Surveying terminology — the branch of surveying that determines both horizontal and vertical positions of points by instrumental observations is called: Select the correct term.

Introduction / Context:
In engineering surveying, rapid determination of positions often requires simultaneous horizontal distances and elevation differences from a single setup. The specialized method that accomplishes this with stadia or subtense principles is widely used for topographic detail and preliminary works.

Given Data / Assumptions:

• The method relies on angular measurements and staff intercepts.
• Heights and distances are derived from instrument readings using fixed constants.
• Common spellings vary (tacheometry/tachymetry), but the meaning is the same.

Concept / Approach:
Tacheometry (also spelled tachymetry) uses a theodolite or tacheometer with stadia lines to compute horizontal distance and elevation from a single observation. Telemetry refers to remote measurement and data transmission and is not a surveying branch. Hence, the correct term in surveying is tacheometry.

Step-by-Step Solution:

Verification / Alternative check:
Surveying manuals define tacheometry/tachymetry as the method for rapid distance and height determination using optical readings.

Why Other Options Are Wrong:

• tachemetry: A variant spelling sometimes used, but the standard in the options is “tacheometry.”
• telemetry: Remote sensing/data transmission; not surveying layout.
• all the above: Incorrect because telemetry does not fit.

Common Pitfalls:
Spelling confusion between tacheometry and tachymetry; focus on meaning rather than orthography.

Final Answer:
tacheometry