+----+----+----+----+----+----+----+----+----+----+----+----+ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 |108 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |100 |110 |120 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 |110 |121 |132 | +----+----+----+----+----+----+----+----+----+----+----+----+ | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 |108 |120 |132 |144 | +----+----+----+----+----+----+----+----+----+----+----+----+ [8] As it is usual to learn the product of numbers up to 12 times 12, I have extended the table thus far. In my opinion, all pupils who shew a tolerable capacity should slowly commit the products to memory as far as 20 times 20, in the course of their progress through this work. If from this table you wish to know what is 7 times 6, look in the first upright column on the left for either of them; 6 for example. Proceed to the right until you come into the column marked 7 at the top. You there find 42, which is the product of 6 and 7. 51. You may find, in this way, either 6 times 7, or 7 times 6, and for both you find 42. That is, six sevens is the same number as seven sixes. This may be shewn as follows: Place seven counters in a line, and repeat that line in all six times. The number of counters in the whole is 6 times 7, or six sevens, if I reckon the rows from the top to the bottom; but if I count the rows that stand side by side, I find seven of them, and six in each row, the whole number of which is 7 times 6, or seven sixes. And the whole number is 42, whichever way I count. The same method may be applied to any other two numbers. If the signs of (23) were used, it would be said that 7 × 6 = 6 × 7. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●