#### f. Vertical Circle Index

##### (1) Description

Reducing slope distance to horizontal requires measurement of a vertical or zenith angle. An index error exists if the vertical circle is not correctly oriented to gravity when the instrument is leveled. TSIs and most theodolites use a gravity actuated vertical circle which should orient itself automatically. Some earlier theodolites, particularly control quality instruments, had a separate bubble for vertical circle orientation. Those instruments types aren't addressed in this chapter.

##### (2) Check

Analog theodolites measure zenith angles; digital theodolites and TSIs can be set up to measure zenith or vertical (from the horizon) angles. We'll use zenith angles*****

Measure the Direct and Reverse zenith angles to a point and check their sum.

In Figure G-36 the Direct zenith angle is 70°.

Figure G-36 Direct Zenith Angle |

In Figure G-37 the Reverse zenith angle is 290°.

Figure G-37 Reverse Zenith Angle |

Adding them: 290°+70°=360°

The sum of the Direct and Reverse angles should be 360°

What happens if the vertical circle is rotated 4°, for example?

The Direct zenith angle to the same point, Figure G-38, is 66°, 4° too small.

Figure G-38 Direct Zenith Angle with Error |

The Reverse zenith angle, Figure G-39, is 286°, also 4° too small.

Figure G-39 Reverse Zenith Angle with Error |

66°+286°=352°, which is 8° less than 360° and twice the circle rotation. Per Reversion Principle, the error is doubled.

##### (3) Compensation

###### (a) Procedure/Mathematical

Because a zenith angle is a continuous angle, the correct angle is *not* the direct and reverse average. Instead, an Index error is computed and applied to the measured zenith angle.

The Vertical Circle Index error is (360°-[Z_{D}+Z_{R}])/2. It is added to the direct (Z_{D}) and/or reverse (Z_{R}) angle.

For this example, the Index error is (360°-352°)/2 = +4°.

66°+4°=70° *check*

286°+4°=290° *check*

70°+290° = 360° *check*

###### (b) Adjustment

Neither a theodolite nor a TSI can be mechanically adjusted. Although a theodolite has a physical circle, it is not user accessible. A TSI uses a digital circle which can have 0° anywhere and axes compensation which should account for an index error. An error with either should be only be adjusted by a qualified repair facility