333 



GEODESY. 



GEODESY. 



364 



p and Q : but tbe effect is insensible. The distance between the two 

 bars was about 2 inches, and the distance of p from the iron bar or p c 

 (which was "determined experimentally) about 3J inches. The bars 

 were enclosed in strong wooden cases, having only the ends of the 

 steel tongues exposed, and the cases laid upon trestles. Five or six 

 sets of bars were used together ; and when levelled and adjusted in 

 the line of the base, the interval between the point Q on one set, and 

 the point p on the next, instead of being variable (as in the methods of 

 Delambre and Bessel), was made equal to a given constant quantity. 

 This was effected by means of a microscopic apparatus, constructed on 

 the same principle as the measuring apparatus, two microscopes taking 

 the place of the steel tongues, and their foci being the points whose dis- 

 tance remains invariable. The microscopes were 6 inches apart, and 

 between them, at the same distance from each, was a small telescope 

 invariably connected with the two metallic bars, to which the micro- 

 scopes were attached ; the whole being so disposed that the three 

 optical axes were in the same plane, and parallel to each other at the 

 normal temperature. This apparatus being placed parallel to the line 

 of the base, over the end of one of the sets of measuring bars, in such 

 a manner that the point Q on the steel tongue was bisected by the 

 cross-wires of one of the microscopes, the next set of bars was moved 

 backwards or forwards until the point P was bisected by the cross- 

 wires of the/other microscope. A very delicate level fixed on the 

 upper bar of the microscopic apparatus gave the means of adjusting 

 the optical axis of the telescope exactly in the vertical ; by which means, 

 when it was necessary to suspend the operations, or to change the level 

 of the line of the base, the point from which the measurement was to 

 be resumed could be determined with much greater precision than by 

 the usual means of a plummet. The measuring-bars were compared 

 (daily, we believe) with a standard iron bar, which therefore is the unit 

 of the distances ; and the interval between the foci of the microscopes 

 was in like manner verified by comparison with a scale. 



The question of superiority among these different modes of measur- 

 ing a base must be decided with reference to practical convenience : 

 it cannot be affirmed that the results of any one of them are decidedly 

 more accurate than those of the others. Col. Colby's apparatus is ex- 

 ceedingly beautiful in theory, but the play of the joints by which the 

 tongues are connected with the bars, and the uncertainty there must be 

 about the determination of the invariable points, and that their dis- 

 tance remains unaltered while changes of temperature are taking place, 

 are obvious disadvantages. Bessel's apparatus is the most compact, 

 and in fewest pieces ; and we should imagine that his mode of 

 measuring the intervals between the successive bars would be found 

 easier in practice than making the distance between them constant. 

 On the other hand, the adjustment of a point under the focus of a 

 mi'.T'i.icope is an operation which can probably be executed with 

 greater precision than the measurement of the distance between two 

 solid bodies, whether by a scale and vernier, according to the method 

 of Delambre, or by a finely-divided wedge, as used by Gauss and 



The length of the base is a matter of some importance. Theoretically 

 speaking, it cannot be too long. If a distance on the earth's surface 

 (an arc or meridian, for example) deduced from a trigonometrical 

 operation be m times the length of the base, then, putting errors of 

 observations and calculation out of view, the probable error in the 

 distance is to the probable error of the base in the ratio of </m : 1 ; 

 consequently the longer the base the less is the probable error of the 

 result. On the other hand, the probable error in the measurement of 

 the base increases as the square root of its length ; so that a distance 

 deduced from a base of three miles measured only once would have 

 as great a probable error as if it had been deduced from a base of only 

 one mile measured three times, and the mean result taken as its true 

 length. The bases measured in connection with the British survey, vary 

 in length from 4'6 miles to about 8 miles. In the Indian survey the 

 bases averaged about 7 miles. The two bases at Melun and Perpignan, 

 on which the great French arc of meridian depends, were both upwards 

 of 7 miles, and each consisted of two parts inclined to each other. In 

 the Irish base 8 miles were directly measured with the compensation 

 bars, and 2 miles were added by triangulation. Struve's base was 

 2315 toises, or about 2'8 miles. The Prussian base, measured by 

 Bessel, was only 935 toises, or about 1} mile. Baron Zach, who 

 measured some small bases in Italy, contends that long bases, such as 

 were measured in the French, English, and Indian surveys, are attended 

 with no advantages corresponding to the expense they occasion ; and 

 I sor Schwerd, in an interesting account of a base of 2818 feet, 

 i red by him near Spire, reasons to the same effect. (' Die Kleine 

 rer Basis,' Speyer, 1822.) Although we cannot subscribe to 

 opinions, it must be admitted that as instruments and the 

 niethoMs of observing and computing the observations have been im- 

 proved, the necessity for frequent verification by the measure of new 

 bases has been proportionably diminished. 



When the measuring-rods have been applied to the whole line of the 

 base, and the proper reductions made for expansion, inclination, &c., 

 the distance is obtained between the terminal points, in terms of the 

 standard to which the measuring-rods are referred, on an arc of a great 

 circle of the earth. In order that the results of different surveys may 

 be comparable with each other, this circle must have a determinate 

 radius ; and hence it is usual to substitute for the arc actually measured, 



ARTS AND SCI. DIV. VOL. IV. 



the corresponding arc on the surface which coincides with the mean 

 level of the sea. Let I denote the measured length of the base, I' its 

 length reduced to the mean level of the sea, h its height above that 

 level, and ) the radius of the earth ; the I' is found from this propor- 

 tion, r + K : r : :l:l'. 



