Optics of instruments: The ratio of the distances at which a stated small length (detail) can be just distinguished by a telescope and by the unaided human eye, respectively, is called the:

Introduction / Context:
When specifying surveying telescopes (levels, theodolites, total stations), several optical performance terms arise: magnification, resolving power, brightness, field of view. Understanding these distinctions ensures correct selection for sight length and target size.

Given Data / Assumptions:

• A “stated small length” is the test object (e.g., a line pair or small graduated mark).
• Two distances are compared: at which the telescope resolves it and at which the unaided eye resolves it.
• All other conditions (illumination, contrast) are comparable.

Concept / Approach:

Magnification (angular magnifying power) can be defined operationally as the ratio of distances at which a given fine detail is just distinguishable through the telescope and by the naked eye. If the eye can just distinguish a mark at distance d_e, and with the telescope it can be distinguished at d_t, then magnification M ≈ d_t / d_e (equivalently, the ratio of the angular sizes after the instrument). Resolving power, in contrast, concerns the smallest angular separation between two points the instrument can distinguish, and brightness deals with exit pupil/illumination—not with this distance ratio definition.

Step-by-Step Solution:

Verification / Alternative check:

Instrument datasheets list magnification (e.g., 20x, 30x), which aligns with practical “distance ratio” tests for just resolvable details.

Why Other Options Are Wrong:

Resolving power is a threshold angular separation, not a distance ratio against the eye. Brightness/field factor describe other optical qualities.

Common Pitfalls:

Interchanging magnification with resolution; assuming higher magnification always improves resolution irrespective of optical quality and seeing conditions.

Final Answer:

Magnification of the telescope