1911 Encyclopædia Britannica/Surveying

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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 measure- ment is often supplemented by angular measurement which enables distances to be determined by principles of geometry over areas which cannot be conveniently measured directly, such, for instance, as hilly or broken ground. The nature of the survey depends on the proportion which the linear and angular measures bear to one another and is almost always a combination of both.

History. — The art of surveying, i.e. the primary art of map- making 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 B.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 {q.v.), 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 Earth, Figure of the), which is the fundamental basis of all scientific surveying.

Classification. — For convenience of reference surveying may be considered under the following heads — involving very distinct branches of the art dependent on different methods and instruments[1]: —

1. Geodetic triangulation.

2. Levelling.

3. Topographical surveys.

4. Geographical surveys.

5. Traversing, and fiscal or revenue surveys.

6. Nautical surveys.

1. 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 Earth, Figure 02 the; 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 Jbe 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 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 of stones or poles. The chains were somewhat rude and their units of length had not been determined originally, and could not be afterwards ascertained. The results were good of their kind and sufficient for geographical pur- poses; but the central meridional arc — the "great arc" — was eventually deemed inadequate for geodetic requirements. A superior instrumental equipment was introduced, with an improved

Trigonometrical Survey of India.

Fig. 1.

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 repre- sented 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 ob- servatory 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 com- pensation bars and microscopes. The bars, 10 ft. long, were set up horizontally on tripod stands; the microscopes, 6 in. apart, were mounted in pairs revolving round a vertical axis and were set up on tribrachs fitted to the ends of the bars. Six bars and five central and two end pairs of microscopes — the latter with their vertical axes perforated for a 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 tempe- rature; 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 thermo- meters 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 il 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 snowed that, for any base-line measured as usual without thermometers in the compound bars, the p.e. may be taken as *i«5 millionth parts of the length, excluding unascertainable constant errors, and that on introducing thermometers into these bars the p.e. was diminished to =*= 0-55 millionths.

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 com- ponents and its influence would consequently be eliminated. Thus the most perfect base-line apparatus would seem to be one of com- pensation bars with thermometers attached to each component; then the comparisons with the standard need only be taken at the times when the temperature is constant, and there is no lagging.

The plan of triangulation was broadly a system of internal meridional and longitudinal chains with an external border of oblique chains following the course of the frontier and the coast lines. The design of each chain was necessarily much influenced by the physical features of the country over which it was carried. The most difficult tracts were plains, devoid of any commanding points of view, in some parts covered with forest and jungle, malarious and almost uninhabited, in other parts covered with towns and villages and umbrageous trees. In such tracts triangulation was impossible except by constructing towers as stations of observation, raising them to a sufficient height to overtop at least the earth's curvature, and then either increasing the height to surmount all obstacles to mutual vision, or clearing the lines. Thus in hilly and open country the chains of triangles were generally made " double " throughout, i.e. formed of polygonal and quadrilateral figures to give greater breadth and accuracy; but in forest and close country they were carried out as series of single triangles, to give a minimum of labour and expense. Symmetry was secured by restricting the angles between the limits of 30° and: 90 °. The average side length was 30 m. in hill country and 11 in the plains; the longest principal side was 62-7 m., though in the secondary tri- angulation to the Himalayan peaks there were sides exceeding 200 m. Long sides were at first considered desirable, on the prin- ciple 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 = § 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 tnangulation, 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 1 , 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 normalof 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

Erecaution against deflection was neglected and the pillars were uilt 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 inter- vened 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 ob- server. 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 char- acter 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 excentncally when the light falls on one side. A signal- ling party of three men was usually found sufficient to manipulate a pair of heliotropes — one for single, two for double reflection, according to the sun's position — and a lamp, throughout the night and_ day. Heliotropers were also employed at the observing stations to flash instructions to the signallers.

The theodolites were invariably set up under tents for protection against sun, wind and rain, and centred, levelled and adjusted for Measuring t ' le runs °^ t ' le microscoP 65 - Then the signals were Horizontal 0Dserve d m regular rotation round the horizon, alter- Angles. nately from right to left and vice versa; after the pre- scribed minimum number of rounds, either two or three, had been thus measured, the telescope was turned through 180 , both in altitude and azimuth, changing the position of the face of the vertical circle relatively to the observer, and further rounds were measured; additional measures of single angles were taken if the prescribed observations were not sufficiently accordant. As the microscopes were invariably equidistant and their number was always odd, either three or five, the readings taken on the azi- muthal circle during the telescope pointings to any object in the two positions of the vertical circle, " face right " and " face left," weremade 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 pre- viously 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 elimin- ated, 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 atmosphere. Terrestrial refraction is capricious, more par- a 1 ticularly when the rays of light graze the surface of the " s es °

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 RetracHon 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 consider- able negative magnitude during the heat of the day, for the lower strata of the atmosphere are then less dense than the strata imme- diately 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 + r*2l down to -0-09 parts of the contained arc on plains; the normal " coefficient of refraction " for free rays between hill stations below 6000 ft. was about 0-07, which diminished to 0-04 above 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 ob- servations 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 surround- ing stations are deduced from the actual observations, but from the " included angles " in the Indian survey. The webhts 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 -j- o 2 -*- n — g, being a value of the e.m.s.[2] of graduation derived empirically from the differences between the general mean and the mean for each setting, 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 mi, wi, . . . for the weights of the single means, w for the weight of the con- cluded angle, M for the general mean, C for the concluded angle, and d\, dt, . . . for the differences between M and the single means, we have

C=M + Wid - 1 + w ' d \+ (1)

and w = v>\ + Wi -f- (2)

C — M vanishes when n is constant ; it is inappreciable when g is much larger than o; it is significant only when the graduation errors are more minute than the errors of observation; but it was always small, not exceeding 0-14' with the system of two rounds of measures and 0-05' with the system of three rounds.

_ The weights of the concluded angles thus obtained were employed in the primary reductions of the angles of single triangles and polygons which were made to satisfy the geometrical conditions 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 in three ways — (i.) from the squares of the reciprocals Theoretical of the we ; g h t w deduced as above from the measures Errors of f sucn angle, (ii.) from the magnitudes of the excess of Annies. the sum of t]le a „gi es f cacn triangle above i8o°+ 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 Harmon- an g] es f every triangle having been observed, their ~'" g sum had to be made = i8o° + the spherical excess;

Angles. j n networ k s it W as also necessary that the snm 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 com- puted. These are called the triangular, central, toto-partial and side conditions; they present n geometrical equations, which contain t unknown quantities, the errors of the observed angles, t being always > n. When these equations are satisfied and the deduced values of errors are applied as corrections to the observed angles, the figure becomes consistent. Primarily the equations were treated by a method of successive approximations; but afterwards they were all solved simultaneously by the so-called method of minimum squares, which leads to the most probable of any system of correc- tions.

The angles having been made geometrically consistent inter se in each figure, the side-lengths are computed from the base-line onwards by Legendre's theorem, each angle being dimin- i>ldes or j shed by one _third of the spherical excess of the triangle mangles. tQ wh ; cJ) j t 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 Latitude and surrounding stations, the differences of latitude and £,on£«'u<feo/longitude and the reverse azimuths are calculated in Stations; succession, for all the stations of the triangulation, Azimuth of by Puissant's formulae (Traite de giodesie, 3rd ed., Paris, Sides. 1842).

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 6 the

minor semi-axis; e the excentricity, = j — -p — [ ; p the radius of curvature to the meridian in latitude X, = 1 1— e 2 sin 2 Xil ' " t ' le norrna '

to the meridian in latitude X, = 1 j_ e 2 sm 2xlj' ^ anc ^ ^ tne S' ven

latitude and longitude of A; X + AX and L + &L the required latitude and longitude of B; A the azimuth of B at A; B the azimuth of A at B; &A =B — {x+A); c the distance between AandB. Then, all azimuths being measured from the south, we have

cos A cosec

sinM. tan X coscc 1"

cosM sin 2X cosec 1"

4 p.v 1 — e 1

+g , sinM cos i4(l+3 tan'X) cosec 1'

Ai" =

v cos X

cosec 1

1 c 1 sin 2j4 tan X .

+- -. — — -r cosec r

2 ** cos X

1 c* ( 1 +3 tan'X) sin 2A cosA ~6 v* cos X . I C* sinM tan 5 X .

+:,-! r - ^ cosec 1

3 v* cos X

cosec 1

(3)

f(4)

A/l"or

B-(TT+A):

— sin A tan X coscc 1" v

, I c 2 \ , ... e'cos'X ) . . ,

+tt j 1+2 tan 2 X-| — - a \ sin 2/lcosec 1

— -jfg+tan'X J — - — sin 2A cos/1 coscc 1"

+|-jsinM tan X (1+2 tan'X) cosec 1" H5 (5)

Each A is the sum of four terms symbolized by S\, 8», it and S t ; the calculations arc so arranged as to produce these terms in the order S\, 6L, and SA, each term entering as a factor in calculating the following term. The arrangement is shown below in equations in which the symbols P, Q, . . . Z represent the factors which depend on the adopted geodetic constants, and vary with the latitude; the logarithms of their numerical values are tabulated in the Auxiliary Tables to Facilitate the Calculations of the Indian Survey. S 1 \ = —P.cosA.c 6iL=-HiX.Q.secX.taa4 SiA = +SiL.sin\]

«,X=+M. R.sinA.c S 1 L = -S,\.S. cot A M = +«ii.r \ (t ~,

6 s \=-i,A.V. cotA «ji = +« 3 X . Z/.sin/l x i*A = +«»i . W [ w «,X = -«jA.X.tanA SiL = +S t \. Y.tanA &A = +««£,. Z J

The calculations described so far suffice to make the anglesof 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. Reduction The results are amply sufficient for the requirements ot Principal of the topographer and land surveyor, and they are Triangula- published in preliminary charts, which give full numerical " on ' 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 com- pletion 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 com- prising seventy-two single triangles with a base-line at each corner as

shown in the accompanying C B

diagram (fig. 2); 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 cal- culation are carried through D the triangulation to obtain the _

lengths and azimuths of the f ig. 2.

