II. BEGIN WITH THE FIRST order, add AND, IF THE AMOUNT BE NOT MORE THAN 9, COLUMN. EACH COLUMN BY ITSELF, III. IF THE AMOUNT BE MORE THAN 9, THINK WHAT FIGURES YOU WOULD USE TO WRITE THAT AMOUNT. OF THOSE FIGURES, SET DOWN THE RIGHT HAND ONE, UNDER THE ORDER ADded. CARRY ALL TO THE LEFT OF THIS, TO THE NEXT HIGHER ORDER. You have now been learning to perform ADDITION. You find it is a process, which makes one number out of several. For brevity's sake, THE SINGLE NUMBER, THUS FOUND, IS CALLED AMOUNT OR SUM. Then, ADDITION IS FINDING A SINGLE NUMBER, WHICH SHALL EXPRESS THE AMOUNT OF TWO OR MORE NUMBERS. + between two numbers, or sets of numbers, signifies that they are to be added. This sign is commonly called plus, which is a Latin word, meaning more. between two numbers, or sets of numbers, signifies that they are equal to each other. Thus, 6+8=14 signifies that 6 and 8 equal 14. EXAMPLES FOR PRACTICE. 1. A man bought 4 pieces of cloth, containing, the first, 60 yards; the second, 97; the third, 82; and the fourth, 112. How many yards in all ? Ans. 351. 2. For the cloth, he paid as follows: for the first piece, 235 dollars; for the second, 384; for the third, 327; and for the fourth, 486 dollars. How much did he give for all? Ans. 1,432. 3. A man had a sum of money to pay. He paid at one time, 693 dollars; at another, 75; at another, 1,421; at another, 1,652; and at another, 542. There were 964 dollars left unpaid. whole sum? 4. A merchant bought wine to the amount of 437 it so as to gain 94 dollars. What did he sell it for? 5. There are four bags, containing, the first, second, 178; the third, 356; and the fourth, 69. lars in all ? What was the Ans. 5,347. dollars, and sold Ans. 531. 275 dollars; the How many dol. Ans. 878. 6. A merchant bought at one time, 543 barrels of beef; at another, 407; at another, 152; and at another, 414. How many did he buy in all Ans. 1,516. 7. From the creation of the world to the food, were 1,656 years; thence, to the building of Solomon's temple, 1,344 years; thence, to the birth of our Saviour, 1,004; thence, to the present time, 1,830. How old is the world? Ans. 5,834 years. 8. A man left his estate, at his death, to four children. They paid debts to the amount of 1,476 dollars, and then had 4,768 dollars each. How much was the whole estate? Ans. 20,548 dollars. 9. A man paid debts, as follows: one of 427 dollars; another of 763 dollars; another of 654; another of 500; another of 325; and another of 3,250. How much did he pay? Ans. 5,919 dollars. 10. In 1820, the following was the population of the several New-England States. How many in all? Maine, 298,335 11. How many in all the Middle States, containing severally, 1,372,812 New-York, 12. The following States, called the Southern States, contained, severally, the following numbers. Virginia, S. Carolina, How many in all ? 1,065,366 638,829 490,309 13. How many inhabitants in the following States and Territories ? Michigan Ter. 8,896 14. Then, how States, in 1820 ? Dist. of Columbia, 33,039 Ans. 906,617 many inhabitants were there in the whole United Ans. 9,625,734. The best method of ascertaining whether you have performed your addition correctly, is, to LOOK OVER THE EXAMPLE CAREFULLY A SECOND TIME, ADDING THE COLUMNS IN a reversed ORDER; THAT IS, DOWNWARDS, IF YOU FISRT ADDED THEM UPWARDS, AND UPWARDS, IF YOU FIRST ADDED THEM DOWNWARDS. The ascertaining whether an operation in Arithmetic is correctly performed, is called proving the operation, and the process by which this is done, is called PROOF: EXAMPLES FOR PRACTICE. 1. A man owns five farms. The first is worth 16,750 dollars; the second, 12,875; the third, 4,387; the fourth, 9,321; and the fifth, 6,223 dollars. What is the value of all? 2. A man possesses property to the amount of 11,764 dollars. He draws a prize in a lottery of 20,000; he obtains by marriage, 8,650; his uncle dies and leaves him 14,825; in a speculation he gains 5,346; and by his regular trade, he acquires 7,432 dollars. How much has he in all ? 3. Add the following numbers: 1,827,554; 9,200,305; 723; 5; 68; 23; 537,452; 12,900,020; 15; 804. 4. Add 12; 37,853,000,503; 2; 91; 16,020,755; 81; 307; 205; 7,033; 15,000,333; 16,275,850,905,306. 5. Add 555,777,333,222,111; 888,222,666,444,999; 678,678,678, 678,678; 324,324,324,324,324; 123,123,123,123,123; 987,987,987, 987,987; 111,222,333,444,555. 6. Add 891,011,123; 12,765,892; 987,654; 321,234; 12,621; 13,789; 15,910; 64; 7,835,876; 2,867,955. 7. Add 272; 5,004; 10,000; 20,000,002; 59; 892,736,552; 11,111,222; 901; 87; 5,600,222. OBSERVATIONS ON ADDITION, FOR ADVANCED PUPILS. IX. The rule directs, that units of the same order should be written under each other, in the same column, before the process of adding is commenced. But it is manifestly of little importance, what arrangement is adopted, if we only add together units of the same orders, and avoid adding those of different orders to each other. The arrangement prescribed by the rule, is adopted, because most convenient, since it brings as near together as possible, the numbers to be added to one another. The rule directs to begin with the units' order. That this is not necessary, the following example will show. Add 2357 2357 8468 8468 9116 9116 19941 19 12. 