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Surveying
1911 Encyclopedia Britannica
or the technical term for the art of determining the position of prominent points and other objects on the surface of the ground, for the purpose of making therefrom a graphic representation of the area surveyed. The general principles on which surveys are conducted and maps computed from such data are in all instances the same; certain measures are made on the ground, and corresponding measures are protracted on paper on whatever scale may be a convenient fraction of the natural scale. The method of surveying varies with the magnitude of the survey, which may embrace an empire or represent a small plot of land. All surveys rest primarily on linear measurements for the direct determination of distances; but linear measurement is often supplemented by angular measurement which enables distances to be determined by principles of geometry over areas which cannot be conveniently measured directly, such, for instance, as hilly or broken ground. The nature of the survey depends on the proportion which the linear and angular measures bear to one another and is almost always a combination of both.
History
The art of surveying, i.e. the primary art of mapmaking from linear measurements, has no historical beginning. The first rude attempts at the representation of natural and artificial features on a ground plan based on actual measurements of which any record is obtainable were those of the Romans, who certainly made use of an instrument not unlike the plane-table for determining the alignment of their roads. Instruments adapted to surveying purposes were in use many centuries earlier than the Roman period. The Greeks used a form of log line for recording the distances run from point to point along the coast whilst making their slow voyage from the Indus to the Persian Gulf three centuries s.c.; and it is improbable that the adaptation of this form of linear measurement was confined to the sea alone. Still earlier (as early as 1600 B.C.) it is said that the Chinese knew the value of the loadstone and possessed some form of magnetic compass. But there is no record of their methods of linear measurements, or that the distances and angles measured were applied to the purpose of map-making (see Compass and MAP). The earliest maps of which we have any record were based on inaccurate astronomical determinations, and it was not till medieval times, when the Arabs made use of the Astrolabe, that nautical surveying (the earliest form of the art) could really be said to begin. In 1450 the Arabs were acquainted with the use of the compass, and could make charts of the coast-line of those countries which they visited. In 1498 Vasco da Gama saw a chart of the coast-line of India, which was shown him by a Gujarati, and there can be little doubt that he benefited largely by information obtained from charts which were of the nature of practical coast surveys. The beginning of land surveying (apart from small plan-making) was probably coincident with the earliest attempts to discover the size and figure of the earth by means of exact measurements, i.e. with the inauguration of geodesy (see Geodesy and Figure of the Earth), which iS the fundamental basis of all scientific surveying.
Classification
For convenience of reference surveying may be considered under the following heads - involving very distinct branches of the art dependent on different methods and instruments 1: 1. Geodetic triangulation. 5. Traversing, and fiscal or revenue 2. Levelling. surveys.
3. Topographical surveys. 6. Nautical surveys.
4. Geographical surveys.
I. Geodetic Triangulation Geodesy, as an abstract science dealing primarily with the dimensions and figure of the earth, may be found fully discussed in the articles Geodesy and Figure of the Earth; but, as furnishing the basis for the construction of the first framework of triangulation on which all further surveys depend (which may be described as its second but most important function), geodesy is an integral part of the art of surveying, and its relation to subsequent processes requires separate consideration. The part which geodetic triangulation plays in the general surveys of civilized countries which require closely accurate and various forms of mapping to illustrate their physical features for military, political or fiscal purposes is best exemplified by reference to some completed system which has already served its purpose over a large area. That of India will serve as an example.
The great triangulation of India was, at its inception, calculated to satisfy the requirements of geodesy as well as geography, because the latitudes and longitudes of the points of the triangulation had to be determined for future reference by process of calculation combining the results of the triangulation with the elements of the earth's figure. The latter were not then known with much accuracy, for so far geodetic operations had been mainly carried on in Europe, and additional operations nearer the equator were much wanted; the survey was conducted with a view to supply this want. Thus high accuracy was aimed at from the first.
