date of the year be divided by 4, and there be no remainder, that year will be leap year; and 1, 2, or 3, will remain, according as the year is 1, 2, or 3 years after leap year. The number expressed by the last two figures only of the date may be used instead of the complete date, as the remainder arising from division by 4 will be the same: thus, the year 1852 is leap year, since 52 is exactly divisible by 4. The addition of a day every fourth year is rendered necessary on account of the ordinary year being taken as 365 days, instead of 365 days, which it is within a few minutes; so that in 4 years a whole day would otherwise be overlooked. Even as it is, the few minutes just noticed, by which 365 days differs from the true year, as shown by the sun, accumulate to an error of about 3 days 23 hours in 400 years. This error is an error of excess ; for the true solar year is 365 d. 5 h. 48 m. 49 sec. To remove the effect of this error, it has been fixed that when the year consists of complete centuries (a century being 100 years), although the date would be exactly divisible by 4, yet the year is not to be considered as leap year, unless the date, omitting the two final noughts or zeros, is also divisible by 4: thus, the years 1800, 1900, 2100, &c., are to be reckoned as common years, since 18, 19, 21, &c., are not divisible by 4; but 1600, 2000, &c., are leap years. With this correction the civil reckoning so far agrees, on the average, with the astronomical determination of the year, as to be only one day in advance of the strict truth in 3546 years,—an error which for the purposes of life it is unnecessary to make any provision for.

The year is divided into twelve parts,-January, February, March, April, May, June, July, August, September, October, November, and December. These are called the 12 calendar months. Each of these, except February in a common year, contains more than 4 weeks, or 28 days; yet, in ordinary language, 4 weeks is called a month. There are 13 of these months in a calendar year. The number of days in each calendar month may be easily recollected by aid of the following lines:

Thirty days have September,
April, June, and November,
February has twenty-eight alone,
And all the rest have thirty-one;
But leap-year, coming once in four,
February then has one day more.

By this weight coarse and bulky goods are weighed, and all the common necessaries of life. The weight called a stone is also much used for like purposes; but it is not fixed, like the other weights given above; in general, however, by a stone is meant 14 lbs. avoirdupois. In London, a stone of butcher's meat is only 8 lbs.; but in many country places it is 14 lbs., and in some 16 lbs. Unless, however, the contrary be stated (butchers' meat excepted), by a stone 14 lbs. is always understood, so that 8 stone make a cwt. ; 2 stone of 14 lbs. make 1 tod of wool; 6 tods, 1 wey; 2 weys, 1 sack; and 12 sacks 1 last: 12 sacks also make a chaldron of coals,-a measure now discontinued.

This weight is for the precious metals, and for ingredients used in philosophical experiments. The grain troy is subdivided into 20 parts, called mites; so that 20 mites = 1 grain. The avoirdupois ounce is less than the troy ounce; for 1 oz. avoir. = 175 oz. troy; but the avoirdupois pound is greater than the troy pound; for 1 lb. avoir. = 172 lb.

12 inches

3 feet

6 feet

5 yards

Apothecaries and chemists use this weight in mixing medicines; but drugs are bought and sold by avoirdupois weight.

MEASURES

OF LENGTH, SURFACE, AND SOLID OR CUBIC CONTENTS. LENGTH (or LONG MEASURE).

marks.

(ft.)

(yd.)

(fath.)

1 rod, pole, or perch (per.)

4 perches, or 100 links 1 chain (22 yards)

40 poles

1 furlong

(fur.)

8 furlongs

(lea.)

A hand, in measuring horses, is 4 in.; a span, 9 in.; and a pace, 5 feet. A rood of fencing or ditching is 7 yards.

Ale and Beer.*

1728 cubic inches make 1 cubic foot; and 27 cubic feet make

1 cubic yard.

MEASURES for WINE, SPIRITS, ALE, BEER, &c.

* These measures are inserted here chiefly because the learner may know the number of gallons meant whenever he meets with the names of them; but he must be here apprised, that these gallons are not imperial gallons, but gallons according to the old measure, which has been abolished. Old gallons, wine measure, are converted into imperial measure by multiplying by ; and old gallons, ale measure, are converted into imperial measure by multiplying by gg, which fractions are sufficiently accurate for practical purposes. The names of the measures for wine and ale, marked above, are now merely the names of casks, and do not denote imperial measures; indeed, some of these names were never un

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 these shillings by the number 12, which reduces the 2480 shillings to 29760 pence; and, lastly, since 4 farthings make one penny, we multiply these pence by the number 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.