21 8.. 19941 Add, first the tens; then the thousands; then the units; and then the hundreds. Then add these amounts, as they stand. It will be seen that the answer is the same, as when the numbers are added by the rule. The example, however, shows that it would be impossible to carry, mentally, unless the addition were to commence with the first order. Hence the advantage of following the rule. The learner will perceive, that, in order to add with ease, it is necessary to commit to memory an Addition table as far as 9; that is, a table containing the sums of all the possible combinations of two digits. For, thus far, a separate character is used for every number. Numbers greater than 9, are expressed by more than one figure, and the addition is made in each order with as much simplicity as in that of units. Now if we could not add the number expressed by any single figure, all at once, we should be obliged to add unit after unit, until we arrived at the number of units, expressed by the figure. This is the manner in which children always add. For this reason, they often count on their fingers, in order to ascertain when they have added the proper number of units. All uneducated persons do the same. These remarks go to show the advantage of our scheme of nota. tion. If we had an independent name and character for every num. ber, we should find it very difficult to perform examples, where large numbers were concerned, and we should either be obliged to commit to memory long and tedious addition tables, or we should be under the necessity of adding unit after unit, as children do. With our present scheme of notation, the largest numbers are managed as easily as units under nine. If the ratio of increase towards the left hand had been twelve fold, instead of ten fold, it would have been necessary to extend our addition tables as high as 11. 1f it had been thirty fold, it would have been necessary to continue them to 29, &c. Of course, we may extend them farther than necessity requires, if we choose. Our common tables extend to 12. It only remains that we consider the mode of proof. No method can be devised, which is a perfect verification of the accuracy of the operation. By none of those, which have been proposed, will an error be necessarily detected, though there is a strong probability that it may. The most natural mode of proving an operation, is to reverse the process, and go back, from the final result, to the starting point. Hence, the natural mode of proving Addition, is by Subtraction. As the student, who peruses these observations, is supposed to be acquainted, at least, with the ground rules, we will explain this method, a little more fully. If two numbers be added, we obtain a third equal to both together. If from this sum then, we take away, or subtract, one of the numbers added, the remainder ought to be the other. This affords us an easy proof, when but two numbers are added. But it usually happens, that several are to be united into one sum. It would be a tedious process, to subtract these, one by one, from the sum; and hence the proof by subtraction, is not used. We might, it is true, add the given numbers into two sums, and take one of these from the total. The remainder would be the other. But there are easier modes than this. We will briefly mention all, that have been suggested. The first, and simplest, is, to add the numbers over again in the same manner as at first. But, if there was an error, in the first process, we shall be likely to fall into the same error again; because all the numbers recur in the same order of succession. It is better, therefore, to reverse the order, as directed in the last section. Yet, in this case, we cannot positively assert, that we have made no error ; though, if the two results agree, it is very probable that the addition has been correctly performed. A third method is, to divide the example into parts, and to find the amount of each part, separately. Then, add these several amounts, and, if the first result was correct, and this is correct likewise, the two will agree. Thus, In this example, we made, by way of proof, four separate additions; dividing the numbers added, into as many different portions. Adding the results, thus obtained, we have a total, which agrees with that first found. Whence we conclude the operation right. This method of proof is the same, in principle, with that found in common Arithmetics; viz., cut off the upper, or lower line, add the other figures, and, to the result, add the line cut off. It may be objected to this proof, as to the one first mentioned, that the recur. rence of the numbers in the same order, renders it liable to error. This objection, however, is not very strong, when the first row is cut off, since, though the others recur in the same order, each figure is added to a different number, from |