III. ADDITION. DEMONSTRATION OF THE RULE. This rule is founded on the known axiom-" The whole is equal to the sum of all its parts." It has been shown that Numeration reduces every number to a certain number of places or orders, each of which is tenfold less in value than its preceding place: consequently, one is carried for every ten, in Addition, because common consent adopts the tenfold ratio in Arithmetical computations. EXPLANATION.-Here it will be seen that the column of units amounts to 23 units. The column of tens amounts to 170 units, or 17 tens. The column of hundreds amounts to 1500 units, 150 tens, or 15 hundreds. But to save the trouble of setting down and adding up so many separate amounts, a way has been contrived of carrying the left hand figure immediately, and uniting it with the next column; from which the Rule is derived. TO PROVE ADDITION. RULE.-Set the excess of nines in each row of figures to the right of its row, and if the excess of nines in the sums result, and the column made by setting out the several excesses, are alike, the work is right. Example. 9876 3 8765 8 7654 4 26295 6 REMARK. The figure 9 has a peculiar property, which, except 3, belongs to no other figure whatever, viz: that any number divided by 9 will leave the same remainder as the sum of its figures divided by 9. NOTE 1.-Compound Addition differs from Simple Addition only in the succession of its orders: consequently, to the ingenious mind, its demonstration seems to be unnecessary. NOTE 2. Multiplication being in substance Addition, the remarks just made respecting Compound Addition are applicable to Compound Multiplication. (10) IV. SUBTRACTION. DEMONSTRATION OF THE RULE. 1. When all the places of the least number are less than their correspondent places in the larger, the difference of the figures in the several like places, must, taken together, make the true difference sought; because, "as the sum of the parts is equal to the whole," so is "the sum of the differences, of all the similar parts, equal to the differences of the whole.” 2. Borrowing from a preceding place to increase an upper place when its correspondent lower place is largest, is only resolving the upper number into such parts, as are, each, greater than, or equal to, the similar parts of the less number. EXAMPLE. From 76254 take 28786. Operation. 70000+6000+200+50+4=76254 Let the numbers be 20000+8000+700+80+6=28786 decomposed and arrang ed as in the margin. 40000+7000+400+60+8-47468 EXPLANATION.--Here I begin at the right hand, and finding that I cannot take 6 from 4, I borrow 10 from 50, and add it to 4, which makes 14. From this I take 6 and set down 8. As 10 is borrowed from 50, there is 40 left. I cannot take 80 from 40; I therefore borrow 100 from 200, and add it to 40, making 140; from which I take 80, and set down 60; and so on through the whole. In borrowing to add to an upper figure, or place, instead of considering the next upper figure, or place, diminished, it has been found most convenient to increase the next lower figure, or place, which brings the result just the same. On this principle was founded the Rule for Subtraction. TO PROVE SUBTRACTION. RULE. 1.-Having subtracted as usual, cast out the 9s from the minuend, and place the ex- | cess at the right hand. 2. Cast out the nines from the subtrahend and remainder, and add their excesses together; and if the work is right, the ex Example. From 46875 subtract 34789. Operation. 46875 cess of 9s in their sum will be 12086 the same as the excess of 9s in 4, excesses. the minuend. 12, sum. 3, excess. REMARK.-As the subtrahend and remainder form a sum in Addition, of which the minuend is the amount, the reason of the proof is obvious. NOTE 1.-Compound Subtraction differs from Simple Subtraction only in the succession of its orders: consequently, its demonstration seems to be unnecessary. NOTE 2.-Division being in substance Subtraction, the remarks just made respecting Compound Subtraction are applicable to Compound Division. V. MULTIPLICATION. 1. DEMONSTRATION OF THE RULE. When the multiplier is a single digit, it is plain that we find the product; for by multiplying every part of the multiplicand, it is evident we multiply the whole; and in writing down the products, which are less than ten, or the excess of tens, under the place of the figures multiplied, and carrying the tens to the product of the next place, is only gathering together the similar parts of the respective products, and is therefore the same in effect, as though we wrote down the multiplicand as often as the multiplier expresses, and added them together; for the sum of every column is the product of the figures in the place of that column and the products, collected together are evidently equal to the whole required product. 2. When the multiplier consists of several figures, we find the product of the multiplicand by the unit figure, and then suppose the multiplier divided into parts, and, after the same manner, find the product of the multiplicand by the second figure of the multiplier; but, as the figure by which we are multiplying, stands in the place of tens, the product must be ten times its simple value; and, therefore, the first figure in this product, must be noted in the place of tens, or, which is the same, directly under the figure we are multiplying by. And proceeding in the same manner with all the figures in the multiplier, separately, it is evident we shall multiply all the parts of the multiplicand by all the parts of the multiplier; therefore, these several products being added together, will be equal to the whole required product. 3. The reason of the method of proof, depends upon this proposition, that if two numbers are to be multiplied together, either of them may be made the multiplier or multiplicand, and the product will be the same. EXAMPLE.-In order to illustrate the demonstration, let 568 be multiplied by 476. These numbers may be decomposed and multiplied thus: You see, in the above process, we multiplied through, first by 6 units, then through by 7 tens, or 70, and then by 4 hundreds, or 400, placing the several products underneath, and adding them up. Lastly, the sums of these products are added, making 270368, the total product. The preceding shews, that the multiplicand is taken as many times as there are units in the multiplier. On this principle, was founded the Rule of Multiplication. REMARK.-Multiplication may be proved by casting out the 9s; but is liable to this inconvenience, viz.: The work will always prove right when it is so; but it will not always be right when it proves so. BRIEF METHODS OF MULTIPLYING. 1. When the multiplier is any number of 9s. RULE. To the right of the multiplicand write as many Os as there are 9s in the multiplier-under this new multiplicand write the given one, units, &c. under units, &c.—then subtract, and this difference is just the same as if the general method had been pursued. Operation. 987654000 987654 986666346 REASON. If a number be multiplied by 9, the product is but nine-tenths of the product of the same sum, multiplied by 10; and, as annexing a cypher, to the right hand of the multiplicand, supposes it to be increased tenfold; therefore, subtracting the given multiplicand from the tenfold multiplicand, it is evident that the remainder will be ninefold the given multiplicand,and equal to the product of the same by 9; this will hold true of any number of nines, and this principle may be extended to other numbers. Example. 75964×13 2. When the multiplier is 13, 14, &c. to 19. RULE. Place the multiplier at the right of the multiplicand, with the sign of multiplication between them,-multiply the multiplicand by the unit figure of the multiplier, removing the product one place to the right of the multiplicand; this product and the multiplicand make the total product. 987532 3. When the multiplier is 101, 102, &c. to 109. 227892 RULE. Multiply by the unit figure of the multiplier, remove the product two places to the right of the multiplicand— add together as before for the product. 4. When the multiplier is 111, 112, 113, &c. to 119. RULE. Multiply by the unit figure only of the multiplier, and add to each multiplication the two figures, which stand next at the right hand of that which is multiplied, and to the two last figures, separately, add what you carry. Example. 9417 119 1120623 5. When the multiplier is 21, 31, &c. to 91. RULE.-Multiply by the ten's figure, only, of the multiplier; and set the unit figure of the product under the place of tens; add them all together, and their sum is the total product. Example. 6. When the multiplier is 22, 23, &c. to 29. RULE.-Multiply every figure of the multiplicand by the unit figure of the multiplier, and add to each product twice that figure which stands next at the right hand of the figure, you multiplied; and to twice the last figure add what you carry. 7657 29 222053 27X27-729 29X29-841 |