III. 9 × 8 means that 8 is to be taken 9 times, and is the same thing as 72. This is the _product_ of 9 and 8, and is read nine _into_ eight. IV. 108/6 means that 108 is to be divided by 6, or that you must find out how many sixes there are in 108; and is the same thing as 18. This is the _quotient_ of 108 and 6; and is read a hundred and eight _by_ six. V. When two numbers, or collections of numbers, with the foregoing signs, are the same, the sign = is put between them. Thus, that 7 and 5 make 12, is written in this way, 7 + 5 = 12. This is called an _equation_, and is read, seven _plus_ five _equals_ twelve. It is plain that we may construct as many equations as we please. Thus: 12 7 + 9 - 3 = 12 + 1; --- - 1 + 3 × 2 = 11, 2 and so on. 24. It often becomes necessary to speak of something which is true not of any one number only, but of all numbers. For example, take 10 and 7; their sum[4] is 17, their difference is 3. If this sum and difference be added together, we get 20, which is twice the greater of the two numbers first chosen. If from 17 we take 3, we get 14, which is twice the less of the two numbers. The same thing will be found to hold good of any two numbers, which gives this general proposition,--If the sum and difference of two numbers be added together, the result is twice the greater of the two; if the difference be taken from the sum, the result is twice the lesser of the two. If, then, we take _any_ numbers, and call them the first number and the second number, and let the first number be the greater; we have [4] Any little computations which occur in the rest of this section may be made on the fingers, or with counters. (1st No. + 2d No.) + (1st No. - 2d No.) = twice 1st No. (1st No. + 2d No.) - (1st No. - 2d No.) = twice 2d No. The brackets here enclose the things which must be first done, before the signs which join the brackets are made use of. Thus, 8-(2 + 1) × (1 + 1) signifies that 2 + 1 must be taken 1 + 1 times, and the product must be subtracted from 8. In the same manner, any result made from two or more numbers, which is true whatever numbers are taken, may be represented by using first No., second No., &c., to stand for them, and by the signs in (23). But this may be much shortened; for as first No., second No., &c., may mean any numbers, the letters _a_ and _b_ may be used instead of these words; and it must now be recollected that _a_ and _b_ stand for two numbers, provided only that _a_ is greater than _b_. Let twice _a_ be represented by 2_a_, and twice _b_ by 2_b_. The equations then become (_a_ + _b_) + (_a_ - _b_) = 2_a_, and (_a_ + _b_) - (_a_ - _b_) = 2_b_. This may be explained still further, as follows: