"The observer's station is the centre of circle whose radius is the distance required, which is obtained by measuring the length, that is, the tangent or subtense, of the small arcs whose limits are defined by viewing their image in the focus of a telescope between two points there placed, and moving them up and down until they appear to touch the very extremities of said limits exactly. The manner of seeing is natural and by practice will become habitual, and therefore continually approach nearer to perfection.
"Thus may any surveyor in less than two hours take all the dimensions of an irregular polygon necessary for obtaining its area, if it be as much as 80 or 100 acres and limited by twenty or thirty unequal sides."
Green points out that if the subtense angle is taken horizontally, atmospheric refraction error is eliminated. He proposes to use both reflecting and refracting telescopes. With the reflector he possibly obtained accurate results, but with the refracting telescope he does not appear to have recognised a constant correction which is necessary and important.
It has since been found that in 1778 the Danish Academy of Sciences awarded a prize to G. F. Brander for a similar device, which he had applied to his plane-table, six years before. Its real discoverer was James Watt, who used it in 1771 for measuring distances in the surveys for the Tarbert and Crinan Canals. In James Patrick Muirhead's Life of James Watt, he gives a statement by Watt himself that he constructed his instrument in 1770 and showed it to Smeaton in 1772.
556.—Subtense Instruments, as that originally made by Green, are of some form of theodolite, the telescopes of which are constructed to measure either the angle subtended by the chord of a small arc or the tangent of the same. For convenience the tangent is more generally taken upon a graduated stadium or staff, which is erected for measurement perpendicularly to the horizon, the principle of which is shown in the following scheme:—
Fig. 239.—Diagram of tangential angle measurement.
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Let AC, Fig. 239, be a horizontal line; BC a stadium set up vertically. Then if the angle BAC and the height BC are known, the distance of AC can be easily calculated. For any intermediate distance between A and C a vertical will be in length proportional to this distance. Let de be at one-third the distance from A; then the line de will be one-third the length of BC. If we divide BC into three parts and place the stadium at fg two-thirds the distance from A, the angle dAe given by an instrument subtending a fixed angle will cut the staff at the second division, equal to two thirds the staff, which demonstrates the principle of all tacheometers, Cleps, etc.