162. The process of the last article is as follows:
| 7,61,76 | (200 | 7,61,76(276 | |||
| 4 00 00 | 70 | 4 | |||
| 400 | ) | 3,61,76 | 6 | 47) | 361 |
| 70 | 3 29 00 | 329 | |||
| 400 | ) | 32 76 | 546) | 3276 | |
| 140 | 32 76 | 3276 | |||
| 6 | 0 | 0 |
In the first of these, the numbers are written at length, as we found them; in the second, as in (79), unnecessary ciphers are struck off, and the periods 61, 76, are not brought down, until, by the continuance of the process, they cease to have ciphers under them. The following is another example, to which the reasoning of the last article may be applied.
| 34,86,78,44,01 | (50000 | 34,86,78,44,01(59049 | |||
| 25 00 00 00 00 | 9000 | 25 | |||
| 100000 | ) | 9 86 78 44 01 | 40 | 109) | 986 |
| 9000 | 9 81 00 00 00 | 9 | 981 | ||
| 100000 | ) | 5 78 44 01 | 1180 | 4)57844 | |
| 18000 | 4 72 16 00 | 47216 | |||
| 40 | —————— | ———— | |||
| 100000 | ) | 5 78 44 01 | |||
| 18000 | 4 72 16 00 | ||||
| 40 | —————— | ||||
| 100000 | ) | 1 06 28 01 | 1180 | 89)1062801 | |
| 18000 | 1 06 28 01 | 1062801 | |||
| 80 | —————— | ————— | |||
| 9 | 0 | 0 |
163. The rule is as follows: To extract the square root of a number;—
I. Beginning from the right hand, cut off periods of two figures each, until not more than two are left.
II. Find the root of the nearest square number next below the number in the first period. This root is the first figure of the required root; subtract its square from the first period, which gives the first remainder.
III. Annex the second period to the right of the remainder, which gives the first dividend.
IV. Double the first figure of the root; see how often this is contained in the number made by cutting one figure from the right of the first dividend, attending to IX., if necessary; use the quotient as the second figure of the root; annex it to the right of the double of the first figure, and call this the first divisor.
V. Multiply the first divisor by the second figure of the root; if the product be greater than the first dividend, use a lower number for the second figure of the root, and for the last figure of the divisor, until the multiplication just mentioned gives the product less than the first dividend; subtract this from the first dividend, which gives the second remainder.