SURVEYING is the art of determining the relative positions of prominent points and other objects on the surface of the ground and making a graphical delineation of the included area. The general principles on which it is conducted are in all instances the same: certain measures are made on the ground and corresponding measures are protracted on paper, on a scale which is fixed at whatever fraction of the natural scale may be most appropriate in each instance. The method of operation varies with the magnitude and importance of the survey, which may embrace a vast empire or be restricted to a small plot of land. All surveys rest primarily on linear measures for direct determinations of distance; but these are usually largely supplemented by angular measures, to enable distances to be deduced by the principles of geometry which cannot be conveniently measured over the surface of the ground where it is hilly or broken. The nature of a survey depends on the proportion which the linear and the angular measures bear to each other; it may be purely linear or even purely angular, but is generally a combination of both methods. Thus in India there are numerous instances of large tracts having been surveyed by the purely linear method, in the course of the revenue surveys which were initiated by the native Governments. The operations were conducted by men who had no knowledge of geometry or of any other measuring instrument than the rod or chain, and whose principal object was the determination of fairly accurate areas; their methods sufficed for this purpose and were accepted and perpetuated for many years by the European officers to whom the revenue assessments became entrusted after the subversion of the native rule. In India, too, there are extensive tracts of country which have been surveyed by the purely angular method, either because the ground did not permit of the chain being employed with advantage, as in the Himalayan mountains and hill tracts generally, or because the chain was considered politically objectionable, as in native states where it would have been regarded with suspicion.
Surveys of any great extent of country were formerly constructed on a basis of points whose positions were fixed astronomically, and in some countries this method of operation is still of necessity adopted. But points whose relative positions have been fixed by a triangulation of moderate accuracy present a more satisfactory and reliable basis; for astronomical observations are liable, not only to the well-known intrinsic errors which are caused by uncertainties in the catalogued places of the moon and stars, but to external errors arising from deflexions of the plumb line under the influence of local attractions, and these of themselves materially exceed the errors which would be generated in a fairly executed triangulation of a not excessive length, say not exceeding 500 miles. The French Jesuits who made a survey of China for the emperor about 1730 appear to have been the first deliberately to discard the astronomical and adopt the trigonometrical basis. In India the change was made in 1800, when what is known as the Great Trigonometrical Survey was initiated by Major Lambton—with the support of Colonel Wellesley, afterwards Duke of Wellington—as a means of connecting the several surveys of routes and districts which had already been made in various parts of the country, and as a basis for future topography. This necessitated the inception of the survey as an undertaking calculated to satisfy the requirements of geodesy as well as geography, because the latitudes and longitudes of the points of the triangulation had to be determined for future reference,—as in the case of the discarded astronomical stations, though in a different manner,—by processes 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 a high order of accuracy was aimed at from the very first. In course of time the operations were extended over the entire length and breadth of Hindustan and beyond, to the farthest limits of British sway; they cover a larger area than any other national survey as yet completed, and are very elaborate and precise. Thus, as triangulation constitutes the most appropriate basis for survey operations generally, a short account will be given of (1) the methods of the Great Trigonometrical Survey of India. This will be followed by accounts of (2) traversing as a basis for survey, (3) levelling, (4) survey of interior detail, (5) representation of ground, (6) geographical reconnaissance, (7) nautical surveying, (8) mapping, (9) map printing, (10) instruments.
I. Great Trigonometrical Survey of India.
1. General Outlines.— 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-foot theodolite, which, however, was badly damaged almost at the outset by an accident to the azimuthal circle. 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 purposes; but the central meridional arc—the “great arc”—was eventually deemed inadequate for geodetic requirements. A superior instrumental equipment was introduced, with an improved modus operandi, under the direction of Colonel Everest in 1832. The network system of triangulation was superseded by meridional and longitudinal chains taking the form of gridirons, and resting on base-lines at the angles of the gridirons, as represented in fig. 1. For convenience of reduction and nomenclature the triangulation west of meridian 92° E. has been divided into five sections,—the lowest a trigon, the other four quadrilaterals distinguished by cardinal points which have reference to an observatory in Central India, the adopted origin of latitudes. In the north-east quadrilateral, which was first measured, the meridional chains are about one degree apart; this distance was latterly much increased, and eventually certain chains—as on the Malabar coast and on meridian 84° in the south-east quadrilateral—were dispensed with, because good secondary triangulation for topography had been accomplished before they could be commenced.
2. Modern Base-Lines.— All these were measured with the Colby apparatus of compensation bars and microscopes. The bars, 10 feet long, were set up horizontally on tripod stands; the microscopes, 6 inches 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 feet between the ground pins or registers. For explanation of compensation see Earth, Figure of the, vol. vii. p. 599. Compound bars are necessarily 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-inch scale. At the very first base-line the compensated bars were found to be liable to sensible variations of length with the diurnal variations of temperature; these were supposed to be due, not to error in effecting the compensation, but 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 very precisely, for which reason they were systematically compared with the standard before and after, and sometimes at the middle of, the base-line measurement throughout the entire day for a space of three days, and under conditions as nearly similar as possible to those obtaining during the measurement. Eventually thermometers were applied experimentally to both components of a compound bar, when it was found that the diurnal variations in length were principally due to difference of position relatively to the sun, not to difference of conductivity,—the component nearest the sun acquiring heat most rapidly or parting with it most slowly, notwithstanding that both were in the same box, which was always kept under the cover of a tent and carefully 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 at the Cape Comorin base showed that, for any baseline measured as usual without thermometers in the compound bars, the p.e . may be taken as ± 1·5 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 Fahr., corresponding to an error of - 2 millionths in the length of a base-line measured with iron bars. With compound bars lagging would be much the same for both components and its influence would consequently be eliminated. Thus the most perfect base-line apparatus would seem to be one of compensation bars with thermometers attached to each component; then the comparisons with the standard need only be taken at the times when the temperature is constant, and there is no lagging.
3. Factor of Expansion of Standard Bar.— This was first determined in 1832 by measuring the increment in length between temperatures of 76° and 212° Fahr.; in 1870 the increment between 52° and 96° was measured; the results indicated an increase of expansion with temperature. They were therefore combined on the empirical assumption that the expansion is the sum of two terms,— the first x times the temperature, the second y times the square of the temperature; x and y were then determined from the two equations of condition given by the two sets of measurements. The resulting value of the expansion at 62° was found to be 5 per cent. less than the previously derived value at the mean temperature of 144°, thus showing the importance of employing a factor varying with the mean temperature of each base-line; and this was done in the final reductions.
4. Plan of Triangulation.— This 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 of great extent, devoid of any commanding points of view, in some parts covered with dense forest and jungle, malarious and deadly, and almost uninhabited, in other parts covered with towns and villages and umbrageous trees,—the adjuncts and concomitants of a teeming population. In such tracts triangulation was impossible except by constructing lofty 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, both of which were laborious and expensive processes. 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 tracts of 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 miles in hill country and 11 in the plains; the longest principal side was 62·7 miles, though in the secondary triangulation to the Himalayan peaks there were sides exceeding 200 miles. Long sides were at first considered desirable, on the principle that the fewer the links the greater the accuracy of a chain of triangles; but it was eventually found that good observations on long sides could only be obtained under exceptionally favourable atmospheric conditions, which were of rare occurrence. The sides were therefore shortened, whereby the observations were much improved and accelerated. 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 — 2 / 3 × square of the distance in miles; thus 24 feet of height was needed at each end of a side to overtop the curvature in 12 miles, 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.
5. Selection of Sites for Stations.— This, a very simple matter in hills and open country, is often very 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 feet 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 feet high, with arrangements for displaying blue lights by night at 90 feet, were erected at the spots where station sites were wanted. But the cost of transport was great; the rate of progress was slow; and the results were unsatisfactory. Eventually a method of touch rather than sight was adopted, feeling the ground to search for the obstacles to be avoided, rather than attempting to look over them; the “rays” were traced either by a minor triangulation, or by a traverse with theodolite and perambulator, or by a simple alignment of flags. The first method gives the direction of the new station most accurately; the second searches the ground most closely; the third is best suited for tracts of uninhabited forest in which there is no choice of either line or site, and the required station may be built at the intersection of the two trial rays leading up to it. As a rule it has been found most economical and expeditious to raise the towers only to the height necessary for surmounting the curvature, and to remove the trees and other obstacles on the lines.
6. Structure of the Principal Stations.— Each has a central masonry pillar, circular and 3 to 4 feet in diameter, for the support of a large theodolite, and around it a platform 14 to 16 feet square for the observatory tent, observer, and signallers. The pillar is carefully isolated from the platform, and when solid carries the station mark—a dot surrounded by a circle—engraved on a stone at its surface, and on additional stones or the rock in situ, in the normal of the upper mark; but, if the height is considerable and there is a liability to deflexion, the pillar is constructed with a central vertical shaft to enable the theodolite to be plumbed over the ground-level mark, to which access is obtained through a passage in the basement. In early years this precaution against deflexion was neglected and the pillars were built solid throughout, whatever their height; the surrounding platforms, being usually constructed of sun-dried bricks or stones and earth, were liable to fall and press against the pillars, some of which thus became deflected during the rainy seasons that intervened between the periods during which operations were arrested or the commencement and close of the successive circuits of triangles. In some instances displacements of mark occurred of which the magnitudes were not ascertainable, but were estimated as equivalent to p.e.' s of about ±9 inches in the length and± 2"·4 in the azimuth of the side between any two deflected towers; and, as these theoretical errors are identical with what may be expected at the end of a chain of 36 equilateral triangles in which all the angles have been measured with ap.e. = ± 0"·5, the old triangulation over solid towers had evidently suffered much more from the deflexions of the towers than from errors in the measurements of the angles. 7. Instruments for Measurign Principal Angles. Large theodolites were invariably employed. Repeating circles were highly thought of by French geodesists at the time when the operations in India were being commenced; but they were not used in the survey, and have now been generally discarded.
The principal theodolites are 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 have mostly diameters of either 36 or 24 inches, the vertical circles having a diameter of 18 inches. In all the theodolites the base is 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 vary. In some the vertical axis is fixed on the tribrach and projects upwards; in others it revolves in the tribrach and projects downwards. . In the former the azimuthal circle is fixed to the tribrach, while the telescope pillars, the microscopes, the clamps, and the tangent screws are attached to a drum revolving round the vertical axis; in the latter the microscopes, clamps, and tangent screws are fixed to the tribrach, while the telescope pillars and the azimuthal circle are attached to a plate fixed at the head of the rotatory vertical axis. The former system—called that of flying microscopes— permits the vertical axis to be readily opened out and cleaned, and presents the same clamp and tangent screw for employment during a round of angles; the latter—the system of fixed microscopes— necessitates the removal and replacement of all the microscopes, clamps, and tangent screws whenever the axis is cleaned, which is very troublesome, and it presents three sets of clamps and tangent screws for successive employment during a round of angles, which is a departure from true differentiality. The vertical axis is perforated for centring over the station mark with the aid of a “look-down telescope” instead of a plummet. The azimuthal circle is invariably read by an odd number of microscopes, either three or five, at equal intervals apart. The telescope rests with its pivots in Y’s at the head of two pillars of a sufficient height to enable it to be completely turned round in altitude. The vertical circle is fixed to the transit axis of the telescope, and is read by two microscopes 180° apart, at the extremities of arms projecting from one of the pillars. The stand is a well-braced tripod, carrying an iron ring on which the theodolite rests and may be turned round bodily whenever desired, as for shifting the position of the zero of the azimuthal circle relatively to the points under observation. The ring is 3 inches broad and of the same diameter as the circle of the foot-screws of the theodolite. In some instruments the foot-screws rest directly on the ring; but the instrument can be raised off the ring and turned round with the aid of an apparatus in the centre of the stand. In others they rest in grooves at the angles of an iron triangle which sits on the ring and can be shifted in position by hand; thus with the stand well levelled in the first instance the circle may be set within 1' of any required reading. The centring over the station mark is performed by pushing screws placed either in the drum of the stand or at the angles of the triangle.
For travelling the theodolites were packed in two cases, the larger containing the body of the instrument, the smaller the telescope and the vertical circle; the stand constituted a third package. Each was carried on men’s shoulders as the safest method of transport; the weights, of the heaviest 36-inch and of the lightest 24-inch instruments, as packed with ropes and bamboos, were, respectively, as follows:—body, 649 lb; telescope, 130; stand, 232; total, 1011 lb; and 300, 135, and 185, total 620 lb.
