Angular Measurement
 Field Use of the
    Theodolite
 Taking Measurements
 Errors of Construction and
    Adjustment
 Collimation Error
 Horizontal Collimation
 Vertical Circle Index
 Plate Level
 Optical Plummet
 Miscentring
 Parallax in Telescope

Plate Level

Plate level error is an error of adjustment that affects horizontal angles observed to elevated targets. You may also hear this error referred to as Dislevelment. The formulae used to correct angle readings effected by plate level error are derived from spherical trigonometry. Although they may seem unfamiliar at the moment it is important to at least understand the phenomenon.

Plate level error occurs when the theodolite is not levelled properly, using the plate level bubble.

In the image to the right, the brown perpendicular lines represent true horizontal and vertical axis. The blue lines represent the telescope in a raised position and the horizontal and vertical axis with respect to the dislevelled theodolite.

This error is not normally eliminated during the set-up procedure, only if it is symmetrical does this occur. The effect is proportional to the tangent of the altitude, so it does not affect angles close to horizontal but is significant when elevated targets are observed.

While the vertical reading is 0° no error is shown but once the telescope raises or lowers, it's line of sight does not follow a true vertical line. In the example to the right, the line of sight of the raised telescope does not align with its intended target. If the theodolite is rotated so that the line of sight reaches the target, this creates an error in the horizontal angle reading. The greater the angle of elevation, the more the line of sight diverges from the true vertical plane and the greater the plate level error becomes.

Ignoring inclination of circle to horizontal, the error in the horizontal circle reading is e.
From spherical trigonometry:

e" = i" six a tan h

The quantity ( i" six a) can be measured using the plate level:

L = 1 R = 3

The correction is then:

The centre of the bubble is displaced by (1-3)/2 i.e. - 1 division (to the right if negative). The value v" is the sensitivity of the bubble, and is often shown on the instrument.

Example. Traverse from ground level down a series of ramps.

(note that the verticals shown here are altitudes, not circle readings)

If dislevelment is not allowed for, the angle would be 32 11' 32.5" Allowing for the error caused by the plate bubble error:

Correction to FL to B:	0.5 (3 - 2) x 20" tan (38° 12') = +7.9"
Correction to FL to D:	0.5 (4 - 1) x 20" tan (-26° 35') = -15.0"
Correction to FR to D:	0.5 (3.5 - 2) x 20" tan (- 26 35') = - 7.5"
Correction to FR to B:	0.5 (3 - 1.5) x 20" tan (38°12') = +11.8

Corrected readings are FL to B 8°27' 29.9" } Corrected readings are FL to D 40 38', 42.0" }	32° 11' 12.1"
Corrected readings are FR to D 180° 27' 07.5" } Corrected readings are FR to B 220° 38' 56.8" }	32° 11' 10.7"

The corrected angle is therefore 32° 11' 11.4", a difference of 21.1" which is quite significant.