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.
- 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