The imperial gallon is a measure of the same uniform capacity, whether for wine, ale, beer, corn, or any other commodity; it must contain exactly 10 lb. avoirdupois of distilled water. The barrel, hogshead, &c.

differs in capacity, according as it is used for wine or beer.

The standard weight of a sack of coals is 2 cwt.; so that 10 sacks weigh 1 ton. A ship-load is 4240 sacks, or 8480 cwt. A sack contains 3 bushels, heaped measure; but heaped measure is now abolished.

DIVISION OF THE CIRCUMFERENCE OF A CIRCLE.

The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; these, of course, are longer or shorter, according as the circle is greater or less. Each degree is divided into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds: the marks for degrees, minutes, and seconds, are a small for degrees, a dash' for minutes, and two dashes "' for seconds,

thus:

60" 1'; 60':

1°; 360° = a whole circumference;
90° a quadrant.

A degree of the circle round the earth at the equator, or a degree of a meridian, is about 69 miles; so that the length of 1' is the 60th part of this, which is the length of a seamile, or, as it is frequently called, of a geographical mile; a geographical, or nautical mile, being the length of 1' of the equator or meridian; it exceeds a land mile by about that mile.

of

alterably fixed in meaning: a pipe of wine of one kind often differed considerably in measure from a pipe of another kind. Whatever name be still retained for the cask, the liquor contained in it is always gauged or measured, and valued in imperial gallons accordingly. It is useful to remember, however, that if a person were now to purchase a runlet of wine,—that is, a cask of that name full of wine, he would get only of 18 imperial gallons; that is, only 15 gallons imperial measure. The word gallons is printed in italics above to imply that old measure is meant.

REDUCTION.

(36.) REDUCTION is the name given to the operations by which a quantity is reduced to another of the same value, but of different denomination. The operation, for instance, by which pounds in money are reduced to shillings, pence, or farthings; or farthings to pounds, years to hours, &c. You see, therefore, that Reduction is of two kinds: the reduction of a higher denomination to a lower, and the reduction of a lower to a higher; it is therefore comprised in two rules.

1. To Reduce a Quantity to one of a Lower Denomination.

RULE. See by the tables how many quantities of the next lower denomination make one of the higher, and multiply the proposed quantity by that number; the product will be the quantity in the next lower denomination.

If it is to be reduced still lower, see how many quantities of that next lower denomination make one of the denomination already reached, and, as before, multiply by that number; and so on till you reach the denomination required.

Ex. 1. Let it be required to reduce £124 to farthings.

As 20 shillings make one pound, we first multiply by the number 20; this reduces the £124 to 2480 shillings; and, since 12 pence make one shilling, we then multiply the number 2480 by the number 12, which reduces the 2480 shillings to 29760 pence; and, lastly, since 4 farthings make one penny, we multiply the number 29760 by 4, which finally reduces the £124 to 119040 farthings.

£124

20

2480 shillings. 12

29760 pence. 4

119040 farthings.

If any quantities of the lower denominations are connected with the quantity to be reduced, we must, of course, add them in with the products which give the same denominations; thus, if shillings had been connected with the £124 above, these shillings must have been added in with the product 2480, which gives shillings; and if pence had also been connected with the pounds, we must have added them in with 29760, the product which gives pence; and so on.

2. Suppose we had to reduce £124. 13s. 44d. to farthings.

Then, multiplying the 124 by 20, and taking in the 13, we have 2493 shillings; and multiplying 2493 by 12, and taking in the 4, we have 29920 pence; and, lastly, multiplying 29920 by 4, and taking in the 2 farthings, we have finally 119682 farthings.

£. S. d. 124 13 4 20

2493 shillings

12

29920 pence

4

119682 farthings

It is proper that I should notice here, that in reducing pounds to shillings you do not multiply the pounds by 20, but only the number of pounds; if pounds be multiplied by any number the product must be pounds. In like manner, in reducing to pence, it is not the shillings you multiply, but only the number of them. It would be tedious to be always making this distinction in rules and examples, though it is right that you should not be misled by the brief language in which rules are sometimes expressed.

3. Reduce £372. 15s. 7 d. to farthings. Here we multiply by 20, and take in the 15; then by 12, and take in the 7; and, lastly, by 4, and take in the 3; as in the margin.

