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the upper one, and then take the lower figure from that sum, setting down the remainder, and carrying 1, for what was borrowed, to the next lower figure, with which proceed as before; and so on till the whole is finished.
TO PROVE SUBTRACTION.
Add the remainder to the less number, or that which is just above it; and if the sum be equal to the greater or uppermost number, the work is right *.
4. From 5331806 take 5073918. 5. From 7020974 take 2766809. 6. From 8503402 take 574271. 7. Sir Isaac Newton was born in the year 1642, and he died in 1727 how old was he at the time of his decease? Ans. 85 years.
8. Homer was born 2568 years ago, and Christ 1835 years ago: then how long before Christ was the birth of Homer? Ans. 733 years.
9. Noah's flood happened about the year of the world 1656, and the birth of Christ about the year 4000: then how long was the flood before Christ?
Ans. 2344 years.
10. The Arabian or Indian method of notation was first known in England about the year 1150: then how long is it since to this present year 1840?
Ans. 690 years.
11. Gunpowder was invented in the year 1330: how long was that before the invention of printing, which was in 1441 ? Ans. 111 years.
12. The mariner's compass was invented in Europe in the year 1302: how long was that before the discovery of America by Columbus, which happened in 1492? Ans. 190 years.
MULTIPLICATION is a compendious method of Addition, teaching how to find the amount of any given number when repeated a certain number of times; as, 4 times 6, which is 24.
The number to be multiplied, or repeated, is called the Multiplicand.—The number you multiply by, or the number of repetitions, is the Multiplier.— And the number found, being the total amount, is called the Product. Also, both the multiplier and multiplicand are, in general, named the Terms or Factors.
* The reason of this method of proof is evident; for if the difference of two numbers be added to the less, it must manifestly make up a sum equal to the greater.
Before proceeding to any operations in this rule, it is necessary to commit thoroughly to memory the following Table, of all the products of the first 12 numbers, commonly called the Multiplication Table, or sometimes the Table of Pythagoras, from its alleged inventor.
To multiply any Given Number by a Single Figure, or by any Number not
* Set the multiplier under the units figure or right hand place of the multiplicand, and draw a line below it. Then, beginning at the right hand, multiply every figure in this by the multiplier. Count how many tens there are in the product of every single figure, and set down the remainder directly under the figure that is multiplied; and if nothing remains, set down a cipher. Carry as many units or ones as there are tens counted, to the product of the next figures; and proceed in the same manner till the whole is finished.
To multiply by a Number consisting of Several Figures.
Set the multiplier below the multiplicand, placing them as in Addition, namely, units under units, tens under tens, &c. drawing a line below it. Multiply the whole of the multiplicand by each figure of the multiplier, as in the last article; setting down a line of products for each figure in the multiplier, so as that the first figure of each line may stand straight under the figure multiplying by. Add all the lines of products together, in the order in which they stand, and their sum will be the answer or whole product required. It will, of course, be always best to take that number as the multiplier which has the fewest effective figures.
TO PROVE MULTIPLICATION.
THERE are three different ways of proving multiplication, which are as below: First Method. Make the multiplicand and multiplier change places, and multiply the latter by the former in the same manner as before. Then if the product found in this way be the same as the former, the number is right.
Second Method. † Cast all the 9s out of the sum of the figures in each of the two factors, as in Addition, and set down the remainders. Multiply these two remainders together, and cast the 9s out of the product, as also out of the whole product or answer of the question, reserving the remainders of these last two, which remainders must be equal when the work is right.
Note. It is common to set the four remainders within the four angular spaces of a cross, as in the example below.
Third Method. Multiplication is also very naturally proved by Division; for the product divided by either of the factors, will evidently give the other. But this cannot be practised till the rule of division is learned.