sides and the latitudes and longitudes of the stations, say in the fol- lowing 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 con- structed 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 s 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 o, 6 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 YandZ, so that these equations are simply [cot Y.y— cot JZ.z]=E. But three out of four of the circuit equations are geodetic, corre- sponding to the closing errors in latitude, longitude and azimuth, and in them the coefficients are very complicated. They are ob- tained as follows. The first term of each of the three expressions for AX, A£, and B is differentiated in terms of c and A, giving

rf.AX = AX- \^-dA tan A sinl"|

d.AL~ AL\ -j + dA cot A sin 1" |

dB=dA+AA \ ~+dA cot A sin 1" | (7) in which dc and dA represent the errors in the length and azimuth of any side c which have been generated 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 d\, = [d.A\], and dL, =\d.AL]. Let station I be the origin, and let 2, 3, ... be the succeeding stations taken along a predetermined line of traverse, which may either run from vertex to vertex of the successive triangles, zigzagging between the flanks of the chain, as in fig- 3 (1)1 or be carried directly along one of the flanks, as in fig. 3 (2). For the general symbols of the differential equa- tions substitute AX„, AL„, AA n , Cn, An, and Bn, for the side between stations n and n-J-i of the traverse; and let Sc n and SA n be the errors generated between the sides Cn_i and Cn ; then

dAi = SAi; dA i =dB 1 +&A 1 ; ... dA n =d3 n -.i+SA n . Performing the necessary substitutions and summations, we get

Ci ' 2 l ~'C2 ' "Cn

+ (l+"[M cot A] sin l'^Ai + d+^lAA cot A] sin i")SA 2

+ .. . +(i+Ai4»cot An sin i")SA n -

[AX]f + >X]g+...+AA^

-|"[AX tan 4]&4i-r£[AX tan A]SA,+ . . . +&K, tan A n SAn) sin 1*

+|"[AL cot A]SAi+ n 2 [AL cot A]SA,+ . . . +ALn cot AnSAn] sin I*. Thus we have the following expression for any geodetic error: —

(8)

where 11 and <j> 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 differ- ent 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 =*=</>*+coty./i'y— cot Z./x'z,

in which 1/ =/x sin 1 *, and the upper sign of tj> 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 ix's and <t>'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 /th station, the form is

+ <l>i.x p + cot Y„ . n\ . y„-cot Z p . nl . g,.

If the gth triangle has a side between the /th and the (/+l)th stations of the traverse, the form is

cot X,(n[ — /i'j+i)*, + (<#>i + ii't+i cot Y q )y q — (<t>i+i — nl cot Z„)z q .

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+cz] r — [ax+by+cz]i =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 seventy- two 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 equations under the condition that

[u ' V ' WJ

is made a minimum,

«, 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, are

x r = u f [a v \] ; y p = v P [b p \] ; z p = w„[c„X]. (9)

ii. One of the unknown quantities in every triangle, as x, may be eliminated from each of the eleven circuit and base-line equa- tions by substituting its equivalent — (y+z) for it, a similar substi- tution being made in the minimum. Then the equations take the form [(&— a)y+(c— a)z]=E, while the minimum becomes

r (y+z)« . y* . zn

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 : —

yr = Z^£f^\fo+Wr)l(h-<h)>]-w P Kc p -a I ,)\])\ Up ^l +w J i(^+ v p)[^p-a p )\]-v P l(.b p -a p ) (10)

The second method has invariably been adopted, originally be- cause it was supposed that, the number of the factors X being re- duced 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 differ- ent weight, their errors should be introduced as unknown quantities of which the most probable values are to be. determined in a simul- taneous 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 simul- taneous solution of a number of equations of condition = that of all the geometrical conditions of every figure +f our times the number of circuits formed by the chains of triangles +the number of base- lines— 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 simul- taneous_ 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 base- lines. This arrangement had the advantage of enabling the final reductions to be taken in hand as soon as convenient after the completion of any 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 01 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 number of “side " and other geometrical equations of condition, which entered irregularly and caused great entanglement. Equations 9 and 10 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=0, 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 ro 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 ri orous 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 in-Huence 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 t observed angles of any figure, and n the number of geometrical conditions presented for satisfaction; then the average weight of the angles after adjustment f .

may be taken as w. 7:71, the factor thus being I-5 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 A 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 0 .L

"z-n

p being the modulus.

The second of the two processes for a plying 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 (Io) by substituting u for the reciprocal final mean weight as above determined:-

yr 3 '1§ Bl(2bP " ap "' 511)>l ()

II

Zp = 1§ 'il(25z1 ' ap " bP))l

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:

Simultaneous. Y External Figural.

Equations. Equations.

Section. m 'ei ' ~ ~ § '=.

6 -= 'BQ .1 -, A "a§ 99

Q35 'ig 'fi E Sid 3-5 wb QE:

@3§ 5 55° 'Ya 5 e iii”

1. N.W. Quad.” C23 550 165b 267 104 152 6 761 110 2. S.E. Quad. 15 277 831 164 64 92 2 476 68 3. Quad. . 49 573 1719 112 56 69 0 341 50 4. Trigon. . 22 303 909 192 79 101 2 547 77 75. S.W. Quad. 24 172 516 83 32 52 1 237 40 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 observahon. 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 A I th and after elongation, and usually at both elongations 0; mu in order to eliminate the error of the star's place. System- H jgrva atic changes of “ face " and of the zero settings of the 0 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 dishevelment. 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 reflexions of the plumb-line and only partially to error in the triangulation, and each was multiplied by the factor P 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 reflexion 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=[(observed Evciomputed) Q w] UZ);

the value of x' thus obtained was - I-1”. 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 ri ht angle at the elongation, the hour angle and the azimuth at that time are found from the equations,

cosP = tanPSc otPZ,

cosZ =cosPSsinP.

The interval, 6P, between the time of any observation and that of the elongation being known, the corresponding azimuthal angle, 6Z, between the two positions of the star at the times of observation and elongation is given rigorously by the following expression -tan 6Z

2sin2%5P (

cotPSsinPZsinP{ I +tan2PScos6P+sec2PScotPsin5P} I3) which is expressed as follows for logarithmic computation.tanZcos2PS 5Z=""T-TT-W"

6F - . BP

where m -= 2 SIII2? cosec 1', n = 2 s1n2PS s1n2?, and l=cot P sin 6P; Z, 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 Dg=Ap, then BD is HeLght and b 3

tae difference of height; draw

t e tangents Aa and Bb at

A and B, then @1419 is the "e'°"“"°”', , depression of B at A and bBA that of D A at B; join AD, then BD is determined fromthetriangleABD. Thetriangulation "I gives the distance between A and B at

the sea-level, whence pg=c; thus, A

putting Ap, the height of A above the 9 sea-level, ==H, and pC=r,

H C2

AD=CI-l-7-Q (14).

Putting D.. and D1, for the actual depressions at A and B, S for the angle at A,

usually called the “ subtended angle, ” and h for BD-5

= i(D1>fD<») (15).

V sm S

and h = ((16)U

FIG. 4.

The angle at C being=D1>, -l-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. C, the “contained arc, ”=c%'Ef'cosec 1" in seconds. Putting DQ, and D'1, for the observed vertical angles, and dn., ¢b for the amounts by which they are affected by refraction, D., =D', ,+¢, , and Db=D'1, +¢;, ; ¢, , and qhi, may differ in amount, but as they number of " side " and other geometrical equations of condition, which entered irregularly and caused great entanglement. Equa- tions 9 and 10 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 pre- sent 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 har- monized with the fixed angles, which was readily done for each figure per se.

This departure from rigorous accuracy was not of material im- portance, 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 in- fluence on the included angles. The figural adjustments had, how- ever, 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 111 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 betaken as w . , _ , the factor thus being 1-5 for a triangle,

1-8 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

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 (10) by substituting u for the reciprocal final mean weight as above determined : —

?, = -T*[(2&„ -a„ -c P )X]\ z p = "z[( 2Cp -a„- 6„)Mj

The following table gives the number of equations of condition and unknown quantities — the angular errors — in the five gre^t sections of the triangulation, which were respectively included in the simultaneous general reductions and relegated to the sub- sequent adjustments of each figure per se: —

Section. Simultaneous. External Figural. Equations. as 3$ Equations. Side.


o'fi


i. N.W. Quad. .

2. S.E. Quad. .

3. N.E. Quad. .

4. Trigon.

5. S.W. Quad. .

23 15

49 22

24


550 277

573 303 172


1650

831

1719

909

516


267 164 112 192 83


104 64 56 79 32


152 92 69

IOI 52


6 2


2 1


761 476 34i 547 237


no 68 50 77 40

The corrections to the angles were generally minute, rarely ex- ceeding 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 triangula- tion 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 ab in ordertoeliminate the error of the star's place. System- </ * erva " atic 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 astronomi- cally determined azimuths, because they are liable to be vitiated by local attractions ; but the azi muths 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 ob- served 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 prob- able value of the error of the observed azimuth at the origin was taken as

(12);

[(observed— computed) p w]

  • E^l

the value of x thus obtained was — i-l .

_ The formulae employed in the reduction of the azimuth observa- tions were as follows. In the spherical triangle PZS, in which P is the pole, Z the zenith and 5 the star, the co-latitude PZ and the polar distance PS are known, and, as the angle at 5 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, BP, 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 observa- tion and elongation is given rigorously by the following expression — tan SZ

2sin 2 $5P ^^ , .

cotPSsinPZsinPI 1 +tan 2 PScos«P +sec 2 P5cotPsin«P| (I V' which is expressed as follows for logarithmic computation — „ m tan Z cos' PS SZ =' 1 -n+l '

where m = 2 sin 2 — cosec 1 *, n = 2 sin 2 PS sin 2 —, and

/ = cot P sin SP; I, 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. Hel . hl aad t> ^s

the tangents , Aa and Bb at Refracttoa . 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 =c(i +f-^) (14).

Putting D a and Db for the actual depres- sions at A and B, S for the angle at A, usually called the " subtended angle," and h for BD-

S = l(Db-D* h = AD-

and

COS Vb

The angle at C being =Db+D a 05). (16).

5 may

Fig. 4.

be expressed in terms of a single vertical angle and C when observations have been taken at only one of the two points. C, the " contained arc,"

P + " •

c c— — cosec 1

in seconds. Putting D' a

and D'b for the observed vertical angles, and fa, <j>b for the amounts by which they are affected by refraction, D a = D' a +<t>a and Db=D'b-\-<fn; <j> a and <j>b may differ in amount, but as they 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

4,~l[C-D'.+D' h )} 07)-

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," <#>-*■ C, is computed for the district, and is employed when heights have to be deter- mined from observations at a single station only. When either of the vertical angles is an elevation— £ must be substituted for D in the above expressions.[3]

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 prolonga- tion 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 deter- mination 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 instru- ment 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. Instru- mental 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.

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 10, the other black with white divisions from 5-55 to 15.55. 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 orrising refraction— from morning to midday or from midday to evening — or when the instru- ment 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 Fines 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 dis- crepancies 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.Sea-level. 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 spring-tides 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.[4] 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 semi- annual 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 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 aeoidor surface, but may be raised or lowered, here and there, Deformed under the influence of local and abnormal attrac- Stirface, tions, presenting an equipotential surface — an ellip- soid 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 attrac- tion 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 travers- ing. 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 reflect- ing 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 field- book 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 correspond- ing 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 (=-ft?irof 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 100 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 S1 ta>.

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. Be- ginning 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 theodo- lite 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 stationsby 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 ex- tremities 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 con- spicuous trees and delineates the woods, footpaths, rocks, precipices, steep slopes, embankments, &c, and supplies the requisite infor- mation regarding minor objects to enable a draughtsman to make a perfect representation according to the scale of the map. In ex- amining 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 ui. scale ( = 2sW) 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 co-ordinates, 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 dis- tance 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 Base good results more rapidly, and with less expenditure

Measure- on equipment and on the staff necessary for transport meats. j n tne fieidj 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 align- ment 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 experi- ments 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 Triangula- acquire such geographical knowledge as will enable tlon. us t0 j a y d own 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 geo- graphical surveys only.

The governing principles of geographical surveys are rapidity and economy. Accuracy is, of course, a recognized necessity, but Principles t ^ ie term must admit of a certain elasticity in geo- which graphical work which is inadmissible in geodetic

govern Geo- or cadastral functions. It is obviously foolish to graphical ex p en d as much money over the elaboration of topo- graphy in the unpeopled sand wastes which 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 $**„%** of 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 determina- _. _.

„ . , 1 , . .,1, Ineij&se*

tions, 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 db 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: (1) latitude; i a it] al n a ta (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 (1) 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 . . .. Surveying in the Field (J. Mackay & Co., Chatham, mu " 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 , anvit de accurate _ enough for combination with triangulation, sm> ' 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 Doints. 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 longi- tude is constant, and can easily be applied. With reference to such absolute determinations of longitude, Major S. Grant's " Dia- gram for determining the parallaxes in declination and right ascen- sion of a heavenly body and its application to the prediction of occultations " (Roy. Geog. Soc. Jotirn. 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 Teiegrapa advantage even to tne checking of long lines of tri- tJoas "' angulation. No effort should be spared to introduce a

telegraphic longitude value into any scheme of geo- graphical 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 determina- tion) 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 sea- level, 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 observa- tions 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 rele- gated 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 under t^fifcft survevs at a ^- Perfect freedom of action, and the (leographi- recognition of such work as a public benefit, are not cat Surveys often attainable. Far more frequently the oppor- oo< Ca tunity offers itself to the surveyor with the progress

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, compre- hensive 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 necessi . tv i a " d 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 deter- minations, 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 topo- graphy 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 con- sistent series of triangulation for the " control " (to use an American 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 deter- Triangula- mining the exact position of those points in latitude i j? nor , as one by one they are passed by the moving force, CoD<n > t 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 opera- tions are consequently restricted, it is usually possible to obtain the necessary ' control" (i.e. a few well-fixed points determined by triangulation) for topography in advance Military of a position securely held. With a very little assist- Oeography. 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 (vide 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. Geo- graphical surveying and military surveying are convertible terms, and itis 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) 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 1 m. = 1 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 is specially prepared, resting on a graduated horizontal Pnoto ' t opo- plate which is read with verniers, and with a small P^P>nr- 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 tri- angulation, as obtained from the primary - triangulation, with the theodolite (which precedes the photo-topographical survey), or from the horizontal plate of the camera. I 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 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 hori- zontal 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 arc 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 P'*" e ' methods of adapting the plane-table to geographical tabic. requirements diner with those varying climatic con-

ditions 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 triangulat- ing 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 tempera- ture, no less than of atmospheric humidity, tend to vitiate the ac- curacy 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 neces- sarily small. It may be reckoned at from l : 500000 to 1 : 125000, or from 1 in. = 8 m. to I in.=2 m. The I in. = i m. Sc^e- scale is the normal scale for rigorous topography, and

although it is impossible to fix a definite line beyond which geo- graphical scales merge into topographical (for instance, the i-in. scale is classed as geographical in America whenever the con- tinuous 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 1 : 250000 an experienced geographical surveyor, in favourable country, will complete an area of mapping from day to day which will practically cover Out-turn. near ]y a n t hat f a n s 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 topo- Cost graphical survey in India, when carried over districts

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 districtswhich probably present far more facilities for survey- ing 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 geo- graphical 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. \m. to the in.), should be one-sixteenth of the cost per mile of the i-in. survey over similar ground. A geographical recon- naissance on the scale of 1 : 500000 (8 m. = I in.) should be one-sixty- fourth 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, arid 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 method a considerable quantity of work has been turned out in much less time and at a much lower cost than would bejHapRspro- involved by any process of photo-zincography or ductl0Or 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 draw- back, 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 work- ing 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 necessi- tated 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 10" 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 obser- vations; 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 follow- ing stations; summing these together, the " inclinations " of the lines between the stations to the meridian of the origin are succes- sively determined. The distances between the stations, multi- plied by the cosines and sines of the inclinations, give the distance of each station from the one preceding it, resolved in the direc- tions 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 (*i+*j) (yz—yi) (18),

  • i, yi being the co-ordinates of the first, and £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 arc 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 geacycf are known, is that of B-(ir+A) of equation (5); Meridians, but, as the first term is sufficient for a traverse, we have

coseci *" convergency = x tan X

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- . tude; those of a traverse survey are always rectangular, Adjustment lane {or a small area but sp h e rical for a large one. ofTra- f t ; s oftcn necessary) therefore, for purposes of com-

versesto p ar i son and check at stations common to surveys of Triangula- g otn d escr ip t ions, to convert either rectangular co- Uoa - ordinatcs 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 arc 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 latitude and longitude; from B draw BP 7Vans/orma " perpendicular to Aa; then AP, BP Hon of Co- w ;ji ^ e tne rec tangular spherical co-ordin- ordlnaies. ates of B relatively to A. Put BP = x, AP = y, the arc Pb=v, and the arc Bb, the differ- ence of longitude, = w ; also let X«, X& and X, be the latitudes of A, B, and the point P, p p the radius of curvature of the meridian, and v, the normal ter- minating in the axis minor for the latitude X p ; and Fig. 5. Jet Po be the radius of curvature for the latitude Jfaa+Xp). Then, when the rectangular co-ordinates are given, we have, taking A as the origin, the latitude of which is known,

X p = X„ + A;oscc r

2p p v p

tan X p cosec 1' <I9>.

(20).

Xt — X =*— cosec l"—rf, u=— sec(\b+iv) cosec 1" po v p

And, when the latitude and longitude are given, we have [5]

'"=(f)S sin2 ^ sin, l

y =Po{Xi — X„ + r;)sin l" [

x= a>KpCOs(X&-t-$7;)sin '"J When a hill peak or other prominent object has been observed from a number of stations whose co-ordinates are already fixed, the . converging rays may be projected graphically, and from

tiiiHd an exam ' nat ' on of their several intersections the most ofunvtsited p ro t, a ble position of the object may be obtained almost Point as accura tely as by calculations by the method of least

squares, which are very laborious and out of place for the deter- mination 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 $1, St, . . . be stations whose rectangular co-ordinates, Xi, * 2 , . . . perpendicular, and y\, yt, . . . parallel, to the meridian of the origin are given; let Ci, ch, 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 T \ 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 indicate the directions of north, south, cast and west. Calculate accurately (y P — yi) tan a u and compare with (x,—xi); the differ- ence will show how far the direction of the ray from si falls to the east or west of p. Or calculate (x p — x{) cot oi, and compare with (y P — 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

Fig. 6. the point thus fixed draw the direction ai 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 £=*=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.

Authorities. — Clarke, Geodesy (London) ; Waller, _ " India's Contribution to Geodesy," Trans. Roy. Soc, vol. clxxxvi. (1895); Thuillier, Manual of Surveying for India (Calcutta) ; Gore, Hand- book 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 Paral- laxes," &c, Geog. Journ. (June 1896); Pierce, "Economic Use of the Plane-Table," vol. xcii. pt. ii., Pro. Inst. Civ. Eng.; Bridges- Lee, Photographic Surveying (1899); London Society of Engineers; Laussedat, Recherches sur les instruments les mithodes el 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 1. by MacDonnell (R.E. Institute, Chatham).

(T. H. H.*)

6. Nautical Surveying

The great majority of nautical surveys are carried out by H.M. surveying vessels under the orders of the hydrographer of the admiralty. Plans of harbours and anchorages are also received from H.M. ships in commission on foreign stations, but surveys of an extended nature can hardly be executed except by a ship specially fitted and carrying a trained staff of officers. The introduction of steam placed means at the disposal of nautical surveyors which Jargely modified the conditions under which they had to work in the earlier days of sailing vessels, and it has enabled the ship to be used in various ways previously impracticable. The heavy draught of ships in the present day, the growing increase of ocean and coasting traffic all over the world, coupled with the desire to save distance by rounding points of land and other dangers as closely as possible, demand surveys on larger scales and in greater detail than was formerly necessary; and to meet these modern requirements resurveys of many parts of the world are continually being called for. Nautical surveys vary much in character according to the nature of the work, its importance to navigation, and the time available. The elaborate methods and rigid accuracy of a triangulation for geodetic purposes on shore are unnecessary, and are not attempted; astronomical observations at intervals in an extended survey prevent any serious accumulation of errors consequent upon a triangulation which is usually carried out with instruments, of which an 8-in. theodolite is the largest size used, whilst 5-in. theodolites generally suffice, and the sextant is largely employed for the minor triangulation. The scales upon which nautical surveys are plotted range from \ in. to 2 or 3 in. to the sea-mile in coast surveys for the ordinary purposes of navigation, according to the requirements; for detailed surveys of harbours or anchorages a scale of from 6 to 12 in. is usually adopted, but in special cases scales as large as 6o in. to the mile are used.

