GEODETIC TRIANGULATION]
latitude in order to form a geodetic arc, with the addition of
astronomically determined latitudes at certain of the stations. The
baselines were measured with chains and the principal
angles with a 3ft. theodolite. The signals were cairns
of stones or poles. The chains were somewhat rude and
their units of length had not been determined originally,
and could not be afterwards ascertained. The results
were good of their kind and sufficient for geographical pur
poses; but the central meridional arc—the " great arc "—was
eventually deemed inadequate for geodetic requirements. A
superior instrumental equipment was introduced, with an improved
modus operandi, under the direction of Colonel Sir G. Everest in 1832. The network system of triangulation was superseded by meridional and longitudinal chains taking the form of gridirons and resting on baselines at the angles of the gridirons, as represented in fig. 1. For convenience of reduction and nomenclature the triangulation west of meridian 92° E. has been divided into five sections—the lowest a trigon, the other four quadrilaterals distinguished by cardinal points which have reference to an observatory in Central India, the adopted origin of latitudes. In the northeast quadrilateral, which was first measured, the meridional chains are about one degree apart; this distance was latterly much increased and eventually certain chains—as on the Malabar coast and on meridian 84° in the southeast quadrilateral—were dispensed with because good secondary triangulation for topography had been accomplished before they could be begun.
All baselines were measured with the Colby apparatus of compensation bars and microscopes. The bars, lo ft. long, were set up horizontally on tripod stands; the microscopes, 6 in. apart, were mounted in pairs revolving round a vertical axis and were set up on tribrachs fitted to the ends of the bars. Six bars and five central and two end pairs of microscopes—the latter with their vertical axes perforated for a lookdown telescope—constituted a complete apparatus, measuring 63 ft. between the ground pins or registers. Compound bars are more liable to accidental changes of length than simple bars; they were therefore tested from time to time by comparison with a standard simple bar; the microscopes were also tested by comparison with a standard 6in. scale. At the first baseline the compensated bars were found to be liable to sensible variations of length with the diurnal variations of temperature; these were supposed to be due to the different thermal conductivities of the brass and the iron components. It became necessary, therefore, to determine the mean daily length of the bars precisely, for which reason they were systematically compared with the standard before and after, and sometimes at the middle of, the baseline measurement throughout the entire day for a space of three days, and under conditions as nearly similar as possible to those obtaining during the measurement. Eventually thermometers were applied experimentally to both components of a compound bar, when it was found that the diurnal variations in length were principally due to difference of position relatively to the sun, not to difference of conductivity—the component nearest the sun acquiring heat most rapidly or parting with it most slowly, notwithstanding that both were in the same box, which was always sheltered from the sun's rays. Happily the systematic comparisons of the compound bars with the standard were found to give a sufficiently exact determination of the mean daily length. An elaborate investigation of theoretical probable errors (p.e.) at the Cape Comorin base showed that, for any baseline measured as usual without thermometers in the compound bars, the p.e. may be taken as t I.5 millionth parts of the length, excluding unascertainable constant errors, and that on introducing thermometers into these bars the p.e. was diminished tot 0.55 millionths.143
In all baseline measurements the weak point is the determination of the temperature of the bars when that of the atmosphere is rapidly rising or falling; the thermometers acquire and lose heat more rapidly than the bar if their bulbs are outside, and more slowly if inside the bar. Thus there is always more or less lagging, and its effects are only eliminated when the rises and falls are of equal amount and duration; but as a rule the rise generally predominates greatly during the usual hours of work, and whenever this happens lagging may cause more error in a baseline measured with simple bars than all other sources of error combined. In India the probable average lagging of the standardbar thermometer was estimated as not less than o.3° F., corresponding to an error of — 2 millionths in the length of a baseline measured with iron bars. With compound bars lagging would be much the same for both components and its influence would consequently be eliminated. Thus the most perfect baseline apparatus would seem to be one of compensation bars with thermometers attached to each component; then the comparisons with the standard need only be taken at the times when the temperature is constant, and there is no lagging.
