Surveying - Encyclopedia Britannica 1911

GEOGRAPHICAL NAMES

SURVEYING, 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.

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.

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.

But the reduction of each of these great sections was in itself a very formidable undertaking, necessitating some departure from a purely rigorous treatment. For the chains were largely composed of polygonal networks and not of single triangles only as assumed in the illustration, and therefore cognizance had to be taken of a dB n = dA,:+1= (ro) number of " side " and other geometrical equations of condition, which entered irregularly and caused great entanglement. Equations 9 and I 0 of the illustration are of a simple form because they have a single geometrical condition to maintain, the triangular, which is not only expressed by the simple and symmetrical equation x+y+z=o, but - what is of much greater importance - recurs in a regular order of sequence that materially facilitates the general solution. Thus, though the calculations must in all cases be very numerous and laborious, rules can be formulated under which they can be well controlled at every stage and eventually brought to a successful issue. The other geometrical conditions of networks are expressed by equations which are not merely of a more complex form but have no regular order of sequence, for the networks present a variety of forms; thus their introduction would cause much entanglement and complication, and greatly increase the labour of the calculations and the chances of failure. Wherever, therefore, any compound figure occurred, only so much of it as was required to form a chain of single triangles was employed. The figure having previously been made consistent, it was immaterial what part was employed, but the selection was usually made so as to introduce the fewest triangles. The triangulation for final simultaneous reduction was thus made to consist of chains of single triangles only; but all the included angles were " fixed " simultaneously. The excluded angles of compound figures were subsequently harmonized with the fixed angles, which was readily done for each figure per se. This departure from rigorous accuracy was not of material importance, for the angles of the compound figures excluded from the simultaneous reduction had already, in the course of the several independent figural adjustments, been made to exert their full influence on the included angles. The figural adjustments had, however, introduced new relations between the angles of different figures, causing their weights to increase caeteris paribus with the number of geometrical conditions satisfied in each instance. Thus, suppose w to be the average weight of the I observed angles of any figure, and n the number of geometrical conditions presented for satisfaction; then the average weight of the angles after adjustment may be taken as w. t t n , the factor thus being 15 for a triangle, I. 8 for a hexagon, 2 for a quadrilateral, 2.5 for the network around the Sironj base-line, &c.

In framing the normal equations between the indeterminate factors X for the final simultaneous reduction, it would have greatly added to the labour of the subsequent calculations if a separate weight had been given to each angle, as was done in the primary figural reductions; this was obviously unnecessary, for theoretical requirements would now be amply satisfied by giving equal weights to all the angles of each independent figure. The mean weight that was finally adopted for the angles of each group was therefore taken as w.pt - n' P being the modulus.

The second of the two processes for applying the method of minimum squares having been adopted, the values of the errors y and z of the angles appertaining to any, the pth, triangle were finally expressed by the following equations, which are derived from (I o) by substituting u for the reciprocal final mean weight as above determined y P = u p [(2b p - a v - cv)A] J P [(2c, - a 5 - b5)X] The following table gives the number of equations of condition and unknown quantities - the angular errors - in the five great sections of the triangulation, which were respectively included in the simultaneous general reductions and relegated to the subsequent adjustments of each figure per se: - The corrections to the angles were generally minute, rarely exceeding the theoretical probable errors of the angles, and therefore applicable without taking any liberties with the facts of observation.

Azimuth observations in connexion with the principal triangulation were determined by measuring the horizontal angle between a referring mark and a circumpolar star, shortly before and after elongation, and usually at both elongations in order to eliminate the error of the star's place. Systematic changes of " face " and of the zero settings of the azimuthal circle were made as in the measurement of the principal angles; but the repetitions on each zero were more numerous; the azimuthal levels were read and corrections applied to the star observations for dislevelment. The triangulation was not adjusted, in the course of the final simultaneous reduction, to the astronomically determined azimuths, because they are liable to be vitiated by local attractions; but the azimuths observed at about fifty stations around the primary azimuthal station, which was adopted as the origin of the geodetic calculations, were referred to that station, through the triangulation, for comparison with the primary azimuth. A table was prepared of the differences (observed at the origin - computed from a distance) between the primary and the geodetic azimuths; the differences were assumed to be mainly due to the local deflexions of the plumb-line and only partially to error in the triangulation, and each was multiplied by the factor tangent of latitude of origin, tangent of latitude of comparing station in order that the effect of the local attraction on the azimuth observed at the distant station - which varies with the latitude and is = the deflexion in the prime vertical X the tangent of the latitude - might be converted to what it would have been had the station been situated in the same latitude as the origin. Each deduction was given a weight, w, inversely proportional to the number of triangles connecting the station with the origin, and the most probable value of the error of the observed azimuth at the origin was taken as x - w] p w] - (12); [w] the value of x thus obtained was - Pi".

The formulae employed in the reduction of the azimuth observations were as follows. In the spherical triangle PZS, in which P is the pole, Z the zenith and S the star, the co-latitude PZ and the polar distance PS are known, and, as the angle at S is a right angle at the elongation, the hour angle and the azimuth at that time are found from the equations cosP = tanPScotPZ, cosZ = cosPSsinP.

The interval, SP, between the time of any observation and that of the elongation being known, the corresponding azimuthal angle, SZ, between the two positions of the star at the times of observation and elongation is given rigorously by the following expression - tan SZ 2sin22SP = - cotPSsinPZsinP[ I +tan 2 PScosSP +sec 2 PScotPsinOP } which is expressed as follows for logarithmic computation - SZ = - n+1 ' m tan Z cos' PS where m = 2 sin' S P cosec I", n sin'PS sin 2 P, and 1= cot P sin SP;1, m, and n are tabulated.

Let A and B (fig. 4) be any two points the normals at which meet at C, cutting the sea-level at p and q; take Dq=Ap, then BD is the difference of height; draw the tangents Aa and Bb at Refraction. A and B, then aAB is the depression of B at A and bBA that of A at B; join AD, then BD is determined from the triangle ABD. The triangulation gives the distance between A and B at the sea-level, whence pq=c; thus, putting Ap, the height of A above the sea-level, =H, and pC = r, AD (1 + - 4 r2 } (1 4).

Putting D a and Db for the actual depressions at A and B, S for the angle at A, usually called the " subtended angle," and h for BD S = 2(D b - Da) (15), and h = AD s CO S in DbS (16).

The angle at C being =D b +Da, S may be expressed in terms of a single vertical angle and C when observations have been taken at only one of the two points.

v C, the " contained arc," =c° cosec 1" in seconds. Putting D'a 2pv and D'b for the observed vertical angles, and Oa, 4b for the amounts by which they are affected by refraction, Da=D'a+Y'a and Db = D'b+cl)b; Oa and Or, may differ in amount, but as they (13), FIG. 4.

cannot be separately ascertained they are always assumed to be equal; the hypothesis is sufficiently exact for practical purposes when both verticals have been measured under similar atmospheric conditions. The refractions being taken equal, the observed verticals are substituted for the true in (15) to find S, and the difference of height is calculated by (16); the third term within the brackets of (14) is usually omitted. The mean value of the refraction is deduced from the formula 0 = z(C - D'd-I-D'b)i (17). An approximate value is thus obtained from the observations between the pairs of reciprocating stations in each district, and the corresponding mean " coefficient of refraction," 0+ C, is computed for the district, and is employed when heights have to be determined from observations at a single station only. When either of the vertical angles is an elevation - E must be substituted for D in the above expressions.' 2. Levelling Levelling is the art of determining the relative heights of points on the surface of the ground as referred to a hypothetical surface which cuts the direction of gravity everywhere at right angles. When a line of instrumental levels is begun at the sea-level, a series of heights is determined corresponding to what would be found by perpendicular measurements upwards from the surface of water communicating freely with the sea in underground channels; thus the line traced indicates a hypothetical prolongation of the surface of the sea inland, which is everywhere conformable to the earth's curvature.

The trigonometrical determination of the relative heights of points at known distances apart, by the measurements of their mutual vertical angles - is a method of levelling. But the method to which the term " levelling " is always applied is that of the direct determination of the differences of height from the readings of the lines at which graduated staves, held vertically over the points, are cut by the horizontal plane which passes through the eye of the observer. Each method has its own advantages. The former is less accurate, but best suited for the requirements of a general geographical survey, to obtain the heights of all the more prominent objects on the surface of the ground, whether accessible or not. The latter may be conducted with extreme precision, and is specially valuable for the determination of the relative levels, however minute, of easily accessible points, however numerous, which succeed each other at short intervals apart; thus it is very generally undertaken pari passu with geographical surveys to furnish lines of level for ready reference as a check on the accuracy of the trigonometrical heights. In levelling with staves the measurements are always taken from the horizontal plane which passes through the eye of the observer; but the line of levels which it is the object of the operations to trace is a curved line, everywhere conforming to the normal curvature of the earth's surface, and deviating more and more from the plane of reference as the distance from the station of observation increases. Thus, either a correction for curvature must be applied to every staff reading, or the instrument must be set up at equal distances from the staves; the curvature correction, being the same for each staff, will then be eliminated from the difference of the readings, which will thus give the true difference of level of the points on which the staves are set up.

Levelling has to be repeated frequently in executing a long line of levels - say seven times on an average in every mile - and must be conducted with precaution against various errors. Instrumental errors arise when the visual axis of the telescope is not perpendicular to the axis of rotation, and when the focusing tube does not move truly parallel to the visual axis on a change of focus. The first error is eliminated, and the second avoided, by placing the instrument at equal distances from the staves; and as this procedure has also the advantage of eliminating the corrections for both curvature and refraction, it should invariably be adopted.

In topographical and levelling operations it is sometimes convenient to apply small corrections to observations of the height for curvature and refraction simultaneously. Putting d for the distance, r for the earth's radius, and for the coefficient of refraction, and expressing the distance and radius in miles and the correction to height in feet, then correction for curvature =3 d 2; correction for refraction = d2; correction for both -4K d2 3 Errors of staff readings should be guarded against by having the staves graduated on both faces, but differently figured, so that the observer may not be biased to repeat an error of the first reading in the second. The staves of the Indian survey have one face painted white with black divisions - feet, tenths and hundredths - from o to io, the other black with white divisions from 5.55 to 1 5.55. Deflexion from horizontality may either be measured and allowed for by taking the readings of the ends of the bubble of the spirit-level and applying corresponding corrections to the staff readings, or be eliminated by setting the bubble to the same position on its scale at the reading of the second staff as at that of the first, both being equidistant from the observer.

Certain errors are liable to recur in a constant order and to accumulate to a considerable magnitude, though they may be too minute to attract notice at any single station, as when the work is carried on under a uniformly sinking or rising refraction - from morning to midday or from midday to evening - or when the instrument takes some time to settle down on its bearings after being set up for observation. They may be eliminated (i.) by alternating the order of observation of the staves, taking the back staff first at one station and the forward first at the next; (ii.) by working in a circuit, or returning over the same line back to the origin; (iii.) by dividing a line into sections and reversing the direction of operation in alternate sections. Cumulative error, not eliminable by working in a circuit, may be caused when there is much northing or southing in the direction of the line, for then the sun's light will often fall endwise on the bubble of the level, illuminating the outer edge of the rim at the nearer end and the inner edge at the farther end, and so biasing the observer to take scale readings of edges which are not equidistant from the centre of the bubble; this introduces a tendency to raise the south or depress the north ends of lines of level in the northern hemisphere. On long lines, the employment of a second observer, working independently over the same ground as the first, station by station, is very desirable. The great lines are usually carried over the main roads of the country, a number of " bench marks " being fixed for future reference. In the ordnance survey of Great Britain lines have been carried across from coast to coast in such a manner that the level of any common crossing point may be found by several independent lines. Of these points there are 166 in England, Scotland and Wales; the discrepancies met with at them were adjusted simultaneously by the method of minimum squares.

The sea-level is the natural datum plane for levelling operations, more particularly in countries bordering on the ocean.