Primarily a network was thrown over the southern peninsula. The triangles on the central meridian were measured with extra care and checked by base-lines at distances of about 2° apart in 1 The subject of tacheometry is treated under its own heading.
] latitude in order to form a geodetic arc, with the addition of astronomically determined latitudes at certain of the stations. The base-lines were measured with chains and the principal angles with a 3-ft. theodolite. The signals were cairns metrical of stones or poles. The chains were somewhat rude and their units of length had not been determined originally, India. and could not be afterwards ascertained. The results were good of their kind and sufficient for geographical purposes; but the central meridional arc - the " great arc " - was eventually deemed inadequate for geodetic requirements. A superior instrumental equipment was introduced, with an improved FIG. I.
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 base-lines 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 north-east 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 south-east quadrilateral - were dispensed with because good secondary triangulation for topography had been accomplished before they could be begun.
All base-lines were measured with the Colby apparatus of compensation bars and microscopes. The bars, to ft. long, were set up horizontally on tripod stands; the microscopes, 6 in. apart, were mounted in pairs revolving round a vertical axi 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 look-down 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 6-in. scale. At the first base-line 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 base-line 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 base-line measured as usual without thermometers in the compound bars, the may be taken as =1.5 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 base-line 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 base-line measured with simple bars than all other sources of error combined. In India the probable average lagging of the standard-bar thermometer was estimated as not less than 0.3° F., corresponding to an error of - 2 millionths in the length of a base-line 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 base-line 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, r.t'sing 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 on 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 well-known condition, height in feet = a 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 theodolite to be plumbed over the ground-level 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 sun-dried 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 alt-azimuth 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 foot-screws, 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 day, 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 dust-laden, 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 the runs of the microscopes. Then the signals were observed in regular rotation round the horizon, alter 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 180°, both in altitude and 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 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 twenty-four 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 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 so-called " 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 I21 down to - o09 parts of the contained arc on plains; the normal " coefficient of refraction " for free rays between hill stations below 6000 ft. was about o07, which diminished to 0.04 above 18,000 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 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 g 2 0 2 - g, being a value of the e.m.s. 1 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 w 1, 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 d 1, 2 for the differences between M and the single means, we have C=M -{- w l d l -? w 2 d 2 -I- (I) wl T and w =w1 ± (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 o05" 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 1 The theoretical " error of mean square " = 1.48 X " probable error." ] of each figure, because they were strictly relative for all angles measured 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 of three ways - (i.) from the squares of the reciprocals of of the weight w deduced as above from the measures Errors of such angle, (ii.) from the magnitudes of the excess of the sum of the angles of each triangle above 180 0 + 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 