The plan of triangulation was broadly a system of internal meridional and longitudinal chains with an external border of oblique chains following the course of the frontier and the coast lines. The design of each chain was necessarily much influenced by the physical features of the country over which it was carried. The most difficult tracts were plains, devoid of any commanding points of view, in some parts covered with forest and jungle, malarious and almost uninhabited, in other parts covered with towns and villages and umbrageous trees. In such tracts triangulation was impossible except by constructing towers as stations of observation, raising them to a sufficient height to overtop at least the earth's curvature, and then either increasing the height to surmount all obstacles to mutual vision, or clearing the lines. Thus in hilly and open country the chains of triangles were generally made " double " throughout, i.e. formed of polygonal and quadrilateral figures to give greater breadth and accuracy; but in forest and close country they were carried out as series of single triangles, to give a minimum of labour and expense. Symmetry was secured by restricting the angles between the limits of 30° and 90°. The average side length was 30 M. in hill country and II in the plains; the longest principal side was 62.7 m., though in the secondary triangulation to the Himalayan peaks there were sides exceeding 200 M. Long sides were at first considered desirable, on the principle that the fewer the links the greater the accuracy of a chain of triangles; but it was eventually found that good observations qn long sides could only be obtained under exceptionally favourable atmospheric conditions. In plains the length was governed by the height to which towers could be conveniently raised to surmount the curvature, under the wellknown condition, height in feet =
X square of the distance in miles; thus 24 ft. of height was needed at each end of a side to overtop the curvature in 12 m., and to this had to be added whatever was required to surmount obstacles on the ground. In Indian plains refraction is more frequently negative than positive during sunshine; no reduction could therefore be made for it.
The selection of sites for stations, a simple matter in hills and open country, is often difficult in plains and close country. In the early operations, when the great arc was being carried across the wide plains of the Gangetic valley, which are covered with villages and trees and other obstacles to distant vision, masts 35 ft. high were carried about for the support of the small reconnoitring theodolites, with a sufficiency of poles and bamboos to form a scaffolding of the same height for the observer. Other masts 70 ft. high, with arrangements for displaying blue lights by night at 90 ft., were erected at the spots where station sites were wanted. But the cost of transport was great, the rate of progress was slow, and the results were unsatisfactory. Eventually a method of touch rather than sight was adopted, feeling the ground to search for the obstacles to be avoided, rather than attempting to look over them: the " rays " were traced either by a minor triangulation, or by a traverse with theodolite and perambulator, or by a simple alignment of flags. The first method gives the direction of the new station most accurately; the second searches the ground most closely; the third is best suited for tracts of uninhabited forest in which there is no choice of either line or site, and the required station may be built at the intersection of the two trial rays leading up to it. As a rule it has been found most economical and expeditious to raise the towers only to the height necessary for surmounting the curvature, and to remove the trees and other obstacles on the lines.
Each principal station has a central masonry pillar, circular and 3 to 4 ft. in diameter, for the support of a large theodolite, and around it a platform 14 to 16 ft. square for the observatory tent, observer and signallers. The pillar is isolated from the platform, and when solid carries the station mark—a dot surrounded by a circle—engraved on a stone at its surface, and on additional stones or the rock in situ, in the normal of the upper mark; but, if the height is considerable and there is a liability to deflection, the pillar is constructed with a central vertical shaft to enable the
Trigonometrical Survey of India.