The earliest surveys of coasts were made for the use of navigators and, as it was considered very important that the charts should everywhere show the minimum depth of water which a vessel would meet with, low water of springtides was adopted as the datum. But this does not answer the requirements of a land survey, because the tidal range between extreme high and low water differs greatly at different points on coast-lines. Thus the generally adopted datum plane for land surveys is the mean sea-level, which, if not absolutely uniform all the world over, is much more nearly so than low water. Tidal observations have been taken at nearly fifty points on the coasts of Great Britain, which were connected by levelling operations; the local levels of mean sea were found to differ by larger magnitudes than could fairly be attributed to errors in the lines of level, having a range of 12 to 15 in. above or below the mean of all at points on the open coast, and more in tidal rivers. 2 But the general mean of the coast stations for England and Wales was practically identical with that for Scotland. The observations, however, were seldom of longer duration than a fortnight, which is insufficient for an exact determination of even the short period components of the tides, and ignores the annual and semiannual components, which occasionally attain considerable magnitudes. The mean sea-levels at Port Said in the Mediterranean and at Suez in the Red Sea have been found to be identical, and a similar identity is said to exist in the levels of the Atlantic and the Pacific oceans on the opposite coasts of the Isthmus of Panama. This is in favour of a uniform level all the world over; but, on the other hand, lines of level carried across the continent of Europe make the mean sea-level of the Mediterranean at Marseilles and Trieste from 2 to 5 ft. below that of the North Sea and the Atlantic at Amsterdam and Brest - a result which 2 In tidal estuaries and rivers the mean water-level rises above the mean sea-level as the distance from the open coast-line increases; for instance, in the Hooghly river, passing Calcutta, there is a rise of 10 in. in 42 m. between Sagar (Saugor) Island at the mouth of the river and Diamond Harbour, and a further rise of 20 in. in 43 m. between Diamond Harbour and Kidderpur.

] it is not easy to explain on mechanical principles. In India various tidal stations on the east and west coasts, at which the mean sea-level has been determined from several years' observations, have been connected by lines of level run along the coasts and across the continent; the differences between the results were in all cases due with greater probability to error generated in levelling over lines of great length than to actual differences of sea-level in different localities.

The sea-level, however, may not coincide everywhere with the geometrical figure which most closely represents the earth's surface, but may be raised or lowered, here and there, under the influence of local and abnormal attractions, presenting an equipotential surface - an ellipsoid or spheroid of revolution slightly deformed by bumps and hollows - which H. Bruns calls a " geoid." Archdeacon Pratt has shown that, under the combined influence of the positive attraction of the Himalayan Mountains and the negative attraction of the Indian Ocean, the sea-level may be some 560 ft. higher at Karachi. than at Cape Comorin; but, on the other hand, the Indian pendulum operations have shown that there is a deficiency of density under the Himalayas and an increase under the bed of the ocean, which may wholly compensate for the excess of the mountain masses and deficiency of the ocean, and leave the surface undisturbed. If any bumps and hollows exist, they cannot be measured; instrumentally; for the instrumental levels will be affected by the local attractions precisely as the sea-level is, and will thus invariably show level surfaces even should there be considerable deviations from the geometrical figure.

3. Topographical Surveys The skeleton framework of a survey over a large area should be triangulation, although it is frequently combined with traversing. The method of filling in the details is necessarily influenced to some extent by the nature of the framework, but it depends mainly on the magnitude of the scale and the requisite degree of minutiae. In all instances the principal triangles and circuit traverses have to be broken down into smaller ones to furnish a sufficient number of fixed points and lines for the subsequent operations. The filling in may be performed wholly by linear measurements or wholly by direction intersections, but is most frequently effected by both linear and angular measures, the former taken with chains and tapes and offset poles, the latter with small theodolites, sextants, optical squares or other reflecting instruments, magnetized needles, prismatic compasses and plane tables. When the scale of a survey is large, the linear and angular measures are usually recorded on the spot in a fieldbook and afterwards plotted in office; when small they are sometimes drawn on the spot on a plane table and the field-book is dispensed with.

In every country the scale is generally expressed by the ratio of some fraction or multiple of the smallest to the largest national units of length, but sometimes by the fraction which indicates the ratio of the length of a line on the paper to that of the corresponding line on the ground. The latter form is obviously preferable, being international and independent of the various units of length adopted by different nations (see MAP). In the ordnance survey of Great Britain and Ireland and the Indian survey the double unit of the foot and the Gunter's link (_of a foot) are employed, the former invariably in the triangulation, the latter generally in the traversing and filling in, because of its convenience in calculations and measurements of area, a square chain of Ioo Gunter's links being exactly one-tenth of an acre.

In the ordnance survey all linear measures are made with the Gunter's chain, all angular with small theodolites only; neither magnetized nor reflecting instruments nor plane tables are ever employed, except in hill sketching. As a rule the filling in is done by triangle-chaining only; traverses with theodolite and chain are occasionally resorted to, but only when it is necessary to work round woods and hill tracts across which right lines cannot be carried.

Detail surveying by triangles is based on the points of the minor triangulation. The sides are first chained perfectly straight, all the points where the lines of interior detail cross the sides being fixed; the alignment is effected with a small theodolite, and marks are established at the crossing points and at any other points on the sides where they may be of use in the subsequent operations. The surveyor is given a diagram of the triangulation, but no side lengths, as the accuracy of his chaining is tested by comparison with the trigonometrical values. Then straight lines are carried across the intermediate detail between the points established on the sides; they constitute the principal " cutting up or split lines"; their crossings of detail are marked in turn and straight lines are run between them. The process is continued until a sufficient number of lines and marks have been established on the ground to enable all houses, roads, fences, streams. railways, canals, rivers, boundaries and other details to be conveniently measured up to and fixed. Perpendicular offsets are limited to eighty and twenty links for the respective scales of 6 in. to a mile and 2, When a considerable area has to be treated by traverses it is divided into a number of blocks of convenient size, bounded by roads, rivers or parish boundaries, and a " traverse on the meridian of the origin " is carried round the periphery of each block. Beginning at a trigonometrical station, the theodolite is set to circle reading o° o' with the telescope pointing to the north, and at every " forward " station of the traverse the circle is set to the same reading when the telescope is pointed at the " back " station as was obtained at the back station when the telescope was pointing to the forward one. When the circuit is completed and the theodolite again put up at the origin and set on the last back station with the appropriate circle reading, the circle reading, with the telescope again pointed to the first forward station, will be the same as at first, if no error has been committed. This system establishes a convenient check on the accuracy of the operations and enables the angles to be readily protracted on a system of lines parallel to the meridian of the origin. As a further check the traverse is connected with all contiguous trigonometrical stations by measured angles and distances. Traverses are frequently carried between the points already fixed on the sides of the minor triangles; the initial side is then adopted, instead of the meridian, as the axis of co-ordinates for the plotting, the telescope being pointed with circle reading o° o' to either of the trigonometrical stations at the extremities of the side.

