt. in. 574 2 4 quarts. 1 galion, gal.

231

8 315 gallons 1 barrel, bar. 72764, 252 126 315 1; 2 barrels “ 1 hogshead, hhd. 145531 5041 252 63 2 1 2 hogsheads

p. 29106) 1008 504'126 4 2 2 pipes

t. 58212 2016'1008 252 81 41 21 1

1 pipe, 1 ton,

tion of the diagram. The cord of wood is sometimes called eight feet. In this case four feet in length, four.in brcadth, and one in height=16 solid feet, is called one foot; or eight feet in length, four in breadth, and usix inches in height, a foot, that is, 1-8th of a cord is called one foot, 2-3ths, two feet, &c. In measuring lands, roads, &c. the distances are usually taken in chains and links. In ordinary business, feet and inches are the most common measures. Many mechanics, however, now take dimensions in feet and tenths of a foot, instead of inches, and if all would do the same, they would find all their caleulations much more simple and easy. By forty feet of round timber, in the table of solid measure, iR meant so muchf round timber, as will make forty feet after it is squared,

* Four pounds Troy weight of wheat gathered from the middle of the ear, and well dried, were called one gallon, and this was the original standard of all English measures, both liquid and dry, and this was the same as the present wine gallon.

But in tine it becaine customary, to use a larger measure in selling cheap liquors, and this custoin at length established the beer measure, which bears about the same proportion 1.0 wine measure that avoirdupois does to troy weight.

The dry measure *as also made larger than the wine measure, and was at length established at about a mean hetween wine and beer mcasure. By wine measure are measured wine, all kinds of spirits, cider, vinegar, oil, &c. By beer measure are measured ale and beer, and by dry measure are measured al

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XII. BEER MEASURE. 2 pints, pts, make 1 quart, qt. cubic 70 qt. 1 gal./bar. bhd. 4 quarts

1

gallon, gal. inches 282 86 gallons

1 barrel, bar. 10152 144 S6 } 54 gallons 1 hogshead, hhd.li 152281 216 54

1 XIII. DRY MEASURE. 2 pints, pts. make 1 quart, qt. cu. 33.6 pt.1 qts. 4 quarts

1 gallon,

gal. in.67.2 2 11 8 quarts

1 peck, pk. 268.81 8 4j 1 4 pecks

I bushet,

bu. 537.6 16 8 2 8 bushels

1 quarter,

gr. 2150.4 64 32 8 4 quarters

1 chaldron, ch. 17203.2 512 256164 32 8 1

XIV. CIRCULAR MEASURE.. 1 60 seconds, "I make 1 miuute,

11601

1 60 minutes

1 degree,

3600 60 30 degrees

1 sign,

108000 1800 30
12 signs, or 360° “ 1 circle. 12960001 21600 360 12/1

XV. MISCELLANEOUS.
12 things make 1 dozen, doz. 5 feet make 1 pace.
12 dozen 1 gross. 8$

BOOKS. 1 great gross.

When a sheet is folded into two 20 things l score,

leaves, it is called Folio. 24 sheets of paper, 1 quire.

When folded into 4 leaves, it is 20 quires make 1 ream.

called Quarto. i 112 pounds 1 quintal.

When folded into 8 leaves, it is 10 things I desin.

called Octavo. 10 desms 1 gross.

When folded into 12, it is called 1 great gross. Duodecimo, or 12mo. 6 points 1 Hime.

When folded into 18, it is called 12 lines I inch.

18mo, 4 inches 1 hand.

When folded into 24, it is called 6 feet 1 fathom.

24s.

circle.

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12 gross

10 gross

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kinds of dry goods, corn, grain, salt, roots, fruit, &e. A standard bushol is 18} inches diamoter and 8 inches deep. The statute bushel for measuring coal, ashes and lime, in Vermont, contains 38 quarts, or 2553.6 cubic inches.

* Every circle, without regard to its size, is supposed to be divided into 360 equal parts, callod degrees, and these again to be subdivided into minutes and seconds; so that the absolute quantity expressed by any of these denominations must always depend upon the size of the circle. In this ineasure are reckoned latitude, longitude, the planetary motions, &ca.

4 The habit of reckoning by the dozen is well adapted to the English method of reckoning money; articles which were 4s. a dozen, being 4d. apjece, 7s. a dozen,

7d. apiece, &c. Points, lines and inches are used in measuring the length of clock pendulums. Hands are used in measuring dhe height of horses, and fathoms in measuring depths at sea.

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138, 139, 140.

REDUCTION.

1. REDUCTION. 138. Reduction is the method of changing numbers from one denomination to another,without altering their value.(40) 1. In £4 8s. 5d. 3qrs. how many farthings?

As £1=208. there are 20 times as many shillings £ s. d. qrs.

as there are pounds; 4 8 5 3

we therefore multiply the pounds by 20, and to the product, 80s. join the 8s. 20

making 88. Then because 18.=120. there are 12 888.

times as many perce as there are shillings; we there12

fore multiply the 886. by 12, joining the 5d. to the proguct, and thus find £4 88. 50.=1061d. Again,

as id.=4qr. we multiply the pence by 4, joining the 181

3qr. to the product, and thus find £4 8s. 5d. 3qr.=

4247 farthings. This process is called Reduction 1061d.

Descending, because by it numbers of a higher denomination are brought into a lower denomination.

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4247qr. Ans. 2. In 4247 farthings, how many pounds? 4) 4247 As it takes 4qr. to make 1 there are evidently

penny, as many pence in 4247gr. as there are times 4 in that

number. We therefore divide by 4, and the quotient 12) 1061–3qr.

is 1061 d. and 3qr. over. Then, as it takes 12 pence

to make ls. there will be as upany shillings as there are 210)818-5d.

times 12 in 1061=885. 5d. Again, as it takes 208. to

nake £1, there will be as many pounds as there are £4 88.

times 20 in 883, *£4 83. Thus we find 42470r.= £4 8s. 5d. Sqr. This process is called Reduction Ascending, because by it a lower denomination is brought into a bigher. By these examples it will be seen that Reduction Ascending and Descending mutually prove each other.

As a process similar to the above may be employed in the reduction of time, weights and measures, as well as moneys, it may be stated in the following general terms: 1.39.--REDUCTION DESCENDING. 140.-REDUCTION ASCENDING.

RULE.-Multiply the high- RULE.-Divide the lowest est denomination by that denomination by the number number which it takes of the which it takes of that to make next lower to make one in one in the next higher deno the higher, adding the num mination; and so continue to ber, if any, of the lower de-do, till you have brought it nomination; and so proceed into the denomination 10to do, till it is brought as low quired. as the question requires.

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