millions, seventy-six thousand and thirty-four; one hundred and eleven millions, six hundred and fifty thousand and fifty; three hundred and twenty-six millions, seven thousand, nine hundred and ninety-one; one thousand seven hundred and ten millions, one thousand seven hundred and ten; one billion, three hundred thousand and five.

SUBTRACTION.

16. Subtraction is the method of finding what number remains when a smaller number is taken from a greater number.

The number found by subtracting the smaller of two numbers from the greater is called the Remainder.

17. There are two kinds of Subtraction, SIMPLE and COMPOUND, which differ from each other in precisely the same way, in which Simple and Compound Addition differ from each other..

18. The sign-, minus, placed between two numbers, signifies that the second number is to be subtracted from the first number.

SIMPLE SUBTRACTION.

19. RULE. Place the less number under the greater number, so that units may come under units, tens under tens, hundreds under hundreds, and so on; then draw a straight line under the lower line.

Take, if possible, the number of units in each figure of the lower line from the number of units in each figure of the upper line which stands immediately over it, and put the remainder below the line just drawn, units under units, tens under tens, and so on: but if the units in any figure in the lower line exceed the number of units in the figure above it, add ten to the upper figure, and then take the number of units in the lower figure from the number in the upper figure thus increased; put the remainder down as before, and then carry one to the next figure of the lower line. The entire difference or remainder, so marked down, will be the difference or remainder of the given numbers.

20. Ex. Subtract 4938 from 5123.

Proceeding by the Rule given above, we obtain

5123

4938

so that the remainder is one hundred and eighty-five (185).

The reason for the Rule will appear from the following considerations. We cannot take 8 units from 3 units, we therefore add 10 units to the 3 units, which are thus increased to 13 units; and taking 8 units from 13 units we have 5 units left; we therefore place 5 under the column of units: but having added 1 ten units to the upper number, we must add the same number of units (1 ten units) to the lower number, so that the difference between the two numbers may not be altered; and adding 1 ten units to the 3 ten units in the lower number, we obtain 4 tens or 40 instead of 3 tens or 30.

Again, we cannot take 4 tens from 2 tens; we therefore add 10 tens or 1 hundred to the 2 tens, which thus become 12 tens or 120; and then taking 4 tens or 40 from 12 tens or 120, we have 8 tens or 80 remaining; we therefore place 8 under the column of tens: but having added 1 hundred to the upper number, we must add 1 hundred to the lower number for the reason given above; and adding 1 hundred to the 9 hundreds in the lower number, we obtain 10 hundreds or 1000 instead of 900.

Again, we cannot take 10 hundreds from 1 hundred, and we therefore add 10 hundreds or 1 thousand to the 1 hundred, which thus becomes 11 hundreds or 1100: and taking 10 hundreds or 1000 from 11 hundreds or 1100, we have 1 hundred or 100 left; we therefore place 1 under the column of hundreds: but having added 10 hundreds or 1 thousand to the upper number, we must add 1 thousand to the lower number for the reason given above; and adding 1 thousand to the 4 thousands in the lower number, we obtain 5 thousands or 5000;

5000 taken from 5000 leaves 0;

therefore the whole difference or remainder is 185.

21. The above Example might have been worked thus, putting down at full length the local values of the figures:

5123= 5000 +100+ 20 +3

(collecting the first 10 with the 100, and the second 10 with the 3,) 4938-4000+900+30 +8.

Therefore subtracting the columns, thousands from thousands, &c. we get the remainder or difference

=100+80+5

Note. The truth of all results in Subtraction may be proved by adding the less number to the difference or remainder; if this sum equals the larger number, the result obtained by subtraction may be presumed to be correct.

(12) Find the difference between 6543756 and 412848; 7863927 and

826957; 303233334 and 192001222.

(13) How much greater is 164326289 than 48476798?

10000001000 than 7077070077 ? 7559030640021 than 6990040005679 ?

(14) Take two thousand and nine, from ten thousand and ninetysix; three thousand and eight, from seven thousand, nine hundred and forty-four.

(15) Required the difference between four and four millions; also between one hundred millions and three hundred thousand.

(16) Subtract five hundred and eighty-four thousand and seventy-six, from fifteen millions, one hundred thousand and three.

22. The following method of expressing numbers was used by the Romans, and it is still in occasional, though not in common use, among ourselves. They represented the number one by the character I; five by V; ten by X; fifty by L; one hundred by C; five hundred by D or Iɔ; one thousand by M or CIɔ.

All other numbers were formed by a combination of the above characters, subject to the following Rules:

First; When a character was followed by one of equal or less value, the whole expression denoted the sum of the values of the single characters; for instance, II stood for 2; III for 3; VI for 6; VIII for 8; LV for 55; LXXVII for 77; CCXI for 211.

Secondly; When a character was preceded by one of less value, the

whole expression denoted the difference of the values of the single characters; for instance, IV stood for 5-1, or 4; IX for 10-1, or 9; XIX for 10+10-1, or 19; XL for 50-10, or 40; XC for 100-10, or 90. Thirdly; Every annexed to Io increased the value of the latter tenfold; for instance, I stood for 5000; Ɔ for 50000; and so forth. And every C prefixed and annexed to CIO increased the value of the latter tenfold; for instance, CCIɔɔ stood for 10000; CCC for 100000; and so forth.

Fourthly; A line drawn over a character or characters increased the value of the latter a thousandfold; for instance, V stood for 5000; °C for 100000; IX for 9000; and so forth.

It follows then that either XXXXVI or XLVI will represent 46: and that either M.DCCC.LIV, or CI.CCCLIV, or I.DCCCLIIII will represent 1854.

Ex. IV.

(1) Express in Roman characters, thirty; forty-eight; fifty-nine; 222; 6000; 1843.

(2) Express in words, and also in Arabic figures, the values of XXIII; LXIX; CCXVIII; VI; CLDCIII; MMC.

MULTIPLICATION.

23. MULTIPLICATION is a short method of finding the sum of any given number repeated as often as there are units in another given number: thus, when 3 is multiplied by 4, the number produced by the multiplication is the sum of 3 repeated 4 times, which sum is equal to 3+3+3+3 or 12.

The number to be repeated or added to itself, is called the MULTIPLI

CAND.

The number which shews how often the multiplicand is to be repeated or added to itself, is called the MULTIPLIER.

The number found by multiplication is called the PRODUCT.

The multiplicand and multiplier are sometimes called 'FACTORS,' because they are factors or makers of the product.

24. Multiplication is of two kinds, SIMPLE and COMPOUND. It is termed Simple Multiplication, when the multiplicand is either an abstract number, or a concrete number of one denomination.

It is termed Compound Multiplication, when the multiplicand contains numbers of more than one denomination, but all of the same kind.

25. The sign ×, placed between two numbers, signifies that the numbers are to be multiplied together.

26. The following Table ought to be learned correctly:

12

24 36 48 60 72 84 96 108 120 132 144

In the above Table, the second line from the top shews the product of each of the numbers, 1, 2, 3, 4, &c. 11, 12, in the first line, when multiplied by 2; the several products being placed under the respective numbers of the line above, from the multiplication of which they arise: the third line shews the several products, when the figures in the first line are respectively multiplied by 3: and so on.

Note. One of the factors, namely the multiplier, must necessarily be an abstract number'; since it would be absurd to speak of 6 shillings multiplied by 4 shillings. We can multiply 6 shillings by 4, i. e. we can find how many shillings there are in four times six shillings; but there is no meaning in 6 shillings multiplied by 4 shillings.

SIMPLE MULTIPLICATION.

27. RULE. Place the multiplier under the multiplicand, units under units, tens under tens, and so on. Multiply each figure of the multiplicand, beginning with the units, by the figure in the units' place of the multiplier (by means of the table given for Multiplication); set down and carry as in Addition. Then multiply each figure of the multiplicand,