4. How many minutes are there in 29 days, 3 hours, and 21 minutes?

Since 24 hours make one day, and 60

£. Ꭶ. d. 372 15 72 20

7455 shillings

12

89467 pence

4

minutes one hour, we have to multiply, first 357871 farthings by 24, taking in the 3 hours, and then by 60, taking in the 21 minutes, as in the margin.

d. h. m.

29

3 21

24

119

58

699 hours

You will of course understand, when it is said that 24 hours make a day, that what in common language is called a day and a night is meant. People in general consider a day to end at 12 o'clock at night, and then a new day to commence, which lasts till 12 o'clock the following night, thus completing 24 hours; yet it is customary to call that part of the 24 hours usually devoted to sleep, night; and to apply the term day more especially to the other portion. Astronomers begin their day at noon, and end it at the noon following. What they would

60

41961 minutes.

call Jan. 5, at 18 h. 15 m., we should call
Jan. 6, at a past 6 in the morning.
5. How many grains are there in a
lump of gold, weighing 17 lb. 6 oz.
14 dwt. 21 gr.?

Here we have to multiply, first by 12, taking in the 6; then by 20, taking in the 14; and, lastly, by 24, taking in the 21. As the first figure, arising from multiplying by the 4, expresses units, we shall add the units in 21, namely the 1, to this figure; and as the figure arising from multiplying by the 2 is tens, we shall add the tens in 21, namely the 2, to this figure.

lb. oz. awt. gr. 17 6 14 21

12

210

20

4214

24

16857

8430

101157 grains.

NOTE. You may sometimes have to multiply by a number with a fraction joined to it, as, for instance, by 5, in order to reduce perches to yards; to do this you may first multiply by the 5, and then to the product add half the multiplicand; or you may multiply by twice 5, that is by 11, and then divide the product by 2. If the fraction be one-fourth, or three-fourths, you may reduce all to fourths, by multiplying the number to which the fraction is joined by 4, and taking the odd fourths in; the result will then be four times the true multiplier, which you may use instead of the true one; but then you must remember to divide the product by 4, to get the true product. In general, however, the best way will be, when you have to multiply by 4, to take a fourth part of the multiplicand; and when you have to multiply by 2, to take half the multiplicand for two fourths, and then half of this for the remaining fourth. These portions of the multiplicand, added to the product you get by using the multiplier without the fraction, will give the complete product. Thus: suppose 327 is to be multiplied by 5, and 304, respectively, the work is as follows:

3

In the first of these operations, 327 perches are reduced to yards; in the second, 327 square perches, or rods, are reduced to square yards.

Exercises.

1. Reduce £865. 17s. 5d. to pence.

2. Reduce £397. 16s. 4 d. to farthings.

3. How many minutes are there in 365 days?

4. How many pounds are there in 5 cwt. 3 qr. 18 lb. of cheese?

5. Reduce 73 oz. 17 dwt. 11 gr. of gold to grains.

6. How many inches are there in 237 yards of length? 7. Reduce 47 miles 5 furlongs 9 perches 3 yards to yards. 8. How many square yards are there in 7 acres?

9. How many pounds are there in 3 tons 13 cwt. 2 qrs. 22 lb.? 10. Reduce 46 barrels of beer to quarts (old measure). 11. The middle arch of the Southwark Iron Bridge weighs about 1523 tons: what is its weight in pounds?

12. The great bell of St. Paul's weighs 5 tons 2 cwt. 1 qr. 22 lb. what is its weight in pounds?

13. The largest bell in the world is that of Moscow; its weight is 192 tons 17 cwt. 16 lb. : reduce this to pounds.

14. The money taken in silver alone at the doors of the Great Exhibition weighed about 35 tons: how many avoirdupois ounces did it weigh; and how much silver money was taken, allowing 58. to weigh an avoirdupois ounce, as is very nearly the case?

15. A pipe of wine is to be drawn off in an equal number of quart, pint, and half-pint bottles: how many of each will there be (old measure)?

16. How many grains are there in three dozen of table-spoons, each spoon weighing 2 oz. 4 dwt.?

17. The ground occupied by St. Paul's Cathedral measures 2 acres 16 perches: how many square feet are there in this extent?

18. A vat, or large cask for preserving beer, was built for Mr. Meux, the brewer, so capacious that 400 men stood without inconvenience inside of it: it held twelve thousand barrels of beer; how many quarts did it contain (old measure)?

19. The total receipts of the Great Exhibition were £505107: if this sum were reduced to shillings, and a person were to begin counting them as soon as Jan. 1, 1852, com