theodolite to be plumbed over the groundlevel mark, to which access is obtained through a passage in the basement. In early years this precaution against deflection was neglected and the pillars were built solid throughout, whatever their height; the surrounding platforms, being usually constructed of sundried bricks or stones and earth, were liable to fall and press against the pillars, some of which thus became deflected during the rainy seasons that intervened between the periods during which operations were arrested or the beginning and close of the successive circuits of triangles. Large theodolites were invariably employed. Repeating circles were highly thought of by French geodesists at the time when the operations in India were begun; but they were not used in the survey, and have now been generally discarded. The principal theodolites were somewhat similar to the astronomer's altazimuth instrument, but with larger azimuthal and smaller vertical circles, also with a greater base to give the firmness and stability which are required in measuring horizontal angles. The azimuthal circles had mostly diameters of either 36 or 24 in., the vertical circles having a diameter of 18 in. In all the theodolites the base was a tribrach resting on three levelling footscrews, and the circles are read by microscopes; but in different instruments the fixed and the rotatory parts of the body varied. In some the vertical axis was fixed on the tribrach and projected upwards; in others it revolved in the tribrach and projected downwards. In the former the azimuthal circle was fixed to the tribrach, while the telescope pillars, the microscopes, the clamps and the tangent screws were attached to a drum revolving round the vertical axis; in the latter the microscopes, clamps and tangent screws were fixed to the tribrach, while the telescope pillars and the azimuthal circle were attached to a plate fixed at the head of the rotary vertical axis.
Cairns of stones, poles or other opaque signals were primarily employed, the angles being measured by day only; eventually it was found that the atmosphere was often more favourable for observing by night than by clay, and that distant points were raised well into view by refraction by night which might be invisible or only seen with difficulty by day. Lamps were then introduced of the simple form of a cup, 6 in. in diameter, filled with cotton seeds steeped in oil and resin, to burn under an inverted earthen jar, 30 in. in diameter, with an aperture in the side towards the observer. Subsequently this contrivance gave place to the Argand lamp with parabolic reflector; the opague day signals were discarded for heliotropes reflecting the sun's rays to the observer.. The introduction of luminous signals not only rendered the night as well as the day available for the observations but changed the character of the operations, enabling work to be done during the dry and healthy season of the year, when the atmosphere is generally hazy and dustladen, instead of being restricted as formerly to the rainy and unhealthy seasons, when distant opaque objects are best seen. A higher degree of accuracy was also secured, for the luminous signals were invariably displayed through diaphragms of appropriate aperture, truly centred over the station mark; and, looking like stars, they could be observed with greater precision, whereas opaque signals are always dim in comparison and are liable to be seen excentrically when the light falls on one side. A signalling party of three men was usually found sufficient to manipulate a pair of heliotropes—one for single, two for double reflection, according to the sun's position—and a lamp, throughout the night and day. Heliotropers were also employed at the observing stations to flash instructions to the signallers.
The theodolites were invariably set up under tents for protection
against sun, wind and rain, and centred, levelled and adjusted for
measuring the runs of the microscopes. Then the signals were
observed in regular rotation round the horizon, alter
errors are practically cancelled and any remaining error is most probably due to lateral refraction, more especially when the rays of light graze the surface of the ground. The three angles of every triangle were always measured.
The apparent altitude of a distant point is liable to considerable variations during the twentyfour hours, under the influence of
changes in the density of the lower strata of the atmo vertical
sphere. Terrestrial refraction is capricious, more par Angles.
ticularly when the rays of light graze the surface of the
ground, passing through a medium which is liable to extremes of rarefaction and condensation, under the alternate influence of the sun's heat radiated from the surface of the ground and of chilled atmospheric vapour. When the back and forward verticals at a pair of stations are equally refracted, their difference gives an exact measure of the difference of height. But the atmospheric conditions are not always identical at the same moment everywhere on long rays which graze the surface of the ground, and the ray between two reciprocating stations is liable to be differently refracted at its extremities, each end being influenced in a greater degree by the conditions prevailing around it than by those at a distance.; thus instances are on record of a station A being invisible from another B, while B was visible from A.
When the great arc entered the plains of the Gangetic valley, simultaneous reciprocal verticals were at first adopted with the hope of eliminating refraction; but it was soon found Refraction. that they did not do so sufficiently to justify the ex
pense of the additional instruments and observers. Afterwards the back and forward verticals were observed as the stations were visited in succession, the back angles at as nearly as possible the same time of the day as the forward angles, and always during the socalled " time .of minimum refraction," which ordinarily begins about an hour after apparent noon and lasts from two to three hours. The apparent zenith distance is always greatest then, but the refraction is a minimum only at stations which are well elevated above the surface of the ground; at stations on plains the refraction is liable to pass through zero and attain a considerable negative magnitude during the heat of the day, for the lower strata of the atmosphere are then less dense than the strata immediately above and the rays are refracted downwards. On plains the greatest positive refractions are also obtained—maximum values, both positive and negative, usually occurring, the former by night, the latter by day, when the sky is most free from clouds. The values actually met with were found to range from + 1.21 down to —o•oq parts of the contained arc on plains; the normal " coefficient of refraction " for free rays between hill stations below 6000 ft. was about 0.07, which diminished to 0.04 above i8,000 ft., broadly varying inversely as the temperature and directly as the pressure, but much influenced also by local climatic conditions.