Triangulation. In commencing the Triangulation. the first step is 

 to make choice of the points or stations which are to form the summits 

 of the- principal triangles. The choice of stations must be determined 

 in some measure by the nature of the country, and with reference to 

 the objects of the survey ; but care must be taken to avoid very acute 

 angles, because small errors in the measurement of such angles will 

 give rise to large errors in the lengths of the sides deduced from them. 

 The best-conditioned triangles are those which are nearly equilateral. 

 The principal triangles should be of considerable magnitude, for the 

 probable error of a distance deduced from a base through a series of 

 triangles increases with the number of intermediate triangles. Sides 

 averaging from 20 to 50 miles may be considered as the most con- 

 venient ; but in mountainous countries, or for connecting stations 

 separated by the sea, the magnitude of the triangles will sometimes be 

 limited only by the distance at which the signals cease to be visible 

 from each other. When the object of the survey is the topography of 

 the country, the geographical positions (the latitudes, longitudes, and 

 altitudes above the sea), as well as the mutual distances of these 

 primary stations, should be determined with all the precision it is 

 possible to attain. The more remarkable features of the country are 

 afterwards connected with the principal stations, by tecondary triangles, 

 which, being liable only to small relative errors, may be determined 

 more expeditiously by less precise observations or with inferior instru- 

 ments; and the intermediate points are filled in by means of the 

 surveyor's compass and chain. [SURVEYING.] 



Siynals. When the stations have been chosen, the next point to be 

 considered ia the erection of signals. In the earlier surveys, the usual 

 practice was to select such conspicuous objects as the country 

 presented, as church spires, windmills, &c. ; but experience has shown 

 that objects of this kind, even when found (which will seldom be the 

 case) in those positions where it is desirable that the angular points of 

 the triangles should be established, are not well adapted for signals, 

 and that in general the most advantageous course is to construct them 

 for the jexpress purpose. In the earlier part of the English survey, 

 the observations were chiefly made by night, and the signals were 

 reverberatory lamps with concave metallic reflectors supported by Hag- 

 staffs, and enclosed in tin cases, having plates of glass in front to 

 screen the light from the action of the wind. Such signals answer 

 well enough for distances under 30 miles. Biot and Arago, in the 

 prolongation of the French arc of meridiau, also employed reverberatory 

 lamps and concave reflectors; and in one case the distance between 

 the station and the signal exceeded 100 miles. Bengal lights, blue- 

 lights, and other contrivances have also been used as night-signals. 

 Delambre constructed his signals of wood in the form of truncated 

 four-sided pyramids, and observed by day. For the large triangles in 

 Ireland and the west of Scotland, Colonel Colby built up conical piles 

 of dry stone, which were thrown down when the instrument was taken 

 to the spot, and again built up when it was necessary to observe the 

 same signal from other stations. Such signals were found to be visible 

 in the telescope of the great theodolite at the distance of 90 or 100 

 miles in favourable weather. Plates of polished metal, placed so as to 

 reflect the light of the sun in the proper direction, have been found a 

 powerful means of rendering a station visible. Gauss proposed the 

 heliotrope, in which the reflecting surface is silvered glass ; and this 

 was the signal which was principally used by Struve and Bessel. 

 Another method, adopted by Bessel for short distances, was the 

 reflexion of light from a hemisphere of polished copper. These two 

 last methods have the disadvantage of rendering the observer depen- 

 dent on sunshine, but in other respects they afford excellent signals, 

 for as the light proceeds from a point, the observation is made with the 

 greatest precision. In the case of the hemisphere indeed, the luminous 

 point is not in the axis of the signal, but as the radius of the 

 hemisphere and the azimuth of the sun at the time of the observation 

 are known, its position with reference to the axis can be accurately 

 computed, and a correction applied if the deviation is sensible. But 

 all solid bodies used as signals render a similar correction necessary 

 when the light falls upon them obliquely. To avoid this incon- 

 venience, Svanberg observed the light of the sky through a rectangular 

 opening in a blackened board which turned about a vertical axis, so 

 that its plane could always be placed perpendicular to the visual ray. 

 Night-signals are found inconvenient by reason of the unsteadiness and 

 the scintillations of the light ; and accordingly geodetical observations 

 are now generally made by day ; nevertheless, under peculiar circum- 

 stances, night observations may be advantageous, or even necessary. 

 Thus, in India, Colonel Everest found that the greater refraction 

 during the night sometimes rendered stations visible which could not 

 be seen by day, being hid by the intervening ground. 



With respect to instruments and the methods of observing in geo- 

 detical surveys, ample information is given in the articles THEODOLITE, 

 REPEATING CIRCLE, &c. We may here remark, however, that as each 

 signal (speaking generally) is the common vertex of several triangles, 

 an angle required for the calculation of a triangle may frequently be 

 obtained from the sum or difference of other angles at the same point, 