angles to satisfy its geometrical conditions. The three an g les of every triangle having been observed, their izing sum had to be made = 180° + the spherical excess; Angles. in networks it was also necessary that the sum 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, toto-partial and side conditions; they present n geometrical equations, which contain 4 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 so-called 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 side-lengths are computed from the base-line Sides of onwards by Legendre's theorem, each angle being dimin. ished by one-third 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 surrounding stations, the differences of latitude and longitude and the reverse azimuths are calculated in succession, for all the stations of the triangulation, by Puissant's formulae (Traite de geodesie, 3rd ed., Paris, Sides. 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 sea-level; 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 semi-axis; e the excentricity, _ a2, b°; p the radius of curvature to the meridian in latitude X - all - e2) ' - { 2 2 ? v the normal to the meridian in latitude X, = a t, X and L the given - {1 - esin-X}z latitude and longitude of A; X + AX 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 - (,r+A); c the distance between A and B. Then, all azimuths being measured from the south, we have c - - cos A cosec I" A I c2 - y=sin 2 A tan X cosec I" OX" = c2 2 3 c e 4 p.v I - e2 cos 2 A sin 2X cosec I" 3 +6-7;2 sin2A cos A (1 +3 tan 2 X) cosec I" c sin A v cos X cosec I" I c 2 sin 2A tan X + 2 cosec 1" 1 c 3 (1+3 tan 2 X) sin 2A cosA „ ((4) 6 v 3 cos X cosec I 1 c3 sin3 A tang X + 3 v 3 cos X cosec 1" sin A tan A cosec v -? v +2 tan 2 X + e IC sin 2A cosec I" - Y 3(6 / 1 (5) +tan 2 X I ta 2 A sin 2A cosA cosec I" I c 3 / +6 v3 sin 3 A tan A +2 tan 2 X) cosec I" Each A is the sum of four terms symbolized by S i, 0 2, 0 3 and 04; the calculations are so arranged as to produce these terms in the order 6X, 61, and OA, 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. I - P. cosA. c S1L = +0 1 X. Q.secX.tanA 3 1 A = +S 1 L. 6 2 X = +6 1 A. R. sinA .c 8 2 L = - 3 2 X. S.cotA 62A= +6 2 L. T II (6) 6 3 X = - 8 2 A. V V. 8 3 L = +8 3 X. U.sinA.c 8 3 A = +8 3 L. W 0 4 X = - S 3 A. X . tanA 6 4 L = +6 4 X. Y.tanA S4A = +64L. Z The calculations described so far suffice to make the angles of the several trigonometrical figures consistent inter se, and to give preliminary values of the lengths and azimuths of the sides and the latitudes and longitudes of the stations. The results are amply sufficient for the requirements of Principal of the topographer and land surveyor, and they are Triangula- published in preliminary charts, which give full numerical details of latitude, longitude, azimuth and side-length, and of height also, for each portion of the triangulation - secondary as well as principal - as executed year by year. But on the completion of the several chains of triangles further reductions became necessary, to make the triangulation everywhere consistent inter se and with the verificatory base-lines, so that the lengths and azimuths of common sides and the latitudes and longitudes of common stations should be identical at the junctions of chains and that the measured and computed lengths of the base-lines should also be identical.
As an illustration of the problem for treatment, suppose a combination of three meridional and two longitudinal chains comprising seventy-two single triangles with a base-line at each corner as shown in the accompanying C diagram (fig. suppose the, three angles of every triangle to have been measured and made consistent. Let A be the origin, with its latitude and longitude given, and also the length and azimuth of the adjoining base-line. With these data processes of calculation are carried through the triangulation to obtain the lengths and azimuths of the sides acid the latitudes and longitudes of the stations, say in the following order: from A through B to E, through F to E, through F to D, through F and E to C, and through F and D to C. Then there are two values of side, azimuth, latitude and longitude at E - one from the right-hand chains via B, the other from the left-hand chains via F; similarly there are two sets of values at C; and each of the base-lines at B, C and D has a calculated as well as a measured value. Thus eleven absolute errors are presented for dispersion over the triangulation by the application of the most appropriate correction to each angle, and, as a preliminary to the determination of