38. The use of the brackets must here be noticed. They mean, that the expression contained inside them must be used exactly as a single letter would be used in the same place. Thus, _pa_ signifies that _a_ is taken _p_ times, and (_m_ + _n_)_a_, that _a_ is taken _m_ + _n_ times. It is, therefore, a different thing from _m_ + _na_, which means that _a_, after being taken _n_ times, is added to _m_. Thus (3 + 4) × 2 is 7 × 2 or 14; while 3 + 4 × 2 is 3 + 8, or 11. 39. When one number is taken away from another, the number which is left is called the _difference_ or _remainder_. The process of finding the difference is called SUBTRACTION. The number which is to be taken away must be of course the lesser of the two. 40. The process of subtraction depends upon these two principles. I. The difference of two numbers is not altered by adding a number to the first, if you add the same number to the second; or by subtracting a number from the first, if you subtract the same number from the second. Conceive two baskets with pebbles in them, in the first of which are 100 pebbles more than in the second. If I put 50 more pebbles into each of them, there are still only 100 more in the first than in the second, and the same if I take 50 from each. Therefore, in finding the difference of two numbers, if it should be convenient, I may add any number I please to both of them, because, though I alter the numbers themselves by so doing, I do not alter their difference. II. Since 6 exceeds 4 by 2, and 3 exceeds 2 by 1, and 12 exceeds 5 by 7, 6, 3, and 12 together, or 21, exceed 4, 2, and 5 together, or 11, by 2, 1, and 7 together, or 10: the same thing may be said of any other numbers. 41. If _a_, _b_, and _c_ be three numbers, of which _a_ is greater than _b_ (40), I. leads to the following, (_a_ + _c_) - (_b_ + _c_) = _a_ - _b_. Again, if _c_ be less than _a_ and _b_, (_a_ - _c_) - (_b_ - _c_) = _a_ - _b_. The brackets cannot be here removed as in (36). That is, _p_- (_q_-_r_) is not the same thing as _p_-_q_- _r_. For, in the first, the difference of _q_ and _r_ is subtracted from _p_; but in the second, first _q_ and then _r_ are subtracted from _p_, which is the same as subtracting as much as _q_ and _r_ together, or _q_ + _r_. Therefore _p_-_q_-_r_ is _p_-(_q_ + _r_). In order to shew how to remove the brackets from _p_ -(_q_-_r_) without altering the value of the result, let us take the simple instance 12-(8-5). If we subtract 8 from 12, or form 12-8, we subtract too much; because it is not 8 which is to be taken away, but as much of 8 as is left after diminishing it by 5. In forming 12-8 we have therefore subtracted 5 too much. This must be set right by adding 5 to the result, which gives 12-8 + 5 for the value of 12-(8-5). The same reasoning applies to every case, and we have therefore, _p_ - (_q_ + _r_) = _p_ - _q_ - _r_. _p_ - (_q_ - _r_) = _p_ - _q_ + _r_. By the same kind of reasoning,