The plotting is done from the field-books of the surveyors by a separate agency. Its accuracy is tested by examination on the ground, when all necessary addenda are made. The examiner - who should be surveyor, plotter and draughtsman - verifies the accuracy of the detail by intersections and productions and occasional direct measurements, and generally endeavours to cause the details under examination to prove the accuracy of each other rather than to obtain direct proof by remeasurement. He fixes conspicuous trees and delineates the woods, footpaths, rocks, precipices, steep slopes, embankments, &c., and supplies the requisite information regarding minor objects to enable a draughtsman to make a perfect representation according to the scale of the map. In examining a coast-line he delineates the foreshore and sketches the strike and dip of the stratified rocks. In tidal rivers he ascertains and marks the highest points to which the ordinary tides flow. The examiner on the 25.344 in. scale (= 5 1 - R1) is required to give all necessary information regarding the parcels of ground of different character - whether arable, pasture, wood, moor, moss, sandy - defining the limits of each on a separate tracing if necessary. He has also to distinguish between turnpike, parish and occupation roads, to collect all names, and to furnish notes of military, baronial and ecclesiastical antiquities to enable them to be appropriately represented in the final maps. The latter are subjected to a double examination - first in the office, secondly on the ground; they are then handed over to the officer in charge of the levelling to have the levels and contour lines inserted, and finally to the hill sketchers, whose duty it is to make an artistic representation of the features of the ground.

In the Indian survey all filling in is done by plane-tabling on a basis of points previously fixed; the methods differ simply in the extent to which linear measures are introduced to supplement the direction rays of the plane-table. When the scale of the survey is small, direct measurements of distance are rarely made and the filling is usually done wholly by direction intersections, which fix all the principal points, and by eye-sketching; but as the scale is increased linear measures with chains and offset poles are introduced to the extent that may be desirable. A sheet of drawing paper is mounted on cloth over the face of the plane-table; the points, previously fixed by triangulation or otherwise, are projected on it - the collateral meridians and parallels, or the rectangular coordinates, when these are more convenient for employment than the spherical, having first been drawn; the plane-table is then ready for use. Operations are begun at a fixed point by aligning with the sight rule on another fixed point, which brings the meridian line of the table on that of the station. The magnetic needle may now be placed on the table and a position assigned to it for future reference. Rays are drawn from the station point on the table to all conspicuous objects around with the aid of the sight rule. The table is then taken to other fixed points, and the process of ray-drawing is repeated at each; thus a number of objects, some of which may become available as stations of observation, are fixed. Additional stations may be established by setting up the table on a ray, adjusting it on the back station - that from which the ray was drawn - and then obtaining a cross intersection with the sight rule laid on some other fixed point, also by interpolating between three fixed points situated around the observer. The magnetic needle may not be relied on for correct orientation, but is of service in enabling the table to be set so nearly true at the outset that it has to be very slightly altered afterwards. The error in the setting is indicated by the rays from the surrounding fixed points intersecting in a small triangle instead of a point, and a slight change in azimuth suffices to reduce the triangle to a point, which will indicate the position of the station exactly. Azimuthal error being less apparent on short than on long lines, interpolation is best performed by rays drawn from near points, and checked by rays drawn to distant points, as the latter show most strongly the magnitude of any error of the primary magnetic setting. In this way, and by self-verificatory traverses " on the back ray " between fixed points, plane-table stations are established over the ground at appropriate intervals, depending on the scale of the survey; and from these stations all surrounding objects which the scale permits of being shown are laid down on the table, sometimes by rays only, sometimes by a single ray and a measured distance. The general configuration of the ground is delineated simultaneously. In checking and examination various methods are followed. For large scale work in plains it is customary to run arbitrary lines across it and make an independent survey of the belt of ground to a distance of a few chains on either side for comparison with the original survey; the smaller scale hill topography is checked by examination from commanding points, and also by traverses run across the finished work on the table.

4. Geographical Surveying The introduction by mechanical means of superior graduation in instruments of the smaller class has enabled surveyors to effect good results more rapidly, and with less expenditure on equipment and on the staff necessary for transport in the field, than was formerly possible. The 12-in. theodolite of the present day, with micrometer adjustments to assist in the reading of minute subdivisions of angular graduation, is found to be equal to the old 24-in. or even 36-in. instruments. New Methods for the measurement of bases have largely superseded the laborious process of measurement by the alignment of " compensation " bars, though not entirely independent of them. The Jaderin apparatus, which consists of a wire 25 metres in length stretched along a series of cradles or supports, is the simplest means of measuring a base yet devised; and experiments with it at the Pulkova observatory show it to be capable of producing most accurate results. But there is a measurable defect in the apparatus, owing to the liability of the wires to change in length under variable conditions of temperature. It is therefore considered necessary, where base measurements for geodetic purposes are to be made with scientific exactness, that the Jaderin wires should be compared before and after use with a standard measurement, and this standard is best attained by the use of the Brunner, or Colby, bars. The direct process of measurement is not extended to such lengths as formerly, but from the ends of a shorter line, the length of which has been exactly determined, the base is extended by a process of triangulation.

There are vast areas in which, while it is impossible to apply the elaborate processes of first-class or " geodetic " triangulation, Secondary it is nevertheless desirable that we should rapidly acquire such geographical knowledge as will enable us to lay down political boundaries, to project roads and railways, and to attain such exact knowledge of special localities as will further military ends. Such surveys are called by various names - military surveys, first surveys, geographical surveys, &c.; but, inasmuch as they are all undertaken with the same end in view, i.e. the acquisition of a sound topographical map on various scales, and as that end serves civil purposes as much as military, it seems appropriate to designate them geographical surveys only.

The governing principles of geographical surveys are rapidity and economy. Accuracy is, of course, a recognized necessity, but the term must admit of a certain elasticity in geo- which graphical work which i s i nadmissible in geodetic or cadastral functions. It is obviously foolish to expend as much money over the elaboration of topo- Surveys. graphy in the unpeopled sand wastes wh i ch border the Nile valley, for instance (albeit those deserts may be full of topographical detail), as in the valley itself - the great centre of Egyptian cultivation, the great military highway of northern Africa. On the other hand, the most careful accuracy attainable in the art of topographical delineation is requisite in illustrating the nature of a district which immediately surrounds what may prove hereafter to be an important military position. And this, again, implies a class of technical accuracy which is quite apart from the rigid attention to detail of a cadastral survey, and demands a much higher intelligence to compass.