The following arc the principal instruments required for use in the field: Theodolite, 5 in., fitted with large telescope of high power, with coloured shades to the eye-piece for observing the sun for true bearings. Sextant, 8 in. observing, stand and artificial horizon. Chronometers, eight box, and two or three pocket, are usually supplied to surveying vessels. Sounding sextants, differing from ordinary sextants in being lighter and handier. The arc is cut only to minutes, reading' to large angles of as much as 140 , and fitted with a tube of bell shape so as to include a large field in the telescope, which is of high power. Measuring chain 100 ft. in length. Ten-foot pole for coast-lining, is a light pole carrying two oblong frames, 18 in. by 24 in., covered with canvas painted white, with a broad vertical black stripe in the centre and fixed on the pole 10 ft. apart. Station-pointer, an instru- ment in constant requisition either for sounding, coast-lining, or topographical plotting, which enables an observer's position to be fixed by taking two angles between three objects suitably situated. The movable legs being set to the observed angles, and placed on the plotting sheet, the chamfered edges of the three legs are brought to pass through the points observed. The centre of the instrument then indicates the observer's position. Heliostats, for reflecting the rays of the sun from distant stations to indicate their position, are invaluable. The most convenient form is Galton's sun signal ; but an ordinary swing mirror, mounted to turn horizontally, will answer the purpose, the flash being directed from a hole in the centre of the mirror. Pocket aneroid barometer, required for topographical purposes. Prismatic compass, patent logs (taffrail and harpoon), Lucas wire sounding machine (large and small size), and James's submarine sentry are also required. For chart-room use are pro- vided a graduated brass scale, steel straight-edges and beam com- passes of different lengths, rectangular vulcanite or ivory protractors of 6-in. and 12-in. length, and semicircular brass protractors of 10-in. radius, a box of good mathematical drawing instruments, lead weights, drawing boards and mounted paper.

Every survey must have fixed objects which are first plotted on the sheet, and technically known as " points." A keen eye is Marks and ret l u ' re d f° r natural marks of all kinds, but these must Beacons. °^. ten be supplemented by whitewash marks, cairns, tripods or bushes covered with white canvas or calico, arid flags, white or black according to background. On low coasts, flagstaffs upwards of 80 ft. high must sometimes be erected in order to get the necessary range of vision, and thereby avoid the evil of small triangles, in working through which errors accumulate so rapidly. A barling spar 35 ft. in length, securely stayed and carrying as a topmast (with proper guys) a somewhat lighter spar, lengthened by a long bamboo, will give the required height. A fixed beacon can be erected in shallow water, 2 to 3 fathoms in depth, by constructing a tripod of spars about 45 ft. long. The heads of two of them are lashed together, and the heels kept open at a fixed distance by a plank about 27 ft. long, nailed on at about 5 ft. above the heels of the spars. These are taken out by three boats, and the third tripod leg lashed in position on the boats, the heel in the opposite direction to the other two. The first two legs, weighted, are let go together; using the third leg as a prop, the tripod is hauledjnto position and secured by guys to anchors, and by addi- tional weights slipped down the legs. A vertical pole with bamboo can now be added, its weighted heel being on the ground and lashed to the fork. On this a flag 14 ft. square may be hoisted. Floating beacons can be made by filling up flush the heads of two 27-gallon casks, connected by nailing a piece of thick plank at top and bottom. A barling spar passing through holes cut in the planks between the casks, projecting at least 20 ft. below and aDout 10 ft. above them, is togg ed securely by iron pins above the uoper and below the lower plank. To the upper part of the spar is lashed a bamboo, 30 to 35 ft. long, carrying a black flag 12 to 16 ft. square, which will be visible from the ship 10 m. in clear weather. The ends of a span of t-in. chain are secured round the spar above and below the casks with a long link travelling upon it, to which the cable is attached by a slip, the end being carried up and lightly stopped to the bamboo below the flag. A wire strop, kept open by its own stiffness, is fitted to the casks for convenience in slipping and picking up. The beacon is moored with chain and rope half as long again as the depth of water. Beacons have been moored by sounding line in as great depth as 3000 fathoms with a weight of 100 lb.


"Fixing."


There is nothing in a nautical survey which requires more attention than the " fix "; a knowledge of the principles involved is essential in order to select properly situated objects. The method of fixing by two angles between three fixed points is generally known as the " two-circle method," but there are really three circles involved. The " station-pointer " is the instrument used for plotting fixes. Its contraction depends upon the fact that angles subtended by the chord of a segment of a circle measured from any point in its circumference are equal. The lines joining three fixed points form the chords of segments of three circles, each of which passes through the observer's position and two of the fixed points. The more rectangular the angle at which the circles intersect each other, and the more sensitive they are, the better will be the fix; one condition is useless without the other. A circle is " sensitive " when the angle between the two objects responds readily to any small movement of the observer to wards or away from the centre of the circle passing through the observer's position and the objects. This is most markedly the case when one object is very close to the observer and the other very distant, but not so when both objects are distant. Speaking generally, the sensi- bility of angles depends upon the relative distance of the two objects from the observer, as well as the absolute distance of the nearer of the two. In the accompanying diagram A, B, C are the objects, and X the observer. Fig. 7 shows the circle passing through C, B and X, cutting the circle ABX at a good angle, and therefore fixing X independently of the circle CAX, which is less sensitive than either of the other two. In fig. & the two first circles are very sensitive, but being nearly tangential

Fig. 7.

Fig. 8. they give no cut with each other. The third circle cuts both at right angles; it is, however, far less sensitive, and for that reason if the right and left hand objects are both distant the fix must be bad. In such a case as this, because the angles CXB, BXA are both so sensitive, and the accuracy of the fix depends on the precision with which the angle CXA is measured, that angle should be observed direct, together with one of the other angles composing it. Fig. represents a case where the points are badly disposed, approaching the condition known as " on the circle," passing through the three points. All three circles cut one another at such a fine angle as to give a very poor fix. The centre of the station-pointer could be moved considerably without materially affecting the coincidence of the legs with the three points. To avoid a bad fix the following rules are safe: —

1 . Never observe objects of which the central is the furthest unless it is very distant relatively to the other two, in which case the fix is admis- sible, but must be used with caution.

2. Choose objects disposed as follows: (a) One outside object distant and the other two near, the angle between the two near

Fig. 9. objects being not less than 30 or more than 140 . The amount of the angle between the middle and distant object is immaterial. (6) The three objects nearly in a straight line, the angle between any two being not less than 30 . (c) The observer's position being inside the triangle formed by the objects.

A fix on the line of two points in transit, with an angle to a third point, becomes more sensitive as the distance between the transit points increases relatively to the distance between the front transit point and the observer; the more nearly the angle to the third point approaches a right angle, and the nearer it is situated to the observer, the better the fix. If the third point is at a long distance, small errors either of observation or plotting affect the result largely. A good practical test for a fix is afforded by noticing whether a very slight movement of the centre of the station-pointer will throw one or more of the points away from the leg. If jt can be moved without appreci- ably disturbing the coincidence of the leg and all three points, the fix is bad.

Tracing-paper answers exactly the same purpose as the station- pointer. The angles are laid off from a centre representing the position, and the lines brought to pass through the points ■as before. This entails more time, and the angles are not so accurately measured with a small protractor. Nevertheless this has often to be used, as when points are close together on a small scale the central part of the station-pointer will often hide them and prevent the use of the instrument. The use of tracing-paper permits any number of angles to different points to be laid down on it, which under certain conditions of fixing is sometimes a great advantage.

Although marine surveys are in reality founded upon triangula- tion and measured bases of some description, yet when plotted Bases irregularly the system of triangles is . not always apparent. The triangulation ranges from the rough triangle of a running survey to the carefully formed triangles of detailed surveys. The measured base for an extended survey is provisional only, the scale resting ultimately mainly upon the astronomical positions observed at its extremes. In the case of a plan the base is absolute. The main triangulation, of which the first triangle contains the measured base as its known side, establishes a series of points known as main stations, from which and to which angles are taken to fix other stations. A sufficiency of secondary stations and marks enables the detail of the chart to be filled in between them. The points embracing the area to be worked on, having been plotted, are transferred to field boards, upon which the detail of the work in the field is plotted; when complete the work is traced and re-transferred to the plotting-sheet, which is then inked in as the finished chart, and if of large extent it is graduated on the gnomonic projection on the astronomical positions of two points situated near opposite corners of the chart.

The kind of base ordinarily used is one measured by chain on flat ground, of % to 15 m. in length, between two points visible from one another, and so situated that a triangulation can be readily extended from them to embrace other points in the survey forming well-conditioned triangles. The error of the chain is noted before leaving the ship, and again on returning, by com- paring its length with the standard length of 100 ft. marked on the ship's deck. The correction so found is applied to obtain the final result. If by reason of water intervening between the base stations it is impossible to measure the direct distance between them, it is permissible to deduce it by traversing.

A Masthead Angle Base is useful for small plans of harbours, &c, when circumstances do not permit of a base being measured on shore. The ship at anchor nearly midway between two base stations is the most favourable condition for employing this method. Theodolite reading of the masthead with its elevation by sextant observed simultaneously at each base station (the mean of several observations being employed) give the necessary data to calculate the distance between the base stations from the two distances resulting from the elevation of the masthead and the simultaneous theodolite-angles between the masthead and the base stations. The height of the masthead may be temporarily increased by secur- ing a spar to extend 30 ft. or so above it, and the exact height from truck to netting is found by tricing up the end of the measuring


chain. The angle of elevation should not be diminished below about l° from either station.

Base by Sound. — The interval in seconds between the flash and report of a gun, carefully noted by counting the beats of a watch or pocket chronometer, multiplied by the rate per second at which sound travels (corrected for temperature) supplies a means of obtaining a base which is sometimes of great use when other methods are not available. Three miles is a suitable distance for such a base, and guns or small brass Cohorn mortars are fired alternately from either end, and repeated several times. The arithmetical mean is not strictly correct, owing to the retardation of the sound

against the wind exceeding the acceleration when travelling with

2//' it; the formula used is therefore T = j-r-p where T is the mean

interval required, t the interval observed one way, I' the interval the other way. The method is not a very accurate one, but is suffi- ciently so when the scale is finally determined by astronomical observations, or for sketch surveys. The measurement should be across the wind if possible, especially if guns can only be fired from one end of the base. Sound travels about 1090 ft. per second at a temperature of 32° F., and increases at the rate of 1-15 ft. for each degree above that temperature, decreasing in the same proportion for temperatures below 32°.

Base by Angle of Short Measured Length. — An angle measured by sextant between two well-defined marks at a carefully measured distance apart, placed at right angles to the required base, will give a base for a small plan.

Astronomical Base. — The difference of latitude between^ two stations visible from each other and nearly in the same meridian, combined with their true bearings, gives an excellent base for an extended triangulation; the only drawback to it is the effect of local attraction of masses of land in the vicinity on the pendulum, or, in other words, on the mercury in the artificial horizon. _ The base stations should be as far apart as possible, in order to minimize the effect of any error in the astronomical observations. The obser- vation spots would not necessarily be actually at the base stations, which would probably be situated on summits at some little distance in order to command distant views. In such cases each observation spot would be connected with its corresponding base station by a subsidiary triangulation, a short base being measured for the pur- pose. The ship at anchor off the observation spot frequently affords a convenient means of effecting the connexion by a masthead angle base and simultaneous angles. If possible, the observation spots should be east or west of the mountain stations from which the true bearings are observed.

If the base stations A and B are so situated that by reason of distance or of high land intervening they are invisible from one another, but both visible from some main station C between them, when the main triangulation is completed, the ratio of the sides AC, BC can be determined. From this ratio and the observed angle ACB, the angles ABC, BAC can be found. The true bearing of the lines AC or BC being known, the true bearing of the base stations A and B can be deduced.

Extension of Base. — A base of any description is seldom long enough to plot from directly, and in order to diminish errors of plotting it is necessary to begin on the longest side possible so as to work inwards. A short base measured on flat ground will give a better result than a longer one measured over inequalities, provided that the triangulation is carefully extended by means of judiciously selected triangles, great care being taken to plumb the centre of each station. To facilitate the extension of the base in as few triangles as possible, the base should be placed so that there are two stations, one on each side of it, subtending angles at them of from 30° to 40°, and the distances between which, on being calculated in the triangles of the quadrilateral so formed, will constitute the first extension of the base. Similarly, two other stations placed one on each side of the last two will form another quadrilateral, giving a yet longer side, and so on.

The angles to be used in the main triangulation scheme must be very carefully observed and the theodolite placed exactly over the centre of the station. Main angles are usually repeated several times by resetting the vernier g ™, atlo ' n . at intervals equidistant along the arc, in order to eliminate instrumental errors as well as errors of observation. The selection of an object suitable for a zero is important. It should, if possible, be another main station at some distance, but not so far or so high as to be easily obscured, well defined, and likely to be permanent. Angles to secondary stations and other marks need not be repeated so many times as the more important angles, but it is well to check all angles once at least. Rough sketches from all stations are of great assistance in identifying objects from different points of view, the angles being entered against each in the sketch.

False Station. — When the theodolite cannot for any reason be

placed over the centre of a station, if the distance be measured

Fig. 10.

and the theodolite reading of it be noted, the observed angles may be reduced to what they would be at the centre of the station. False stations have frequently to be made in practice; a simple rule to meet all cases is of great assistance to avoid the possibility of error in applying the correction with its proper sign. This may very easily be found as follows, without having to bestow a moment's thought beyond applying the rule, which is a matter of no small gain in, time when a large number of angles have to be corrected.

Rule. — Put down the theodolite reading which it is required to correct (increased if necessary by 360 ), and from it subtract the theodolite reading of the centre of the station. Call this remainder 6. With 6 as a " course " and the number of feet from the theodolite to the station as a " distance," enter the traverse table and take out the greater increment if 6 lies between 45 and 135 , or between 225 ^ncemsnz an u d 315°. and the lesser increment for other angles. The accompanying dia- gram (fig. 10) will assist the memory. Refer this increment to the " table of subtended angles by various lengths at different distances ' (using the distance of the object observed) and find the corresponding correction in arc, which mark + or — according as 6 is under or over 180 . Apply this correction to the observed theodolite angle. A " table of subtended angles " is unnecessary if the formula

number of feet subtended X 3 4

Angle in seconds = -p-r 1 — . r ", T be used instead.

6 distance of object in sea-miles

Convergency of Meridians. — The difference of the reciprocal true bearings between two stations is called the " convergency." The formula for calculating it is : Conv. in minutes = dist. in sea-miles X sin. Merc, bearing X tan. mid. lat. Whenever true bearings are used in triangulation, the effect of convergency must be con- sidered and applied. In north latitudes the southerly bearing is the greater of the two, and in south latitudes the northerly bearing. The Mercatorial bearing between two stations is the mean of their reciprocal true bearings.

After a preliminary run over the ground to note suitable positions for main and secondary stations on prominent head- Triaagu- lands, islands and summits not too far back from lated Coast the coast, and, if no former survey exists, to make Survey. at ^ same ^ me a rougn p i t f them by compass

and patent log, a scheme must be formed for the main triangulation with the object of enclosing the whole survey in as few triangles as possible, regard being paid to the limit of vision of each station due to its height, to the existing meteoro- logical conditions, to the limitations imposed by higher land intervening, and to its accessibility. The triangles decided upon should be well-conditioned, taking care not. to introduce an angle of less than 30 to 35 , which is only permissible when the two longer sides of such a triangle are of nearly equal length, and when in the calculation that will follow one of these sides shall be derived from the other and not from the short side. In open country the selection of stations is comparatively an easy matter, but in country densely wooded the time occupied by a triangulation is mainly governed by the judicious selection of stations quickly reached, sufficiently elevated to command distant views, and situated on summits capable of being readily cleared of trees in the required direction, an all-round view being, of course, desirable but not always attainable. The positions of secondary stations will also generally be decided upon during the preliminary reconnaissance. The object of these stations is to break up the large primary triangles into triangles of smaller size, dividing up the distances between the primary stations into suitable lengths; they are selected with a view to greater accessibility than the latter, and should therefore usually be near the coast and at no great elevation. Upon shots from these will depend the position of the greater number of the coast-line marks, to be erected and fixed as the detailed survey of each section of the coast is taken in hand in regular order. The nature of the base to be used, and its position in order to fulfil the con- ditions specified under the head of Bases must be considered, the base when extended forming a side of one of the main triangles. It is immaterial at what part of the survey the base is situated, but if it is near one end, a satisfactory check on the accuracy of the triangulation is obtained by comparing the length of a


side at the other extreme of the survey, derived by calculation through the whole system of triangles, with its length deduced from a check base measured in its vicinity. It is generally a saving of time to measure the base at some anchorage or harbour that requires a large scale plan. The triangulation involved in extending the base to connect it with the main triangulation scheme can thus be utilized for both purposes, and while the triangulation is being calculated and plotted the survey of the plan can be proceeded with. True bearings are observed at both ends of the survey and the results subsequently compared. Astronomical observations for latitude are obtained at observa- tion spots near the extremes of the survey and the meridian distance run between them, the observation spots being connected with the primary triangulation; they are usually disposed at intervals of from 100 to 150 m., and thus errors due to a tri- angulation carried out with theodolites of moderate diameter do not accumulate to any serious extent. If the survey is greatly extended, intermediate observation spots afford a satis- factory check, by comparing the positions as calculated in the triangulation with those obtained by direct observation.

Calculating the Triangulation— -The triangles as observed being tabulated, the angles of each triangle are corrected to bring their sum to exactly 180 . We must expect to find errors in the triangles of as much as one minute, but under favourable conditions they may be much less. In distributing the errors we must consider the general skill of the observer, the size of his theodolite relatively to the others, and the conditions under which his angles were observed; failing any particular reason to assign a larger error to one angle than to another, the_ error must be divided equally, bearing in mind that an alteration in the small angle will make more difference in the resulting position than in either of the other two, and as it approaches 30 (the limit of a receiving angle) it is well to change it but very slightly in the absence of any strong reason to the contrary. The length of base being determined, the sides of all the triangles involved are calculated by the ordinary rules of trigonometry. Starting from the true bearing observed at one end of the survey, the bearing of the side of each triangle that forms the immediate line of junction from one to the other is found by applying the angles necessary for the purpose in the respective triangles, not forgetting to apply theconvergency between each pair of stations when reversing the bearings. The bearing of the final side is then compared with the bearing obtained by direct observa- tion at that end of the survey. The difference is principally due to accumulated errors in the triangulation ; half of the difference is then applied to the bearing of each side. Convert these true bearings into Mercatorial bearings by applying half the convergency between each pair of stations. With the lengths of the connecting sides found from the measured base and their Mercatorial bearing, the Mercatorial bearing of one observation spot from the other is found by middle latitude sailing. Taking the observed astronomical positions of the observation spots and first reducing their true difference longitude to departure, as measured on a spheroid from

the formula Dep.=T. D. long. "o. ft. in 1 m. of long. h ^ fc h

r ° no. ft. in 1 m. of lat.

d. lat. and dep. the Mercatorial true bearing and distance between the observation spots is calculated by middle latitude sailing, and compared with that by _ triangulation and measured base. To adjust any discrepancy, it is necessary to consider the probable error of the observations for latitude and meridian distance; within those limits the astronomical positions may safely be altered in order to harmonize the results; it is more important to bring the bearings into close agreement than the distance. From the amended astronomical positions the Mercatorial true bearings and distance between them are re-calculated. The difference between this Mercatorial bearing and that found from the triangulation and measured base must beapplied to the bearing of each side to get the final corrected bearings, and to the logarithm of each side of the triangulation as originally calculated must be added or sub- tracted the difference between the logarithms of the distance of the amended positions of the observation spots and the same distance by triangulation.

Calculating Intermediate Astronomical Positions. — The latitude and longitude of any intermediate main station may now be calculated from_ the finally corrected Mercatorial true bearings and lengths of sides. The difference longitude so found is what it would be if measured on a true sphere, whereas we require it as measured on a spheroid, which is slightly less. The correction

= d. long.

cos 1 mid. lat. must therefore be subtracted; or the true

difference longitude may be found direct from the formula

no. ft. in 1 m. of lat. -^ ., , ... „u_»

dep. zi — n • rrom the foregoing it is seen that

v no. ft. in 1 m. of long. s b

in a triangulation for hydrographical purposes both the bearings of the sides and their lengths ultimately depend almost entirely upon the astronomical observations at the extremes of the survey; the observed true bearings and measured base are consequently more in the nature of checks than anything else. It is obvious, therefore, that the nearer together the observation spots, the greater effect will a given error in the astronomical positions have upon the length and direction of the sides of the triangulation, and in such cases the bearings as actually observed must not be altered to any large extent when a trifling change in the astronomical positions might perhaps effect the required harmony. For the reasons given under Astronomical Base, high land near observation spots may cause very false results, which may often account for discrepancies when situated on opposite sides of a mountainous country.

Great care is requisite in projecting on paper the points of a survey. The paper should be allowed to stretch and shrink Plotting. as it pleases until it comes to a stand, being exposed to the air for four or five hours daily, and finally well flattened out by being placed on a table with drawing boards placed over it heavily weighted. If the triangulation has been calculated beforehand throughout, and the lengths of all the different sides have been found, it is more advantageous to begin plotting by distances rather than by chords. The main stations are thus got down in less time and with less trouble, but these are only a small proportion of the points to be plotted, and long lines must be ruled between the stations as zeros for plotting other points by chords. In ruling these lines care must be taken to draw them exactly through the centre of the pricks denoting the stations, but, however carefully drawn, there is liability to slight error in any line projected to a point lying beyond the distance of the stations between which the zero line is drawn. In plotting by distances, therefore, all points that will subsequently have to be plotted by chords should lie well within the area covered by the main triangulation. Three distances must be measured to obtain an intersection of the arcs cutting each other at a sufficiently broad angle; the plotting of the main stations once begun must be completed before distortion of the paper can occur from change in the humidity of the atmosphere. Plotting, whether by distance or by chords, must be begun on as long a side as possible, so as to plot inwards, or with decreasing distances. In plotting by chords it is impor- tant to remember in the selection of lines of reference (or zero lines), that that should be preferred which makes the smallest angle with the line to be projected from it, and of the angular points those nearest to the object to be projected from them.

Irregular Methods of Plotting. — In surveys for the ordinary purposes of navigation, it frequently happens that a regular cystem of triangulation cannot be carried out, and recourse must be had to a variety of devices; the judicious use of the ship in such cases is often essential, and with proper care excellent results may be obtained. A few examples will best illustrate some of the methods used, but circumstances vary so much in every survey that it is only possible to meet them properly by studying each case as it arises, and to improvise methods. Fixing a position by means of the ." back-angle " is one of the most ordinary expedients. Angles having- been observed at A, to the station B, and certain other fixed points of the survey, C and D for instance; if A is shot up from B, at which station angles to the same fixed points have been observed, then it is not necessary to visit those points to fix A. For instance, in the triangle ABC, two of the angles have been observed, and there- fore the third angle at C is known (the three angles of a triangle being equal to 180 ), and it is called the " calculated or back-angle from C." A necessary condition is that the receiving angle at A, between any' two lines (direct or calculated), must be sufficiently broad to give a good cut; also the points from which the " back-angles " are calculated should not be situated at too great distances from A, relatively to the distance between A and B. A station may be plotted by laying down the line to it from some other station, and then placing on tracing-paper a number of the angles taken at it, including the angle to the station from which it has been shot up. If the points to which angles are taken are well situated, a good position is obtained, its accuracy being much strengthened by being able to plot on a line to it, which, moreover, forms a good zero line for laying off other angles from the station when plotted. Sometimes the main stations must be carried on with a point plotted by only two angles. An effort must be made to check this subsequently by getting an " angle back " from stations dependent upon it to some old well-fixed point; failing this, two stations being plotted with two angles, pricking one and laying down the line to the other will afford a check. A well-defined mountain peak, far inland and never visited, when once it is well fixed is often invaluable in carrying on an irregular triangulation, as it may remain visible when all other original points of the survey have disappeared, and " back-angles " from it may be continually obtained, or it may be Used for plotting on true bearing lines of it. In plotting the true bearing of such a peak, the convergency must be found and applied to get the reversed bearing, which is then laid down from a meridian drawn through it; or the reversed bearing of any other line already drawn through the peak being known, it may simply be laid down with that as a zero. A rough position of the spot from which the true bearing was taken must be assumed in order to calculate (the convergency.

Fig. II will illustrate the foregoing remarks. A and Bare astronomical observation spots at the extremes of a survey, from both of which the high, inaccessible peak C is visible. D, E, F are intermediate stations; A and D, D and E, E and F, F and B being respectively visible from each other. G is visible from A and D, and C is visible from all stations. The latitudes of A and B and meridian distance between them being determined, and the true bearing of C being observed from both observation spots, angles are observed at all the stations. Calcu- lating the spheroidal correction (from the formula, correction =

d. long. C0S 'j ' ) and adding it to the true (or chronometric)

difference longitude between A and B to obtain the spherical d. long.; with this spherical d. long, and the d. lat., the Mer- catorial true bearing and distance is found by middle latitude sailing (which is an equally correct but shorter method than by spherical trigonometry, and may be safely used when dealing with the distances usual between observation spots in nautical surveys). The convergency is also calculated, and the true bearing of A from B and B from A are thus determined. In the plane triangle ABC the angle A is the difference between the calculated bearing of B and the observed bearing of C from A ; similarly angle B is the difference between calculated bearing of A and observed bearing of C from B. The distance AB having been also calculated, the side AC is found. Laying down AC on the paper on the required scale, D is plotted on its direct shot from A, and on the angle back from C, calculated in the triangle ACD. G is plotted on the direct shots from A and D, and on the angle back from C, calculated either in the triangle ACG or GCD. The perfect intersection of the three lines at G assures these four points being correct. E, F and B are plotted in a similar manner. The points are now all plotted, but they depend on calculated angles, and except for the first four points we have no check whatever either on the accuracy of the angles observed in the field or on the plotting. Another well-defined object in such a position, for instance as Z, visible from three or more stations, would afford the necessary check, if lines laid off to it from as many stations as possible gave a good intersection. If no such point, however, exists, a certain degree of check on the angles observed is derived by applying the sum of all the calculated angles at C to the true bearing of A from C (found by reversing observed bearing of C from A with convergency applied), which will give the bearing of B from C. Reverse this bearing with convergency applied, and compare it with the observed bearing of C from B. If the discrepancy is but small, it will be a strong presumption in favour of the substantial accuracy of the work. If the calculated true bearing of B from A be now laid down, it is very unlikely that the line will pass through B, but this is due to the discrepancy which must always be expected between astronomical positions and trian- gulation. If some of the stations between A and B require to be placed somewhat closely to one another, it may be desirable to obtain fresh true bearings of C instead of carrying on the original bearing by means of the calculated angle.

In all cases of irregular plotting the ship is very useful, especially if she is moored taut without the swivel, and angles are observed from the bow. Floating beacons may also assist an irregular triangulation.

Surveys of various degrees of accuracy are included among sketch surveys. The roughest description is the ordinary Sketch Surveys. running survey, when the work is done by the ship steaming along the coast, fixing points, and sketching in the coast-line by bearings and angles, relying for her position upon her courses and distances as registered by patent log, necessarily regardless of the effect of wind and current and errors of steerage. At the other extreme comes the modified running survey, which in point of practical accuracy falls little short of that attained by irregular triangulation. Some of these modifications will be briefly noticed. A running survey of a coast-line between two harbours, that have been surveyed independently and astronomically fixed, may often be carried out by fixing the ship on the points already laid down on the harbour surveys and shooting up prominent intermediate natural objects, assisted possibly by theodolite lines from the shore stations. Theodolite»Iines to the ship at any of her positions are particu- larly valuable, and floating beacons suitably placed materially increase the vaiue of any such work. A sketch survey of a coast upon which it is impossible to land may be well carried out by dropping beacons at intervals of about 10 m., well out from the land and placed abreast prominent natural objects called the " breastmarks," which must be capable of recognition from the beacons anchored off the next " breastmark " on either side. The distance between the beacons is found by running a patent log both ways, noting the time occupied by each run; if the current has remained constant, a tolerably good result can be obtained. At the first beacon, angles are observed between the second beacon and the two " breastmarks," an " intermediate " mark, and any other natural object which will serve as " points." At the second beacon, angles are observed between the first beacon and the same objects as before. Plotting on the line of the two beacons as a base, all the points observed can be pricked in on two shots. At a position about midway between the beacons, simultaneous angles are observed to all the points and laid off on tracing-paper, which will afford the necessary check, and the foundation is thus laid for filling in the detail of coast-line, topography, and soundings off this particular stretch of coast in any detail desired. Each section of coast is complete in itself onitsown base; the weak pointliesin the junction of the different sections, as the patent log bases can hardly be expected to agree precisely, and the scales of adjacent sections may thus be slightly different. This is obviated, as far as possible, by fixing on the points of one section and shooting up those of another, which will check any great irregularity of scale creeping in. The bearing is preserved by getting occasional true bearing lines at the beacons of the most distant point visible. Space does not here permit of dwelling upon the details of the various pre- cautions that are necessary to secure the best results the method is capable of; it can only be stated generally that in all cases of using angles from the ship under weigh, several assistants are necessary, so that the principal angles may be taken simul- taneously, the remainder being connected immediately after- wards with zeros involving the smallest possible error due to the ship not being absolutely stationary, these zeros being included amongst the primary angles. When close to a beacon, if its bearing is noted and the distance in feet obtained from its elevation, the angles are readily reduced to the beacon itself. Astronomical positions by twilight stars keep a check UDon the work.

Sketch Surveys by Compass Bearings and Vertical Angles. — In the case of an island culminating in a high, well-defined summit visible from all directions, a useful and accurate method is to steam round it at a sufficient distance to obtain a true horizon, stopping to make as many stations as may be desirable, and fixing by compass bearing of the summit and its vertical angle. The height is roughly obtained by shooting in the summit, from two positions on a patent log base whilst approaching it. With this approximate height and Lecky's vertical danger angle tables, each station may be plotted on its bearing of the summit. From these stations the island is shot in by angles between its tangents and the summit, and angles to any other natural features, plotting the work as we go on any convenient scale which must be con- sidered only as provisional. On completing the circuit of the island, the true scale is found by measuring the total distance in inches on the plotting-sheet from the first to the last station, and dividing it by the distance in miles between them as shown by patent log. The final height of the summit bears to the rough height used in plotting the direct proportion of the provisional scale to the true scale. This method may be utilized for the sketch survey of a coast where there are well-defined peaks of sufficient height at convenient intervals, and would be superior to an ordinary running survey. From positions of the ship fixed by bearings and elevations of one peak, another farther along the coast is shot in and its height determined; this second peak is then used in its turn to fix a third, and so on. The smaller the vertical angle the more liability there is to error, but a glance at Lecky's tables will show what effect an error of say i' in altitude will produce for any given height and distance, and the limits of distance must depend upon this consideration.

Surveys of Banks out of Sight of Land. — On striking shoal soundings unexpectedly, the ship may either be anchored at once and the shoal sounded by boats starring round her, using prismatic com- pass and masthead angle; or if the shoal is of large extent and may be prudently crossed in the ship, it is a good plan to get two Deacons laid down on a bearing from one another and patent log distance of 4 or 5 m. With another beacon (or mark-boat, carrying a large black flag on a bamboo 30 ft. high) fixed on this base, forming an equilateral triangle, and the ship anchored as a fourth point, soundings may be_ carried out by the boats fixing by station-pointer. The ship's position is determined by observations of twilight stars.

In a detailed survey the coast is sketched in by walking along it, fixing by theodolite or sextant angles, and plotting by tracing- paper or station-pointer. A sufficient number of fixed marks along the shore afford a constant check lining, on the minor coast-line stations, which should be plotted on, or checked by, lines from one to the other wherever possible to do so. When impracticable to fix in the ordinary way, the ten-foot pole may be used to traverse from one fixed point to another. With a coast fronted by broad drying, coral reef or flats over which it is possible to walk, the distance between any two coast-line stations may be found by measuring at one of them the angle subtended by a known length placed at right angles to the line joining the stations. There is far less liability to error if the work is plotted at once on the spot on field board with the fixed points pricked through and circled in upon it; but if circumstances render it necessary, the angles being registered and sketches made of the bits of coast hetween the fixes on a scale larger than that of the chart, they may be plotted after- wards; to do this satisfactorily, however, requires the surveyor to appreciate instinctively exactly what angles are necessary at the time. It is with the high-water line that the coast-liner is concerned, delineating its character according to the admiralty symbols. The officer sounding off the coast is responsible for the position of the dry line at low-water, and on large scales this would be sketched in from a small boat at low-water springs. Heights of cliffs, rocks, islets, &c, must be inserted, either from measurement or from the formula,

height in feet

angle of elevation in seconds X distance in miles, 34

and details of topography close to the coast, including roads, houses and enclosures, must be shown by the coast-liner. Rocks above water or breaking should be fixed on passing them. Coast- line may be sketched from a boat pulling along the shore, fixing and shooting up any natural objects on the beach from positions at anchor.

The most important feature of a chart is the completeness with which it is sounded. Small scale surveys on anything less than one inch to the mile are apt to be very misleading; such a survey may appear to have been closely sounded, but in reality the lines are so far apart that they often fail to disclose indications of shoal-water. The work of sounding may be proceeded with as soon as sufficient points for fixing are plotted; but off an intricate coast it is better to get the coastline done first. The lines of soundings are run by the boats parallel to one another and perpendicular to the coast at a distance apart which is governed by the scale; five lines to the inch is about as close as they can be run without overcrowding; if closer lines are required the scale must generally be increased. The distance apart will vary with the depth of water and the nature of the coast; a rocky coast with shallow water off it and projecting points will need much closer examination than a steep-to coast, for instance. The line of prolongation of a point under water will require special care to ensure the fathom lines being drawn correctly. If the soundings begin to decrease when pulling off-shore it is evidence of something suspicious, and intermediate lines of soundings or lines at right angles to those previously run should be obtained. Whenever possible lines of soundings should be run on transit lines; these may often be picked up by fixing when on the required line, noting the angle on the protractor between the line and some fixed mark on the field board, and then placing the angle on the sextant, reflecting the mark and noting what obiects are in line at that angle. On large scale surveys whitewash marks or flags should mark the ends of the lines, and for the back transit marks natural objects may perhaps be picked up; if not, they must be placed in the required positions. The boat is fixed by two angles, with an occasional third angle as a check; the distance between the fixes is dependent upon the scale of the chart and the rapidity with which the depth alters; the 3, 5 and 10 fathom lines should always be fixed, allowing roughly for the tidal reduction. The nature of the bottom must be taken every few casts and recorded. It is best to plot each fix on the sounding board at once, joining the fixes by straight lines and numbering them for identification. The tidal reduction being obtained, the reduced soundings are written in the field-book in red underneath each sounding as originally noted; they are then placed in their proper position on the board between the fixes. Suspicious ground should be closely examined; a small nun buoy anchored on the shoal is useful to guide the boat while trying for the least depth. Sweep- ing for a reported pinnacle rock may be resorted to when sounding fails to discover it. Local information from fishermen and others is often most valuable as to the existence of dangers. Up to depths of about 15 fathoms the hand lead-line is used from the boats, but beyond that depth the small Lucas machine for wire effects a great saving of time and labour. The deeper soundings of a survey are usually obtained from the ship, but steamboats with wire sounding machines may assist very materi- ally. By the aid of a steam winch, which by means of an endless rounding line hauls a 100-lb lead forward to the end of the lower boom rigged out, from which it is dropped by a slipping apparatus which acts on striking the water, soundings of 40 fathoms may be picked up from the sounding platform aft, whilst going at a speed of 45 knots. In deeper water it is quicker to stop the ship and sound from aft with the wire sounding machine. In running long lines of soundings on and off shore, it is very essential to be able to fix as far from the land as possible. Angles will be taken from aloft for this purpose, and a few floating beacons dropped in judiciously chosen positions will often well repay the trouble. A single fixed point on the land used in conjunc- tion with two beacons suitably placed will give an admirable fix. A line to the ship or her smoke from one or two theodolite stations on shore is often invaluable; if watches are compared, observations may be made at stated times and plotted after- wards. True bearings of a distant fixed object cutting the line of position derived from an altitude of the sun is another means of fixing a position, and after dark the true bearing of a light may be obtained by the time azimuth and angular dis- tance of a star near the prime vertical, or by the angular distance of Polaris in the northern hemisphere.

A very large percentage of the bugbears to navigation denoted by vigias[6] on the charts eventually turn out to have no existence, but before it is possible to expunge them a large area has to be examined. No-bottom soundings are but little use, but the evidence of positive soundings should be conclusive. Submarine banks rising from great depths necessarily stand on bases many square miles in area. Of recent years our knowledge of the angle of slope that may be expected to occur at different depths has been much extended. From depths of upwards of 2000 fathoms the slope is so gradual that a bank could hardly approach the surface in less than 7 m. from such a sounding; therefore anywhere within an area of at least 150 sq. m. all round a bank rising from these depths, a sounding must show some decided indications of a rise in the bottom. Under such circumstances, soundings at intervals of 7 m., and run in parallel lines 7 m. apart, enclosing areas of only 50 sq. m. between any four adjacent soundings, should effectually clear up the ground and lead to the discovery of any shoal; and in fact the soundings might even be more widely spaced. From depths of 1500 and 1000 fathoms, shoals can scarcely occur within 35 m. and 2 m. respectively; but as the depth decreases the angle of slope rapidly increases, and a shoal might occur within three-quarters of a mile or even half a mile of such a sounding as 500 fathoms. A full appreciation of these facts will indicate the distance apart at which it is proper to place soundings in squares suitable to the general depth of water. Contour lines will soon show in which direction to prosecute the search if any irregularity of depth is manifested. When once a decided indication is found, it is not difficult to follow it up by paying attention to the contour lines as developed by successive soundings. Discoloured water, ripplings, fish jumping or birds hovering about may assist in locating a shoal, but the submarine sentry towed at a depth of 40 fathoms is here invaluable, and may save hours of hunting. Reports being more liable to errors of longitude than of latitude, a greater margin is necessary in that direction. Long parallel lines east and west are preferable, but the necessity of turning the ship more or less head to wind at every sounding makes it desirable to run the lines with the wind abeam, which tends to disturb the dead reckoning least. A good idea of the current may be obtained from the general direction of the ship's head whilst sounding considered with reference to the strength and direction of the wind, and it should be allowed for in shaping the course to preserve the parallelism of the lines, but the less frequently the course is altered the better. A good position in the morning should be obtained by pairs of stars on opposite bearings, the lines of position of one pair cutting those of another pair nearly at right angles. The dead reckoning should be checked by lines of position from observations of the sun about every two hours throughout the day, preferably whilst a sounding is being obtained and the ship stationary. Evening twilight stars give another position.

Tides.—The datum for reduction of soundings is low-water ordinary springs, the level of which is referred to a permanent bench mark in order that future surveys may be reduced to the same datum level. Whilst sounding is going on the height of the water above this level is observed by a tide gauge. The time of high-water at full and change, called the " establishment," and the heights to which spring and neap tides respectively rise above the datum are also required. It is seldom that a sufficiently long series of observations can be obtained for their discussion by har- monic analysis, and therefore the graphical method is preferred; an abstract form provides for the projection of high and low waters, lunitidal intervals, moon's meridian passage, declination of sun and moon, apogee and perigee, and mean time of high-water following superior transit, and of the highest tide in the twenty-four hours. A good portable automatic tide gauge suitable for all requirements is much to be desired.

