set their sum under the number that is to be proved.—Lastly, add this last found number and the uppermost line together; then if their sum be the same as that found by the first addition, it may be presumed the work is right. This method of proof is founded on the plain axiom, that “The whole is equal to all its parts taken together,"

EXAMPLE 1.

5

5

3197

6512

8295

Excess of nines

Third Method. Add the figures in the uppermost line together, and find how many nines are contained in their sum. Reject those nines, and set down the remainder towards the right hand directly even with the figures in the line, as in the next example.-Do the same with each of the proposed lines of numbers, setting all these excesses of nines in a column on the right hand, as here 5, 5, 6. excess of 9's in this sum, found as before, be excess of 9's in the total sum 18304, the work is right. Thus, the sum of the right hand column 5, 5, 6, is 16, the excess of which above 9 is 7. Also the sum of the figures in the sum total 18304 is 16, the excess of which above 9 is also 7, the same as the former.*

Then, if the 18304
equal to the

Ex. 5. Add 3426; 9024; 5106; 8390; 1204 together.

Ans. 27150.

6. Add 509267; 235809; 72910; 8392; 420; 21; and 9 together.

Ans. 826828

7. Add 2; 19; 817; 4298; 50916; 730205; 9120634 together. Ans. 9906891. 8. How many days are in the twelve calendar months? Ans. 365.

• This method of proof depends upon a property of the number 9, which, except the number 3, beongs to no other digit whatever; namely, "that any number divided by 9, will leave the same remainder as the sum of its figures or digits divided by 9;" which may be demonstrated in this manner. Demonstration-Let there be any number proposed, as 4658. This, separated into its several parts, becomes 4000+690 + 50 +8. But 4000 4 X 1000 4 X (999 + 1) = 4 × 999 +4. In like manner 6006 × 99 +6; and 50 = 5 × 9+5. Therefore the given number 4658 4 X 999 + 4 + 6 × 99 +6+5×9+5 + 8 = 4 × 999 + 6 × 99+ 5 × 9+4+6+5+8; and 4658÷9= (4 × 999 + 6 × 99+5×9+ 4 + 6 + 5 + 8) ÷ 9. But 4 × 999 +6 × 99 +5 × 9 is evidently divisible by 9, without a remainder; therefore if the given number 4658 be divided by 9, it will leave the same remainder as 4+6+5+8 divided by 9. And the same, it is evident, will hold for any other number whatever. In like manner, the same property may be shown to belong to the number 3; but the preference is usually given to the number 9, on account of its being more convenient in practice.

Now, from the demonstration above given, the reason of the rule itself is evident; for the excess of 9's in two or more numbers being taken separately, and the excess of 9's taken also out of the sum of the former excesses, it is plain that this last excess must be equal to the excess of 9's contained in the total sum of all these numbers; all the parts taken together being equal to the whole.-This rule was first given by Dr Wallis in his Arithmetic, published in the year 1657

9. How many days are there from the 15th day of April to the 24th day of November, both days included?

Ans. 224. 10. An army consisting of 52714 infantry* or foot, 5110 horse, 6250 dragoons, 3927 light-horse, 928 artillery or gunners, 1410 pioneers, 250 sappers, and 406 miners; what is the whole number of men?

Ans. 70995.

OF SUBTRACTION.

SUBTRACTION teaches to find how much one number exceeds another, called their difference or the remainder, by taking the less from the greater. The method of doing which is as follows :

Place the less number under the greater, in the same manner as in Addition, that is, units under units, tens under tens, and so on; and draw a line below them.-Begin at the right hand, and take each figure in the lower line or number from the figure above it, setting down the remainder below it. But if the figure in the lower line be greater than that above it, first borrow or add 10 to 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.†

Ans. 7929231.

6. From 8503602 Take 574371. 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 2573 years ago, and Christ 1840 years ago; then how long before Christ was the birth of Homer? Ans. 733 years.

* The whole body of foot soldiers is denoted by the word Infantry; and all those that charge on horseback, by the word Cavalry.-Some authors conjecture, that the term infantry is derived from a certain Infanta of Spain, who, finding that the army commanded by the king her father had been defeated by the Moors, assembled a body of the people together on foot, with which she engaged and totally routed the enemy. In honour of this event, aud to distinguish the foot soldiers, who were not before held in much estimation, they received the name of Infantry, from her own title of Infanta.

+ 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.

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 number 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.

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

32=

* 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.

280 = 2400=

8 X 4 70 X 4 600 X 4

20000 5000 X 4 22712 5678 X 4

1234567

the multiplicand.

4567

8641969 7 times the mult. 7407402 = 60 times ditto. 6172835 = 500 times ditto. 4938268 4000 times ditto.

+ 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 all the parts of the multiplicand 5638267489 = 4567 times ditto. by all the parts of the multiplier, or the whole of the multipli

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 peculiar 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 demon

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.

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 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 multiplied, and a and b what remain; then 9P + a and 9Q+b will be the numbers themselves, 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 × b is their product; therefore the rule is true.