8. Signals.— 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 inches in diameter, filled with cotton seeds steeped in oil and resin, to burn under an inverted earthen jar, 30 inches in diameter, with an aperture in the side towards the observer. Subsequently this contrivance gave place to the Argand lamp with parabolic reflector; the opaque day signals were discarded for heliotropes reflecting the sun’s rays to the observer. The introduction of luminous signals not only rendered the night as well as the day available for the observations but changed the character of the operations, enabling work to be done during the dry and healthy season of the year, when the atmosphere is generally hazy and dust-laden, instead of being restricted as formerly to the rainy and unhealthy seasons, when distant opaque objects are best seen. À higher degree of accuracy was also secured, for the luminous signals were invariably displayed through diaphragms of appropriate aperture, truly centred over the station mark; and, looking like stars, they could be observed with greater precision, whereas opaque signals are always dim in comparison and are liable to be seen excentrically when the light falls on one side.
A signalling party of three men was usually found sufficient to manipulate a pair of heliotropes—one for single, two for double reflexion, according to the sun’s position— and a lamp, throughout the night and day. Heliotropem were also employed at the observing stations to flash instructions to the signallers.
9. Measuring Horizontal Angles.— The theodolites were invariably set up under tents for protection against sun, wind, and rain, and centred, levelled, and adjusted for the runs of the microscopes. Then the signals were observed in regular rotation round the horizon, alternately from right to left and vice versa ; after the prescribed 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 azimuthal circle during the telescope pointings to any object in the two positions of the vertical circle, “face right” and “face left,” were made on twice as many equidistant graduations as the number of microscopes. The theodolite was then shifted bodily in azimuth, by being turned on the ring on the head of the stand, which brought new graduations under the microscopes at the telescope pointings; then further rounds were measured in the new positions, face right and face left. This process was repeated as often as had been previously prescribed, the successive angular shifts of position being made by equal arcs bringing equidistant graduations under the microscopes during the successive telescope pointings to one and the same object. By these arrangements all periodic errors of graduation were eliminated, the numerous graduations that were read tended to cancel accidental errors of division, and the numerous rounds of measures to minimize the errors of observation arising from atmospheric and personal causes.
The following table (I.) gives details of the procedure at different times; in the headings M stands for the number of microscopes over the azimuthal circle of the theodolite, Z for the number of the zero settings of the circle, N for the number of graduations brought under the microscopes, A = 360°÷N, the arc between the graduations, R the prescribed number of rounds of measures, and P = R × Z, the minimum number of telescope pointings to any station, excluding repetitions for discrepant observations :— [table]
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.
10. Vertical Angles. Refraction.— 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 very capricious, more particularly when the rays of light graze the surface of the ground, passing through a medium which is liable to extremes of rarefaction and condensation, under the alternate influence of the sun’s heat radiated from the surface of the ground and of chilled atmospheric vapour. When the back and forward verticals at a pair of stations are equally refracted, their difference gives an exact measure of the difference of height. But the atmospheric conditions are not always identical at the same moment everywhere on long rays which graze the surface of the ground, and the ray between two reciprocating stations is liable to be differently refracted at its extremities, each end being influenced in a greater degree by the conditions prevailing around it than by those at a distance; thus instances are on record of a station A being invisible from another B, while B was visible from A.
When the great arc entered the plains of the Gangetic valley, simultaneous reciprocal verticals were at first adopted with the hope of eliminating refraction; but it was soon found that they did not do so sufficiently to justify the expense 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 commences about an hour after apparent noon and lasts from two to three hours. The apparent zenith distance is always greatest then, but the refraction is a minimum only at stations which are well elevated above the surface of the ground; at stations on plains the refraction is liable to pass through zero and attain a considerable negative magnitude during the heat of the day, for the lower strata of the atmosphere are then less dense than the strata immediately above and the rays are refracted downwards. On plains the greatest positive refractions are also obtained,—maximum values, both positive and negative, usually occurring, the former by night, the latter by day, when the sky is most free from clouds. The values actually met with were found to range from + 1·21 down to - 0·09 parts of the contained arc on plains; the normal “coefficient of refraction” for free rays between hill stations below 6000 feet was about ·07, which diminished to ·04 above 18,000 feet, broadly varying inversely as the temperature and directly as the pressure, but much influenced also by local climatic conditions.
In measuring the vertical angles with the great theodolites, graduation errors were regarded as insignificant compared with errors arising from uncertain refraction; thus no arrangement was made for effecting changes of zero in the circle settings. The observations were always taken in pairs, face right and left, to eliminate index errors, only a few daily, but some on as many days as possible, for the variations from day to day were found to be greater than the diurnal variations during the hours of minimum refraction.
11. Results deduced from Observations of Horizontal Angles; Weights.— In the Ordnance and other surveys the bearings of the surrounding stations are deduced from the actual observations, but from the “included angles” in the Indian Survey. The observations of every angle are tabulated vertically in as many columns as the number of circle settings face left and face right, and the mean for each setting is taken. For several years the general mean of these was adopted as the final result; but subsequently a “concluded angle” was obtained by combining the single means with weights inversely proportional to g 2 + o 2 ÷ n , —g being a value of the e.m.s. of graduation derived empirically from the differences between the general mean and the mean for each setting, o the e.m.s. of observation deduced from the differences between the individual measures and their respective means, and n the number of measures at each setting. Thus, putting w 1 , w 2 ,, . . . for the weights of the single means, w for the weight of the concluded angle, M for the general mean, C for the concluded angle, and d 1 , d 2 , . . . for the differences between M and the single means, we have
(1)
W 1 + W⅛ +
and to=w 1 + w 2 + (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.
12. Determination of Theoretical Absolute Errors of Observed Angles.— 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 of the weight w deduced as above from the measures of such angle, (ii.) from the magnitudes of the excess of the sum of the angles of each triangle above 180° + 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 (indicated in the next section). Let e 1 , e 2 , and e 3 be the values of the e.m.s. thus obtained; then, putting n for the number of angles grouped together, we have e 1 2 = n / [w] and e 2 2 = [squares of triangular errors] / n also, putting W for the mean of the weights of the t angles of a polygonal figure having m geometrical equations of condition, and x for the most probable value of the error of any observed angle, we have = = n for a single figure, 3 IV mm σ = tfh3 for a group of figures,
[wι]
the brackets [] in each case denoting the sum of all the quantities involved, e 3 usually gives the best value of the theoretical error, then e 2 . As a rule the value by e 1 is too small; but to this there are notable exceptions, in which it was found to be much too great. The instrument with which the angles were measured in these instances gave very discrepant results at different settings of the circle; but this was caused by large periodic errors of graduation which did not affect the “concluded angles,” because they were eliminated by the systematic changes of setting, so the results were really more precise than was apparent.
When weights were determined for the final simultaneous reduction of triangulations executed by different instruments, it became necessary to find a factor p to be applied as a modulus to each group of angles measured with the same instrument and under similar conditions, to convert the as yet relative weights into absolute measures of precision. ρ was made = e 1 ÷ e 3 whenever data were available, if not to e 1 ÷ e 2 ; then the absolute weight of an observed angle in any group was taken as wρ 2 and the e.m.s. of the angle as 1 ÷ρ√w. The average values of the e.m.s. thus determined for large groups of angles, measured with the 36-inch and the 24-inch theodolites, ranged from ± 0"·24 to ± 0"·67, the smaller values being usually obtained at hill stations, where the atmospheric conditions were most favourable.
13. Harmonizing Angles of Trigonometrical Figures.— 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 angles of every triangle having been observed, their sum had to be made = 180° + the spherical excess; in networks it was also necessary that the sum of the angles measured round the horizon at any station should be exactly = 360°, that the sum of the parts of an angle measured at different times should equal the whole, and that the ratio of any two sides should be identical, whatever the route through which it was computed. These are called the triangular, central, toto-partial, and side conditions; they present n geometrical equations, which contain 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 corrections; it is demonstrated under Earth, Figure of the (vol. vii. p. 599). The following is a general outline of the process :—
Let x be the most probable value of the error and u the reciprocal of the weight of any observed angle X, and let α, b,. .. n be the coefficients of x in successive geometrical equations of condition whose absolute terms are e a , e b ,. . . e n ; then we have the following group of n equations containing t unknown quantities to be satisfied, the significant coefficients of x being 1 in the triangular, toto-partial, and central, and ± cot X in the side equations :—
a 1 x 1 + a 2 x 2 + . . . + a t x t =e a } b]Xl +b2r2 + . . . ÷ btXt = eb I (g)
n 1 x 1 + >½a½ +. · · +n i ¾ = e n )
The values of x will be the most probable when [{{}}}] is a minimum, a condition which introduces n indeterminate factors λ a , . . . λ n , whose values are obtained by the solution of the following equations :— [oa.u]λ a + [α⅛4i]λ ft +. . . + [an. u]λ n = e t A
[aδ. w]λ a + [δδ. w]λ 4 +. . . + [⅛n,w]λ n = e k p
[a>ι.n]λ lj + [δπ.a]λ ft +.. . + [nn. ιz]λ n = e n J the brackets indicating summations of t terms as to left of (3).
Then the value of any, the p th, x is xp —- up {iq,X α -f- bp∖ k -f-. . , + (5)*
The minimum or [3 is =[eλ] (6).
In the application to a single triangle we have x 1 +x 2 +x 3 =e, λ = e ÷(u 1 + u 2 + u 3 ); x 1 = u 1 λ; x 2 = u 2 λ; x 3 = u 3 λ .
In the application to a simple polygon, by changing symbols and putting X and Y for the exterior and Z for the central angles, with errors x, y, and z and weight reciprocals u, v, and w, a for cot X and b for cot Y, e for any triangular error, e c and e s for the central and side errors, λ c and λ s for the factors for the central and side equations, and W for u + v + w, the equations for obtaining the factors become
Γ w 2- k i - w(αii-δ-r)Ι. Γwe~] ∖ L w -7f> -L—^J x ∙= i ∙-W l m
-[⅛^⅛] x . + [A+δ⅛-S^⅛]x.= e .-[⅛∑⅛J∫'' λ and the general expressions for the errors of the angles are—
x=^,{e + {aW - au + bvj∖ t - w∖ c } 1
y= - (δ JK+αii-frυ)λ,-λw> c } J- (8).
w I
2=^jp∙ {c-(αw - bv y )∖ s + {u + r)λ c J∙ j
14. Calculation of Sides of Triangles.— 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 diminished by one-third of the spherical excess of the triangle to which it appertains. The theorem is applicable without sensible error to triangles of a much larger size than any that are ever measured.
15. Calculation of Latitudes and Longitudes of Stations and Azimuths of Sides.— A station of origin being chosen of which the latitude and longitude are known astronomically, and also the azimuth of one of the surrounding stations, the differences of latitude and longitude and the reverse azimuths are calculated in succession, for all the stations of the triangulation, by Puissant’s formulae (Traité de Géodésie, Paris, 1842, 3d ed.).
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 α the major and b the minor semi-axis; e the excentricity, = - f- ; p the radius of tι( 1 - c% y } curvature to the meridian in latitude λ, = —,: v the normal
{l-e⅝in 2 λ}U
to the meridian in latitude λ, = 1 ; λ and L the given
{l-e 2 sin 2 λ}1°
latitude and longitude of A; λ + ∆λ and L + Δ L the required latitude and longitude of B; A the azimuth of B at A; B the azimuth of A at B; ∆ A=B-(π + A) ; c the distance between A and B. Then, all azimuths being measured from the south, we have r c
- cos A cosec 1"
1 2
- - — sin 2 √4 tan X cosec 1"
δx "H 3 c ' 2 c 2 . r ox
- - r —- ι ⅛ cos 2 √4 sin 2λ cosec 1"
4 ρ.v 1 - e i
1
+ - — 3 sin 3 √i cos A(l + 3 tan 2 λ) cosec 1"
I θ P∙ v J
c sin A ... Ί
r-cosec 1
v cos λ
1 e 2 sin 2 A tan λ ,,,
+ s —, r cosec 1
λ τ"- i * v cθs ^ lorn
~ 1 c 3 (1 + 3 tan 2 λ) sin 2A cos√∕ „ i '∙ ' ’
- 5 ~ ½ cosec 1
o v j cos λ
, 1 c 3 sin 3 ∠f tan 2 λ ,,,
+ s -⅞; cosec 1
3 v i cos X j ’ C .! * ’
— sin A tan X cosec 1"
V
1 e n - l . o , ,. 1 e 2 cos 2 X ) . s,A" or + y — ) 1 + 2 tanX + ——-- 2 1 sin 2 A cosec 1
Z-(ιr + √4)=- c 3^ 5 ∖ tanX . . . „ ^(11 )∙
— ( - + tanX I— s - sin 2 A cos A cosec 1
r 3 ∖ 6 ∕ I
+ - A* sin 3 J tan X (1 + 2 tan 2 X) cosec 1"
„ 6 v 3
Each Δ is the sum of four terms symbolized by δ 1 , δ 2 , δ 3 , and δ 4 ; the calculations are so arranged as to produce these terms in the order δλ, δ L , and δA, 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, δ,∖ = - P.cοsA.c δ 1 L = +δ∣λ.<2.secλ.tan A δ 1 ∠∕ = +δ 1 Z.sinλλ
δ.,λ= + δ l A.BsinA.c δ 2 L = - δ. 2 λ.S.cotA δ 2 A=+δ 2 L.T I ,- ζ >.