The technical principles of procedure, however, are the same in geographical as in other surveys. A geographical survey must equally start from a base and be supported by triangulation, or at least by some process analogous g ? Y P g Su rv ey. to triangulation, which will furnish the necessary skeleton on which to adjust the topography so as to ensure a complete and homogeneous map.

This base may be found in a variety of ways. If geodetic triangulation exists in the country, that triangulation should of course include a wide extent of secondary determinations, the fixing of peaks and points in the landscape far away to either flank, which will either give the data for further extension of geographical triangulation, or which may even serve the purposes of the map-maker without any such extension at all. In this manner the Indus valley series of the triangulation of India has furnished the basis for surveys across Afghanistan and Baluchistan to the Oxus and Persia.

Should no such preliminary determinations of the value of one or two starting-points be available, and it becomes necessary to measure a base and to work ab initio, the Jaderin wire apparatus may be adopted. It is cheap (cost about £50), and far more accurate than the process of measuring either by any known " subtense " system (in which the distance is computed from the angle subtended by a bar of given length) or by measurement with a steel chain. This latter method may, however, be adopted so long as the base can be levelled, repeated measurements obtained, and the chain compared with a standard steel tape before and after use.

The initial data on which to start a comprehensive scheme of triangulation for a geographical survey are: (i) latitude; (2) longitude; (3) azimuth; and (4) altitude, and this data should, if possible, be obtained pari passu with the measurement of the base.

A 6-in. transit theodolite, fitted with a micrometer eyepiece and extra vertical wires, is the instrument par excellence for work of this nature; and it possesses the advantages of portability and comparative cheapness.

The method of using it for the purposes of determining values for (i) and (3), i.e. for ascertaining the latitude of one end of the base and the azimuth of the other end from it, are fully explained in Major Talbot's paper on Military Latitude and A zi Surveying in the Field (J. Mackay & Co., Chatham, muth. 1889), which is not a theoretical treatise, but a practical illustration of methods employed successfully in the geographical survey of a very large area of the Indian transfrontier districts. It should be noted that these observations are not merely of an initial character. They should be constantly repeated as the survey advances, and under certain circumstances (referred to subsequently) they require daily repetition.

The problems connected with the determination of (2) longitude have of late years occupied much of the attention of scientific surveyors. No system of absolute determination is accurate enough for combination with triangulat i on, as affording a check on the accuracy of the latter, and the spaces in the world across which geographical surveying has yet to be carried are rapidly becoming too restricted to admit of any liability to error so great as is invariably involved in such determinations. It is true that absolute values derived from the observation of lunar distances, or occultations, have often proved to be of the highest value; but there remains a degree of uncertainty (possibly due to the want of exact knowledge of the moon's position at any instant of time), even when observations have been taken with all the advantages of the most elaborate arrangements and the most scientific manipulation, which renders the roughest form of triangulation more trustworthy for ascertaining differential longitude than any comparison between the absolute determination of any two points. Consequently, if an absolute determination is necessary it should be made once, with all possible care, and the value obtained should be carried through the whole scheme of triangulation. It rests with the surveyor to decide at what point of the general survey this value can best be introduced, provided he can estimate the probable longitudinal value of his initial base within a few minutes of the truth. A final correction in longitude is constant, and can easily be applied. With reference to such absolute determinations of longitude, Major S. Grant's " Diagram for determining the parallaxes in declination and right ascension of a heavenly body and its application to the prediction of occultations " (Roy. Geog. Soc. Journ. for June 1896) will afford the observer valuable assistance.

But the recognized method of obtaining a longitude value in recent geographical fields is by means of the telegraph - a method so simple and so accurate that it may be applied with Telegraph advantage even to the checking of long lines of tri- Determina- angulation. No effort should be spared to introduce a lions. telegraphic longitude value into any scheme of geographical survey. It involves a clear line and an instructed observer at each end, but, given these desiderata, the interchange of time signals sufficient for an accurate record only requires a night or two of clear weather. But inasmuch as rigorous accuracy in the observations for time is necessary, it would be well for the surveyor in the field to be provided with a sidereal chronometer. Under all other circumstances demanding time observations (and they are an essential supplement to every class of astronomical determination) an ordinary mean time watch is sufficient.

With reference to altitude determinations, there has lately been observable amongst surveyors a growing distrust of barometric Altitude. results and a reaction in favour of direct levelling, or of differential results derived from direct observation with the theodolite (or clinometer) rather than from comparison of those determined by aneroid or hypsometer. It is indeed impossible to eliminate the uncertainties due to the variable atmospheric pressure introduced by " weather " changes from any barometric record. A mercurial barometer advantageously placed and carefully observed at fixed diurnal intervals throughout a comparatively long period may give fairly trustworthy results if a constant comparison can be maintained throughout that period with similar records at sealevel, or at any fixed altitude. Yet observations extending over several months have been found to yield results which compare most unfavourably with those attained during the process of triangulation by continued lines of vertical observations from point to point, even when the uncertainties of the correction for refraction are taken into account. Errors introduced into vertical observations by refraction are readily ascertainable and comparatively unimportant in their effect. Those due to variable atmospheric conditions on barometric records are still indefinite, and are likely to remain so. The result has been that the latter have been relegated to purely local conditions of survey, and that whenever practicable the former are combined with the general process of triangulation.

The conditions under which geographical surveys can be carried out are of infinite variety, but those conditions are rare which absolutely preclude the possibility of any such expression) of the topography, even when the configuration of the land surface is favourable. In such circumstances the method of observing azimuths to points situated approximately near to the probable route in advance, and of determining the exact position of those points in latitude " 1 " .r as one by one they are passed by the moving force, Contiol. has been found to yield results which are quite sufficiently accurate to ensure the final adjustment of the entire route geography to any subsequent system of triangulation which may be extended through the country traversed, without serious discrepancies in compilation. It is, however, obvious that as accuracy depends greatly on the exact determination of absolute latitude values, this method is best adapted to a route running approximately parallel to a meridian, and is at complete disadvantage in one running east and west. Where the conditions are favourable to its application, it has been adopted with most satisfactory results; as, for instance, on the route between Seistan and Herat, where the initial data for the Russo-Afghan boundary delimitation was secured by this means, and more recently on the boundary surveys of western Abyssinia.

