Historical information

Used to measure the rate of rise of helium balloons
The telescope is mounted on two movable axes. One axis (vertical) rotates to change elevation, the other (horizontal) azimuth. There are vernier scales and in some cases micrometres that give precise readouts of the relative position of the telescope to each axis.
The instrument is set up so that it is level and it is pointed towards true north with both scales reading 0 degrees exactly. A balloon is released in front of the theodolite. It is sighted at timed intervals (usually one minute apart) and the position of the theodolite's telescope (azimuth and elevation) is recorded.
It can chart the direction and velocity of winds at various altitudes
The rate of ascent of a balloon is mostly dependant on the balloon's drag and its "free lift" (the vertical pull of the balloon).
There is some degree of control over these these factors, and as a result, it possible to know approximately how high our balloon will be at any given time after its release. Given a known height and an angular direction (read off the theodolite) to the balloon, a fix is made of the horizontal movement component of the balloon's travel as it moves through different altitudes.
The horizontal movement is due to the winds blowing the balloon around at the altitudes that the balloon is traveling through

Physical description

Theodolite used to measure the rate of rising helium balloons
The rate of rise is used in atmospheric calculations such as upper winds and determining inversion layers