How stadia tacheometry measures distance
The optics behind stadia hairs, where the constants K and C come from, and how the inclined-sight formulas reduce a rod reading to true distance.
The geometry of the stadia hairs
A stadia telescope carries two extra horizontal cross hairs set a fixed tiny distance apart, one above and one below the main line of sight. Because the spacing is fixed, the rays leaving those hairs form a fixed angle, so the further the rod stands the more of it falls between the hairs. Reading the top hair and the bottom hair and taking the difference gives the intercept, the length of rod the instrument sees. That intercept is directly proportional to distance, which is the whole idea behind the method.
Where K and C come from
The proportion between intercept and distance is the multiplying constant K, fixed by the manufacturer at the ratio of the focal length to the hair spacing. On virtually every modern instrument that ratio is engineered to be exactly 100, so distance in feet is one hundred times the intercept in feet. The additive constant C accounts for the gap between the instrument center and the front of the objective lens. External focusing scopes had a small nonzero C, but internally focusing designs drive it to zero, which is why the tool defaults K to 100 and C to 0.
Reducing an inclined line of sight
A level sight gives distance straight away as K times the intercept plus C, but most shots tilt up or down a slope. When the line of sight leans by an angle, the rod is no longer perpendicular to it, so the raw reading overstates the flat distance. The horizontal reduction multiplies by the cosine of the angle squared, and the vertical component uses the sine times the cosine. Feeding a 2.0 ft intercept at 5 degrees through those formulas yields about 198.48 ft horizontal and 17.36 ft of rise.
Sources of error and good practice
Stadia distance is only as good as the intercept, and a rod held out of plumb, heat shimmer near the ground, or a coarse rod graduation each add error that grows with range. Keeping shots under a few hundred feet, holding the rod vertical with a level, and reading to the nearest hundredth of a foot all tighten the result. Because the method leans on the cosine of the angle, a small mistake in the vertical angle matters less on a nearly level sight than on a steep one.