When an active enemy is in the field, and topographical operations are consequently restricted, it is usually possible to obtain the necessary " control " (i.e. a few well-fixed points M il i tary determined by triangulation) for topography in advance of a position securely held. With a very little assist- Geography.. ance from the triangulator an experienced topographer will be able to sketch a field of action with far more certainty and rapidity than can be attained by the ordinary so-called " military surveyor,'" and he may, in favourable circumstances, combine his work with that of the military balloonist in such a way as to represent every feature of importance, even in a widely extended position held by the enemy. The application of the camera and of telephotography to the evolution of a map of the enemy's position is well understood in France (ride Colonel Laussedat's treatise on " The History of Topography "), as it is in Russia, and we must in future expect that all advantages of an expert and professional map of the whole theatre of a campaign will lie in the hands of the general who is best supplied with professional experts to compass them. Geographical surveying and military surveying are convertible terms, and it is important to note that both equally require the services of a highly trained staff of professional topographers. During the war between Russia and. Turkey (1877-78)78) upwards of a hundred professional geographical surveyors were pressed into military service, besides the regular survey staff which is attached to every army corps. Triangulation was carried across the Balkans by eight different series; every pass and every notable feature of the Balkans and Rhodope Mountains was accurately surveyed, as well as the plains intervening between the Balkans and Constantinople. Surveys on a scale which averaged about I m. = I in. were carried up to the very gates of the city.

The use of the camera as an accessory to the plane table (i.e. the art of photo-topography) has been applied almost exclusively to geographical or exploratory surveys. The camera -i"° is specially prepared, resting on a graduated horizontal Photo plate which is read with verniers, and with a small graphs. telescope and vertical arc attached. Cross wires are fixed in the focal plane of the camera, which is also fitted with a magnetic. needle and a scale so placed that the magnetic declination, the scale, and the intersection of the cross wires are all photographed on the plate containing the view. A panoramic group of views. (slightly overlapping each other) is taken at each station, and the angular distance between each is measured on the horizontal circle. The process of constructing the horizontal projection from these perspective views involves plotting the skeleton triangulation, as obtained from the primary triangulation, with the theodolite (which precedes the photo-topographical survey), or from the horizontal plate of the camera. With several stations so plotted, the view from each of them of a certain portion of the country may be projected on the plane of the map, and salient points seen in perspective may be fixed by intersection. The field work of a photo-topographic party consists primarily in execution of a triangulation by the usual methods which would be adapted to any ordinary topographical survey. To this is. added a secondary triangulation, which is executed pari passu with the photography for the purpose of fixing the position of the camera stations. From such stations alone the topographical details are finally secured with the aid of the photographs. Great care is necessary in the selection of stations that will be suitable both for the extension of triangulation and the purposes of closely overlooking topographical details. In order to obtain means for correctly orienting the photographic views when plotting the map from them, it is usual, whilst making the exposures, to observe two or three points in each view with the alt-azimuth attached to the camera, in order to ascertain the horizontal and vertical angles between them. It is also advisable to keep an outline sketch of the landscape for the purpose of recording names of roads, buildings, &c.

The process of projecting the map from the photographs involves, the use of two drawing-boards, on one of which the graphical determination of the points is made, and on the other the details C o y Conditions surveys at all. Perfect freedom of action, and the under which Geographi- recognition of such work as a public benefit, are not cal Surveys often attainable. Far more frequently the oppor are carried tunity offers itself to the surveyor with the progress out. of a political mission or the advance of an army in the field. It cannot be too strongly insisted on that geographical surveys are functions of both civil and military operations. Very much of such work is also possible where a country lies open to exploration, not actively hostile, but yet unsettled and adverse to strangers. The geographical surveyor has to fit himself to all such conditions, and it may happen that a continuous, comprehensive scheme of triangulation as a map basis is impossible. Under such circumstances other expedients must be adopted to ensure that accuracy of position which cannot be attained by the topographer unaided.

. During a long-continued march extending through a line of country generally favourable for survey purposes - a condition Route which frequently occurs - when forward movement is Surveying. a necessity, and an average of 10 to 15 m. of daily progress is maintained, one officer and an assistant can measure a daily base, obtain the necessary astronomical determinations, triangulate from both ends so as to fix the azimuth and distance from the base of points passed yesterday and those to be passed to-morrow; project those points on to the topographer's plane-table to be ready for the next day's work, and check each day's record by latitude; whilst a second assistant runs the topography through the route, basing his work on points so fixed, on the scale of 2 or 4 m. to the inch, according to the amount of detail. Occasionally a hill can be reached in the course of the day's march, or during a day's halt, which will materially assist to consolidate and strengthen the series.

It may, however, frequently be impossible to maintain a consistent series of triangulation for the " control " (to use an American 152 of the final topography are drawn. The principal trigonometrical points are plotted on both these boards by their co-ordinates, and the camera stations either by their co-ordinate values or by intersection. Intermediate points, selected as appearing on two or more negatives, are then projected by intersection. The horizontal projection of a panorama consisting of any given number of plates is a regular geometrical figure of as many sides as there are plates, enclosing an inscribed circle whose radius is the focal length of the camera. Having correctly plotted the position of one plate, or view, with reference to the projected camera station by means of the angle observed to some known point within it, it is possible to plot the position of the rest of the series, with reference to the camera station and the orienting triangulation point, by the angular differences which are dependent on the number of photographs forming the sides of the geometrical figure. Having secured the correct orientation of the horizontal plan, direction lines are drawn from the plotted camera station to points photographed, and the position of topographical features is fixed by intersection from two or more camera stations.

