Similarly to the preceding examples, HEIGHTS, AND DISTANCES may be rapidly (and for military purposes, sufficiently accurately) computed in the field, by means of the foregoing trigonometrical table, if proper attention is paid to the principles by which the unknown angles of triangles may be ascertained: a base line, and requisite angle, or angles, having been given.
It will, however, be necessary to use advantageously the methods in Cases 1, 2 (vide [Trigonometry]), and also the properties in the subsequent theorems, and corollaries.[50]
Table,
Showing the reduction in feet, and decimals upon 100 feet, for the following angles of elevation, and depression.
| Angle. | Reduction. | Angle. | Reduction. | Angle. | Reduction. | |||
| ° | ′ | ° | ′ | ° | ′ | |||
| 3 | 0 | ·14 | 9 | 0 | 1·22 | 15 | 0 | 3·40 |
| 9 | 30 | 1·38 | 15 | 30 | 3·64 | |||
| 4 | 0 | ·25 | 10 | 0 | 1·52 | 16 | 0 | 3·88 |
| 10 | 30 | 1·68 | 16 | 30 | 4·12 | |||
| 5 | 0 | ·38 | 11 | 0 | 1·84 | 17 | 0 | 4·37 |
| 11 | 30 | 2·01 | 17 | 30 | 4·63 | |||
| 6 | 0 | ·55 | 12 | 0 | 2·19 | 18 | 0 | 4·90 |
| 6 | 30 | ·65 | 12 | 30 | 2·37 | 18 | 30 | 5·17 |
| 7 | 0 | ·76 | 13 | 0 | 2·56 | 19 | 0 | 5·44 |
| 7 | 30 | ·86 | 13 | 30 | 2·77 | 19 | 30 | 5·74 |
| 8 | 0 | .98 | 14 | 0 | 2·97 | 20 | 0 | 6·08 |
| 8 | 30 | 1·10 | 14 | 30 | 3·18 | 20 | 30 | 6·33 |
The reduction for 100 feet (from the above table) multiplied by the number of times 100 feet measured, will give the quantity to be subtracted from the measured length of an inclination, to reduce it to a horizontal position.
Table,
showing the rate of inclination of inclined planes, for the following angles of elevation.
| Angle. | One in | Angle. | One in | Angle. | One in | |||
| ° | ′ | ° | ′ | ° | ′ | |||
| 0 | 15 | 228 | 3 | 30 | 17 | 7 | 0 | 8 |
| 0 | 30 | 114 | 3 | 45 | 16 | 7 | 30 | 7¼ |
| 0 | 45 | 76 | 4 | 0 | 15 | 8 | 0 | 7 |
| 1 | 0 | 56 | 4 | 15 | 14 | 9 | 0 | 6½ |
| 1 | 15 | 46 | 4 | 30 | 13 | 10 | 0 | 6 |
| 1 | 30 | 38 | 4 | 45 | 12 | 11 | 0 | 5¾ |
| 1 | 45 | 32 | 5 | 0 | 11½ | 12 | 0 | 5½ |
| 2 | 0 | 28 | 5 | 15 | 11 | 13 | 0 | 5 |
| 2 | 15 | 26 | 5 | 30 | 10½ | 14 | 0 | 4½ |
| 2 | 30 | 23 | 5 | 45 | 10 | 15 | 0 | 4 |
| 2 | 45 | 21 | 6 | 0 | 9½ | 16 | 0 | 3¾ |
| 3 | 0 | 19 | 6 | 30 | 9 | 17 | 0 | 3½ |
| 3 | 15 | 18 | 6 | 45 | 8½ | 18 | 0 | 3¼ |