beginning with the units, by the figure in the tens' place of the multiplier, placing the first figure so obtained under the tens of the line above, the next figure under the hundreds, and so on. Proceed in the same way with each succeeding figure of the multiplier. Then add up all the results thus obtained, by the rule of Simple Addition.

Note. If the multiplier does not exceed 12, the multiplication can be effected easily in one line, by means of the Table given above.

28. Ex. Multiply 7654 by 397.

Proceeding by the Rule given above, we obtain

7654

397

53578

68886

22962

3038638

The reason for the Rule will appear from the following considerations.

When 7654 is to be multiplied by 7, we first take 4 seven times, which by the Table gives 28, i.e. 8 units and 2 tens; we therefore place down 8 in the units' place and carry on the 2 tens: again, 5 tens taken 7 times give 35 tens, to which add 2 tens, and we obtain 37 tens, or 7 tens and 3 hundreds; we put down 7 in the tens' place, and carry on 3 hundreds: again, 6 hundreds taken 7 times give 42 hundreds, to which add 3 hundreds, and we obtain 45 hundreds, or 4 thousands and 5 hundreds; we put down 5 in the hundreds' place, and carry on the 4 thousands: again, 7 thousands taken 7 times give 49 thousands, to which we add the 4 thousands, thus obtaining 53 thousands, which we write down.

Next, when we multiply 7654 by the 9, we in fact multiply it by 90; and 4 units taken 90 times give 360 units, or 3 hundreds, 6 tens, and 0 units; therefore, omitting the cypher, we place the 6 under the tens' place, and carry on the 3 to the next figure, and proceed with the operation as in the line above.

When we multiply 7654 by the 3, we in fact multiply by 300; and 4 multiplied by 300 gives 1200, or 1 thousand, 2 hundreds, 0 tens, and O units; therefore, omitting the cyphers, we place the first figure 2 under the hundreds' place, and proceed as before. Then adding up the three lines of figures which we have just obtained, we obtain the product of 7654 by 397.

29. The above Example might have been worked thus, putting down

at full length the local values of the figures:

63 × 10000+

21 × 100000+18 x 10000+

49 × 1000+ 42 × 100+ 35 × 10+28
54 × 1000+45 × 100+36 × 10
15× 1000+ 12 × 100

21 x 100000+81 x 10000+118 × 1000+ 99 × 100 + 71 × 10+28

20 x 100000+ 1×100000

+8x100000+1 x 10000

+ 1x 100000+1 × 10000+ 8×1000

+ 9×1000+ 9×100

+7x100+1x10

+2×10+8

2000000+10 x 100000 +2 × 10000+17 × 1000+ 16 × 100 +3 × 10+8

=2000000+1000000 + 2 x 10000+10 × 1000+7 × 1000+ 10 × 100+ 6 × 100+3×10+8

=

3000000 + 2 × 10000 + 1 × 10000 + 7 × 1000 +1 × 1000 + 6x 100 + 3 × 10+8 =3000000+3 × 10000+8 × 1000+ 600 +30 +8

30. If the multiplier or multiplicand, or both, end with cyphers, we may omit them in the working; taking care to affix to the product as many cyphers as we have omitted from the end of the multiplier or multiplicand, or both. Thus, if 263 be multiplied by 6200, and 570 be multiplied by 3200, we have

The reason is clear: for, in the first case, when we multiply by the 2, in fact we multiply by 200; and 3 multiplied by 200 gives 600: in the second case, the 7 multiplied by the 2 is the same as 70 multiplied by 200; and 70 multiplied by 200 gives 14000.

31. If the MULTIPLIER contain any cypher in any other place, then, in multiplying by the different figures of the multiplier, we may pass over the cypher; taking care, however, when we multiply by the next figure, to place the first figure arising from that multiplication, under the

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 I; one thousand by M or CIO.

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 Ɔ annexed to I increased the value of the latter tenfold; for instance, I stood for 5000; IƆƆƆ for 50000; and so forth. And every C prefixed and annexed to CI increased the value of the latter tenfold; for instance, CCI stood for 10000; CCC

100000; and so forth.

for

Fourthly; A line drawn over a character or characters increased the value of the latter a thousand-fold; 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 Cl ̧.ƆCCCLIV, 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.

It is

24. Multiplication is of two kinds, SIMPLE and COMPOUND. 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.