ARTICLE IV.

ADDITION.

1. ADDITION means increase, and in arithmetic it teaches to join two or more quantities, having units of the same kind, into one number, which is called their sum; hence,

2. A sum is a single number, which is found by addition, and is always equal in quantity, to all the given numbers, added together, or the whole; hence,

3. The whole is equal to the sum of all its parts.

Remark. The operations of Addition, Subtraction, Multiplication, and Division, are best represented by the use of equations, the nature and properties of which will be more fully explained in Articles XIII, and XIV.

4. An equation is an expression containing two dissimilar quantities, or distinct members, which are equal in value; as, 4+3—2—6—1; here the quantity on the left of the sign of equality (=), (4+3-2), constitutes the left hand member, and that on the right, (6—1), constitutes the right hand member of the equation.

Remark. In every equation there is an unknown quantity either expressed or implied. When the unknown quantity can be represented in an equation, it is generally expressed by the use of an alphabetical letter; as, 5+3+6+4=s; where s represents the sum of 5, 3, 6, & 4; or, 8—4—1=d; where d represents the difference, or what remains after 4 & 1 have been subtracted from 8, and 4× 3=p&6÷3=q; where p represents the product of 4 multiplied by 3, and q represents the quotient of 6 divided by 3.

We are now prepared to enter upon the solution of equa tions containing quantities to be added.

5. The solution of an equation is the process of finding the unknown quantity; thus, 468562+567856+367259+ 968297-3675684=s, is an equation in which the sum (s) is the unknown quantity, and may be found in the following

manner:

First, we will set down the numbers in the left hand member of the equation, so that the same order of units in each number may fall exactly under one another; thus,

We have now as many perpendicular columns as there are places (or figures,) in the numbers set down. Under the bottom number let a line be drawn, and we will add the column containing the first order of units, first,

Thus, 4 and 7 are 11; 11 and 9 are 20; 20 and 6 are 26; 26 and 2 are 28. We will now set down the first order of units (8), of the sum, (28), (found by adding the right hand column,) and carry the second order of units, (2), (as so many units of the same kind,) to the next column, which we will add in a like manner, beginning with the Two, carried from the sum of the first column, as follows: 2 and 8 are 10; 10 and 9 are 19; 19 and 5 are 24; 24 and 5 are 29; 29 and 6 are 35; (i. e., 3 units of the first order, and 5 units of the second order.) Set down the first order of units, (under the column added,) and carry the second order of units, (3), as before, to the next column, which we will add up, setting down and carrying as before, and so on, for each successive column, till all are added, taking care to set down the whole amount of the last, or left hand column, and the number below the line, will be the sum required, (6047658=s); hence, 468562+567856+367259+968297+3675684= 6047658, which is the proper solution of the given equation.

Before proceeding further, the pupil should learn to add, without a moment's hesitation, any or all the numbers less than ten; as, 3 and 5 are 8; 7 and 3 are 10; 4 and 3 are 7; 6 and 5 are 11; 9 and 7 are 16; 8 and 9 are 17; 7 and 8 are 15; 5 and 7 are 12; 5 and 4 are 9; 3 and 5 are 8; 9 and 8 are 17; 8 and 8 are 16; 5 and 9 are 14; 8 and 5 are 13; 7 and 6 are 13; 1 and 2 are 3; 3 and 2 are 5; 5 and 3 are 8; 8 and 4 are 12; 12 and 5 are 17; 17 and 6 are 23; 23 and 7 are 30; 30 and 8 are 38, &c.

To solve equations in Addition, we have only to add the numbers in the left hand member, or find their sum according to the following

RULE for Addition.

First, set down the given quantities, so that the same orders of units in each number may fall exactly under one another, so as to form perpendicular columns, and draw a line underneath.

Second, add the right hand column, and set under it the first order of units, in the sum (last found,) and carry the NUMBER formed by the remaining figure, or figures, (i. e., by the remaining orders of units,) to the next column, which add, setting down and carrying, as before, and so on till all are added; observing to set down the whole amount of the last column; the number thus formed, below the line, will be the sum required.

PROOF.

