52. To take any quantity a number of times, it will be enough to take every one of its parts the same number of times. Thus, a sack of corn will be increased fifty-fold, if each bushel which it contains be replaced by 50 bushels. A country will be doubled by doubling every acre of land, or every county, which it contains. Simple as this may appear, it is necessary to state it, because it is one of the principles on which the rule of multiplication depends. 53. In order to multiply by any number, you may multiply separately by any parts into which you choose to divide that number, and add the results. For example, 4 and 2 make 6. To multiply 7 by 6 first multiply 7 by 4, and then by 2, and add the products. This will give 42, which is the product of 7 and 6. Again, since 57 is made up of 32 and 25, 57 times 50 is made up of 32 times 50 and 25 times 50, and so on. If the signs were used, these would be written thus: 7 × 6 = 7 × 4 + 7 × 2. 50 × 57 = 50 × 32 + 50 × 25. 54. The principles in the last two articles may be expressed thus: If _a_ be made up of the parts _x_, _y_, and _x_, _ma_ is made up of _mx_, _my_, and _mz_; or, if _a_ = _x_ + _y_ + _z_. _ma_ = _mx_ + _my_ + _mz_, or, _m_(_x_ + _y_ + _z_) = _mx_ + _my_ + _mz_. A similar result may be obtained if _a_, instead of being made up of _x_, _y_, and _z_, is made by combined additions and subtractions, such as _x_ + _y_-_z_, _x_- _y_ + _z_, _x_-_y_-_z_, &c. To take the first as an instance: Let _a_ = _x_ + _y_ - _z_, then _ma_ = _mx_ + _my_ - _mz_. For, if _a_ had been _x_ + _y_, _ma_ would have been _mx_ + _my_. But since _a_ is less than _x_ + _y_ by _z_, too much by _z_ has been repeated every time that _x_ + _y_ 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_. 55. 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 | ● ● ● ● ● ● ● | | ● ● ● ● ● ● ● | | ● ● ● ● ● ● ● | +---------------+