Angles
A closed polygon's interior angles sum to (n − 2) × 180°. Seven angles sum to 900°. The difference between the measured sum and that figure is the angular misclosure, and the allowable amount is K √n, with K set by the order of survey.
Two faces
Observing on both faces and averaging removes every error that reverses sign when the telescope is plunged: horizontal collimation, trunnion axis tilt, vertical circle index, and circle eccentricity. It does not remove a mis-levelled vertical axis, which is the same on both faces. Level the instrument; averaging will not save you.
The vertical index error falls out of a two-face zenith angle: i = (zdirect + zreverse − 360°) / 2. Divide by two; the sum alone is the distractor.
Centering
A small setup error becomes a large angle on a short sight. The angular error in arc-seconds is 206,265 × e / D. Five millimetres over a 50 m sight is 21″; the same 5 mm at 500 m is 2″.
EDM
An EDM's stated accuracy is ±(a mm + b ppm). The atmosphere shifts the result by roughly 10 ppm for every 10 °C or 1 in Hg you get wrong, and 1 ppm is 1 mm per kilometre, so a 10 °C slip on a 3 km line is 30 mm.
The system constant for an instrument and prism pair comes from three collinear points: K = AC − (AB + BC). A −30 mm answer is the classic result of mixing prism conventions.
Trigonometric leveling
With slope distance S and zenith angle z: horizontal distance is S sin z, and the elevation difference is
Δelev = S cos z + hi − hr + c&r
where hi is the instrument height, hr the reflector height, and c&r the curvature and refraction term (0.0675 K² m). Zenith 88°30′ is a gentle upward sight, so S cos z is a small positive number.