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"In the zenith telescope, or Horrebow-Talcott, method of determining the
latitude, there is substituted for the measurement of the absolute
zenith distance of a star the measurement of the small difference of
meridional zenith distances of two stars culminating at about the same
time, and on opposite sides of the zenith. The effect of this
substitution is the attainment of a much higher degree of precision,
arising from the increased accuracy of a differential measurement, in
general, over the corresponding absolute measurement; from the
elimination of the use of a graduated circle for the essential part of
the measurement; and from the fact that the computed result is affected,
not by the error in estimating the absolute value of the astronomic
refraction, but simply by the error in estimating the very small
difference of refraction of two stars at nearly the same altitude."

"A sufficient number of pairs should be observed at a station to make it
reasonably certain that the probable error of the mean result is not
greater than ±0″.10." (Typically, about 15 pairs were enough.)

"The stars observed upon should be taken from "The Preliminary General
Catalogue of 6188 Stars for the Epoch 1900" by Lewis Boss, which was
published by the Carnegie Institution at Washington in 1910."

[https://archive.org/details/preliminarygener00carnrich/page/n15]

"Among the requisites for a pair of stars for an observing list, are,
that their right ascensions shall not differ by more than 20m to avoid
too great errors arising from the instability in the relative positions
of different parts of the instrument; nor by less than about 1m, that
interval being required to take the readings upon the first star and
prepare for the second star of the pair; that their difference of zenith
distances shall not exceed the half length of the micrometer comb, 20′
for most instruments; that each star shall be bright enough to be seen
distinctly, not fainter than the seventh magnitude for the larger
instruments; and that no zenith distance shall exceed 45°, to guard
against too great an uncertainty in the refraction."

William Bowie, "Determination of Time, Longitude, Latitude, and
Azimuth," U.S. Coast and Geodetic Survey Special Publication No. 14, 1917.

(In later years that was replaced by Special Publication No. 237,
"Manual of Geodetic Astronomy.")

Since a transit instrument is free to move only in altitude in the plane
of the meridian, the reversal to observe the second star of each pair
was accomplished with a mechanism which lifted the telescope trunnions
from their V-blocks, rotated 180° about a vertical axis, and lowered
them in the opposite blocks. In the bottom photo on this page the
telescope assembly has been lifted and partly rotated.

http://amhistory.si.edu/surveying/object.cfm?recordnumber=758936

The only graduated circles on that instrument are the Talcott-Horrebow
levels attached to the sides of the telescope tube. Those are set to the
expected altitude of the first star. No great accuracy is necessary, it
being sufficient that the star is well positioned in the field of view.
What is necessary is that the telescope is accurately returned to the
same inclination for the second star.

Exact centering of the bubble is not assumed. Its position is recorded
immediately after each observation, and the corresponding dislevelment
factored into the computation.

Both stars are measured with a movable horizontal wire of a micrometer
as they pass through the meridian. From the difference in micrometer
readings, the difference in zenith distance is computed, from which
latitude is computed. For example, if the readings are identical,
latitude equals the mean declination of the pair. Note that zenith
distance itself is not measured in this technique.

The illustrated instrument is quite old, but I chose it because the
reversing mechanism is shown so well. By the time of the 1917
publication, one of the favored instruments of the Survey had a 3-inch
f/15 telescope and was normally operated at 100x.

Somebody mentioned the polar motion. Corrections for that were applied:
"About the middle of each year the Latitude Service of the International
Geodetic Association publishes in the Astronomische Nachrichten
provisional values of the coordinates of the instantaneous pole for the
preceding calendar year, together with tables to reduce observed
latitudes, longitudes and azimuths to the mean position of the pole."