δ 3 λ=-δ s J.r.cot∠i δ 3 Z = + δ 3 λ.f∕.sin∠∕.c δ 3 A = + δ 3 L.JV P ιz >∙
δ 4 λ = -δ 3 Λ.A.tan∠∕ δ 4 L = + δ 4 λ. F. tan√f δ 4 A=+δ 4 L.Z J
By this artifice the calculations are rendered less laborious and made susceptible of being readily performed by any persons who are acquainted with the use of logarithm tables.
16. Limits within which Geodetic Formulae may be employed without Sensible Error.— Each Δ is expressed as a series of ascending differentials in which all terms above the third order are neglected; for the side length c in no case exceeded 70 miles, nor was the latitude ever higher than 36°, and for these extreme values the maximum magnitudes of the fourth differential are only 0"·002 in latitude and 0"·004 in longitude and azimuth.
Far greater error may arise from uncertainties regarding the elements of the earth’s figure, which was assumed to be spheroidal, with semi-axes a = 20,922,932 feet and b = 20,853,375 feet. The changes in ∆λ, Δ L , and ΔA which would arise from errors da and db in a and b are indicated by the following formulae :—
, .. .. dp „ . dv -,(dv 2de ∖ . dv^ d.∆λ = - ∆λ. — - δ 2 λ. —-δ .λ( - 5r I - 2δ 4 λ.-
p 2 V i ∖v (1 - e 2 )e∕ v d.ΔL = -ΔL.^- δ 2 L.^ - (δ 3 Z + δ 4 Z)2^
d∆A= - ∆∠i-δu ⅛ - -⅛>j—1— i " ,(13) ’
" l " ⅛∕ ^2tan 2 λ + 4
-(δ 3 J + δ 4 J)2- P j J in which
⅛= - O00,000,0478 {dα - 2db - ⅛{dα - db) sin 2 λ} ' y = + '000,000,0478 {da + {da - db) sin 2 λ} . ...(14).
, 2de - = + ‘000,0145 {da - db}
(1 - e 2 )e 7
The adopted values of the semi-axes were determined by Colonel Everest in an investigation of the figure of the earth from such data as were available in 1826. Forty years afterwards an investigation was made by Captain (now Colonel) A. R. Clarke with additional data, which gave new values, both exceeding the former. Accepting these as exact, the errors of the first values are da = -3130 feet and db= - 1746 feet, the former being 150, the latter 84 millionth parts of the semi-axis. The corresponding changes in arcs of 1° of latitude and longitude, expressed in seconds of arc and in millionth parts (μ ) of arc-length, are as follows :—
In lat. 5° rf.∆λ= - "Ό69 or 19 g and dΛL= -" - 540or 150 g;
„ 15° „ -"∙113 ,, 31,, „ „ -"∙554 ,, 154,,;
,, 25° „ -"∙195,,54,, „ ,, -"∙581 ,, 161;
„ 35° „ -"∙303 „ 84,, „ „ -" ’ 617 „ 171 „.
These assumed errors in the geodetic latitudes and longitudes are of service when comparisons are made between independent astronomical and geodetic determinations at any points for which both may be available: they indicate the extent to which differences may be attributable to errors in the adopted geodetic constants, as distinct from errors in the trigonometrical or the astronomical operations.
17. Final Reduction of Principal Triangulations— The calculations described so far suffice to make the angles of the several trigonometrical figures consistent inter se, and to give preliminary values of the lengths and azimuths of the sides and the latitudes and longitudes of the stations. The results are amply sufficient for the requirements of the topographer and land surveyor, and they are published in preliminary charts, which give full numerical details of latitude, longitude, azimuth, and side-length, and of height also, for each portion of the triangulation—secondary as well as principal—as executed year by year. But on the completion of the several chains of triangles further reductions became necessary, to make the triangulation everywhere consistent inter se and with the verificatory baselines, 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.
How this was done will now be set forth. But first it must be noted that the triangulation might at the same time have been made consistent with any values of latitude, longitude, or azimuth which had been determined by astronomical observations at either of the trigonometrical stations. This, however, was undesirable, because such observations are liable to errors from deflexion of the plumb-line from the true normal under the influence of local attraction, and these errors are of a much greater magnitude than those that would be generated in triangulating between astronomical stations which are not a great distance apart. The trigonometrical elements could not be forced into accordance with the astronomical without altering the angles by amounts much larger than their probable errors, and the results would be useless for investigations of the figure of the earth. The only independent facts of observation which could be legitimately combined with the angular adjustments were the base-lines, and all these were employed, while the several astronomical determinations—of latitude, differential longitude, and azimuth—were held in reserve for future geodetic investigations.
As an illustration of the problem for treatment, suppose a combination of three meridional and two longitudinal chains comprising seventy-two single triangles, with a base-line at each corner, as shown in the accompanying diagram (fig. 2); suppose the three angles of every triangle to have been measured and made con sistent. Let A be the ori gin, with its latitude and longitude given, and also the length and azimuth of the adjoining base-line.
With these data processes of calculation are carried through the triangulation to obtain the lengths and azimuths of the sides and the latitudes and longi tudes of the stations, say in the following order :—from A through B to E, through F to E, through F to D, through F and E to C, and through F and D to C. Then there are two values of side, azimuth, latitude, and longitude at E,—one from the righthand 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 baselines at B, C, and D has a calculated as well as a measured value. Thus eleven absolute errors are presented for dispersion over the triangulation by the application of the most appropriate correction to each angle, and, as a preliminary to the determination of these corrections, equations must be constructed between each of the absolute errors and the unknown errors of the angles from which they originated. For this purpose assume X to be the angle opposite the flank side of any triangle, and Y and Z the angles opposite the sides of continuation; also let x, y, and z be the most probable values of the errors of the angles which will satisfy the given equations of condition. Then each equation may be expressed in the form [ax+by + cz] = E, the brackets indicating a summation for all the triangles involved. We have first to ascertain the values of the coefficients a, b, and c of the unknown quantities. They are readily found for the side equations on the circuits and between the base-lines, for x does not enter them, but only y and z, with coefficients which are the cotangents of Y and Z, so that these equations are simply [cot Y.y - cot Z.z ] = E. But three out of four of the circuit equations are geodetic, corresponding to the closing errors in latitude, longitude, and azimuth, and in them the coefficients are very complicated. They are obtained as follows. The first term of each of the three expressions for ∆λ, ΔA, and B is differentiated in terms of c and A, giving d.∆λ = ∆λ (—-dA tanH sin 1" j-
d.ΔL= ΔB I c ~+dA cot A sin 1" j- - (15), dB = dA + ΔA ( “ C + dA cot A sin 1" j-
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 λ, = [ d .∆λ], and dL, = [ d . Δ L ].
Let station 1 be the origin, and let 2, 3, . .. be the succeeding stations taken along a predetermined line of traverse, which may either ran from vertex to vertex of the successive triangles, zigzagging between the flanks of the chain, as in fig. 3 (1), or be carried directly along one of the flanks, as in fig. 3 (2). For the general symbols of the differential equations substitute ∆λ n , Δ L n , ΔA n , c n , A n , and B n , for the side between stations n and n + 1 of the traverse; and let δc n and δA n be the errors generated between the sides c n-1 and c n ; then
⅛Cj - δcj i de2-δc i + δc 2 . <fc„_ "Γδc - ]
c l c l c 2 C 1 c 2 C n lL c J
dA 1 = δA i j dA 2 = dB 1 + δΑ 2 ; ... dA n = dΒ n ~ i + δA n .
Performing the necessary substitutions and summations, we get f J∆^⅛ + J∆∠⅛ + . . . +Δ∠⅛
<LB n ={ n ^ 2 n
I +(1+ 1 [∆ricotri]sinl")δ√f 1 + (l + . j [∆ri cot√∕]sinl")δ√i 2 ∖ + . . . + (1+ ∆ri n cot √f n sin l")δri n .
∩[ΔX]⅛ + "[Δλ]⅛ + ... + Δλ ll ^
I <-l t 2 t n (I — ∙∖ ∏
- { 1 [∆λ tan A]δA 1 + J∆λ tan A]δA 2 + ...
∖ + ∆λ n tan A n δA n [ sin 1"
( "[ΔA]- 1 + "[ΔZ]‰ . . . + ΔA n δ ⅛
<‰ h =J q ' 2 n
Ι + { 1 [ΔZ, cot A]δA l + 2 [ΔZ cot A]δA 2 +. . . ∖ + ΔL n cot A n δA n } sin 1"
Thus we have the following expression for any geodetic error—
Z t 17y+ · · · + Mn^τ^ n + φ 1 δA 1 +. . . + φ n δA n = E, ...(16), where μ and φ represent the respective summations which are the coefficients of δc and δA in each instance but the first, in which 1 is added to the summation in forming the coefficient of δA.
The angular errors x, y, and z must now be introduced, in place of δc and δA, into the general expression, which will then take different forms, according as the route adopted for the line of traverse was the zigzag or the direct. In the former, the number of stations on the traverse is ordinarily the same as the number of triangles, and, whether or no, a common numerical notation may be adopted for both the traverse stations and the collateral triangles; thus the angular errors of every triangle enter the general expression in the form ±φx + cot Y.μ!y - cot Z.Vz, in which f=μ sin 1", and the upper sign of φ 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 μs and φ'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 p th triangle has no side on the line of the traverse but only an angle at the 7th station, the form is + φ i .x p + cot Y p -μ'ι-y p - cot Z p .μj.z p .
If the qth triangle has a side between the l th and the (l +1)th stations of the traverse, the form is cot X q [μ' l - √ l+1 )a½ + (≠ i + √, +1 cot Y q }y q - {φ l+χ - μ' l cot Z q '}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 + ez] 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 whieh 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=0 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 λ may be obtained for each of the eighty-three equa-Γ~ ?/* z~~~ j tions under the condition that ——+ — is made a minimum, u , L u v w J
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 λ. The resulting values of the errors of the angles in any, the p th, triangle, are a⅛ = Wp[fl⅛λ]; y p = ^>[^Λ] » ¾>~ Wp[Cpλ] (17).
(ii. ) One of the unknown quantities in every triangle, as x, may be eliminated from each of the eleven circuit and base-line equations by substituting its equivalent - (y + 2) for it, a similar substitution being made in the minimum. Then the equations take the form [(b - a)y + (c - a y )z]= E, while the minimum becomes
Γ(y + 2) 2 y 2 2°—j
Thus we have now to find only eleven values of λ 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 :—
l/p ~ „ 4- J4. μ ' ( u p ^*^ 7 ⅜)[(⅛ - a pl M ~ w p⅛ c p ~ α j>)∖l} I
un ^t^ an - Γ uln I
y...(i8 )∙
⅝= u p+υ p p+Wp * l ⅛+⅞>)[(⅞> - ⅜)M-⅜[(⅛> - ¾)λ]} J
The second method has invariably been adopted, originally because it was supposed that, the number of the factors λ being reduced from the total number of equations to that of the circuit and base-line equations, a great saving of labour would be effected. But subsequently it was ascertained that in this respect there is little to choose between the two methods; for, when x is not eliminated, and as many factors are introduced as there are equations, the factors for the triangular equations may be readily eliminated at the outset. Then the really severe calculations will be restricted to the solution of the equations containing the factors for the circuit and base-line equations, as in the second method.
In the preceding illustration it is assumed that the base-lines are errorless as compared with the triangulation. Strictly speaking, however, as base-lines are fallible quantities, presumably of different weight, their errors should be introduced as unknown quantities of which the most probable values are to be determined in a simultaneous investigation of the errors of all the facts of observation, whether linear or angular. When they are connected together by so few triangles that their ratios may be deduced as accurately, or nearly so, from the triangulation as from the measured lengths, this ought to be done; but, when the connecting triangles are so numerous that the direct ratios are of much greater weight than the trigonometrical, the errors of the base-lines may be neglected. In the reduction of the Indian triangulation it was decided, after examining the relative magnitudes of the probable errors of the linear and the angular measures and ratios, to assume the base-lines to be errorless (see § 19, p. 704 below).
