Heinrich Wild’s optical theodolite, introduced in Switzerland in the 1920s, had several new features, including an auxiliary telescope

Leonard Digges introduced the word "theodolitus" in his Pantometria (London, 1571). This surveying instrument had a circular ring or plate divided into 360 degrees, and a pivoting alidade with sight vanes at either end. Theodolites of this sort, as well as others with a second pair of sight vanes affixed to the graduated circle, were soon in widespread use. In 1791, George Adams Jr. called this instrument a "common theodolet," reserving the term theodolite for the telescopic instruments with horizontal circles and vertical arcs that had been introduced in London in the 1720s. While the telescopic theodolite was popular in England, Americans preferred the surveyor’s compass and, later, the surveyor’s transit, which were cheaper and more robust. In the 18th century form, the telescope is mounted directly on the vertical arc. In the transit theodolite, which originated in London in the 1840s, the telescope is transit mounted, with a vertical circle mounted at one side. Heinrich Wild’s optical theodolite, introduced in Switzerland in the 1920s, had several new features, including an auxiliary telescope that lets the user read either circle without moving away from the station.

Some theodolites measure horizontal angles with geodetic accuracy. The first instrument of this sort was made by Jesse Ramsden in London in 1787, and purchased by the Royal Society for use on the geodetic link between Greenwich and Paris. The first instrument of this sort in America was made around 1815 by Troughton in London for the fledgling United States Coast Survey.

Ref:

J. A. Bennett, The Divided Circle (Oxford, 1987), pp. 40–41, 146–149, 195–200.

George Adams Jr., Geometrical and Graphical Essays (London, 1791), pp. 220–222 and fig. 5.