In measuring the vertical angles with the great theodolites, graduation errors were regarded as insignificant compared with errors arising from uncertain refraction; thus no arrangement was made for effecting changes of zero in the circle settings. The observations were always taken in pairs, face right and left, to eliminate index errors, only a few daily, but some on as many days as possible, for the variations from day to day were found to be greater than the diurnal variations during the hours of minimum refraction.
In the ordnance and other surveys the bearings of the surrounding stations are deduced from the actual observations, but from the " included angles " in the Indian survey. The weights. observations of every angle are tabulated vertically in
as many columns as the number of circle settings face left and face right, and the mean for each setting is taken. For several years the general mean of these was adopted as the final result; but subsequently a " concluded angle " was obtained by combining the single means with weights inversely proportional to g2 + o2 n—g, being a value of the e.n.s.i of graduation derived empirically from the differences between the general mean and the mean for each setting, o the e.m.s. of observation deduced from the differences between the individual measures and their respective means, and n the number of measures at each setting. Thus, putting wi, w2, ..
for the weights of the single means, w for the weight of the concluded angle, M for the general mean, C for the concluded angle, and d2, . . . for the differences between M and the single means, we have
C — M + wadi + w2d2 + (i)
wi+W2+
and w =wi + wz + (2) C — M vanishes when n is constant; it is inappreciable when g is much larger than o; it is significant only when the graduation errors are more minute than the errors of observation; but it was always small, not exceeding 0.14" with the system of two rounds of measures and 0.05" with the system of three rounds.
The weights of the concluded angles thus obtained were employed in the primary reductions of the angles of single triangles and polygons which were made to satisfy the geometrical conditions
I. The theoretical " error of mean square " se 1.48 X " probable error."
Horizontal
nately from right to left and vice versa; after the pre
Angles.
scribed minimum number of rounds, either two or three, had been thus measured, the telescope was turned through 18o°, both in altitude and azimuth, changing the position of the face of the vertical circle relatively to the observer, and further rounds were measured; additional measures of single angles were taken if the prescribed observations were not sufficiently accordant. As the microscopes were invariably equidistant and their number was always odd, either three or five, the readings taken on the azimuthal circle during the telescope pointings to any object in the two positions of the vertical circle, " face right " and " face left," were made on twice as many equidistant graduations as the number of microscopes. The theodolite was then shifted bodily in azimuth, by being turned on the ring on the head of the stand, which brought new graduations under the microscopes at the telescope pointings; then further rounds were measured in the new positions, face right and face left. This process was repeated as often as had been previously prescribed, the successive angular shifts of position being made by equal arcs bringing equidistant graduations under the microscopes during the successive telescope pointings to one and the same object. By these arrangements all periodic errors of graduation were eliminated, the numerous graduations that were read tended to cancel accidental errors of division, and the numerous rounds of measures to minimize the errors of observation arising from atmospheric and personal causes.
Under this system of procedure the instrumental and ordinary
of each figure, because they were strictly relative for all angles me_sured with the same instrument and under similar circumstances and conditions, as was almost always the case for each single figure. But in the final reductions, when numerous chains of triangles composed of figures executed with different instruments and under different circumstances came to be adjusted simultaneously, it was necessary to modify the original weights, on such evidence of the precision of the angles as might be obtained from other and more reliable sources than the actual measures of the angles. This treatment will now be described.