The chains of triangles being largely composed of polygons or other networks, and not merely of single triangles, as has been assumed for simplicity in the illustration, the geometrical harmony to be maintained involved the introduction of a large number of “side,” “central,” and “toto-partial” equations of condition, as well as the triangular. Thus the problem for attack was the simultaneous solution of a number of equations of condition = that of all the geometrical conditions of every figure + four times the number of circuits formed by the chains of triangles + the number of baselines-1, the number of unknown quantities contained in the equations being that of the whole of the observed angles; the method of procedure, if rigorous, would be precisely similar to that already indicated for “harmonizing the angles of trigonometrical figures,” of which it is merely an expansion from single ligures to great groups.
The rigorous treatment would, however, have involved the simultaneous solution of about 4000 equations between 9230 unknown quantities, which was quite impracticable. The triangulation was therefore divided into sections for separate reduction, of which the most important were the live between the meridians of 67° and 92° (see fig. 1, p. 696), 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 of inferior quality. It happened that both the north-east and the south-west quadrilaterals contained several of the older chains; their reduction was therefore made to follow that of the collateral sections containing the modern chains.
But the reduction of each of these great sections was in itself a very formidable undertaking, necessitating some departure from a purely rigorous treatment. For the chains were largely composed of polygonal networks and not of single triangles only as assumed in the illustration, and therefore cognizance had to be taken of a number of “side” and other geometrical equations of condition, which entered irregularly aud caused great entanglement. Equations 17 and 18 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 whieh they can be well controlled at every stage and eventually brought to a successful issue. The other geometrical conditions of networks are expressed by equations which are not merely of a more complex form but have no regular order of sequence, for the networks present a variety of forms; thus their introduction would cause much entanglement and complication, and greatly increase the labour of the calculations and the chances of failure. Wherever, therefore, any compound figure occurred, only so much of it as was required to form a chain of single triangles was employed. The figure having previously been made consistent, it was immaterial what part was employed, but the selection was usually made so as to introduce the fewest triangles. The triangulation for final simultaneous reduction was thus made to consist of chains of single triangles only; but all the included angles were “fixed” simultaneously. The excluded angles of compound figures were subsequently harmonized with the fixed angles, which was readily done for each figure per se.
This departure from rigorous accuracy was not of material importance, for the angles of the compound figures excluded from the simultaneous reduction had already, in the course of the several independent figurai adjustments, been made to exert their full influence on the included angles. The figurai 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 may be taken as w . t / t-n , 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 λ 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 figurai 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 t w.ρ. t / t-n p being the modulus already indicated in section 12.
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 p th, triangle were finally expressed by the following equations, which are derived from (18) by substituting u for the reciprocal final mean weight as above determined :—
Vp =⅛ [(2 ⅛> - ¾ - ⅛) χ ] Ί
„ 1 (19)∙
⅞=⅞[(2⅛-⅝-⅛,)λ]J
The most laborious part of the calculations was the construction and solution of the normal equations between the factors λ. On this subject a few hints are desirable, because the labour involved is liable to be materially influenced by the order of sequence adopted in the construction. The normal equations invariably take the form of (4), the coefficients on the diagonal containing summations of squares of the coefficients in the primary equations, while those above and below contain summations of products of the primary quantities, such that the coefficient of the p th λ in the q th equation is the same as that of the q th λ in the p th equation. In practice, as any single angular error only enters a few of the primary equations of condition, many of the coefficients vanish, both in the primary and in the normal equations; and it is an object of great importance so to arrange the normal equations that most blanks shall occur above and fewest blanks between the significant values on each vertical line of coefficients; in other words, the significant values above and below the diagonal should lie as closely as possible to the diagonal, every value on which is always significant. This advantage is secured when the primary equations are arranged in groups in which each contains a number of angular errors in common and as many as possible of those entering the group on each side. Thus the arrangement must follow the natural succession of the chains of triangles rather than the characteristics of the primary equations; if, for example, all the side equations were grouped together, and all the latitude equations, and so on, great entanglement would arise in the solution of the normal equations, enormously increasing the labour and the chances of failure. The best arrangement was found to be to group the side and the three geodetic equations of each circuit together in the order of sequence of the meridional chains of triangles, and then to introduce the side equations connecting base-lines between the groups with which they had most in common.
The following table (II.) 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 :— [table]
The magnitudes of the 2481 angular errors determined simultaneously in the first two sections were very small, 2240 being under 0"·1, 205 between 0"·1 and 0"·1, 33 between 0"·2 and 0"·3, 2 between 0''·3 and 0"·4, and 1 between 0"·4 and 0"·5. In the third section, which contained a number of old chains, executed with instruments inferior to the 2 and 3 foot theodolites, they were larger: 780 were under 0"·1, 911 between 0"·1 and 1"·0, 27 between 1"·0 and 2''·0, and 1 between 2"·0 and 2"·1. Thus the corrections to the angles were generally very minute, rarely exceeding the theoretical probable errors of the angles, and therefore applicable without taking any liberties with the facts of observation.
18. Theoretical Error of any Function of Angles of a Geometrically corrected Triangulation.— The investigation of such theoretical errors was no easy matter. When first essayed it was generally assumed by mathematicians in England that any attempt to exhibit the theoretical error by a purely algebraical process soon led to results of intolerable complexity, so that it was desirable to introduce numbers as soon as possible for every symbol except the absolute terms of the geometrical or primary equations of condition. But on continuing the algebraical process certain relations were found to exist between the coefficients of the indeterminate factors in the normal equations of the minimum square method and the coefficients of the unknown quantities in the primary equations of condition, which enormously simplified the process and led to a general algebraical expression of no great complexity; it was also found that, the number of primary equations being », the labour of calculation by the formula was reduced to an nth of that involved by resorting at once to numbers.
Let F be any function whatever of the corrected angles (X 1 - x 1 ), X 2 - x 2 ,),. . . of a trigonometrical figure; let , _ clF , _ dF , -' 1 ~dX 2 ^' 2 dX 2 ' · ’ also let u 1 , u 2 , .... symbols hitherto employed to represent the relative reciprocal weights of the observed angles X 1 , X 2 , . . . , in future represent absolute measures of precision, the p.e. 2 of the observed angles; then the following formula expresses the p.e. of any function of the corrected angles rigorously :—
,, z , 2 ,, f p ( + [fa∙U]ttfa.u]A a + tfb.u]Ab+ ... + [fn.u]AA ∖ =fSf-l + lP>.u][[f a .u]B a + [fb.u]B i + ...+[fn.u]BA l (20) .
V + [f>ι. u] {iy⅛. u] X r a + [/ό. u] i v i + ...+[/». u] 2V n }' J The symbols a, b, . . . n have the same signification as in (3) to (6) of section 13. A, B,. . . N are coefficients which must be determined in the process of solving the normal equations as follows :—
∖ a = A a e a + Aι,eb + · · · +
λ⅛ = B a e a + Bbe∣, + . . . + Bnen I (21)
λ n = N a e a + Nbob+ ∙ · · + NHen J where the coefficient represented by any two letters in one order is identical with that represented by the same letters in the reverse order; thus A n = N a . Hence to find the p.e. of any angle, as (X 1 - x 1 ), in a single triangle we have f 1 = l, and A a = r - ,=— —; j1 [aa . w] κ 1 + u. 2 + u 3 all the other factors vanish, and p.e, i of ( W 1 - a∙ 1 )= u 1 - - + + ι - =p.e. n - of X 1 -p.e. 2 of a⅛.
To find the p.e. of the ratio R of either side to the base,—if R = sin (A r 1 - a¾)÷sin (A r 2 - a⅛), then fι = P cot X 1 sin l",f 2 -R cot X 2 sin l",∕ 3 =0, and p.e. 2 of R
7T> · >,„ f .<>ιr ..Tr (‰cotx∖-‰cotA 2 ) 2 ) _
=Λ 2 sm-l {u 1 c 0 t 2 X 1 +u 2 cc>t-X 2 j (22).
When the function of the corrected angles is the ratio of the terminal to the initial side of an equilateral triangle or a regular quadrilateral or polygon (either of two sides being taken if the figure has an odd number of exterior sides), then, assuming all the angles to be of equal weight, we have the following values of the ρ.ejs and the relative weights of the ratios :—
Figure. p.e. Weight. Figure. p.e. Weight.
Triangle ±·82-√ u sin 1" 1·49
Quadrilateral 1·00 ,, 1·00
Trigon 1·05 ,, 0·90
Tetragon ... 1·15 ,, 0·75
Pentagon ±1∙21√ u sin 1" 0·68
Hexagon 1·29 ,, 0·60
Heptagon 1∙41 ,, 0·50
Octagon 1 ’ 57 ,, 0·41
In ordinary ground seven single triangles will span about as much as two hexagons and the weights of the terminal sides would be as twenty-one by the former to thirty by the latter. In a flat country two quadrilaterals would not span more than one hexagon, giving terminal side weights as five to six; but in hills a quadrilateral may span as much as any polygon and give a more exact side of continuation. Thus in the Indian Survey polygons predominate in the plains and quadrilaterals in the hills.
The theoretical errors of the lengths and azimuths of the sides, and of the latitudes and longitudes of the stations, at the termini of the chains of triangles or at the circuit closings, might be calculated with the coefficients a, b, and c of x, y, and z in the circuit and base-line equations as the f s, and the known p.e' s of X, Y , and Z and the other data of the figural reductions. Such calculations are, however, much too laborious to be ordinarily undertaken. Thus the exactitude of a triangulation is very generally estimated merely on the evidence of the magnitudes of the differences between the trigonometrical and the measured lengths of the base-lines; for, though the combined influence of angular precision and geometrical configuration is what really governs the precision of the results, it is not readily ascertainable, and is therefore generally ignored. But, when questions as to the intrinsic value of a triangulation arise, the theory of errors should always be appealed to, and its intimations accepted rather than the evidence of base-line discrepancies, which if very small are certainly accidental, and if seemingly large may be no greater than what we should be prepared to expect. Good work has occasionally been redone unnecessarily, and inferior work upheld, because their merits were erroneously estimated. The following formulae will be found useful in acquiring a fairly approximate knowledge of the magnitude of the errors which theory would lead us to expect, not only in side, but in latitude, longitude, and azimuth also, at the close of any chain of triangles. They indicate rigorously the p.e' s at the terminal end of a chain of equilateral triangles of which all the angles have been measured and corrected and are of equal weight; the results may be made to serve for less symmetrical chains, including networks of varying weight, by the application of certain factors which can be estimated with fair precision in each instance.
Let c be the side length, ε the p.e. of the angles, n the number of triangles, and R the ratio (here=l) of the terminal to the initial side, then p.e. of R = e sin l"β Vf n ) υ.e. of azimuth -c∖Pfιι I ∕qq∖ ι
s in i" ·. Ç p.e. of either coordinate = ec —-—V2ît 3 + 3n 2 + 10?i .....(23).
When the form of the triangles deviates much from the equiangular, the p.e. must be multiplied by a factor increasing up to 1·4 as the angles diminish from 60° to 30°, and a mean value of c must be adopted. When the chain is double throughout, the p.e. must be diminished by a factor taking cognizance of the greater weight of compound figures than of single triangles. When the chain is composed of groups of angles measured with different instruments, a separate value of ε must be employed for each group, and the final result obtained from√[ p . e . 2 ]. The p.e. of R may be determined rigorously for any chain of single triangles, with angles of varying magnitudes and weights, by (22), with little labour of calculation.
19. Relations between Theoretical Errors of Base-lines and those of a Triangulation.— These relations have to be investigated in order to ascertain whether the base-lines may be assumed to be errorless in the general reduction of the triangulation; being fallible quantities, their errors must be included among the unknown quantities to be investigated simultaneously, if their respective p.e.' s differ sensibly, or if the p.e 's of their ratios are not materially smaller than those of the corresponding trigonometrical ratios. By (23) the p.e. of the ratio of any two sides of an equilateral triangle is ε sin 1" √2÷3; but the p.e. of the ratio of two base-lines of equal length and weight is η √2, where η is the p.e. of either base-line; thus weight of trigonometrical ratio: weight of base-line ratio: 3η 2 : ε 2 sin 2 1 ", or as 3:1 when e= ± 0"·3 and η = ± 1·5 millionth parts, which happens generally in the Indian triangulation. But the chains between base-lines were always composed of a large number of triangles, and the average weight of the base-line ratios was about eleven times greater by the direct linear measurements than by the triangulation, even when all the unascertainable constant or accidental errors —as from displacements of mark-stones—which might be latent in the latter were disregarded. Moreover, the baselines were practically all of the same precision; they were therefore treated as errorless, and the triangulation was made accordant with them.
