REMARKS ON APPENDIX. 59
Therefore take the above power and seek for the root of it which corresponds to the quotient of the first Logarithm; thereby you will find the required second sine. Also the Logarithm of the power itself will be the continued product of the quotients and the common divisor.
Thus let the given Logarithms be 8 and 14, and the sine corresponding to the first Logarithm be 3. A common divisor of the Logarithms is 2; this gives the quotients 4 and 7. If 3 multiply itself six times, you will have 2187 for the power which, in a series of continued proportionals from unity, will occupy the seventh place, and hence it may, without inconvenience, be called the seventh power. The same number, 2187, is the fourth power from unity in another series of continued proportionals, in which the first power, , is the required second sine. The product of the quotients 4 and 7 is 28, which, multiplied by the common divisor 2, makes 56, the Logarithm of the power 2187.
| Continued Proportionals. |
Logarithms. | Continued Proportionals. |
Logarithms. | ||
| 2181 | (0) | 0 | 2181 | (0) | 0 |
| 2183 | (1) | 8 | 2186838521 | (1) | 14 |
| 2189 | (2) | 16 | 2146765372 | (2) | 28 |
| 2127 | (3) | 24 | 231980598 | (3) | 42 |
| 2181 | (4) | 32 | 2187 | (4) | 42 |
| 2243 | (5) | 40 | |||
| 2729 | (6) | 48 | |||
| 2187 | (7) | 56 |
It will be observed that these Logarithms differ from those employed in illustration of the previous Proposition:
