As
any surveyor should understand, all measurements are in error. We try to
minimize error and calculate reasonable tolerances, but error will always be
there. Not occasionally; not frequently; always. In the interest of more
accurate measurements, we look for better instruments and better procedures.

The theodolite is an instrument for measuring horizontal and vertical angles. It
has long had the same general look. A sighting telescope rotates on a vertical
axis. A circular scale rotates on this same axis to measure the horizontal
angle. A second axis, the trunnion axis, moves with the instrument and is
perpendicular to the vertical axis. The trunnion axis allows the scope to pivot
up and down, and it has a scale to measure the vertical angle.

One major design improvement came with the invention of the transiting theodolite.
With this innovation, the telescope was able to swing all the way over on the
trunnion axis. This in itself did not reduce any of the inherent
error in the instrument, but it gave surveyors the means of doing so. When the
scope is inverted, the instrument error is still there, but most of the error
reverses direction. By taking the mean of an even number of observations, half
direct and half inverted, the error is turned against itself and greatly
reduced.

In common usage among American surveyors, a transit is an older-style instrument
with a compass and exposed metal scales, while a theodolite usually has no
compass and has enclosed glass-plate scales, which are read with a built-in
microscope or an electronic micrometer. Strictly speaking though, they are both
theodolites and they probably are both transits. Nearly all modern surveyors’
theodolites are transits. Non-transiting theodolites are still manufactured, mostly
for builders working on small sites.

Detailed below are four instrument misadjustments that can lead to significant error. The error formulas below may be indeterminate for certain unrealistic values of the variables. Keep in mind that the calibration errors (*α*, *β*, *γ*, *ε*) are near 0°, and the zenith angle, *φ*, cannot be near 0° or 180°.

**The
trunnion axis is not perpendicular to the vertical axis.**

When
the scope is turning about the trunnion axis, the line of sight should sweep through
a vertical plane. If the trunnion axis is not perpendicular to the vertical
axis, then this plane will be deflected. On level ground, the error may be very
small, but traversing up or down a steep hill will increase the error.

In
this exaggerated illustration, the trunnion axis is listing to the right. The
locus of the line of sight is a plane. It intersects the desired vertical plane
on a horizontal line. Looking uphill, the line of sight is deflected to the
right. Looking downhill, in misses to the left. Imagine having a steep uphill
backsight. The reading on the horizontal scale would be to the left of the
actual sight. Looking at a steep downhill foresight, the scale reading will be
to the right of the sight direction. This results in an angle measurement
(clockwise) that is too large.

In
the formula below, *δ* is the clockwise horizontal deflection of the line of
sight from the scale reading, *φ* is the true zenith angle, and *α* is the clockwise deflection of the trunnion axis from the
perpendicular. As shown above, the line of sight must be projected onto the horizontal plane in order to measure the deflection error. Note that *δ* goes to zero and then changes sign as *φ* passes 90°.

Here is the best reason for reading both sides. When the scope is inverted, the line
of sight still travels through an oblique plane, but now it is listing in the
opposite direction. On the uphill backsight, the scale reading now misses to
the right. On the downhill foresight, it misses to the left. The result of the
inverted reading is an angle that is too small, compensating for the error in
the direct reading.

**The line of sight is not perpendicular to the trunnion axis.**

Suppose now that the trunnion axis is perpendicular to the vertical axis, but the line
of sight is not perpendicular to the trunnion axis. Again the images are exaggerated in order to make the point. This condition is not actually cause by a poorly mounted telescope. It is more likely a misalignment of the crosshair reticule inside the scope.

This time when the scope
turns on the trunnion axis, the line of sight sweeps a cone rather than a
plane. While the scope is in the direct position, the deflection is always in the same direction, but variable in magnitude. On steep ground, it even has a tendency to take care of itself,
because the backsight and foresight may have close to the same deflection. The
worst error occurs when one sight is much steeper than the other.

