40 cubic feet of rough or 50 of hewn timber=1 load or ton. A cubic yard of earth is called a load.

To gallons, inclusive, the measures for dry and for liquid substances are

the same.

In dry measure the multiples of the gallon are,

2 gallons 1 peck (pk.),

8 gallons 4 pecks 1 bushel (bush.),

64 gallons=32 pecks=8 bushels=1 quarter (qr.) ;

And in liquid measure,

9 gallons=1 firkin (fir.) beer measure,

36 gallons=4 firkins=1 barrel (bar.),

54 gallons 6 firkins=1 barrels=1 hogshead (hhd.),
63 gallons=1 hogshead, wine measure,

126 gallons 2 hogsheads=1 pipe,

252 gallons=4 hogsheads=2 pipes=1 tun.

Gold, silver, platina, pearls, and precious stones are weighed by troy weight. Diamonds are weighed with a weight termed a carat, which is the Tth part of the ounce troy.

The term carat, when applied to gold, denotes its degree of fineness. Any quantity of pure gold, or of gold alloyed with some other metal, being supposed to be divided into 24 equal parts; if the gold is pure it is said to be 24 carats fine; if 23 parts of the whole are pure gold and 1 part is copper (or some other alloy), the gold is said to be 23 carats fine; if 22 parts of the compound are pure gold and 2 alloy, it is 22 carats fine, and so on.

M

Apothecaries weight is used for medical prescriptions, but the trader weighs drugs with the commercial or avoirdupois weight.

The troy pound is less than the avoirdupois pound in the proportion of 144 to 175; and the avoirdupois ounce less than the troy ounce in the propor

Years expressed by numbers which are not multiples of 4 are made to consist of 365 days; and those expressed by numbers which are multiples of 4, of 366 days. Three years, therefore, contain always 365 days each, and the fourth 366. The year of 366 days is called leap-year.

The average length of the year is thus assumed to be 365 or 365.25 days, which exceeds the true duration by 00776 of a day.

In 129 years this amounts to a day, or, in round numbers, it makes 3 days in 4 centuries. The error is corrected by making three of the years which close the centuries common years, and the fourth only leap-year. Thus 1700, 1800, 1900 are common years, and 2000 a leap-year.

In the gold coinage of Great Britain, twelve ounces of the metal of which standard gold coins are made contain 11 oz. of pure gold and 1 oz. of alloy (the alloy used is copper).

89

The pound troy is coined into 46 sovereigns; the weight of a sovereign is, therefore, 5760÷46=123111 grs.=123-25 grs. nearly; of which 113 grains are pure gold, and 10-25 grs. alloy.

In the silver coinage, 12 oz. of the metal of which standard silver coins are made contain 11 oz. 2 dwt. of pure silver, and 18 dwt. of alloy.

The pound troy of this metal is coined into 66 shillings; whence the weight of 1 shilling is 5760+66=87,3 grains=87-3 grs. nearly, of which 80-7 grs. are pure silver and 6-6 grs. alloy.

In the present coinage

the actual value of the sovereign is 20 sh.,
that of the shilling is 11-27 d.,

and the relative values of equal weights of gold and silver are nearly as 14.3 to 1.

The penny contains 10 drams avoird. of copper. The penny, shilling, and sovereign are, respectively, the standard coins of copper, silver, and gold.

Multiplication Table, containing the Products of Numbers from 1 to 25.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 8 12 16 20 24 28

4

32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 175 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 231 242 253 264 275 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 252 264 276 288 300 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 273 286 299 312 325 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280 294 308 322 336 350 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 375 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 336 352 368 384 400 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340 357 374 391 408 425 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 399 418 437 456 475 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 21 42 63 84 105 126 147 168 189 210 231 252 273 294 315 336 357 378 399 420 441 462 483 504 525 22 44 66 88 110 132 154 176 198 220 242 264 286 308 330 352 374 396 418 440 462 484 506 528 550 23 46 69 92115 138 161 184 207 230 253 276 299 322 345 368 391 414 437 460 483 506 529 552 575 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625

298. From the formation of compound numbers, it is evident that the same absolute quantity can be expressed by numbers which contain fewer or more units, according as the measure employed is a principal or a subordinate unit. One pound troy weight, for example, can be expressed as 1 lb., or 12 oz., or 240 dwt., or 5760 grs.

