It remains, then, to find how many times 468 contains 18. Proceed exactly as before. Observe that 46 contains 18 more than twice, and less than 3 times; therefore, 460 contains it more than 20, and less than 30 times (77); as does also 468. Subtract 18 20 times from 468, that is, subtract 360; the remainder is 108. Therefore, 468 contains 18 20 times, and as many more as 108 contains it. Now, 108 is found to contain 18 6 times exactly; therefore, 468 contains it 20 + 6 times, and 4068 contains it 200 + 20 + 6 times, or 226 times. If we write down the process that has been followed, without any explanation, putting the divisor, dividend, and quotient, in a line separated by parentheses it will stand, as in example(A).
Let it be required to divide 36326599 by 1342 (B).
| A. | B. | ||||
| 18) | 4068 | (200 + 20 + 6 | 1342) | 36326599 | (20000 + 7000 + 60 + 9 |
| 3600 | 26840000 | ||||
| 468 | 9486599 | ||||
| 360 | 9394000 | ||||
| 108 | 92599 | ||||
| 108 | 80520 | ||||
| 0 | 12079 | ||||
| 12078 | |||||
| 1 | |||||
As in the previous example, 36326599 is separated into 36320000 and 6599; the first four figures 3632 being separated from the rest, because it takes four figures from the left of the dividend to make a number which is greater than the divisor. Again, 36320000 is found to contain 1342 more than 20000, and less than 30000 times; and 1342 × 20000 is subtracted from the dividend, after which the remainder is 9486599. The same operation is repeated again and again, and the result is found to be, that there is a quotient 20000 + 7000 + 60 + 9, or 27069, and a remainder 1.
Before you proceed, you should now repeat the foregoing article at length in the solution of the following questions. What are
| 10093874 | , | 66779922 | , | 2718218 | ? |
| 3207 | 114433 | 13352 |
the quotients of which are 3147, 583, 203; and the remainders 1445, 65483, 7762.
79. In the examples of the last article, observe, 1st, that it is useless to write down the ciphers which are on the right of each subtrahend, provided that without them you keep each of the other figures in its proper place: 2d, that it is useless to put down the right hand figures of the dividend so long as they fall over ciphers, because they do not begin to have any share in the making of the quotient until, by continuing the process, they cease to have ciphers under them: 3d, that the quotient is only a number written at length, instead of the usual way. For example, the first quotient is 200 + 20 + 6, or 226; the second is 20000 + 7000 + 60 + 9, or 27069. Strike out, therefore, all the ciphers and the numbers which come above them, except those in the first line, and put the quotient in one line; and the two examples of the last article will stand thus:
| 18) | 4068 | (226 | 1342) | 36326599 | (27069 |
| 36 | 2684 | ||||
| 46 | 9486 | ||||
| 36 | 9394 | ||||
| 108 | 9259 | ||||
| 108 | 8052 | ||||
| 0 | 12079 | ||||
| 12078 | |||||
| 1 |
80. Hence the following rule is deduced: