25. Suppose a number of sealed packets, marked _a_, _b_, _c_, _d_, &c., on the outside, each of which contains a distinct but unknown number of counters. As long as we do not know how many counters each contains, we can make the letter which belongs to each stand for its number, so as to talk of _the number a_, instead of the number in the packet marked _a_. And because we do not know the numbers, it does not therefore follow that we know nothing whatever about them; for there are some connexions which exist between all numbers, which we call _general properties_ of numbers. For example, take any number, multiply it by itself, and subtract one from the result; and then subtract one from the number itself. The first of these will always contain the second exactly as many times as the original number increased by one. Take the number 6; this multiplied by itself is 36, which diminished by one is 35; again, 6 diminished by 1 is 5; and 35 contains 5, 7 times, that is, 6 + 1 times. This will be found to be true of any number, and, when proved, may be said to be true of the number contained in the packet marked _a_, or of the number _a_. If we represent a multiplied by itself by _aa_,[5] we have, by (23) _aa_ - 1 ------------- = _a_ + 1. _a_ - 1 [5] This should be (23) _a_ × _a_, but the sign × is unnecessary here. It is used with numbers, as in 2 × 7, to prevent confounding this, which is 14, with 27. 26. When, therefore, we wish to talk of a number without specifying any one in particular, we use a letter to represent it. Thus: Suppose we wish to reason upon what will follow from dividing a number into three parts, without considering what the number is, or what are the parts into which it is divided. Let _a_ stand for the number, and _b_, _c_, and _d_, for the parts into which it is divided. Then, by our supposition, _a_ = _b_ + _c_ + _d_. On this we can reason, and produce results which do not belong to any particular number, but are true of all. Thus, if one part be taken away from the number, the other two will remain, or _a_ - _b_ = _c_ + _d_. If each part be doubled, the whole number will be doubled, or 2_a_ = 2_b_ + 2_c_ + 2_d_. If we diminish one of the parts, as _d_, by a number _x_, we diminish the whole number just as much, or _a_ - _x_ = _b_ + _c_ + (_d_ - _x_). 27. EXERCISES. What is _a_ + 2_b_ - _c_, where _a_ = 12, _b_ = 18, _c_ = 7?--_Answer_, 41. _aa_ - _bb_ What is ----------- , _a_ - _b_ where _a_ = 6 and _b_ = 2?--_Ans._ 8. What is the difference between (_a_ + _b_)(_c_ + _d_) and _a_ + _bc_ + _d_, for the following values of _a_, _b_, _c_, and _d_? _a_ | _b_ | _c_ | _d_ | _Ans._ 1 | 2 | 3 | 4 | 10 2 | 12 | 7 | 1 | 25 1 | 1 | 1 | 1 | 1 SECTION II. ADDITION AND SUBTRACTION.