In performing the elementary operations of Arithmetic with compound numbers, it may become necessary to reduce numbers expressed in terms of any unit of a weight or measure to equivalent numbers expressed in terms of any other unit of that weight or measure.

Such reduction must be either from a higher denomination to a lower, or from a lower denomination to a higher.

Since the principal unit of any weight or measure is a certain multiple of the first subordinate unit, this a certain multiple of the second, &c., it follows that the number of units of a lower denomination equivalent to any given number of units of a higher denomination must be the same multiple of the given number that the higher unit is of the lower;

And, conversely, that the number of units of a higher denomination equivalent to a given number of units of a lower denomination must be the same part of the given number that the lower unit is of the higher.

The general inference therefore is, that to reduce a number expressed in terms of any unit of a weight or measure to an equivalent number expressed in terms of an inferior unit of the same weight or measure, it is necessary to multiply the given number by a multiplier expressing the number of units of the lower denomination which make 1 unit of the higher denomination;

And that to reduce from any unit to a higher it is necessary to divide the given number by a divisor expressing the number of units of the given which are equivalent to one of the required denomination.

299. Examples of reduction from units of a given denomination to units of the next lower denomination.

1st. Reduce 5 poles, long measure, to yards:

1

11

1 po.=5, yds.= yds.,

2

[1£.=20sh....=(x) sh.sh. =87
=8sh.

די

3d. Reduce 34 quarters to pounds avoirdupois :

1 qr.=28 lb.. 34 qrs.=(34×28) lbs. 9.52 lbs.

4th. Reduce 5sh. 4d. to pence:

since 1sh.

The last example differs number is compound.

12d., 5sh.=(5 × 12)d.=60d.; and 60d.+4d.=64d.

.. 5sh. 4d.=64d.

from the others in this respect, that the given

When the 5sh. have been reduced to an equivalent number of pence, in order to obtain the value of the compound expression, it is necessary to add the pence to this number; the sum of both is the result required.

300. The reduction of any number, from units of a given denomination to units of any other denomination lower than the next, is made by successive reductions from one unit to the inferior, then from this to the next inferior, by a repetition of the same process, until the units of the required denomination have been attained.

a. If the number is compound, it is indispensable that the reduction be made thus by successive steps, in order that the units of the denominations successively attained may be combined with the sum of the higher units which have been reduced to their measure.

b. But when the given number is a multiple or part of one unit only of a given weight or measure, it may be reduced to any lower denomination of the same weight or measure by multiplying at once by that number of the required which makes one of the given denomination; for it is equivalent to multiply any number by the factors of a product, or at once by that product (Art. 71).

5th. Reduce 3 £. to farthings:

1 £.

1 sh.

1 d.

20 sh... 3 £.=(3 × 20) sh.=60 sh.
12 d... 60 sh.=(60 × 12) d.=720 d.
4 qrs... 720 d.=(720 × 4) qrs.=2880 qrs.
whence 3 £.=2880 qrs.

Otherwise, since 1 £.=960 qrs.,

6th. Reduce

3 £.=(3x960) qrs.=2880 qrs.

1 cwt. 4 qrs. .*. cwt. =

48

•7

4

1 or 48 lbs.

c. When the number to be reduced is expressed by a vulgar fraction the manner (followed in the 2d solution of the last example) of indicating the successive steps, striking out factors common to the numerator and denominator, and then performing the multiplication (as in multiplication of vulgar fractions), seems preferable to the last method, inasmuch as the common factors are detected in small more readily than in large numbers.

7th. Reduce 82 poles to inches, long measure:

Calculation by successive reductions:

Otherwise, multiply 82 poles by 198, which is the number of inches

contained in 1 pole:

.82×198=162.36 inches, the same result as before.