(b.) The HOGSHEAD of 63 gallons is used in estimating the contents of reservoirs, or other large bodies of water; but, in most other cases, the term hogshead is not a definite measure. Casks containing from 50 or 60 to 100

or 200 gallons, are called hogsheads.

(c.) A barrel of cider is usually reckoned at 31 gallons.

(d.) The gallon contains 231 cubic inches; but the BEER GALLON, which is sometimes used in measuring beer, milk, and ale, contains 282 cubic inches.

29. Comparison of Dry, Liquid, and Beer Measures.

(a.) To avoid the inconvenience of reckoning of a day with each year, every fourth year is, with the exception named below, reckoned at 366 days, and called LEAP YEAR. The other years are reckoned at 365 days each.

(b.) The year, in reality, contains but 365 days, 5 h. 48 m. 48 sec.; so that, by reckoning 365 days, we make a slight error each year, which, in 100 years, amounts to about of a day. The centennial years are not, therefore, reckoned as leap years, unless the number of the year be divisible by 400. Thus, the year 2000 will be a leap year, but the year 1900 will not.

(c.) The year is divided into 12 months, which differ somewhat in length, as is seen in the following

(b.) A barrel of beef or pork weighs 200 lb. A barrel of flour

(b.) The metre is regarded as the unit of measure, and equals 39.371 of our inches. It is the twenty-millionth part of the distance measured on a meridian, from one pole to another.

* Except in leap year, when it has 29 days.

(d.) The gramme is regarded as the unit of this weight, and equals 15.434 grains Troy.

(e.) The kilogramme is the weight most frequently used in business transactions, and equals very nearly 2 pounds Avoirdupois.

(g.) The franc equals $.186, and the five-franc piece, often seen in the United States, is equal to 93 cents.

SECTION IV.

ADDITION.

33. Definitions and Explanations.

(a.) ADDITION is a process by which we ascertain how many units there are in two or more numbers taken together.

(b.) The result, or number found, is called the sum or amount.

ILLUSTRATIONS.-In "3 and 4 are 7," or "3 + 4 = 7," 3 is added to 4, giving 7 for the sum or amount.

(c.) In order that numbers may be added, they must be of the same denomination, i. e. they must represent units of the same name or kind.

ILLUSTRATIONS.

6 pens and 3 pens are 9 pens; 6 pencils and 3 pencils are 9 pencils; but 6 pens and 3 pencils are neither 9 pens nor 9 pencils. They cannot be added, because they are not of the same denomination; but, if we should change their name or denomination, by calling them both things, we could add them; thus, 6 things and 3 things are 9 things.

(d.) Numbers consisting of several denominations, as units, tens, and hundreds, can only be added by adding the units with the units, the tens with the tens, etc.; hence, for convenience of addition, they should be so written that figures of the same denomination shall stand under each other, i. e. so that units shall stand under units, tens under tens, etc.

(e.) We add the columns separately, beginning at the right-hand or lowest denomination, and reduce as much as possible of the sum of each column to units of the next higher denomination, writing the remainder in its appropriate place in the amount, and adding the higher units obtained by the reduction to the next column.

(f.) The following example will illustrate this:

What is the value of 827+ 948 + 759?

827

948

759

2534

SOLUTION.Writing the numbers as opposite, we add thus: 9 units and 8 units are 17 units, and 7 units are 24 units. But, as 24 units equal 2 tens and 4 units, we write the 4 units, and add the 2 tens with the tens. 2 tens and 5 tens are 7 tens, and 4 tens are 11 tens, and 2 tens are 13 tens. But, as 13 tens equal 1 hundred and 3 tens, we write the 3 tens and add the 1 hundred with the hundreds. 1 hundred and 7 hundreds are 8 hundreds, and 9 hundreds are 17 hundreds, and 8 hundreds are 25 hundreds, which, being the last sum, we write. The answer, then, is 2534.

(g.) The labor of adding may be lessened by omitting to name the separate numbers added.

ILLUSTRATION. In the above example, instead of saying "9 units and 8 units are 17 units, and 7 units are 24 units," it will be shorter to name only the results; thus, "nine, seventeen, twenty-four, or 'twenty-four units, equal to 2 tens and 4 units." Writing the 4 units, we add the 2 tens with the tens; thus, "two, seven, eleven, thirteen, or thirteen tens, equal,” etc. as before.

(h.) Methods of Proof.-To test the correctness of the work

1st. Add the numbers carefully a second time, as before.

2d. Add the numbers in a reverse direction from that taken in the first addition.

3d. Find the sum of two or more of the numbers, and add to it the remaining numbers.

The results should in each case equal the first result; otherwise, there is an error in one operation or the other, and possibly in both.

34. Examples for Practice.

Add the numbers in each of the following examples, and prove the work.