§. 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, till this present year 1840 ? Ans 690 years. 11. Gunpowder was invented in the year 1330; then how long was this before the invertion of printing, which was in 1441 ? Ans. 111 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 14927 Ans. 190 years.

OF MULTIPLICATION.

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 umes 6, which is 24.

The number to be multiplied, or repeated, is called the Multiplicand.—The rumber you multiply by, or number of repetitions, is the Multiplier.—And the number found, being the total amount, is called the Product.-Also, both the mut pler 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, metimes called the Multiplication Table, or Pythagoras's Table, from its

72 *

To multiply any Given Number by a Single Figure, or by any Number more than 12.

* Set the multiplier under the units figure, or right-hand place, of the plicand, and draw a line below it.—Then, beginning at the right-hand, m every figure in this by the multiplier.-Count how many tens there are product of every single figure, and set down the remainder directly und figure that is multiplied; and if nothing remains, set down a cipher.—Ca many units or ones, as there are tens counted, to the product of the next f 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 A namely, units under units, tens under tens, &c. drawing a line below itply the whole of the multiplicand by each figure of the multiplier, as in article; setting down a line of products for each figure in the multiplie that the first figure of each line may stand straight under the figure mu by. Add all the lines of products together, in the order as they stand, a sum will be the answer or whole product required.

TO PROVE MULTIPLICATION.

THERE are three different ways of proving Multiplication, which are a First Method.-Make the multiplicand and multiplier change pla multiply the latter by the former in the same manner as before. The product found in this way be the same as the former, the number is rig Second Method.―‡ Cast all the 9's out of the sum of the figures in ea

5678

• The reason of this rule is the same as for the process in Addition, in which I is carried for every 10, to the next place, gradually as the several products are produced, one after another, instead of setting them all down below each other, as in the annexed Example.

1234567

4

32 =

280 = 2400: 20000 E

22712 E

the m

4567

+ 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 10 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 thing directly under the figure multiplied by And proceeding in this manner separately with all the figures of the multiplier, it is evident that we shall multiply ail the parts of the multiplicand 5638267489 = 4567 tim by all the parts of the multiplier, or the whole of the multipli

8641969- 7 tim 7407402 = 60 tim 6172835 = 500 tim 4938268 = 4000 t

cand by the whole of the multiplier: therefore these several products being added tog be equal to the whole required product: as in the example annexed.

This method of proof is derived from the pecular property of the number 9, mentione of Addition, and the reason for the one may serve for that of the other. Another more a

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, till this present year 1840 ? Ans 690 years. 11. Gunpowder was invented in the year 1330; then how long was this 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; then how long was that before the discovery of America by Columbus, which happened in 1492? Ans. 190 years.

OF MULTIPLICATION.

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 rumber you multiply by, or 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.

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, sometimes called the Multiplication Table, or Pythagoras's Table, from its inventor.

9 18 27 36 45 54 63 72 | 81 | 90 | 99 |108

56

64 72 80 88 96

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 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 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 the 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

5678

4

The reason of this rule is the same as for the process in Addition, in which I is carried for every 10, to the next place, gradually as the several products are produced, one after another, instead of setting them all down below each other, as in the annexed Example.

1234567

4567

the multiplicand.

+ 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 10 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 thing directly under the figure multiplied by And proceeding in this manner separately with all the figures of the multiplier, it is evident that we shall multiply ail the parts of the multiplicand 5638267489 = 4567 times ditto. by all the parts of the multiplier, or the whole of the multipli

8641969 = 7407402 = 6172835 = 4938268

7 times the mult 60 times ditto. 500 times ditto. 4000 times ditto.

d

cand by the whole of the multiplier: therefore these several products being added together, will be equal to the whole required product: as in the example annexed.

This method of proof is derived from the pecular 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 demou

two factors, as in Addition, and set down the remainders. Multiply these two remainders together, and cast the 9's 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.

CONTRACTIONS In Multiplication.

1. 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 product, as are in both the factors. And 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 the multiplication is made.

stration of this rule may be as follows:-Let P and Q denote the number of 9's in the factors to be multipled, and a and b what remain; then 9P+ a and 92 + 6 will be the numbers them. selves, and their product is (9P × 9Q) + (9P × b) + (9Q × a) + (a × b); but the first three of these products are each a precise number of 9's, because their factors are so, either one or both: these therefore being cast away, there remains only a × b; and if the 9's be also cast out of this, the excess is the excess of 9's in the total product: but a and b are the excesses in the factors themselves, and a × 6 is their product; therefore the rule is true.