these corrections, equations must be constructed between each of the absolute errors and the unknown errors of the angles from which they originated. For this purpose assume X to be the angle opposite the flank side of any triangle, and Y and Z the angles opposite the sides of continuation; also let x, y and z be the most probable values of the errors of the angles which will satisfy the given equations of condition. Then each equation may be expressed in the form [ax+by+cz] =E, the brackets indicating a summation for all the triangles involved. We have first to ascertain, the values of the coefficients a, b and c of the unknown quantities. They are readily found for the side equations on the circuits and between the base-lines, for x does not enter them, but only y and z, with coefficients which are the cotangents of Yand Z, so that these equations are simply [cot Y.y - cot Z.z] =E. But three out of four of the circuit equations are geodetic, corresponding to the closing errors in latitude, longitude and azimuth, and in them the coefficients are very complicated. They are obtained as follows. The first term of each of the three expressions for AL, and B is differentiated in terms of c and A, giving d.AX = d o - dA tan A sin I" fI.OL = AL d o +dA cot A sin I" (7) dB =dA +AA) do +dA cot A sin ," AA" or B - (7r+A)= OL " =1 614 ? PP" PPE 1. F AVA [[[Geodetic Triangulation]] in which dc and dA represent the errors in the length and azimuth of any side c which have been generated 4 in the course of the triangulation up to it from the base-line and the azimuth station at the origin. The errors in the latitude and longitude of any station which are due to the triangulation are 3 dX, = [d. A&], and dL, =[d. AL]. Let station I be the origin, and let 3, ... be the succeeding stations taken along a predetermined line of traverse, which may either run from vertex to vertex 2 of the successive triangles, zigzagging between the flanks of the chain, as in fig. 3 (1), or be carried directly along one of the flanks, as in fig. 3 (2). For the general symbols of the differential equa 1 tions substitute AX., AL„, AA„, c,,, An, and Bn, for the side between stations n and n+i of the traverse; and let and SA n be the errors generated between the sides c„. 1 and c n; then dc i 3c 1 dc, Sc 1 Sc 2 dc. n C1; 77- --? G2; ... 1 [ dA i =8A 1 i dA 2 =dB 1 +SA 2 i ... dAn=dBn_1+5An. Performing the necessary substitutions and summations, we get 11[AA]Scl+ [oA]Sc2+...+AA n (I+i[AA cot A] sin 1 +(I +2[AA cot A] sin i")511 +...+ (1 +AAn cot A n sin i")8.4..
1 [AX] ?' + 2 [AX] s ? 22 -FAXns?n 1 z - {7[AX tan A]SA 1 +2[AX tan A]SA2+... -FAX n tan A n SA n ] sin I" [AL] 11 +2 [AL] s - 2 + ... +ALn Sc ?? 1 z {i[AL cot A]8A 1 +2[AL cot A]5A 2 +.. +AL, cot A.M.) sin 1".
Thus we have the following expression for any geodetic error: - /P1 + ... +/unS n+41 1 +. .. +0noA n = E, (8) where and 49 represent the respective summations which are the coefficients of Sc and SA in each instance but the first, in which I is added to the summation in forming the coefficient of SA. The angular errors x, y and z must now be introduced, in place of Sc and SA, into the general expression, which will then take different forms, according as the route adopted for the line of traverse was the zigzag or the direct. In the former, the number of stations on the traverse is ordinarily the same as the number of triangles, and, whether or no, a common numerical notation may be adopted for both the traverse stations and the collateral triangles; thus the angular errors of every triangle enter the general expression in the form t 4 x +cot Y.u'y - cot Z. p'z, in which p.t.' µ sin 1", and the upper sign of is taken if the triangle lies to the left, the lower if to the right, of the line of traverse. When the direct traverse is adopted, there are only half as many traverse stations as triangles, and therefore only half the number of A 's and 4,'s to determine; but it becomes necessary to adopt different numberings for the stations and the triangles, and the form of the coefficients of the angular errors alternates in successive triangles. Thus, if the pth triangle has no side on the line of the traverse but only an angle at the lth station, the form is + 01.xp+ cot Pl i Z p. i .zp.
If the qth triangle has a side between the lth and the (l+I)th stations of the traverse, the form is / cot Xq(N-i - /-?'i-{-1)x4 + (01 + /-'! +1 cot Y 4) y 4 - 4,l+1 - pi cot Zq)zq. As each circuit has a right-hand and a left-hand branch, the errors of the angles are finally arranged so as to present equations of the general form [ax +by+Czl [ax+by+Lz]1 E.
