If the tangent be made a constant equal to the length of the stadium BC, and this stadium be placed at another position, say de or fg; then the angle subtended by its entire length will vary in a manner that can only be estimated by trigonometrical calculation.
In case of reading two distant marks on the stadium only for the subtense, the single central web of the telescope being directed first to one and then to the other of these webs, the distance is calculated as follows:—
Given the tangent BC and the angle BAC, required the distance AC. Let the angle BAC be represented by D; then—
| CA | = cotan D, or CA = CB × cotan D. |
| CB |
Reducing by logarithms, we have—
log CA = log CB + L cotan D - 10.
For example, make CB 14 feet, and the angle D 2° 45′ 50″, thus:—
| log CB | = 1·146128 | |
| L cotan D - 10 | = 1·316265 | |
| log CA | = 2·462393, | or CA = 290 feet. |
The above gives the principles followed with instruments of the theodolite class simply; but arrangements are made in omnimeters and similar instruments to read the tangent directly and determine the height CB in equal parts, so that observation of the heights BC gives rectangular co-ordinates and thus saves reduction from degrees of arc.
In practice the staff or stadium is made of the greatest length convenient for portability. With a telescopic staff, 14 or 16 feet is commonly used. If a unit tangent be not employed, the foot is divided into 100 parts, each of which parts, with the tacheometer, represents 1 foot of the base, and the whole staff 1400 or 1600 feet. The ordinary Sopwith staff, art. 263, answers the purpose, but art. 268 better.