If a base-line AD be divided at B and C into three equal sections connected together by equilateral triangles, and every angle has been measured with a p.e. = ε, the p.e. of any trigonometrical ratio may be put = κ.ε sin 1", κ being a coefficient which has two values for each ratio,— the greater value when the triangulation has been carried along one flank of the line, the smaller when along both flanks, as follows:—for ratio BC / AB , κ= 1·41 and 1; for CD / AB ,
1·33 and 1·23; for AD / AB , 2·94 and 1·99; for AD / BC , 2·16 and
1·46. The values for the last two ratios show that, when the length of a base-line is determined partly by measurement and partly by triangulation, the p.e. is smallest if the central section rather than an end section is measured.
If, with linear and angular p.e 's as in the Indian operations, a single section is measured once only, and the lengths of the other sections are derived from it by triangulation, the p.e. of the entire length will be greater than that of the whole line once measured; it will be less if the section is measured oftener than once and the mean taken.
20. Azimuth Observations in connexion with Principal Triangulation.— These were invariably determined by measuring the horizontal angle between a referring mark and a circumpolar star, shortly before and after elongation, and usually at both elongations in order to eliminate the error of the star’s place. Systematic changes of “face” and of the zero settings of the azimuthal circle were made as in the measurement of the principal angles (§ 9); but the repetitions on each zero were more numerous; the azimuthal levels were read and corrections applied to the star observations for dislevelment. As already mentioned (§ 17), the triangulation was not adjusted, in the course of the final simultaneous reduction, to the astronomically determined azimuths, because they are liable to be vitiated by local attractions; but the azimuths observed at about fifty stations around the primary azimuthal station, which was adopted as the origin of the geodetic calculations, were referred to that station, through the triangulation, for comparison with the primary azimuth. A table was prepared of the differences (observed at the origin - computed from a distance) between the primary and the geodetic azimuths; the differences were assumed to be mainly due to the local deflexions of the plumb-line and only partially to error in the triangulation, and each was multiplied by the factor 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 deflexion in the prime vertical × 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-computed) p w] / [w] .............. .(24) ; the value of x thus obtained was - 1"∙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 is a right angle at the elongation, the hour angle and the azimuth at that time are found from the equations cos P =tan PS cot PZ , cos Z =cos PS sin P
The interval, δ P , between the time of any observation and that of the elongation being known, the corresponding azimuthal angle, 8Z, between the two positions of the star at the times of observation and elongation is given rigorously by the following expression—tan 8Z
_ 2sin 2 ⅜δP cotftS ’ sin∕ ’ Z sin∕- ’ {1 + tan 2 PZ cosδP + scc, 2 ΡS cotP sinδ∕ ’} ' ’
which is expressed as follows for logarithmic computation— A m tan Z cos 2 PS l-η+l ’ where m = 2 sin 2 y cosec 1", n = 2 sinW sin 2 ^, and
∕ = cotPsin 8P · l j m f and n are tabulated.
21. Calculatioη of Height and Refraction.— Let A and P (fig. 4) be any two points the normals at which meet at C, cutting the sea-level at p and q ; take Dq = Ap, then BD is the difference of height; draw the tangents Aa and Bb at 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 sealevel, whence pq = c ; thus, putting Ap, the height of A above the sealevel, = H, and pC = r ,
^=< 1 + 7-2⅛) < 26 >∙
Putting D α and D b for the actual depressions at A and B, S for the angle at A, usually called the “subtended angle,” and h for BD— S=⅛D t -D a ) (27), and λ =^ b c⅛⅛ < 28 >∙
The angle at C being = D b ,+ D a , 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 — cosec 1'' in seconds. Putting D' a and D' b for the observed vertical angles, and φ a , φ b for the amounts by which they are affected by refraction, D a = D' a + φ a and D b = D' b + φ b ∙, φ a and φ b may differ in amount (see § 10), 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 (27) to find S , and the difference of height is calculated by (28); the third term within the brackets of (26) is usually omitted. The mean value of the refraction is deduced from the formula φ= 1 / 2 {C-(D' a +D' b )} (29)∙
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,” φ ÷ C, is computed for the district, and is employed when heights have to be determined from observations at a single station only. When either of the vertical angles is an elevation - E must be substituted for D in the above expressions.
II. Traversing, as a basis for Survey.—Rectangular Spherical Coordinates.
Traversing is a combination of linear and angular measures in equal proportions: the surveyor proceeds from point to point, measuring the lines between them and at each point the angle between the back and forward lines; he runs his lines as much as possible over level and open ground, avoiding obstacles by working round them. The system is well suited for laying down roads, boundary-lines, and circuitous features of the ground, and is very generally resorted to for filling in the interior details of surveys based on triangulation. It has been largely employed in certain districts of British India, which had to be surveyed in a manner to satisfy fiscal as well as topographical requirements; for, the village being the administrative unit of the district, the boundary of every village had to be laid down, and this necessitated the survey of an enormous number of circuits. Moreover, the traverse system was better adapted for the country than a network of triangulation, as the ground was generally very flat and covered with trees, villages, and other obstacles to distant vision, and was also devoid of hills and other commanding points of view. The principal triangulation had been carried across it, but by chains executed with great difficulty and expense, and therefore at wide intervals apart, with the intention that the intermediate spaces should be provided with points as a basis for the general topography in some other way. A system of traverses was obviously the best that could be adopted under the circumstances, as it not only gave all the village boundaries but was practically easier to execute than a network of minor triangulation.
Procedure of 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 observations; the angle at the origin between the meridian and the next station is measured, and then at each of the successive stations the angle between the immediately preceding and following stations; summing these together, the “inclinations” of the lines between the stations to the meridian of the origin are successively determined. The distances between the stations, multiplied by the cosines and sines of the inclinations, give the distance of each station from the one preceding it, resolved in the directions parallel and perpendicular respectively to the meridian of the origin; and the algebraical sums of these quantities give the corresponding rectangular coordinates of the successive stations relatively to the origin and its meridian. The area included in any circuit is expressed by the formula area = half algebraical sum of products (x 1 + x 2 ) (y 2 -y 1 ) (30), x 1 , y l being the coordinates of the first, and x 2 , y 2 those of the second station, of every line of the traverse in succession round the circuit.
Of geometrical tests there are two, both applicable at the close of a circuit: the first is angular, viz., the sum of all the interior angles of the described polygon should be equal to twice as many right angles as the figure has sides, less four; the second is linear, viz., the algebraical sum of the x coordinates and that of the y coordinates should each be = 0. The astronomical test is this: at any station of the traverse the azimuth of a referring mark may be determined by astronomical observations; the inclination of the line between the station and the referring mark to the meridian of the origin is given by the traverse; the two should differ by the convergency of the meridians of the station and the origin. In practice the angles of the traverse are usually adjusted to satisfy their special geometrical and astronomical tests in the first instance, and then the coordinates 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.
Convergency of Meridians.— The exact value of the convergence, when the distance and azimuth of the second astronomical station from the first are known, is that of B -(π + A ) of equation (11); but, as the first term is sufficient for a traverse, we have convergency =x tan λ cosec 1'' / v , substituting x, the coordinate of the second station perpendicular to the meridian of the origin, for c sin A. Adjustment of a System of Traverses to a Triangulation.— The coordinates of the principal stations of a trigonometrical survey are usually the spherical coordinates of latitude and longitude; those of a traverse survey are always rectangular, plane for a small area but spherical for a large one. It is often necessary, therefore, for purposes of comparison and check at stations common to surveys of both descriptions, to convert either rectangular coordinates into latitudes and longitudes, or vice versa, in order that the errors of traverses may be dispersed by proportion over the coordinates of the traverse stations, if desired, or adjusted in the final mapping. The latter is generally all that is necessary, more particularly when the traverses are referred to successive trigonometrical stations as origins, as the operations are being extended, in order to prevent any large accumulation of error. Similar conversions are also frequently necessary in map projections. The method of effecting them will now be indicated.
Transformation of Latitude and Longitude Coordinates into Rectangular Spherical Coordinates, and vice versa.— 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 perpendi cular to Aa; then AP, BP will be the rect angular spherical coordinates of B relatively to A. Put BP = x, A P = y, the arc Pb = η, and the arc Bb, the difference of longitude, — ω; also let λ a , λ b , and λ p be the latitudes of A, B, and the point P, ρ p the radius of curvature of the meridian, and ν p the normal terminating in the axis minor for the latitude λ p ; and let p 0 be the radius of curvature for the latitude ½(λ α + λ p ). Then, when the rectangular coordinates are given, we have, taking A as the origin, the latitude of which is known, h p = h α +~ cosec 1"; 17 = — tan λ∏ cosec 1"; Ί
„ P " 1(81).
λ⅛-λ α = -cosecl"-y; ω = - sec(λ⅞+ 377) cosec 1" I Pο v p '
And, when the latitude and longitude are given, we have
17 = (^) — sin 2 λ 6 sin 1" )
2∕=Po{ λ 4- λ α + , z} sinl " „ Γ (32 )'
χ=ωv p cos (λ⅛ + ⅜y) sin l" J Graphic Method of Determining the Coordinates of an Unvisited Point observed from Several Stations.— When a hill peak or other prominent object has been observed from a number of stations whose coordinates are already fixed, the converging rays may be projected graphically, and from an examination of their several intersections the most probable position of the object may be obtained almost as accurately as by calculations by the method of least squares, which are very laborious and out of place for the determination of a secondary point. The following is a description of the application of this method to points on a plane surface in the calculations of the Ordnance Survey. Let s 1 , s 2 , . . . be stations whose rectangular coordinates, x 1 , x 2 ,... perpendicular, and y 1 , y 2 , . . . parallel, to the meridian of the origin are given; let a 1 , a 2 , . . . 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 coordinates x p , y p , as determined by graphical projection on a district map or by rough calculation. Construct a diagram of the rays converging around p, by taking a point to represent p and drawing two lines through it at right angles to each other to indicate the directions of north, south, east, and west. Calculate accurately (y p - y 1 ) tan a p and compare with (x p - x 1 ); the difference will show how far the direction of the ray from s 1 falls to the east or west of p. Or calculate (x p — x 1 ) cot a 1 , and compare with (y p -y 1 ) 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 the point thus fixed draw the direction a 1 with a com mon protract or. All the other rays around p may be drawn in like manner; they will in tersect each other in a number of points, the centre of which may be adopted as the most prob able position of P. The co ordinates of P will then be readily obtained from those of p ± the distances on the meridian and perpendicular.
In the annexed diagram
(fig. 6) P is supposed to have been observed from five stations, giving as many intersecting rays, (1, 1), (2, 2), . . .; there are ten points of intersection, the mean position of which gives the true position of P, the assumed position being p. The advantages claimed for the method are that, the bearings being independent, an erroneous bearing may be redrawn without disturbing those that are correct; similarly new bearings may be introduced without disturbing previous work, and observations from a large number of stations may be readily utilized, whereas, when calculation is resorted to, observations in excess of the minimum number required are frequently rejected because of the labour of computing them.
III. 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 commenced at the sea-level, a series of heights is determined corresponding to what would be found by perpendicular measurements upwards from the surface of water communicating freely with the sea in underground channels; thus the line traced indicates a hypothetical prolongation of the surface of the sea inland, which is everywhere conformable to the earth’s curvature.
The trigonometrical determination of the relative heights of points at known distances apart, by the measurements of their mutual vertical angles,—as already described in section I.—is a method of levelling. But the method to which the term “levelling” is always applied is that of the direct determination of the differences of height from the readings of the lines at which graduated staves, held vertically over the points, are cut by the horizontal plane which passes through the eye of the observer. Each method has its own advantages. The former is less accurate, but best suited for the requirements of a general geographical survey, to obtain the heights of all the more prominent objects on the surface of the ground, whether accessible or not. The latter may be conducted with extreme precision, and is specially valuable for the determination of the relative levels, however minute, of easily accessible points, however numerous, which succeed each other at short intervals apart; thus it is very generally undertaken paid 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 (see footnote, page 705) must be applied to every staff reading, or the instrument must be set up at equal distances from the staves; the curvature correction, being the same for each staff, will then be eliminated from the difference of the readings, which will thus give the true difference of level of the points on which the staves are set up.
