Tacheometry – Variation of Staff Intercept with Distance In stadia (tacheometric) observations using a fixed-multiplying telescope, how does the staff intercept (the difference between upper and lower stadia hair readings) change as the staff is held progressively farther from the instrument along the line of sight?

Introduction / Context:
Tacheometry (stadia surveying) determines distance and elevation by reading the intercept between the upper and lower stadia hairs on a levelling staff. Understanding how the staff intercept varies with distance is fundamental to quick field checks and error spotting. This question targets the core proportionality that links staff intercept to line-of-sight distance in a fixed-multiplying stadia setup.

Given Data / Assumptions:

• Fixed stadia hair interval in the telescope (fixed-multiplying instrument).
• Line of sight approximately horizontal or with known vertical angle.
• Standard distance formula: D = k * s + C, where D is distance along the line of sight, s is staff intercept, k is the multiplying constant, and C is the additive constant.

Concept / Approach:

The stadia principle forms similar triangles between the objective, stadia hairs, and staff graduations. For a fixed hair interval, the intercept on the staff increases linearly with the distance to the staff. Hence, as the staff is carried farther along the ray, the intercept grows proportionally. The additive constant C (from instrument optics) shifts the distance-origin slightly but does not change the linear growth of s with D.

Step-by-Step Solution:

Verification / Alternative check:

Field practice: doubling the line-of-sight distance approximately doubles the staff intercept for the same instrument and sighting geometry (neglecting small-angle effects). Crews routinely confirm this when selecting suitable staff ranges for clear intercepts.

Why Other Options Are Wrong:

'Smaller, farther off' inverts the relation. 'Smaller, nearer' misstates the near-field behavior. 'Same, wherever' would imply k = ∞ (impossible). 'Becomes zero' is incorrect; s → 0 only if D → C, i.e., extremely near the instrument, not at long range.

Common Pitfalls:

Confusing the additive constant with a distance threshold where intercept stops changing; ignoring that vertical angles slightly modify the horizontal distance but not the linear dependence of s on D along the line of sight.

Final Answer:

Greater, the farther off the staff is held