In this formula, *δ* is the clockwise horizontal
deflection of the line of sight from the scale reading, *φ* is the true zenith angle, and *β* is the deflection of the line of sight toward the
right end of the trunnion axis in the direct position. The line of sight is again projected onto the level plane.

Here
the deflection is least on level sights, but it is never zero and does not
change sign as *φ* passes the horizon at 90°. Again,
inverting the scope provides a reading with compensating error. The conic locus
is moved to the opposite side, and in the deflection formula, the sign changes.

**The vertical axis is not plumb.**

Of the three horizontal errors, this one is, perhaps, the least significant and the
easiest to correct. It is only a matter of leveling the instrument properly.
Unfortunately though, this is one case in which the transiting capability of
the theodolite does nothing to correct the error.

In
this image, the black scale is level and shows the true direction, but the
tilted blue scale is the one that is actually being read. There is a line of
intersection running through the middle of both scales. This line of
intersection is arbitrarily given a direction of 0° here, but it has no
relationship to the direction in which the instrument is pointed. The error can
be observed by projecting the true scale onto the plane of the instrument
scale. The (red) projection is elliptical, and the scales coincide only when
the line of sight is along or perpendicular to the line of intersection (0°, 90°, 180°, and 270°).

That
is not where it ends. That simplified description assumes that the line of
sight is level. Suppose that the instrument is pointed upward or downward. As
the instrument turns, assuming a constant *φ*, the locus of the line of sight is a cone with a vertical axis. When the true
scale is projected onto this cone, it is still true (the purple scale in the
image). This scale in turn is projected onto the plane of the instrument scale.
The result is a translation of the elliptical scale above, and it is true
only when the line of sight is perpendicular to the line of intersection (90° and 270°).

The
geometric transformation of the scale is not really so complicated, but the
corresponding analytic formula is. This is no bother since the formula has no
practical value anyway. Here *δ* is again the deflection of the line
of sight from the scale reading and *φ* is the true
zenith angle. The variable *θ* represents the true clockwise horizontal
angle from the line of intersection of the two planes. When the instrument is
pointed in the direction of the line of intersection, with the “vertical” axis
listing to the right, then *θ* is 0°.

The condition described here is an issue of leveling, and transiting the instrument does nothing to compensate for it. Check the level vials
frequently.

**The vertical angle collimation is out of adjustment.**

Theodolites
measure vertical angles, usually from the zenith direction, sometimes from the
horizon, rarely from the nadir. This difference affects nothing but the
arithmetic. The vertical axis should point to the zenith, but for greater
precision, theodolites have separate collimation systems so that the angle is
referenced directly to the gravity vector. This system may use a
leveling vial or a pendulum compensator, either of which can go out of
adjustment.

A
theodolite would make a very inefficient level (except for trig leveling,
another topic), but it essentially carries a level in its housing. The leveling
vial is equivalent to a tilting level, and the pendulum compensator does the
work of an automatic level. The theodolite actually has one advantage over most
levels. By inverting the telescope, the collimation can be checked from a
single setup.

If
a vertical angle is measured in both direct and inverted positions, then the
sum of the observations should be 360°. The collimation error, *ε*,
will either add to both observations or subtract from both, so it will show up
in the sum of the two angles. In this picture, two observations are made on the
same stationary target. The measured vertical angle is *φ*_{1} in the
direct position, and *φ*_{2} in the
inverted position. Find *ε* using the formula below, and subtract it
from the direct observation to get the true vertical angle. In the
illustration, both measured angles are too small, and *ε* is
negative.

A
few seconds, or even minutes, of error here makes no appreciable difference in
horizontal distances, but it can play all havoc with elevations. Unlike the
horizontal angle errors, this one is constant, which is to say, it is not
affected by changes in the direction of the sight. That makes it a fairly
simple matter to correct the angle without even adjusting the instrument. In
fact, electronic instruments typically have an onboard routine that will
measure and correct the vertical angle error. Push a few buttons, sight a
target in both positions, and have the instrument store the correction. The
procedure takes only a couple of minutes, so it can be done at the beginning of each work day.