Levelling is an essentially simple operation; but, as it has to be repeated very frequently in executing a long line of levels—say seven times on an average in every mile—it must be conducted with every precaution against errors of various kinds, instrumental and personal, some accidental and tending to cancel each other, others systematic and cumulative. Instrumental errors arise when the visual axis of the telescope is not perpendicular to the axis of rotation, and when the focusing tube does not move truly parallel to the visual axis on a change of focus. The first error is eliminated, and the second avoided, by placing the instrument at equal distances from the staves; and, as this procedure has also the advantage of eliminating the corrections for both curvature and refraction, it should invariably be adopted. 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 biassed 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 0 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 or rising refraction—from morning to midday or from midday to evening—or when the instrument takes some time to settle down on its bearings after being set up for observation. They may be eliminated (i. ) by alternating the order of observation of the staves, taking the back staff first at one station and the forward first at the next; (ii. ) by working in a circuit, or returning over the same line back to the origin; (iii. ) by dividing a line into sections and reversing the direction of operation in alternate sections. Cumulative error, not eliminable by working in a circuit, may be caused when there is much northing or southing in the direction of the line, for then the sun’s light will often fall endwise on the bubble of the level, illuminating the outer edge of the rim at the nearer end and the inner edge at the further end, and so biassing the observer to take scale readings of edges which are not equidistant from the centre of the bubble; this introduces a tendency to raise the south or depress the north ends of lines of level in the northern hemisphere. On long lines, the employment of a second observer, working independently over the same ground as the first, station by station, is very desirable. The great lines are usually carried over the main roads of the country, a number of “bench marks” being fixed for future reference. In the Ordnance Survey of Great Britain lines have been carried across from coast to coast, in such a manner that the level of any common crossing point may be found by several independent lines. Of these points there are 166 in England, Scotland, and Wales; the discrepancies met with at them were adjusted simultaneously by the method of minimum squares.
Sea-Level.— The sea-level is the natural datum plane for levelling operations, more particularly in countries bordering on the ocean. The earliest surveys of coasts were made for the use of navigators, and, as it was considered very important that the charts should everywhere show the minimum depth of water which a vessel would meet with, low water of 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 inches above or below the mean of all at points on the open coast, and more in tidal rivers. 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 feet 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 sealevel in different localities.
The sea-level, however, may not coincide everywhere with the geometrical figure which most closely represents the earth’s surface, but may be raised or lowered, here and there, under the influence of local and abnormal attractions, presenting an equipotential surface—an ellipsoid or spheroid of revolution slightly deformed by bumps and hollows—which Bruns calls a “geoid.” Archdeacon Pratt has shown that, under the combined influence of the positive attraction of the Himalayan Mountains and the negative attraction of the Indian Ocean, the sea-level may be some 560 feet higher at Kurrachee 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 hollow's exist, they cannot be measured instrumentally; for the instrumental levels will be affected by the local attractions precisely as is the sea-level, and will thus invariably show level surfaces even should there be considerable deviations from the geometrical figure.
IV. Survey of Interior Detail.
(1) General Principles.— We have seen that the skeleton framework of a survey may be either a triangulation or a system of traverses; very generally it is a combination of both. The method of filling in the details is necessarily influenced to some extent by the nature of the framework, but it depends mainly on the magnitude of the scale and the requisite degree of minutiae. In all instances the principal triangles and circuit traverses have to be broken down into smaller ones, to furnish a sufficient number of fixed points and lines for the subsequent operations. The filling in may be performed wholly by linear measurements or wholly by direction intersections, but is most frequently effected by both linear and angular measures, the former taken with chains and tapes and offset poles, the latter with small theodolites, sextants, optical squares, or other reflecting instruments, magnetized needles, prismatic compasses, and plane tables. When the scale of a survey is large, the linear and angular measures are usually recorded on the spot in a 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.
(2) The Scale.— In every country the scale is generally expressed by the ratio of some fraction or multiple of the smallest to the largest national units of length, but sometimes by the fraction which indicates the ratio of the length of a line on the paper to that of the corresponding line on the ground. The latter form is obviously preferable, being international and independent of the various units of length adopted by different nations. See table of maps and scales under Map, vol. xv. p. 522. In the Ordnance Survey of Great Britain and Ireland both forms of expression are adopted, the smaller scales being 1 inch and 6 inches to a mile for provinces and counties, the larger 1 / 2500 for parishes and 1 / 500 for towns. In the Indian Survey the standard topographical scale is 1 inch to a mile, diminishing to ½ and ¼ inch for geographical reconnaissance, and rising by multiples of 2 to higher scales, of which the greatest, for other than city surveys, is 32 inches, for cadastral purposes. In both surveys the double unit of the foot and the Gunter’s link (= 66 / 100 th of a foot) is employed, the former invariably in the triangulation, the latter very 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.
(3) Ordnance Survey Methods.— 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, when bearings are taken with the prismatic compass. 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.
(oe) Detail Surveying by Triangles.— This 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 detail to be conveniently measured up to and fixed. Perpendicular offsets are limited to eighty aud twenty links for the respective scales of 6 inches to a mile and 1 / 2500
(b ) Detail Surveying by Traverses.— When a considerable area has to be thus treated 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. Commencing at a trigonometrical station, the theodolite is set to circle reading 0° 0' with the telescope pointing to the north, and at every “forward” station of the traverse the circle is set to the same reading when the telescope is pointed at the “back” station as was obtained at the back station when the telescope was pointing to the forward one. When the circuit is completed and the theodolite again put up at the origin and set on the last back station with the appropriate circle reading, the circle reading, with the telescope again pointed to the first forward station, will be the same as at first, if no error has been committed. This system establishes a convenient check on the accuracy of the operations and enables the angles to be readily protracted on a system of lines parallel to the meridian of the origin. Λs a further check the traverse is connected with all contiguous trigonometrical stations by measured angles and distances. Traverses are frequently carried between the points already fixed on the sides of the minor triangles; the initial side is then adopted, instead of the meridian, as the axis of coordinates for the plotting, the telescope being pointed with circle reading 0° 0' to either of the trigonometrical stations at the extremities of the side.
(c) Plotting and Examination.— 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 both surveyor, plotter, and draftsman—mounts the plot on his sketching block, and verifies the accuracy of the detail by intersections and productions and occasional direct measurements, and generally endeavours to cause the details under examination to prove the accuracy of each other rather than to obtain direct proof by remeasurement. He fixes conspicuous trees and delineates the woods, footpaths, rocks, precipices, steep slopes, embankments, &c., and supplies the requisite information regarding minor objects—whether pit, shaft, level, spring, well, conduit, weir, quarry, refuse heap, waste, orchard, stackyard, railway, canal, manufacturing and mineral works, viaducts, bridges, tramways, plantations, &c.—to enable a draftsman to make a perfect representation according to the scale of the map. In examining a coast-line he delineates the foreshore and sketches the strike and dip of the stratified rocks. In tidal rivers he ascertains and marks the highest points to which the ordinary tides flow. The examiner on the 25·344 inch scale ( = 1 / 2500 ) 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.
(4) Indian Survey Methods.— All filling in is invariably done by plane-tabling on a basis of points previously fixed; the methods differ simply in the extent to which linear measures are introduced to supplement the direction rays of the plane table. When the scale of the survey is small, direct measurements of distance are rarely made and the filling is usually done wholly by direction intersections, which fix all the principal points, and by eye-sketching; but as the scale is increased linear measures with chains and offset poles are introduced to the extent that may be desirable. A sheet of drawing paper is mounted on cloth over the face of the plane table; the points, previously fixed by triangulation or otherwise, are projected on it — the collateral meridians and parallels, or the rectangular coordinates, when these are more convenient for employment than the spherical, having first been drawn; the plane table is then ready for use. Operations are commenced 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.
Checking and Examination. — Various methods are followed. For large scale work in plains it is customary to run arbitrary lines across it and make an independent survey of the belt of ground to a distance of a few chains on either side for comparison with the original survey; the smaller scale hill topography is checked by examination from commanding points, and also by traverses run across the finished work on the table.
V. Representation of Ground.
The master lines of ground are the main ridges and water-partings of the hills, the watercourses, and the horizontal contour lines of the coasts; the subordinate lines are those which define the undulations and minor features falling between the low-lying plains and the crests of the hills. These lines must first be laid down on a horizontal projection to fix the dimensions of each feature of the ground, after which the slopes must be indicated with sufficient relief and character to present a true picture of the corrugations of surface. In ancient maps the hills are represented as seen against the sky in profile by a spectator standing on the ground below at some distance off. This system of “natural representation,” as it was called, was serviceable in enabling persons looking at the hills from the quarter from which they had been sketched to identify them readily, for which reason such views of distant inland hills are still commonly given on the margins of marine charts of coast-lines for the assistance of navigators. But, when all other objects except the hills are shown in a map by their horizontal projections, hills represented in perspective are false to their surroundings, and misleading to all who approach them from other directions than that of the adopted point of view, for the vertical projection of the profile is practically turned over and confused with the horizontal plane. Hence in course of time hills came to be drawn as if seen from a high bird’s-eye point of view, the position of which was shifted until at last the point of sight was supposed to be vertically over them; thus the evils of the perspective system were diminished, whilst something of natural representation was still preserved. About the end of the 18th century the perspective and the bird’s-eye systems gave way to the true method of indicating the forms of hills, viz., by their horizontal projections, like all the other details of the ground, and by adding the requisite shading to bring every feature into proper relief.
Hill-Shading.— There are two rival methods of hill-shading,—one by horizontal contours, the other by vertical hachures. A contour being the line of intersection of a hill by a horizontal plane, contour lines indicate the markings which would be made by the successive risings of a flood to different levels above the sea; vertical hachures indicate the directions which the particles of a volume of water, equally disseminated over the top of a hill, would naturally take in running down the sides and slopes. The most perfect representation of ground is obtained when the shade lines, whether horizontal or vertical, are sufficiently close and well graduated in tone and intensity to imitate good mezzotint shading in Indian ink. A good effect may be and is frequently produced by assuming light to fall on the hills obliquely from a specific direction, illuminating them on one side and throwing the reverse slopes into shadow. But this has the disadvantage of giving similar slopes different intensities of shade according to their position with reference to the assumed direction of the light; on the other hand, vertical lighting, which gives the same intensity to the same slope wherever situated, fails in relief and perspicacity. A commission of citizens appointed by the republican Government of France in 1803 to formulate rules on the subject of topography, condemned the representation of hills in demi-perspective as absurd, but approved the system of oblique side-light; it also condemned contours, except for engineering works, and recommended vertical hachures, under the idea that the slope lines of the fall of water represent a material effect of which the eye is witness every moment, and recalls the general cause, if not of the formation, at least of the figure and characteristics, of the mountains.
Scale of Shade.— For military purposes it is very desirable that maps should be so drawn as to enable the angles of inclination of all slopes to be readily ascertained, with a view to determining what portions of the ground are suited for the manoeuvres of each of the three arms,—infantry, cavalry, and artillery. Thus military topographers of different nationalities have proposed a variety of scales to regulate the thickness and distance apart of the shade lines, and generally the proportion of black to white, for different angles of slope, that the map may convey to the mind as accurate a knowledge of the slopes of the ground as of the horizontal outlines. All slopes, however, are not of equal practical importance, but only those which are of most common occurrence and most liable to be gone over by men and horses and wheeled vehicles, and their inclination rarely exceeds 25°; consequently it is of most importance to be able to distinguish variations of slope below that angle: it is occasionally desirable to know the sharper slopes up to 45° or 50°, but greater inclinations are comparatively of rare occurrence and unimportant. Now in a true scale of shade the intensity increases with the inclination from 0° to 90°; thus, putting black + white = 1, the proportion of black to white for any inclination i by a scale of cosines will be black = 1 — cos i, white = cos i. But that scale does not sufficiently accentuate the lower inclinations, which are the most important, and have therefore to be dealt with more emphatically; this has led to the introduction of a variety of conventional scales, each with the special characteristics which commended themselves to its author. Major Lehmann of the German army supposed light to be admitted in parallel vertical rays and gave the horizontal plane the fullest light, because the reflected coincides with the vertical ray; at an inclination of 45° the reflected ray is perfectly horizontal, and this slope was therefore least illumined. Disregarding all greater slopes, he placed 45° at the head of his scale and represented it by absolute black; the scale was divided into nine equal parts of 5° each, from 0° to 45°, up to which the illumination varies inversely as the angle of inclination. General van Gorkum of the Netherlands army improved on Lehmann’s system: he adopted certain groups of contours arranged according to the slope, making the vertical distances between the contours equal in each group but greater in the higher groups, and between the contours he drew vertical hachures the lengths of which showed by reference to a scale the angles of slope. His lowest group included all angles up to 25°, the vertical distance between the contours being so regulated with reference to the scale of the map as to permit the draftsman to represent the slopes without inconveniently long hachures. For higher angles he doubled and trebled the vertical interval of his contours and the thickness of his hachures. Thus the relative altitudes of any required points might be deduced with comparative facility by noting the thickness and counting the number of the vertical hachures between them. In this respect the system satisfies the requirements of a military map, but the effect is unpleasing and unsuggestive of hill forms. In 1828 a second French commission, having Laplace for its president, was appointed to report on topographical drawing. It reversed the decision of the first commission in favour of oblique side light, as being difficult to execute and inaccurate in giving different intensities to the same angles of slope facing differently; and, after trying various scales of shade, it determined to increase the intensity in proportion to the sines of double the angles of inclination diminished by 1 / 15 , which gives a more rapid increase of shade for the gentle than the steep slopes. In subsequent instructions of the “depot de la guerre” the proportion of black to white is fixed at one and a half times the angle of slope. In England various scales of shade have been proposed, by Colonel Scott and Captain Webber of the Royal Engineers, and by the Council of Military Education. Colonel Scott’s scale is interesting as having been derived from the average of measurements taken from the best examples of hill sketching in the Ordnance and other surveys, whereas all the others were deduced from a conventional application of geometrical principles. The following table (III.) gives the several scales :—
Table showing the Proportion of Black to White on any Unit of Area, in Horizontal Plan. [table]
Of late years the system of shading by lines has been abandoned for the English army, and a method of representing slopes by mezzotint shading over a few governing contour lines, laid down by actual survey, has been introduced instead. The effect aimed at is a transparent shade, dark in proportion to the steepness of the ground represented; its object is to give body and expression to the contours and to explain and develop minor features of the ground which may lie between them. This style of shading, being distinct from all line drawing, may be applied over the most crowded details without causing confusion, such as would be produced by hachure shading. The contours are indicated by continuous red lines of constant thickness, strong enough to be everywhere visible through the shading, which is effected by applying lead with a soft pencil over the parts where it is wanted, and then rubbing it in firmly with a piece of chamois leather folded into a small pad. No pencil marking is allowed; lightening is done with india-rubber; the shading is finally fixed with a wash of thin gum-water.
