Stadia Distance from Rod Intercept

Before electronic distance measurement, a level or transit measured distance optically: two extra cross hairs cut a known angle, and the length of rod they span grows in exact proportion to the distance. James Watt is generally credited with the idea around 1771, and instrument makers standardised the geometry so that the stadia interval factor K equals 100 — one foot of rod intercept means a hundred feet of distance, an arrangement so convenient it outlived the technology. The additive constant C accounted for the gap between the instrument's centre and the front of an external-focusing telescope, typically about 1 ft; every modern internal-focusing instrument has C = 0, which is why the constant has almost disappeared from the textbooks.

A worked example: the upper hair reads 6.42 ft and the lower reads 3.00 ft, so s = 3.42 ft and D = 100 × 3.42 + 0 = 342 ft. Stadia is good to roughly 1:300 to 1:500 — fine for topographic detail, useless for boundary work — and this form assumes a horizontal sight. On an inclined sight the horizontal distance picks up a cos²θ factor and the vertical difference a sin θ cos θ term, which is where the old tacheometric tables came in.