The plane-table is the instrument, par excellence, on which the geographical surveyor must depend for the final mapping of the physical features of the country under survey. The methods of adapting the plane-table to geographical table. requirements differ with those varying climatic conditions which affect its construction. In the comparatively dry climate of Asiatic Russia or of the United States, where errors arising from the unequal expansion of the plane-table board are insignificant, the plane-table is largely made use of as a triangulating instrument, and is fitted with slow-motion screws and with other appliances for increasing the certainty and the accuracy of observations. Such an adaptation of the plane-table is found to be impossible in India, where the great alternations of temperature, no less than of atmospheric humidity, tend to vitiate the accuracy of the projections on the surface of the board by the unequal effects of expansion in the material of which it is composed. The Indian plane-table is of the simplest possible construction, and it is never used in connexion with the stadia for ascertaining the distances of points and features of the ground (as is the case in America); and in place of the complicated American alidade, with its telescope and vertical arc, a simple sight rule is used, and a chirometer for the measurement of vertical angles. The Indian plane-table approximates closely in general construction to the Gannett " pattern of America, which is specially constructed for exploratory surveys.

The scale on which geographical surveys are conducted is necessarily small. It may be reckoned at from I: 500000 to I t 125000, or from i in. =8 m. to I in. =2 m. The i in. = I m. Scale. scale is the normal scale for rigorous topography, and although it is impossible to fix a definite line beyond which geographical scales merge into topographical (for instance, the i-in. scale is classed as geographical in America whenever the continuous line contour system of ground representation gives place to hachuring), it is convenient to assume generally that geographical scales of mapping are smaller than the i-in. scale.

On the smaller scales of I: 500000 or I: 250000 an experienced geographical surveyor, in favourable country, will complete an area of mapping from day to day which will practically cover Out-turn, nearly all that falls within his range of vision; and he will, in the course of five or six months of continuous travelling (especially if provided with the necessary " control ") cover an area of geographical mapping illustrating all important topographical features representable on the small scale of his survey, which may be reckoned at tens of thousands of square miles. But inasmuch as everything depends upon his range of vision, and the constant occurrence of suitable features from which to extend it, there is obviously no guiding rule by which to reckon his probable out-turn. The same uncertainty which exists about " out-turn " manifestly exists about " cost." The normal cost of the I-in. rigorous topographical survey in India, when carried over districts Cost, which present an average of hills, plains and forests, may be estimated as between 35 to 40 shillings a square mile. This compares favourably with the rates which obtain in America over districts which probably present far more facilities for surveying than India does, but where cheap native labour is unknown. The geographical surveyor is simply a topographer employed on a smaller scale survey. His equipment and staff are somewhat less, but, on the other hand, his travelling expenses are greater. It is found that, on the whole, a fair average for the cost of geographical work may be struck by applying the square of the unit of scale as a factor to I-in. survey rates; thus a quarter-inch scale survey (i.e. 4 m. to the in.), should be one-sixteenth of the cost per mile of the i-in. survey over similar ground. A geographical reconnaissance on the scale of I: 500000 (8 m. =I in.) should be one-sixtyfourth of the square-mile cost of the I-in. survey, &c. This is, indeed, a close approximation to the results obtained on the Indian transfrontier, and would probably be found to hold good for British colonial possessions.

In processes of map reproduction an invention for the reproduction of drawings by a method of direct printing on zinc without the intervention of a negative has proved of great value. By this [[[Traversing And Fiscal]] method a considerable quantity of work has been turned out in much less time and at a much lower cost than would be p p involved by any process of photo-zincography or duction. lithography. A large number of cadastral maps have been reproduced at about one-ninth of the ordinary cadastral rate.

For the rapid reproduction of geographical maps in the field in order to meet the requirements of a general conducting a campaign, or of a political officer on a boundary mission, no better method has been evolved than the ferrotype process, by which blue prints can be secured in a few hours from a drawing of the original on tracing-cloth. The sensitized paper and printing-frame are far more portable than any photo-lithographic apparatus. Sketches illustrative of a field of action may be placed in the hands of the general commanding on the day following the action, if the weather conditions are favourable for their development. The necessity for darkness whilst dealing with the sensitized material is a drawback, but it may usually be arranged with blankets and waterproof sheets when a tent is not available.

5. Traversing And Fiscal, Or Revenue, Surveys Traversing is a combination of linear and angular measures in equal proportions; the surveyor proceeds from point to point, measuring the lines between them and at each point the angle between the back and forward lines; he runs his lines as much as possible over level and open ground, avoiding obstacles by working round them. The system is well suited for laying down roads, boundary lines, and circuitous features of the ground, and is very generally resorted to for filling in the interior details of surveys based on triangulation. It has been largely employed in certain districts of British India, which had to be surveyed in a manner to satisfy fiscal as well as topographical requirements; for, the village being the administrative unit of the district, the boundary of every village had to be laid down, and this necessitated the survey of an enormous number of circuits. Moreover, the traverse system was better adapted for the country than a network of triangulation, as the ground was generally very flat and covered with trees, villages, and other obstacles to distant vision, and was also devoid of hills and other commanding points of view. The principal triangulation had been carried across it, but by chains executed with great difficulty and expense, and therefore at wide intervals apart, with the intention that the intermediate spaces should be provided with points as a basis for the general topography in some other way. A system of traverses was obviously the best that could be adopted under the circumstances, as it not only gave all the village boundaries, but was practically easier to execute than a network of minor triangulation.

In the Indian survey the traverses are executed in minor circuits following the periphery of each village and in major circuits comprising groups of several villages; the former are done with 4" to 6" theodolites and a single chain, the latter with 7" to io" theodolites and a pair of chains, which are compared frequently with a standard. The main circuits are connected with every station of the principal triangulation within reach. The meridian of the origin is determined by astronomical observations; the angle at the origin between the meridian and the next station is measured, and then at each of the successive stations the angle between the immediately preceding and following stations; summing these together, the " inclinations " of the lines between the stations to the meridian of the origin are successively determined. The distances between the stations, multiplied by the cosines and sines of the inclinations, give the distance of each station from the one preceding it, resolved in the directions parallel and perpendicular respectively to the meridian of the origin; and the algebraical sums of these quantities give the corresponding rectangular co-ordinates of the successive stations relatively to the origin and its meridian. The area included in any circuit is expressed by the formula area = half algebraical sum of products (x i +x2) (y2 - yi) (18), xl, yi being the co-ordinates of the first, and x 2, y 2 those of the second station, of every line of the traverse in succession round the circuit.