It is to be noted that the several scales of shade above given were devised for military maps to be drawn on a scale of not less than 4 inches to the mile and possibly much greater. The harshness and mannerism to which all line-shading by rule is liable are of less importance in maps of small areas represented on large scales than on maps of large areas on small scales. In the former the sacrifice of pictorial effect is more than compensated by the additional information regarding the slopes of the ground; in the latter any attempt to introduce so much information would tend to crowd the map objectionably, and confuse the vertical with the horizontal details. The smaller the scale of a map of hill country the more necessary it is to abandon mechanical conventionalism, and to aim at achieving an artistic representation which will convey an immediate and accurate impression of the general character of the ground.
In India the topography has been mainly executed on scales of or less than 1 inch = l mile and rarely exceeding 2 inches, and, as the range of altitude varies considerably in different parts of the country, from plains and undulations little above the sea-level to mountains rising to an altitude of 29,000 feet, scales of shade were long deemed wholly unsuitable for employment. The higher mountains had necessarily to be brought into prominence over the lower by giving them a darker shade than was due simply to their slopes, and similarly the elevated plateaus had to be more lightened and illuminated than the low-lying plains. But in course of time, as the number of hands employed in the operations increased more rapidly than the available supply of artistic draughtsmen, the introduction of a scale of shade became necessary, in order that the multitude of workers might be put more nearly on a par with the few. For men who have been accustomed to associate a certain depth of shade with a certain angle of slope will work together within narrower limits of error and divergence than if left entirely to their own unaided judgment and untutored proclivities. The field sketchers should therefore learn to work on a system which gives every hachure line a definite meaning, so that their sketches may be rightly interpreted and appropriately translated and rendered in the final representation of the ground, when it is the duty of the draughtsman to enhance the tone of the map as much as possible while maintaining its truthfulness.
Ordnance Survey System of Delineating Ground.— As a rule the features of the ground are sketched in the field on the 6-inch scale, and afterwards reduced and published on the 1-inch scale. The Highlands of Scotland were sketched partly on the 1-inch and partly on the 2-inch scale; in Ireland the 1-inch scale only was used; and this scale is now being adopted for hill sketching in England and Wales. In the parts where the 6-inch scale was used the ground was first contoured instrumentally; a plan of the contours and of all surveyed outlines was supplied to the sketcher, who proceeded to insert the hill features with the aid of a prismatic compass, protractor, plotting scale, and a “hill-sketcher’s scale,” graduated to show the horizontal intervals between the contours which correspond to various angles of inclination from 0° to 45°. He was required to delineate slopes up to 45° by horizontal hachures, and slopes beyond 45° by vertical hachures. The thickness and number of the strokes, the relation to light and shade, and the character of the touch were left to the skill and experience of the sketcher. The introduction of scales of shade adapted to various inclinations and altitudes was frequently mooted, with a view to securing greater uniformity; but no such scale was adopted, for it was found that, though at first different workmen produced different results, long practice and constant comparison, together with the aid derived from the instrumental contours, effected all desirable uniformity. Thus in good sketches it was found that the maximum breadth of stroke used in the representation of very steep mountain slopes was 1 / 45 inch, and the minimum used in low and nearly flat country, 1 / 500 inch, also that the average proportions of light to shade were I to 3 at the maximum and 25 to 1 at the minimum inclinations. In the field sketches the light is supposed to fall vertically, and all slopes of like altitude and inclination are similarly expressed. The 6-inch sketches are reduced to the 1-inch scale for publication by an artist working with Indian ink and the camel-hair brush on an impression in outline of the 1-inch map. He makes a careful study of the several sketches which he has to combine together, in order to determine which features should be retained and which omitted in the reduction, and he divides the ground into zones of different altitude to guide him in giving a strength of shade proportioned to the altitude rather than to the slope as in the field sketches; and in drawing he increases the contrasts between light and shade and introduces light from a corner of the map to give a stronger relief, and to attract the eye to the highest points and enable it to distinguish readily the higher from the lower ground. His general aim is to produce a more pictorial and less mechanical study of the ground than is supplied by the field sketches. Many exquisite maps have been thus produced and afterwards engraved; see sheets 32, 33, 38, 53, and 64 of Scotland,
38 and 48 of England, 75 of Wales, 93, 94,191, and 192 of Ireland. These sheets, however, though admirable specimens of engraving, fall short of the original drawings in tone and relief, for in them the hill-shading is necessarily shown by line-etching, and it does not produce such effective contrasts and gradations of light and shade as the original brush work. Delineation by Instrumental Contouring.— A very precise knowledge of the configuration of surface may be acquired by carrying true contour lines over the ground and projecting them on the map of the survey. But the contours do not give a true representation of the ground, for they seldom represent actual lines on the surface, as do the lines on the map which indicate roads, watercourses, walls, enclosures, &c.; they give, however, a conventional representation which is sufficient per se for the engineer and the expert, and they furnish guiding lines for all shading, whether by hachures or mezzotint, which may be subsequently executed to produce an artistic delineation of the features of the ground. In instrumental contouring we have first to decide on the vertical intervals to be maintained between the contours. They depend on the scale of the survey and the nature of the ground. In the Ordnance Survey they are made as small as from 5 to 10 feet, when special plans on large scales are being prepared for engineering requirements; but for the general maps they are 50 feet up to an elevation of 100 feet above the sea-level, and 100 beyond up to 900 feet, which elevation, being the practical limit of cultivation, is the highest generally marked, though in the northern counties of England and in parts of Scotland additional contours have been executed at the altitudes of 1000, 1250, 1500, 1750, 2000, 2500, 3000, &c., feet. The intervals having been determined on, instrumental levelling is commenced at either the top or the bottom of those slopes which best define the general lay of the ground, or at some previously established bench mark of which the height above the sea is known. Points are marked out on the slopes with pickets at the prescribed vertical intervals, and then the contour lines of the horizontal planes passing in succession through each of these points are traced with a levelling instrument and staff and surveyed by traverse, the two processes being performed either simultaneously or consecutively as may be most convenient.
The instruments generally used in the Ordnance Survey are a 5-inch theodolite—employed as a levelling instrument—and a contouring staff, 8 feet long, provided with a sliding vane which may be fixed at any required height; the staff is shifted about until the vane is brought into the horizontal plane of the theodolite, when the bottom of the staff will be on the contour line. A serviceable contouring instrument of very simple construction is the waterlevel, which consists of a pair of transparent phials partially filled with water; the phials are placed upside down at the ends of a hollow bar fixed on a rotatory vertical axis, and have their mouths connected with piping of any available material,— brass, tin, or gutta-percha. The water in both phials is in free communication, and the water surfaces indicate the horizontal plane naturally, without any mechanical contrivance. The instrument is well suited for short sights not requiring a telescope, and may be readily manipulated by persons ignorant of the use of instruments of a higher class. Eye-reflecting levels, clinometers, orometers, and other light instruments, which may be held in the hand and do not require a fixed support, are frequently employed for interpolating minor between major contours. In military sketching on large scales hypothenusal inclinations and lengths are sometimes measured; the bases and perpendiculars are deduced on the spot from a table of gradients; and then the contour lines, and the orthogonals also if required, are laid down.
VI. Geographical Reconnaissance.
When a traveller passes through an unknown or little known region the opportunity afforded him of acquiring ' some new geographical knowledge depends largely on the configuration and aspects of the ground, the condition of the atmosphere, the attitude of the inhabitants, and the time available. If hills are numerous and prominent and free from forest, and other conditions are favourable, a large area may be covered in a short time by reconnaissance from the stations of a chain of triangles carried along the line of route, fixing points in advance, some of which become stations of observation whence further points are fixed; and thus the continuity of the operations is maintained. But the ground may be flat and devoid of prominent points, the view circumscribed by forests and other obstacles, the atmosphere dense and unfavourable for distant vision, the inhabitants hostile, and the time short, and the traveller may be restricted to his line of route and unable to deviate from it; he must then endeavour to maintain a continuous traverse of the route, sketching in the ground in its immediate vicinity. Whenever breaks of continuity occur he must resort to astronomical observations to effect a connexion between the dissociated sections of his survey and to obtain an independent check on the general accuracy of the operations. He has therefore to be prepared to measure base-lines, to carry on a triangulation in some regions and a traverse in others, and to make any astronomical observations which may be wanted, and, if possible, to complete his mapping on the ground instead of postponing it to be done elsewhere. He should supply himself with some instruments suited for rough and rapid work and with others for better work when time and opportunity permit, and he should be careful to arrange beforehand the general character of the proposed operations and the scales and projections to be adopted for the mapping; he should also provide himself with blank sheets of paper duly graticulated to scale, for work in detail in the vicinity of the line of route and for general geography. For measures of base-lines and distances on the ground, chains, rolls of crinoline wire, long Assam canes, and perambulators may be employed, also omnimeters and subtense theodolites, to measure the angle subtended by a pole of known length, whence the distance may be deduced. For measures of angles and bearings, either theodolites, or sextants, or prismatic compasses may be used, according as more or less accuracy is required. For the general survey the plane table is a most valuable instrument: it enables bearings to be at once laid down on the paper without previous measurement, and much detail to be sketched in on the spot, instead of being plotted subsequently from a field-book; then the only independent angular measurements which need be taken are those of the principal triangles and of very distant points beyond the range of the table. Rough and rapid route surveys may be made by pacing the distances, taking the magnetic bearings, and combining with the results of astronomical observation. Many thousand miles of itinerary through regions in Central Asia have been surveyed by Asiatic employés of the Indian Government in this way; the northings and southings were controlled by latitude observations, and the factors thus obtained were applied to the eastings and westings, longitudes being impracticable. The theodolite should be employed to fix points on very distant ranges, for it will give good results, even with short bases and very acute angles, provided the objects actually observed are well identified in each instance. Observations should be taken from three stations, giving two triangles with a common side, which will at once show up any mistake, whether of identity, circle reading, or calculation. Whenever a break of continuity occurs in the triangulation or the traversing, astronomical observations must be resorted to. Much may be done by a judicious introduction of latitudes and azimuths, more particularly where there is considerable northing and southing, for then differences of longitude may be obtained from the azimuths and differences of latitude. A prominent peak, visible from great distances all round, may be made to serve as a connecting link between regions which cannot be continuously connected, by measuring its azimuth and distance from a base-line in each region; the addition of latitudes at the azimuth stations will much strengthen the work.
Collateral Astronomical Determinations. — Determinations of azimuth, latitude, time, and longitude may all be required for geographical reconnaissance,—the first two more particularly, as they can be obtained readily with much accuracy; the fourth, being much the most troublesome to get and the least reliable when got, is only re sorted to when it cannot be dispensed with.
