Levelling fundamentals – lines of sight and staff readings: Which statements correctly describe the geometry of back sight/fore sight observations and the reference plane for staff readings?

Introduction / Context:
Differential levelling establishes elevations by reading a staff from a fixed instrument set-up. Knowing how the line of collimation and staff readings relate to the horizontal reference surface prevents conceptual errors in field notes and computations.

Given Data / Assumptions:

• An instrument is properly levelled, so the line of sight is horizontal.
• Back sight (BS) and fore sight (FS) are taken without moving the instrument.
• The level surface through the instrument axis approximates the reference horizontal plane at the set-up.

Concept / Approach:
With the instrument clamped, the BS and FS share the same line of collimation (horizontal plane). A staff reading is effectively the vertical distance between this plane and the staff zero at the observed point. In geometric levelling, we treat the horizontal reference as a level surface passing through the telescope axis for the small area around the instrument.

Step-by-Step Solution:

Verification / Alternative check:
Field-book checks (sum BS − sum FS = last RL − first RL) rely on the common horizontal plane assumption for each set-up and confirm consistency when arithmetic balances.

Why Other Options Are Wrong:

• Choosing any single statement omits the other equally essential elements of levelling geometry.

Common Pitfalls:
Confusing the reference plane when the instrument is moved; ignoring curvature/refraction over long sights (where refined corrections may be required); mixing instrument height with staff readings.

Final Answer:
All of the above