Of geometrical tests there are two, both applicable at the close of a circuit: the first is angular, viz. the sum of all the interior angles of the described polygon should be equal to twice as many right angles as the figure has sides, less four; the second is linear, viz. the algebraical sum of the x co-ordinates and that of the y co-ordinates ,should each be = o. The astronomical test is this: at any station of the traverse the azimuth of a referring mark may be determined by astronomical observations; the inclination of the line between the station and the referring mark to the meridian of the origin is given by the traverse, the two should differ by the convergency of the meridians of the station and the origin. In practice the angles of the traverse are usually adjusted to satisfy their special geometrical and astronomical tests in the first instance, and then the co-ordinates of the stations are calculated and adjusted by corrections applied to the longest, that the angles may be least disturbed, as no further corrections are given them.

The exact value of the convergence, when the distance and azi Conver- muth of the second astronomical station from the first gency of are known, is that of B - (71-+A) of equation (5); Meridians. but, as the first term is sufficient for a traverse, we have convergency = x tan A cosecs 1" substituting x, the co-ordinate of the second station perpendicular to the meridian of the origin, for c sin A. The co-ordinates of the principal stations of a trigonometrical survey are usually the spherical co-ordinates of latitude and longi- Adjustment tude; those of a traverse survey are always rectangular, of Tray plane for a small area but spherical for a large one.

verses to It is often necessary, therefore, for purposes of ccorn versesverses to and check at stations common to surveys of Triangula- both descriptions, to convert either rectangular co tion. ordinates into latitudes and longitudes, or vice versa, in order that the errors of traverses may be dispersed by proportion over the co-ordinates of the traverse stations, if desired, or adjusted in the final mapping. The latter is generally all that is necessary, more particularly when the traverses are referred to successive trigonometrical stations as origins, as the operations are being extended, in order to prevent any large accumulation of error. Similar conversions are also frequently necessary in map projections. The method of effecting them will now be indicated.

Let A and B be any two points, Aa the meridian of A, Bb the parallel of latitude of B; then Ab, Bb will be their differences in Q Transforma- latitude and longitude; from B draw BP perpendicular to Aa; then AP, BP tion of Co- will be the rectangular spherical co-ordin- ordinates. ates of B relatively to A. Put BP = x, AP = y, the arc Pb =7 , and the arc Bb, the differ ence of longitude, = also let %a, and X, be the latitudes of A, B, and the point P, p p the radius of curvature of the meridian, and p p the normal ter minating in the axis minor for the latitude X i ,; and L FIG. 5. let po be the radius of curvature for the latitude 2(Aa+Ap)Then, when the rectangular co-ordinates are given, we have, taking A as the origin, the latitude of which is known, = y cosec 1"; 7 1 = tan A p cosec I"; (19).

Aa = p cosec I" - 7 1 ; w = x sec(A b +-- 37 1) cosec I" Pp And, when the latitude and longitude are given, we have n= (=1-L b -sin = 2 sin I" (20).

y = X Aa + f }sin I" x = wv p cos (Xi + J71)sin I" When a hill peak or other prominent object has been observed from a number of stations whose co-ordinates are already fixed, the Co-ordinates converging rays may be projected graphically, and from Co of Unvisited an examination of their several intersections the most probable position of the object may be obtained almost Point. as accurately as by calculations by the method of least squares, which are very laborious and out of place for the determination of a secondary point. The following is a description of the application of this method to points on a plane surface in the calculations of the ordnance survey. Let sl, s2,. . be stations whose rectangular co-ordinates, x0, x2,. .. perpendicular, and yl, y2,. .. parallel, to the meridian of the origin are given; let al, a2,. be the bearings - here the direction-inclinations with the meridian of the origin - of any point P, as observed at the several stations; and let p be an approximate position of P, with co-ordinates x p, y p , as determined by graphical projection on a district map or by rough calculation. Construct a diagram of the rays converging around p, by taking a point to represent p and drawing two lines through it at right angles to each other to In the Indian survey, tables are employed for these calculations which give the value of I" of arc in feet on the meridian, and on each parallel of latitude, at intervals of 5' apart; also a corresponding table of arc-versines (Pb) of spheroidal arcs of parallel (Bb) I° in length, from which the arc-versines for shorter or longer arcs are obtained proportionally to the squares of the arcs; x is taken as the difference of longitude converted into linear measure.

indicate the directions of north, south, east and west. Calculate accurately (yp - y l) tan a l, and compare with (x p - x i); the difference will show how far the direction of the ray from s 1 falls to the east or west of p. Or calculate (x, - x i)cot al, and compare with (y, - yi) to find how far the direction falls to the north or south of p. Set off the distance on the corresponding axis of p, and through N w FIG. 6.

the point thus fixed draw the direction a l with a common protractor. All the other rays around p may be drawn in like manner; they will intersect each other in a number of points, the centre of which may be adopted as the most probable position of P. The co-ordinates of P will then be readily obtained from those of p the distances on the meridian and perpendicular. In the annexed diagram (fig. 6) P is supposed to have been observed from five stations, giving as many intersecting rays, (1, 1), (2, 2),; there are ten points of intersection, the mean position of which gives the true position of P, the assumed position being p. The advantages claimed for the method are that, the bearings being independent, an erroneous bearing may be redrawn without disturbing those that are correct; similarly new bearings may be introduced without disturbing previous work, and observations from a large number of stations may be readily utilized, whereas, when calculation is resorted to, observations in excess of the minimum number required are frequently rejected because of the labour of computing them.

- Clarke, Geodesy (London); Waller, " India's Contribution to Geodesy," Trans. Roy. Soc., vol. clxxxvi. (1895); Thuillier, Manual of Surveying for India (Calcutta); Gore, Handbook of Professional Instructions for the Topographical Branch Survey of India Department (Calcutta); D'A. Jackson, Aid to Survey Practice (London, 1899); Woodthorpe, Hints to Travellers (Plane-tabling section); Grant, "Diagram for Determining Parallaxes," &c., Geog. Tourn. (June 1896); Pierce, " Economic Use of the Plane-Table," vol. xcii. pt. ii., Pro. Inst. Civ. Eng.; BridgesLee, Photographic Surveying (1899); London Society of Engineers; Laussedat, Recherches sur les instruments les me'thodes et le dessin topographique (Paris, 1898); H. M. Wilson, Topographic Surveying (New York, 1905); Professional Papers Royal Engineers (occasional paper series), vol. xiii. paper v. by Holdich; vol. xiv. paper ii. by Talbot; vol. xxvi. paper i. by MacDonnell (R.E. Institute, Chatham).