Values of theoretical error for groups of angles measured with the same instrument and under similar conditions may be obtained Theoretical in three ways—(i.) from the squares of the reciprocals Errors of of the weight w deduced as above from the measures
of such angle, (ii.) from the magnitudes of the excess of Angles. the sum of the angles of each triangle above 18o°+ the spherical excess, and (iii..) from the magnitudes of the corrections which it is necessary to apply to the angles of polygonal figures and networks to satisfy the several geometrical conditions.
Every figure, whether a single triangle or a polygonal network, was made consistent by the application of corrections to the observed Harmon.' angles to satisfy its geometrical conditions. The three
angles of every triangle having been observed, their
izing
sum had to be made =18o° + the spherical excess;
Angles. in networks it was also necessary that the sim of the angles measured round the horizon at any station should be exactly = 360°, that the sum of the parts of an angle measured at different times should equal the whole and that the ratio of any two sides should be identical, whatever the route through which it was computed. These are called the triangular, central, totopartial and side conditions; they present n geometrical equations, which contain t unknown quantities, the errors of the observed angles, t being always > n. When these equations are satisfied and the deduced values of errors are applied as corrections to the observed angles, the figure becomes consistent. Primarily the equations were treated by a method of successive approximations; but afterwards they were all solved simultaneously by the socalled method of minimum squares, which leads to the most probable of any system of corrections.
The angles having been made geometrically consistent inter se in each figure, the sidelengths are computed from the baseline Sides of onwards by Legendre's theorem, each angle being diminTriaag/es. ished by onethird of the spherical excess of the triangle
to which it appertains. The theorem is applicable without sensible error to triangles of a much larger size than any that are ever measured.
A station of origin being chosen of which the latitude and longitude are known astronomically, and also the azimuth of one of the Latitudeandsurrounding stations, the differences of latitude and Longitudeoflongitude and the reverse azimuths are calculated in Stations; succession, for all the stations of the triangulation, Azimuth of by Puissant's formulae (Traits de geodesie, 3rd ed., Paris,
Sides. 184 2).
Problem.—Assuming the earth to be spheroidal, let A and B be
two stations on its surface, and let the latitude and longitude of A be known, also the azimuth of B at A, and the distance between A and B at the mean sealevel; we have to find the latitude and longitude of B and the azimuth of A at B.
The following symbols are employed: a the major and b the
minor semiaxis; e the excentricity, = a2a2b2 ; p the radius of
a(I —e)
curvature to the meridian in latitude X, 1—e2sin2A; ; z the normal
a
to the meridian in latitude A, = 11e2sin2A 1 l; X and L the given latitude and longitude of A; X + &A and L + AL the required latitude and longitude of B; A the azimuth of B at A; B the azimuth
of A at B ; AA = B — (7r+A) ; c the distance between A and B. Then, all azimuths being measured from the south, we have
c
—cos A cosec 1"
c2
2 —sin2A tan X cosec I"
DA—_
c e
3 c'
2 z ri
4 p v 1 —ez cos2A sin 2X cosec I
3
+ sin2A cos A(1l3 tan2X) cosec 1"
p.v
csin A
7 cos Acosec I"
1 r2 sin 2A tan X
+2 cos x cosec I"
t c3 (I+3 tan2X) sin 2A cosA
6 v3 cos A cosec I"
c3 sin3A tan2 A
+7373 cos x cosec I"
 sin A tan A cosec 1"
/ `` ) +1 v2 i +2 tan2A +eleoC z } sin 2A cosec I"
c3 5
+ tanzA tan2Asi
— V3(6 n 2A cosA cosec I„
c3 .
+6 Y3sin3A tan A (1+2 tan2A) cosec 1"
Each A is the sum of four terns symbolized by Si, S2, S3 and S4; the calculations are so arranged as to produce these terms in the order SA, SL, and SA, each term entering as a factor in calculating the following term. The arrangement is shown below in equations in which the symbols P, Q, Z represent the factors which depend on the adopted geodetic constants, and vary with the latitude; the logarithms of their numerical values are tabulated in the Auxiliary Tables to Facilitate the Calculations of the Indian Survey.
End of Article: GEODETIC 

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