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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*.
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 2543 years ago, and Christ 1810 years ago : then how long before Christ was the birth of Homer?
Ans. 733 years. 9. Noah's dood 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
Ans. 660 years. it since to this present year 1810?
11. Gunpowder was invented in the year 1330 : then how long was this before the invention of printing, which was in 1441 ?
Ans. Il years. 12. The mariner's compass was invented in Europe in the year 1302 : then how long was that before the discovery of America by Columbus, which happened in 1492 ?
Ans. 190 years.
* The reason of this method of proof is evident; for if the differ. ence of two numbers be added to the less, it must manifestly make up a sum equal to the greater.
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 Mul iplicand.–The number you multiply by, or the number of repetitions, is the Multiplier. And the number found, being the otal amount, is called the Produci.-Also, both the multiplier and multiplicand are, in general named the Terms or Factors.
Before proceeding to any operations in this rule, it is necessary to learn off very perfectly the following Table, of all the products of the first 12 numbers, commonly called the Multiplication Table, or sometimes Pythagoras's Table, from its inventor.
To multiply any Given Number by a Single Figure, or by any
Number not more than 12.
* 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 dowli the remainder directly under the figure that is multiplied; and if nothing remains, set down a cipher.-Carry as inany units or nes as there are tens counted, to the product of the next figures ; and proceed in the same inanner till the whole is finished.
Multiply 9876543210 the Multiplicand.
2 the Multiplier.
19753086420 the Product.
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 lens, &c. drawing a line below1.-Multiply the whole of the multiplicand by each figure of the multiplier, as in the last article ;
* The reason of this rule is the same as for the process in Addition, in which 1 is car. ried for every 10, to the next place, gradually as the several products are poduced one after another, instead of setting them all down one below each other, as in the annesed example.
32 8 X 4 280 70 x 4 2400 = 600 X 4, 20000=5000 X 4
22712 =5678 X 4
† After having found the produce 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
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 as they stand, and their sum will be the answer or whole product required.
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 ihe product found in this way be the same as the former, the number is right.
Second Method.-*Cast all the 9's 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 e's out of the product, as also out of
is the same thing, directly under the figure multiplied by. And pro-
7407 402 = GO times ditto.
* This method of proof is derived from the pecnliar property of the number 9, mentioned in the proof of Addition, and the reason for the one may serve for that of the other. Another more ample demonstration of this rule may be as follows:-Let P and Q denote the number of 9's in the factors to be multiplied, and a and b what rmain; then 9 Pta and 9 Q+ b will be the numbers themselves, and their product is (9PX90)
+ (9PX6)+(9QX a) + (a X b); but the first three of these products are each a precise number of O's because their factors are so, either one or both: these therefore being cast away, there remains only a xb; and if the 9's also be cast out of this, the excess is the excess of 9's in the tetai prolust: but a and bare the escesses in the factors themselves, and a x bis their product; therefore the rule is true.
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 Icarned.