has been repeated;—that is, mz too much has been taken; consequently, ma is not mx + my , but mx + my - mz . Similar reasoning may be applied to other cases, and the following results may be obtained:
m(a + b + c - d) = ma + mb + mc - md.
| a ( a - b ) | = | aa - ab . |
|---|---|---|
| b ( a - b ) | = | ba - bb . |
| 3(2 a - 4 b ) | = | 6 a - 12 b . |
| 7 a (7 + 2 b ) | = | 49 a + 14 ab . |
| ( aa + a + 1) a | = | aaa + aa + a . |
| (3 ab - 2 c )4 abc | = | 12 aabbc - 8 abcc . |
- There is another way in which two numbers may be multiplied together. Since 8 is 4 times 2, 7 times 8 may be made by multiplying 7 and 4, and then multiplying that product by 2. To shew this, place 7 counters in a line, and repeat that line in all 8 times, as in figures I. and II.
| I. | |||||||
|---|---|---|---|---|---|---|---|
| A | ● | ● | ● | ● | ● | ● | ● |
| ● | ● | ● | ● | ● | ● | ● | |
| ● | ● | ● | ● | ● | ● | ● | |
| ● | ● | ● | ● | ● | ● | ● | |
| B | ● | ● | ● | ● | ● | ● | ● |
| ● | ● | ● | ● | ● | ● | ● | |
| ● | ● | ● | ● | ● | ● | ● | |
| ● | ● | ● | ● | ● | ● | ● |
| II. | ||||||
|---|---|---|---|---|---|---|
| ● | ● | ● | ● | ● | ● | ● |
| ● | ● | ● | ● | ● | ● | ● |
| ● | ● | ● | ● | ● | ● | ● |
| ● | ● | ● | ● | ● | ● | ● |
| ● | ● | ● | ● | ● | ● | ● |
| ● | ● | ● | ● | ● | ● | ● |
| ● | ● | ● | ● | ● | ● | ● |
| ● | ● | ● | ● | ● | ● | ● |
The number of counters in all is 8 times 7, or 56. But (as in fig. I.) enclose each four rows in oblong figures, such as a and b. The number in each oblong is 4 times 7, or 28,