A land surveying instrument used to very accurately measure angles and distances. Modern theodolites can measure angles to within one half of a second of arc (a second is one sixtieth of a minute, which is one sixtieth of a degree) and distances to one part per million.
A theodolite is set up on a tripod vertically aligned precisely over a point (a nail or spike) on the ground. The instrument itself must be leveled with a bubble level. The tripod must be firmly dug into the ground so that the instrument can't be shaken as it operates.
The instrument contains a very strong telescope (else you wouldn't be able to see accurately enough to measure tiny angles): often 30x or more. On hot days, turbulence in the air limits the accuracy of an instrument, since distant points seem to dance about in the scope.
The scales can be locked and unlocked, allowing angles to be "turned" multiple times in multiple directions to check for error and squeeze the maximum accuracy out of the instrument.
In days gone by, the instrument man used a series of hand signals to direct the rodman at the distant point to plumb the rod, move to the next point, move to the right or left and so on. These days, walkie talkies have made these quaint signals obsolete.
Modern theodolites measure distance as well as angles, using built-in laser range finders that bounce their beams off of reflectors set up over the remote point. They even remember the transit points and can calculate the "closure" of a transit to see if any error has occurred.
See: land surveying.