The eleven circuit and base-line equations of condition having been duly constructed, the next step is to find values of the angular errors which will satisfy these equations, and be the most probable of any system of values that will do so, and at the same time will not disturb the existing harmony of the angles in each of the seventytwo triangles. Harmony is maintained by introducing the equation of condition x+y+z =o for every triangle. The most probable results are obtained by the method of minimum squares, which may be applied in two ways.
i. A factor X may be obtained for each of the eighty-three equa y2 zz tions under the condition that [ + +] is made a minimum, U, v and w being the reciprocals of the weights of the observed angles. This necessitates the simultaneous solution of eighty-three equations to obtain as many values of X. The resulting values of the errors of the angles in any, the pth, triangle, f are j j /!?
x p =u P[ a P X ]; y p =v P[ b P X ]; P P[ ](9) ii. One of the unknown quantities in every triangle, as x, may be eliminated from each of the eleven circuit and base-line equations by substituting its equivalent - (y+z) for it, a similar substitution being made in the minimum. Then the equations take the form [(b - a)y+(c - a)z] =E, while the minimum becomes L u ?)2 + V Thus we have now to find only eleven values of X by a simultaneous solution of as many equations, instead of eighty-three values from eighty-three equations; but we arrive at more complex expressions for the angular errors as / follows: P - 7 v (u W (b - aP) x] - W C aP) Al wp zp up+vp+wpt(+ v p)[(C p - a p) X ] - vP[(b P - aP)X]} The second method has invariably been adopted, originally because it was supposed that, the number of the factors X being reduced from the total number of equations to that of the circuit and base-line equations, a great saving of labour would be effected. But subsequently it was ascertained that in this respect there is little to choose between the two methods; for, when x is not eliminated, and as many factors are introduced as there are equations, the factors for the triangular equations may be readily eliminated at the outset. Then the really severe calculations will be restricted to the solution of the equations containing the factors for the circuit and base-line equations as in the second method.
In the preceding illustration it is assumed that the base-lines are errorless as compared with the triangulation. Strictly speaking, however, as base-lines are fallible quantities, presumably of different weight, their errors should be introduced as unknown quantities of which the most probable values are to be determined in a simultaneous investigation of the errors of all the facts of observation, whether linear or angular. When they are connected together by so few triangles that their ratios may be deduced as accurately, or nearly so, from the triangulation as from the measured lengths, this ought to be done; but, when the connecting triangles are so numerous that the direct ratios are of much greater weight than the trigonometrical, the errors of the base-lines may be neglected. In the reduction of the Indian triangulation it was decided, after examining the relative magnitudes of the probable errors of the linear and the angular measures and ratios, to assume the base-lines to be errorless.
The chains of triangles being largely composed of polygons or other networks, and not merely of single triangles, as has been assumed for simplicity in the illustration, the geometrical harmony to be maintained involved the introduction of a large number of " side," " central " and " toto-partial " equations of condition, as well as the triangular. Thus the problem for attack was the simultaneous solution of a number of equations of condition =that of all the geometrical conditions of every figure+four times the number of circuits formed by the chains of triangles+the number of baselines-1, the number of unknown quantities contained in the equations being that of the whole of the observed angles; the method of procedure, if rigorous, would be precisely similar to that already indicated for " harmonizing the angles of trigonometrical figures," of which it is merely an expansion from single figures to great groups.
The rigorous treatment would, however, have involved the simultaneous solution of about 4000 equations between 9230 unknown quantities, which was impracticable. The triangulation was therefore divided into sections for separate reduction, of which the most important were the five between the meridians of 67° and 92° (see fig. 1), consisting of four quadrilateral figures and a trigon, each comprising several chains of triangles and some baselines. This arrangement had the advantage of enabling the final reductions to be taken in hand as soon as convenient after the completion of ariy section, instead of being postponed until all were completed. It was subject, however, to the condition that the sections containing the best chains of triangles were to be first reduced; for, as all chains bordering contiguous sections would necessarily be " fixed " as a part of the section first reduced, it was obviously desirable to run no risk of impairing the best chains by forcing them into adjustment with others of inferior quality. It happened that both the north-east and the south-west quadrilaterals contained several of the older chains; their reduction was therefore made to follow that of the collateral sections containing the modern chains.