Having placed the multiplier under the multiplicand as in the previous rule, multiply by the left-hand figure, setting down the product as if that figure were
* After having found the product of the multiplicand by the first figure of the multiplier, as in the former case, the multiplier is supposed to be divided into parts, and the product is found for the second figure in the same manner: but as this figure stands in the place of tens, the product must be ten times its simple value; and therefore the first figure of this product must be set in the place of tens; or, which is the same 1234567 the multiplicand. thing, directly under the figure multiplying by. And proceeding in this manner separately with all the figures of the multiplier, it is evident that we shall multiply all the parts of the multiplicand by all the parts of the multiplier, or the whole of the multiplicand by the whole of the multiplier: therefore these several products being added together, will be equal to the whole required product; 5638267489 = 4567 times ditto. as in the example annexed.
7 times the mult. 60 times ditto. 500 times ditto.
This method of proof is derived from the peculiar property of the number 9, mentioned in the proof of Addition, and the reason for the one includes that of the other. Another more ample demonstration of this rule may, however, be as follows:-Let p and q denote the number of 9s in the factors to be multiplied, and a and b what remain; then 9P+ a and 9Q + will be the numbers themselves, and their product is (9P × 9Q) + (9p × b) + (9q × a) +(ab); but the first three of these products are each a precise number of 9s, because their factors, either one or both, are so: these therefore being cast away, there remains only a × b; and if the 9s also be cast out of this, the excess is the excess of 9s in the total product: but a and b are the excesses in the factors themselves, and a × b is their product; therefore the rule is true. This mode of proof, however, is not an entire check against the errors that might arise from a transposition of figures, or other compensation of errors.
the only multiplier. Proceed to the next figure of the multiplier, putting the first figure one place to the right of the right-hand figure of the last product, in the line below proceed to the next, carrying out the first figure one place more to the right; and so on till all the partial products are made. Add up as in the last rule
Multiply 123456789 by 3.
Multiply 123456789 by 12.
CONTRACTIONS IN MULTIPLICATION.
I. When there are Ciphers in the Factors.
If the ciphers be at the right-hand of the numbers; multiply the other figures only, and annex as many ciphers to the right-hand of the whole product, as are in both the factors. When the ciphers are in the middle parts of the multiplier; neglect them as before, only taking care to place the first figure of every line of products exactly under the figure by which you are multiplying.
*This is the eastern mode of putting down the work; and is evidently productive of the same final result, only that the progressive partial multiplications are taken in an inverse order.
3. Multiply 81503600 by 7030. 4. Multiply 9030100 by 2100. 5. Multiply 8057069 by 70050.
II. When the Multiplier is the product of two or more Numbers in the Table ;
* Multiply by each of those parts successively, instead of the whole number
7. There was an army composed of 104 † battalions, each consisting of 500 men; what was the number of men contained in the whole ?
8. A convoy of ammunition ‡ bread, consisting of 250 waggons, and each waggon containing 320 loaves, having been intercepted and taken by the enemy, what is the number of loaves lost? Ans. 80000.
DIVISION is a kind of compendious method of Subtraction, teaching to find how often one number is contained in another, or may be taken from it, which is the same thing.
The number to be divided, is called the Dividend. The number to divide by, is the Divisor: and the number of times the dividend contains the divisor is called the Quotient. Sometimes there is a Remainder left, after the division is finished.
The usual manner of placing the terms, is, the dividend in the middle, having the divisor on the left hand, and the quotient on the right, each separated by a curve line; as, to divide 12 by 4, the quotient is 3,
The chief advantage of this process is, that it assimilates with the method employed in contracted decimals, in the extractions of roots in duodecimals, and in Algebra.
* The reason of this rule is obvious; for any number multiplied by the component parts of another, must give the same product as if it were multiplied by that number at once. Thus, in the 1st example, 7 times the product of 8 by the given number, make 56 times the same number, as plainly as 7 times 8 make 56.
† A battalion is a body of foot, consisting of 500, or 600, or 700 men, more or less.
The ammunition bread is that which is provided for, and distributed to, the soldiers; the usual allowance being a loaf of 6 pounds to every soldier, once in 4 days.