Cut off the top number, by a line, and find the sum of all the rest, to which add the top number, and if their sum agree with that already found, the work is supposed to be right.

EXAMPLES.

806009700+46000561+700809004+70605040321=s.

1. Solve

The proof in this case is plain; first, we find the sum of all the numbers, as already explained; and, second, the sum of all except the top number, which place immediately under the sum of the whole, and call it the partial sum. Third, this partial sum is added to the top number, and their sum placed under the former. Now, since the partial sum, is made up of all the numbers EXCEPT THE TOP ONE, it is plain that it falls short of the sum of the whole, in that precise quantity contained in the top number, and that, if the work be right,

the sum of these two, must make the sum of the whole. Such might be the case if the work were wrong; but must always be the case when the work is right.

2. Solve 468965 +7639785 +587467951 +5836447 + 45891264+32435465=s.

3. Solve 70612897051 + 9630091064 + 763004506 + 9500607091+475006=s.

4. Solve 600050040301 +90160807805+4690001651+ 4286865634+123456=s

5. Solve 102030405060708090+758645000281+468+ 15001+72000006=s.

PROBLEMS PRODUCING EQUATIONS IN ADDITION.

1. George, Thomas, Richard, James, Peter, and John, each had a bushel of apples; after they had counted them, it was found, that George had 958, Thomas 596, Richard 1241, James 563, Peter 327, and John 759; how many apples had they all? Thus, the numbers 958; 596; 1241; 563; 327, and 759, are given, and their sum (s) required; hence, 958+596 +1241 +563+327+759=s.

2. The population of America is 46,500,000; Europe, 233,500,000; Asia, 450,000,000; Africa, 57,000,000; and Oceanica, 4,500,000. Supposing these divisions to comprise the entire globe, what is the population of the world? Given 46,500,000 + 233,500,000 + 450,000,000 + 57,000,000 + 4,500,000 s. to find the value of s.

=

3. The population of New York is 312,000; Philadelphia, 228,000; Baltimore, 102,213; New Orleans, 102,193; Washington, 23,364; and Providence, 23,171.. How large would a city be that contained as many people as all these put together?

4. The population of the State of New York is 2,428,921; New Jersey, 373,306; Pennsylvania, 1,724,033; and Delaware, 78,085. What is the population of the whole?

5. A merchant having failed in business, it was found that he owed A. 7256 dollars, B. 4689 dols., C. 756 dols., D. 5496 dols., E. 362 dols., and F. 1671 dols.; what was the amount of his debts?

6. The State of Connecticut has eight counties, whose populations are as follow: Fairfield, 49,917; Hartford.

55,629; Litchfield, 40,448; Middlesex, 24,879; New Haven, 48,619; New London, 44,463; Tollard, 17,955; and Windham, 28,080; what is the population of the whole State?

NOTE. The pupil should always state the equation before solving it.

ARTICLE V.

SUBTRACTION.

1. SUBTRACTION* means decrease, and teaches to take a less number from a greater, and thereby show the difference, or remainder. The numbers employed in Subtraction are two, called the Minuend, and Subtrahend.

2. The minuend is the number to be decreased; as, in the equation 8-5=3, the number 8 is the minuend.

3. The subtrahend is the number by which subtraction is made; as, in the equation 8-5=3, the number 5 is the subtrahend.

4. The difference is the number which remains after subtraction has been made; as, 8-5-3, the number 3 is the remainder.

To solve the equation 756325-435243-d, we have the minuend and subtrahend given, and the difference (d) required.

First, set down the minuend, and place the subtrahend immediately under it, as in addition, and draw a line below it.

756325

435243

321082

First, Take the units figure (3), of the subtrahend from the units figure (5), of the minuend, and there will be a remainder, (of 2,) which we will place immediately below the figure subtracted, (3), under the line. The figure in the place of tens, is 4, which cannot

*This word comes from sub, under; and Traho, to contract, or decrease, make less, or draw out; hence, the word literally means to make less, or decrease, by drawing, or taking from under, and since, in Subtraction, the minuend is made less by taking the subtrahend from under it, the word is correctly used.