OPTICAL METHODS OF MEASURING DISTANCES The optical methods of distance measurement which are here dealt with are those which depend upon the measurement of a length from a small base at one end and the angle subtended by that base at the other. They include a number of alternatives. For example the base may be fixed or variable, the subtended angle variable or fixed. The base may be at the observer's end or the far end, may be vertical or horizontal, whilst the angle may be measured by hairs or lines in the focal plane, by a mi crometer scale in the focal plane, by repetition measurement on the horizontal arc, or by optical devices actually included in the base apparatus. The earliest form development took was in the use of fixed stadia hairs in the focal plane, reading, according to distance, a variable length on a vertical graduated staff. This method was later called tachymetry, or tacheometry (quick meas urement) by the Italian Porro.
Tacheometry or tachymetry was first outlined and applied by the English astronomer Gascoigne in 1639. Montanari, a Venetian doctor, constructed and used an instrument of similar principle in 1674. James Watt used it for surveying in the West of Scotland from 1771 onwards, and William Green, a London optician, did much to develop the method by the publication of a description in 1778. (Description and use of an improved Re flecting and Refracting Telescope and Scales for Surveying.) The unequal effect of refraction on the top and bottom lines of sight was understood by Green who called attention to the ad vantage of a horizontal, as opposed to a vertical, angular measure ment. Nevertheless the practical convenience of the vertical staff has made it the most popular. Methods in which the base is variable but horizontal, that is a horizontal tachymetry, are sometimes known by the term telemetry. Methods in which the base is fixed (and generally horizontal) and angles read on the theodolite arcs are known as subtense, whilst the fixed base which includes its own system of angular measurement is commonly called rangefinding. The use of the various terms is, however, and must be, elastic.
The theodolite still most commonly used has a biconvex object glass and an eyepiece, and it can be shown that the point at which the subtended angle is constant lies not on the vertical axis but at a distance in front equal to f+c, where f is the focal length at stellar focus, and c is the distance of the centre of the object glass to the diaphragm, then s=k. a+ (f+c).
(k if not known must be determined by experiment.) Used in this general form for many years, the inconvenience of the constant computation of (f +c) has led to the use of an "anallatic," or converging, lens between the object glass and the diaphragm. Due to the Italian Porro the introduction of this lens makes it possible to measure lengths directly from the sta tion as defined by the vertical axis of the instrument. To an in creasing extent the internal focussing telescope, translating for focussing, is now replacing the older style, and in it the point of measurement is so close to the vertical axis as to result in con sequent errors of an order inferior to many others which are unavoidable. (See bibliography—Henrici.) The general equation thus becomes s =k.a.
The length "s" is however the actual distance between instrument and staff, whereas surveying demands the horizontal and vertical components, and the computation of these will depend on the way in which the staff is held. Normally the staff is held vertical. Let 0 be the angle of elevation, or depression, to the centre of the intercept a. Let d be the horizontal component and h the ver tical component. Now the image of a will be reduced, obviously, to equal acos0, and s =kacos0— but d= scos0.