The Stadia


The Stadia is a method of measuring distances rapidly with a telescope (usually on an engineer's transit or an alidade) and a graduated rod. When the telescope is focused on the rod, the distance s intercepted on the vertically-held rod between two stadia hairs seen in the eyepiece gives the distance D as D = ks, where k, the stadia constant is often made to be 100. Therefore, if 6 ft is intercepted on the rod, then the distance from the telescope to the rod is 600 ft. There are small corrections to this that will be mentioned below. If the line of sight is inclined, the vertical angle is also measured and can be used to reduce the results to horizontal and vertical distances. Stadia can give results correct to about 1 ft under the best conditions, which is often sufficient, and can also serve as a check on more precise measurements.

The term stadia comes from the plural of the Greek stadion, the word for a distance of 185 to 192 metres (607-630 ft). A very similar length is the modern furlong, or eighth of a mile, 660 ft. A "stadion" was also an athletic venue, with lengths laid out for competition and seats for spectators. The Latin stadium, stadia was a direct borrowing with the same meaning.

Distance over the ground was traditionally measured by long poles or rods laid successively end to end. The ancient Egyptians used rope for the same purpose. This practice is reflected in the traditional rod, pole or perch of 16.5 feet. This odd length came from dividing down an English mile of 5280 ft, first into furlongs of 1/8 mile or 660 ft, then into tenths, or chains, of 66 ft, and finally into quarters of this, or 16.5 feet. Four rods make a chain, ten chains a furlong, and 80 chains a mile. The Gunter's chain of 100 iron links and length 66 ft was much easier to use and carry than an ungainly pole, and gave more accurate results. 10 square chains is an acre, so Gunter's chain was closely related to traditional measures of distance and land areas. The engineer's chain of 100 links, each 1 ft long, replaced Gunter's chain, and was itself replaced by the 100 ft steel tape, which is an excellent and easily handled way to measure distances. Doing this is still called "chaining," however, and the people who do it are called chainmen. Accurate chaining is subject to many errors, which are largely systematic, but with care they can be overcome. These errors include thermal expansion and elasticity of the tape, as well as ground irregularities.

Distances are now conveniently measured by timing modulated laser beams returned by retroreflectors. Large distances can be covered at one leap, and the intervening ground does not have to be traversed on foot. Stadia shares these advantages. Microwaves were first used for this purpose, but have now been superseded by lasers. The main errors are in estimating propagation conditions, temperature and humidity, which affect the velocity of light, and are often poorly known or vary over the path. Even without consideration of these uncertainties, laser ranging is more accurate than stadia, but is also much more expensive. We also have Global Positioning System location, which is accurate to roughly 1 metre (with special care, centimetre accuracy is possible, but it requires work). In spite of these excellent alternatives, it is still interesting to know the stadia method, which is often applicable in unusual circumstances.

The stadia method is an application of paraxial optics. The telescope consists of an objective (usually one achromatic lens, but sometimes more) that produces an image of the distant scene close to its focal plane, which is then examined by the eyepiece. We will be concerned only with the objective. The action of the telescope objective is described by principal planes, nodal planes and focal lengths. Since the final and initial media are the same, the nodal planes coincide with the principal planes, and the primary and secondary focal lengths are equal. The telescope is mounted so that the outer principal plane of the lens is a distance c from the axis of the instrument, that is vertically over the occupied location. If the distance of the stadia rod from the instrument axis is D, then the object distance is D - c. The corresponding image distance d behind the other principal plane is then given by 1/d + 1/(D - c) = 1/f.

Fine lines are etched on a glass reticle placed approximately at the focal point of the telescope objective. These were once crosshairs made of spider web, and are still called crosshairs for that reason. There are vertical and horizontal crosshairs for sighting purposes, and two shorter stadia hairs at equal distances above and below the horizontal crosshair. The separation of the stadia hairs is denoted by i. The eyepiece is adjusted so that the crosshairs are sharp, and the telescope then focused so that the object viewed is also sharp, so that their images occur at the same point.

We now make use of the unit angular magnification property of the nodal points to establish that the angle s/(D - c) of the rod intercept as seen from the outer nodal point is equal to the angle i/d at the inner nodal point, or 1/d = s/i(D - c). The relations are illustrated in the diagram. When this is substituted in the lens equation, the result is f + fs/i = D - c, or D = (f/i)s + (f + c), which is the fundamental stadia formula. The derivation is confused in Breed and Hosmer; the principles are not clearly stated, and reference is made to a different diagram than the one appearing on the page, possibly one from an earlier edition. I hope that this derivation will make things clear, since they really aren't very difficult. Now, f/i will be a constant determined by the construction of the telescope and reticle, and is usually called k, the stadia constant of the instrument. It is commonly 100, but a more accurate value can be established by experiment if necessary. k = 100 corresponds to an angle of 0.01 radian, or 0.573°. The correction (f + c) is to be added to ks to find D. If f = 200mm and c = 100 mm, then (f + c) is 30 cm, or about 1 ft. This correction is sometimes ignored.

The formula just derived applies to a horizontal sight on a vertical rod, or to an inclined sight on a rod held perpendicular to the direction of view. It is not easy to hold a rod perpendicular to the line of sight, so it is held accurately vertical. If s is the intercept on a vertical rod, then s cos α would be the intercept, approximately, if the rod were held perpendicular to the line of sight. The slant distance is, then, D' = ks cos α = (f + c). Now it is easy to find the horizontal distance D = D' cos α and the vertical distance V = D' sin α. At one time, tables were prepared for performing these calculations, but with pocket calculators they are no longer necessary. A pocket calculator can reduce the data quickly and accurately, including the correction (f + c) without any approximation.

References

C. B. Breed and G. L. Hosmer, The Principles and Practice of Surveying, 11th ed. (New York: John Wiley & Sons, 1977). pp. 100-108.


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Composed by J. B. Calvert
Created 12 August 2003
Last revised