Tidal Streams and Surface Currents are observed from the ship or boats at anchor in different positions, by means of a current log ; or the course of a buoy drifted by the current may be followed by a boat fixing at regular intervals. Tidal streams often run for some hours after high and low water by the shore ; it is important to find out whether the change of stream occurs at a regular time of the tide. Undercurrents are of importance from a scientific point of view. A deep-sea current meter, devised (1876) by Lieut. Pillsbury, U.S.N. , has, with several modifications, been used with success on many occasions, notably by the U.S. Coast and Geodetic Survey steamer " Blake " in the investigation of the Gulf Stream. The instrument is first lowered to the required depth, and when ready is put into action by means of a heavy weight, or messenger, travelling down the supporting Deep-sea line and striking on a metal plate, thus closing the XfT*" jaws of the levers and enabling the instrument to _ meter ' begin working. The rudder is then free to revolve inside the framework and take up the direction of the current; the small cones can revolve on their axis and register the number of revolu- tions, while the compass needle is released and free to take up the north and south line. On the despatch of a second messenger, which strikes on top of the first and fixes the jaws of the levers open, every part of the machine is simultaneously locked. Having noted the exact time of starting each of the messengers, the time during which the instrument has been working at the required depth is known, and from this the velocity of the current can be calculated, the number of revolutions having been recorded, while the direction is shown by the angle between the compass needle and the direction of the rudder.

The instrument is shown in fig. 12. AA are the jaws of the levers through which the first messenger passes and strikes on the metal plate B. The force Of the blow is sufficient to press B down, thus bringing the jaws as close together as possible, and putting the meter into action. The second messenger falling on the first opens the levers again and prevents their closing, thus keeping all parts of the machine locked. C is the rudder which takes up the direction of the current when the levers are unlocked. D is a set of small levers on the rudder in connexion with AA. The outer end on the tail of the rudder fits into the notches on the outer ring of the frame when the machine is locked and thus keeps the rudder fixed, but when the first messenger has started the machine by_ pressing down B and opening the levers AA, this small lever is raised and the rudder can revolve freely. EE are four small cones which revolve on their axis in a vertical plane, similar to an anemometer; the axis is connected by a worm screw to gearcdj'wheels which register the number of revolutions up to 5000, corresponding to about 4 nautical miles. There is a small lever in connexion with AA which prevents the cones revolving when the machine is locked, but allows them to revolve freely when the machine is in action. Below the rudder-post is a compass-bowl F, which is hung in gimbals and capable of removal. The needle is so arranged that it can be lifted off the pivot by means of a lever in connexion with AA ; when the meter is in action the needle swings freely on its pivot, but [when the levers are

Fig. 12.

locked it is raised off its pivot by the inverted cup-piece K placed inside the triple claws on the top of the compass and screwed to the lever, thus locking the needle without chance of moving. The compass bowl should be filled with fresh water before lowering the instrument into the sea, and the top screwed home tightly. The needle should be removed and carefully dried after use, to pre- vent corrosion. The long arm G is to keep the machine steady in one direction; it works up and down a jackstay which passes between two sheaves at the extremity of the long arm. This also assists to keep the machine in as upright a position as possible, and prevents it from being drifted astern with the current. A weight of as much as 8 or 10 cwt. is required at the bottom of the jackstay in a very strong current. An elongated weight of from 60 to 80 lb must be suspended from the eye at the bottom of the meter to help to keep it as vertical as possible. On the outer part of the horizontal notched ring forming the frame, and placed on the side of the machine opposite to the projecting arm G, it has been found necessary to bolt a short arm supported by stays from above, from which is sus- pended a leaden counterpoise weight to assist in keeping the appara- tus upright. This additional fitting is not shown in fig. 12. A f-in. phosphor-bronze wire rope is used for lowering the machine; it is rove through a metal sheave H and india-rubber washer, and spliced round a heart which is attached to metal plate B. The messengers are fitted with a hinged joint to enable them to be placed round the wire rope, and secured with a screw bolt. To obtain the exact value of a revolution of the small cones it is necessary to make experiments when the actual speed of the current is known, by immersing .the meter just below the surface and taking careful observations of the surface-current by means of a current log or weighted pole. From the number of revolutions registered by the meter in a certain number of minutes, and taking the mean of several observations, a very fair value for a revolution can be deduced. On every occasion of using the meter for under-current observations the value of a revolution should be re-determined, as it is apt to vary owing to small differences in the friction caused by want of oil or the presence of dust or grit ; while the force of the current is probably another important factor in influencing the number of revolutions recorded.

The features of the country should generally be delineated as far back as the skyline viewed from seaward, in order to assist _ . the navigator to recognize the land. The summits

Topography. , .... ° , . ° , , . , ,

of hills and conspicuous spurs are fixed either by lines to or by angles at them; their heights are determined by theodolite elevations or depressions to or from stations


whose height above high-water is known. As much of the ground as possible is walked over, and its shape is delineated by contour lines sketched by eye, assisted by an aneroid barometer. In wooded country much of the topography may have to be shot in from the ship; sketches made from different positions at anchor along the coast with angles to all prominent features, valleys, ravines, spurs of hills, &c, will give a very fair idea of the general lie of the country.

Circum-meridian altitudes of stars on opposite sides of the zenith observed by sextant in the artificial horizon is the method adopted wherever possible for observations for latitudes. Arranged in pairs of nearly the same " " e *' altitude north and south of zenith, the mean of each pair should give a result from which instrumental and personal errors and errors due to atmospheric conditions are altogether eliminated. The mean of several such pairs should have a probable error of not more than =±= 1". As a rule the observations of each star should be confined to within 5 or 6 minutes on either side of the meridian, which will allow of from fifteen to twenty observa- tions. Two stars selected to " pair " should pass the meridian within an hour of each other, and should not differ in altitude more than 2° or 3 . Artificial horizon roof error is eliminated by always keeping the same end of the roof towards the observer; when observing a single object, as the sun, the roof must be reversed when half way through the observations. The observa- tions are reduced to the meridian by Raper's method. When pairs of stars are not observed, circum-meridian altitudes of the sun alone must be resorted to, but being observed on one side of the zenith only, none of the errors to which all observa- tions are liable can be eliminated.

Sets of equal altitudes of sun or stars by sextant and artificial horizon are usually employed to discover chronometer errors. Six sets of eleven observations, a.m. and p.m., chrono- observing both limbs of the sun, should give a result meter which, under favourable conditions of latitude and Errors. declination, might be expected to vary less than two-tenths of a second from the normal personal equation of the observer. Stars give equally good results. In high latitudes sextant observations diminish in value owing to the slower movement in altitude. In the case of the sun all the chronometers are compared with the " standard " at apparent noon; the com- parisons with the chronometer used for the observations on each occasion of landing and returning to the ship are worked up to noon. In the case of stars, the chronometer compari- sons on leaving and again on returning are worked up to an intermediate time. A convenient system, which retains the advantage of the equal altitude method, whilst avoiding the necessity of waiting some hours for the p.m. observation, is to observe two stars at equal altitudes on opposite sides of the meridian, and, combining the observations, treat them as rela- ting to an imaginary star having the mean R.A. and mean declination of the two stars selected, which should have nearly the same declination and should differ from 4* to 8* in R.A.

The error of chronometer on mean time of place being obtained, the local time is transferred from one observation spot to another by the ship carrying usually eight box chronometers. The best results are found by using travelling rates, n/^j" a " which are deduced from the difference of tjie errors found on leaving an observation spot and returning to it; from this difference is eliminated that portion which may have accumulated during an interval between two determinations of error at the other, or any intermediate, observation spot. A travelling rate may also be obtained from observations at two places, the meridian distance between which is known; this rate may then be used for the meridian distance between places observed at during the passage. Failing travelling rates, the mean of the harbour rates at either end must be used. The same observer, using the same instrument, must be employed throughout the observations of a meridian distance.

If the telegraph is available, it should of course be used. The error on local time at each end of the wire is obtained, and a number of telegraphic signals are exchanged between the observers, an equal number being transmitted and received at either end. The local time of sending a signal from one place being known and the local time of its reception being noted, the difference is the meridian distance. The retardation due to the time occupied by the current in travelling along the wire is eliminated by sending signals in both directions. The relative personal equation of the observers at either end, both in their observations for time, and also in receiving and transmitting signals, is eliminated by changing ends and repeating the operations. If this is impracticable, the personal equations should be determined and applied to the results. Chronometers keeping solar time at one end of the wire, and sidereal time at the other end, materially increase the accuracy with which signals can be exchanged, for the same reason that comparisons between sidereal clocks at an observatory are made through the medium of a solar clock. Time by means of the sextant can be so readily obtained, and within such small limits of error, by skilled observers, that in hydrographic surveys it is usually employed; but if transit instruments are available, and sufficient time can be devoted to erecting them properly, the Value of the work is greatly enhanced in high latitudes.

True bearings are obtained on shore by observing with theodolite the horizontal angle between the object selected as the zero and the sun, taking the latter in each quadrant as defined by the cross-wires of the telescope. The altitude may be read on the vertical arc of the theodolite; except in high latitudes, where a second observerTrue
Bearings.
with sextant and artificial horizon are necessary, unless the precise errors of the chronometers are known, when the time can be obtained by carrying a pocket chronometer to the station. The sun should be near the prime vertical and at a low altitude; the theodolite must be very carefully levelled, especially in the position with the telescope pointing towards the sun. To eliminate instrumental errors the observations should be repeated with the vernier set at intervals equidistant along the arc, and a.m. and p.m. observations should be taken at about equal altitudes.

At sea true bearings are obtained by measuring with a sextant the angle between the sun and some distant well-defined object making an angle of from 100° to 120° and observing the altitude of the sun at the same time, together with that of the terrestrial object. The sun’s altitude should be low to get the best results, and both limbs should be observed. The sun’s true bearing is calculated from its altitude, the latitude, and its declination; the horizontal angle is applied to obtain the true bearing of the zero. On shore the theodolite gives the horizontal angle direct, but with sextant observations it must be deduced from the angular distance and the elevation.

For further information see Wharton, Hydrographical Surveying (London, 1898); Shortland, Nautical Surveying (London, 1890).  (A. M. F .*) 

  1. 1 The subject of tacheometry is treated under its own heading.
  2. The theoretical " error of mean square " = 1-48 X " probable error."
  3. 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 k for the coefficient of refraction, and expressing the distance and radius in miles and the correction to height in feet, then correction for curvature = %d?; correction for refraction = — JmZ 2 ; correction for both
  4. 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.
  5. 1 In the Indian survey, tables are employed for these calculations which give the value of 1' 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.
  6. A Spanish word meaning " look-out," used of marks on the chart signifying obstructions to navigation.