How a pair of extra hairs measures distance
Look through a stadia-equipped telescope at a levelling rod and you see three horizontal crosshairs. The outer two are fixed a small distance apart in the reticle, so the piece of rod they cut off — the intercept — grows in proportion to how far away the rod is. Space them so that a one foot intercept means a hundred feet of distance and you have an instrument that reads distance directly off the rod, with no tape and no second person at the far end doing anything but holding the rod upright.
That ratio is the stadia interval factor, usually written K. Almost every instrument ever built was designed to put it at 100, and almost every one is close but not exactly there. Determining it is a morning of work: set up on a line with known distances marked at intervals, read the intercept at each, and fit the ratio. Assuming it is exactly 100 introduces a systematic error in every shot the instrument ever takes.
Why the cosine is squared
On a level sight the distance is K times the intercept and there is nothing more to say. Tilt the telescope up and two separate things happen, and the reduction has to account for both.
The first is familiar from any slope measurement: the sighted length is the hypotenuse and the horizontal distance is that times the cosine of the vertical angle. The second is peculiar to stadia. The rod is held plumb, not square to the line of sight, so the two hairs cut a longer piece of it than they would if it were held perpendicular — longer by one over the cosine. The intercept is therefore already too large before any reduction happens, and correcting it costs another cosine.
Multiply the two and the horizontal distance is K times the intercept times the cosine squared, plus the additive constant times a single cosine. At the 3-24-00 in the form the second cosine costs 0.35 percent, which on a 336.4 foot sight is 1.18 feet. At ten degrees it is 3.0 percent. Missing the second cosine gives distances that are consistently long on steep sights and correct on flat ones, which is exactly the pattern that makes a systematic error hard to spot.
The vertical half
The height difference from the instrument axis to the point sighted on the rod is K times the intercept times the sine of the angle times the cosine of the angle, which is more compactly half of K times the intercept times the sine of twice the angle. Getting from that to an elevation is a walk in four steps: start at the station elevation, add the height of instrument to reach the telescope axis, add the vertical component to reach the point on the rod, then subtract the middle hair reading to come back down the rod to the ground.
The height of instrument is the term that causes the most trouble, because an error in it is identical on every shot from that setup. The topography comes out perfectly consistent within itself and sitting at the wrong elevation as a block, and it does not reveal itself until the setup is tied to something else. Measuring it rather than estimating it takes ten seconds.
What stadia is good for and what it is not
Read the rod as carefully as anybody can and you are good to about a hundredth of a foot on each hair. The interval factor multiplies that by a hundred. So a stadia distance carries roughly a foot of uncertainty at any range, before anything else goes wrong. Over 300 feet that is about 1 in 300, which is nowhere near what a traverse wants and perfectly adequate for what stadia was actually used for: contours, spot elevations, the shape of a hillside, the location of a creek, drainage patterns over a few hundred acres. One instrument operator and one rod person could pick up a topographic map in a day.
It was never a boundary tool and nobody in the profession pretended otherwise. The distinction between control-quality measurement and topographic detail is older than any of the instruments involved, and stadia sat firmly on the detail side of it.
Why it is still worth knowing
Two reasons. The first is that older field books are full of it, and reading a stadia book from 1950 requires knowing what the columns mean and what interval factor the instrument had. The second is that the geometry has not gone anywhere: a total station reducing a slope distance to horizontal is doing the first of the two cosines, and anybody who has understood why stadia needs the second one understands what a rod held plumb rather than perpendicular does to a measurement. That comes up again with prism poles that are out of plumb, and the answer has the same shape.
Questions people ask
What is the stadia formula?
For a level sight, distance equals the interval factor times the rod intercept, plus the instrument constant. For an inclined sight the horizontal distance is K times the intercept times the cosine squared of the vertical angle, plus C times the cosine, and the vertical component is K times the intercept times the sine times the cosine, plus C times the sine. The cosine appears squared because the sight is inclined and the rod is held plumb rather than square to it.
Why is the cosine squared in the stadia formula?
Two separate effects. One cosine reduces the sighted distance to horizontal, which is true of any slope measurement. The second is because the rod stands upright rather than perpendicular to the line of sight, so the hairs cut a longer piece of it than they would otherwise, by exactly one over the cosine. Correcting that costs the second cosine. At three degrees the difference is about 0.3 percent, so missing it makes steep sights read long while flat ones stay correct.
Is the stadia interval factor always 100?
It is usually close, because that is what instruments were designed for, but it is a property of a particular telescope and should be determined rather than assumed. The determination is straightforward: set up on a line with known distances, read the intercept at each, and fit the ratio. An instrument that has been dropped or repaired is worth rechecking. This page uses whatever value you enter.
How accurate is a stadia distance?
Roughly a foot, at any range, dominated by how finely the rod can be read. A hundredth of a foot on a hair is about the practical limit and the interval factor multiplies it by a hundred. Over 300 feet that is about 1 in 300. That was fine for the work it was used for — contours, spot heights, drainage — and never good enough for boundary or control work, which is a distinction the profession drew long before electronic distance measurement existed.
What is the instrument constant C?
An additive length that comes from the optical design, accounting for the distance between the instrument centre and the front focal point. Internally focusing telescopes are designed so it is negligible and it is often taken as zero. Externally focusing ones have a real constant, typically around a foot. Because it is a fixed length rather than a proportion, it matters most on short sights and is nearly irrelevant on long ones.
How do I get an elevation from a stadia shot?
Four terms. Start with the elevation of the station you are set up on, add the height of instrument to get to the telescope axis, add the vertical component of the shot to get to the point sighted on the rod, then subtract the middle hair reading to come back down the rod to the ground. An error in the height of instrument shifts every point from that setup by the same amount, so it is worth measuring rather than estimating.