The azimuth of an object may be determined without calculation by observing the angles between the object and a star at equal altitudes on opposite sides of the meridian; but it is generally found by observing the angle in one position of the star and applying thereto the azimuth of the star as obtained by calculation. In the spherical triangle PZS, in which P represents the pole, Z the zenith, and S the star, the angle PZS is the star’s azimuth, which can be computed when any three parts of the triangle are given. PS, the polar distance of the star, is given by the tables, and PZ, the co-latitude, must be previously determined; then, for the third part, we may have either (1) PSZ, a right angle, by observing a circumpolar star at its maximum elongation, or (2) the hour angle P for any star, by taking the time of the observation, or (3) the zenith distance ZS, by measurement simultaneously with the horizontal angle. Of these three methods the first is the most accurate, but it is not always convenient; the second requires, in addition, special observations for time; the third is generally the most convenient, for it may be performed between sunset and dark, when the stars are coming into view, but when there is still sufficient light to illuminate the wires of the telescope and the referring mark, and thus enable lamps to be dispensed with.
The latitude is most readily determined by measures of stars’ zenith distances on the meridian, duly corrected for refraction; then, the polar distance being known, the latitude is at once ascertained. The stars should be observed in pairs of nearly equal zenith distance, north and south, for this eliminates all constant instrumental errors, as of index, excentricity, and graduation, and also errors in the adopted refractions. When a single star is employed, circum-meridian observations of zenith distance may be taken and reduced to the meridian by calculation; tables for the pole star are given in the Nautical Almanac, which enable an observation, taken at any known time in the 24 hours, to be reduced to the pole. The time is usually best determined by measuring the zenith distances of stars situated not far from the prime vertical; then, the latitude and polar distance being known, the hour angle P of the spherical triangle is found by calculation. Time may also be determined by observing the transits of stars over the wires of the telescope of a theodolite set up in the meridian.
The longitude may be determined either absolutely, by purely astronomical methods, as by observations of the moon’s motion, or differentially, with the aid of telegraph lines and travelling chronometers. Absolute longitude is the geographer’s great difficulty; for much time must be devoted to the observations, and much more to their reduction, when undertaken with the object of fixing the relative positions of the stations of a survey. The observations are of various kinds,—(1) lunar distances, i.e ., the distance between the moon and the sun or one of the stars given for this purpose in the Nautical Almanac·, (2) lunar zenith distances, observed at points of the moon’s path where the conditions are favourable; (3) lunar transits over the meridian, observed with transits of the moon-culminating stars given in the Nautical Almanac; (4) lunar occultations of stars; (5) eclipses of the sun and moon; (6) eclipses of Jupiter’s satellites. The first method requires the employment of a sextant or other reflecting instrument; the second may be accomplished with either a reflecting instrument or a theodolite; the third with a theodolite; for the last three a good astronomical telescope is wanted. The first, when carried out strictly, requires three observers,—one to measure the lunar distance, while the others are measuring the zenith distances of the moon and the star; but, as the last two are not wanted with great accuracy, the several observations may be taken in succession by one person, and the observed zenith distances afterwards adjusted to the time of the lunar distance.
The effects of errors of observation in these methods are as follows. In (1) an error in time produces the same error in the longitude, and an error of one second of arc in the distance produces two seconds in time in the longitude. In (2) an error of one second in time produces at least thirty seconds of time error in the longitude, and one second of arc in the zenith distance at least two seconds of time in the longitude. In (3) to (6) an error of time produces the same error in the longitude. The first method is preferred by seamen and travellers, who are more expert in the use of the sextant than of the theodolite. The second method is preferred by those who are more familiar with the theodolite, and who are equipped with one of good telescopic power. It gives very good results when the observations are made at the most favourable time, which occurs when the resultant of the moon’s motion in right ascension and in declination lies in the direction of the observer’s zenith; this time may be readily found by graphical projection on a chart of the heavens.
Differential longitude may be determined chronometri-cally, on land as at sea, by carrying about several wellrated chronometers and comparing their times with the local times deduced from observations of the sun and stars; or electro-telegraphically, by interchanging signals between two stations connected by a telegraph wire, and ascertaining the local times at which the signals are transmitted from and received at each station.
Hypsometry.— Determinations of height form a very necessary part of geographical reconnaissance. Whenever triangulation is possible, vertical angles may be measured and the heights ascertained in regular succession. But in a traverse this is scarcely practicable; breaks of continuity in the verticals are liable to be of frequent occurrence, and then recourse must be had to observations of the pressure and temperature of the atmosphere, or of the temperature of the vapour of boiling water, from either of which fairly correct heights may be deduced differentially under normal atmospheric conditions in settled weather. The instruments employed for this purpose are mercurial and aneroid baro meters and boiling-point thermometers; descriptions of them, and the formulae employed in reducing the observa tions, are given under Barometer (vol. iii. pp. 381-387). Here it is only necessary to add that the date and hour of every barometric observation should be recorded, and the observations referred for reduction to those taken at the same time at one or more of the nearest standard meteorological observatories; otherwise corrections should be given to the barometer readings for the hour of the day and the month of the year, in order to reduce them as nearly as may be to the local mean altitude of the mercury. The index errors of aneroid barometers, being liable to variations, should be determined from time to time by observations at stations of known altitude, or by comparisons with boiling-point thermometers.
VII. Nautical Surveying.
Nautical surveying has for its object the determination of the configuration of land which is covered and concealed from view by water, more particularly along the foreshore of a coast-line, and wherever navigation is carried on in comparatively shallow waters and a knowledge of the depth of water is of great importance; it has likewise to lay down the positions of oceanic islands, shoals, and rocks, and generally to delineate whatever land exists immediately above or below the surface of the ocean. Its methods differ according as they are performed in or out of sight of land. When in the vicinity of land it is preceded by a survey of the coast-line and a belt of the country beyond, which must be of sufficient breadth to furnish suitable points of reference for the survey operations on the water, and may have to be extended inland to embrace those peaks of distant hill ranges which are prominent objects at sea for the guidance of mariners. This done, the nautical survey is carried on in boats, by taking soundings and determining the positions of the boats by observations to some of the points already fixed on land. The observations are necessarily made with sextants and magnetic compasses only. With the former the angles between conspicuous land-marks are measured, and, as the angle between any two points is half the magnitude of the angle between the same points at the centre of the circle which passes through them and through the boat, the measurement of two angles between three points enables two circles to be drawn on the chart, the intersection of which will generally indicate the position of the boat with sufficient accuracy. Occasionally, however, it happens that the positions of all three points on shore and the boat also lie actually, or very nearly, on the circumference of one and the same circle; then a bearing taken with the compass will fix the position of the boat on the circumference of the circle. Time is noted whenever soundings are taken, that due allowance may be made for the rise and fall of the tide. All the sounding stations are not fixed by observations to points on shore, as just indicated, but only a certain proportion, and between them straight lines of sounding are run, with intervals measured either by a patent log, or by time, or by counting the strokes of the oars; whenever possible the lines of sounding are carried parallel to each other. Sounding is the most important part of a nautical surveyor’s duty and that on which his character mainly depends. It is essentially the work of the sailor, for in carrying it out the accidents of wind and water—the direction and force of the wind, the rise and fall of the tide, and the velocity of currents—must be duly taken cognizance of and the work managed to suit wind and weather; on the other hand, the work on land may be done by landsmen. Nautical surveying, out of sight of land, rests on astronomical determinations of latitude and time, chronometric longitudes, and dead reckoning by log. When triangulation is resorted to, base-lines are measured sometimes with a patent log, sometimes by sound, by noting the interval in time between the flash and the report of a gun. The great length of modern ironclads presents a base-line which is occasionally very convenient: points are taken at each end of the ship, as far apart as possible, from which two observers can see each other; they are carefully marked, and the distances between them determined for future reference; then angles between moderately distant objects and observers standing at these points, taken simultaneously from each point, enable the required distances to be obtained. The magnetic variation is determined by observing the azimuth of the sun, when on or near the horizon, with a standard compass fixed amidship, care being taken beforehand to determine any deviation of the needle which may be due to the attraction of the surrounding ironwork, by observing the bearing of a distant mark as the ship is swung round and her head laid on different points of the compass. See also Navigation (Practical), vol. xvii. p. 264.
VIII. Mapping. Graticulation.— The sheets of paper on which the details of the survey of any large area of country are to be laid down must be furnished with a system of conventional lines, drawn with a view to assimilate the margins of contiguous sheets and to form a graticulation within which the details may be accurately inserted. The graticule is sometimes rectangular, sometimes spherical, sometimes a combination of both, as when points of which the latitude and longitude coordinates are given have to be plotted within rectangular marginal lines. Spherical graticules are constructed in various ways, usually in accordance with some specific method of projection; see Geography (Mathematical), vol. x. p. 197. The following convenient method is not referable to any demonstrated projection, but is generally employed on the Indian Survey. Suppose the intersection of two meridians by two parallels to form a small spherical quadrilateral, with sides of aliquot parts of a degree in latitude and in longitude; let m be the length of each of the meridional arcs, p, p the lengths of the arcs on the upper and lower parallels, and let q be a diameter, then q =√ m 2 + p . p' ; thus, m,p, and p ' being given, q is calculated. With these data, which are tabulated for different arcs and scales, the corner points of a number of quadrilaterals are laid off in succession on either side of an adopted meridian, and lines are drawm through the points to indicate the collateral meridians and the parallels of latitude. The latter are always curved, more or less sensibly; the former are also curved, though in a much less degree, being concave to their initial meridian, and the more so the farther they are from it. When the area is small and the scale large, the meridians are practically straight lines, and the several sheets of a map, each projected on its own meridian, will fit together closely when carried on in any direction. But, when the area is large—exceeding 8 or 10 square degrees —and the scale small, the sheets will not fit together continuously unless they are projected with reference to a single meridian for the whole map, to which the meridians on either side will be increasingly concave, or unless all the meridians are made straight lines, by slightly contracting each of the intermediate arcs of parallel to a length which is exactly proportioned to the lengths and relative distances of the upper and lower parallels of the map from it. There must be some distortion in either case: in the first, meridians which are actually straight lines are represented as being curved; in the second, straight meridians are obtained, but the distances between them are exact only on the upper and lower parallels, and are too small elsewhere, more particularly on the middle parallel, the length of which necessarily exceeds the mean length of the upper and lower parallels. But distortion is inevitable whenever a spherical surface is projected on a plane.
When a map is constructed in rectangular sheets, some station is adopted as the origin and its meridian as the principal axis, to which the corner points of the sheets are to be referred; the coordinates of these points are given such dimensions as are most suitable for the size and scale of the map, and are equivalent to the rectangular spherical coordinates of imaginary points on the curved surface of the earth, at corresponding distances from the origin and its meridian. These being given, the distances of the points from the origin in latitude and longitude may be computed, as already shown (p. 706); thus data become available for projecting the graticulation of meridians and parallels within the rectangular marginal lines of each sheet, or for introducing the divisions of latitude and longitude on the marginal lines if preferred. Conversely, when the latitudes and longitudes are given, the rectangular spherical coordinates are computed and the marginal lines projected around the graticulation. Filling in is then commenced: the principal stations are laid down by their coordinates and the topographical details pencilled around them by copying or tracing the field sheets of the survey; the names and the outlines are then inked in; the shading for delineating the features and general configuration of the ground is usually done last of all. The manner in which the details are inked in and rendered permanent depends on whether the map is to be reproduced by hand only—as when it is to be engraved or lithographed—or whether in its reproduction photography is to be employed and the action of light invoked, either in entire supersession of or in partial co-operation with the labour of the draftsman. In the former case the map is made as perfect a pictorial representation of the surface of the ground as possible, the hill features being represented artistically in mezzotint shading with a brush or in chalk drawing, and a variety of colours used to facilitate discrimination of differences of topographical detail. In the latter no colours are used which will not photograph well, nor flat shades of any colour, nor—as a rule—mezzotint shading, but only some substitute therefor in pen and ink. This last condition is essential for the commonly employed processes of photozincography and photo-lithography; but endeavours have recently been made, with some degree of success, to reproduce mapping in middle tones by the processes of photocollotype and photo-gravure.
Photography is much employed as an auxiliary in mapping; for when a map is to be published on various scales the hand-drawn details of the largest scale edition may be reduced by its means as accurately as by the familiar pantagraph, and of course very much more rapidly. Thus in the Ordnance Survey town maps on the scale of 1 / 500 are reduced to the scale of 1 / 2500 for incorporation into the parish maps; the latter are reduced for insertion in the 6-inch maps, and they in turn for the 1-inch map. By limiting the dimensions of each sheet for reduction to 3 feet by 2, and by a judicious use of stops to lenses, the reductions are made without any error in scale or any distortion that can be detected by the most rigid examination. But photography reduces every part of the original alike, the printing of words and names as well as the topographical details, and it reproduces all the minor and less import- ant as well as the more important features; hence a reduction is rarely suited for reproduction without intermediate modification, the printing being generally too small to be easily legible, and the mass of minor detail tending to confuse the principal sub-lines. The draftsman is therefore called in and the procedure so arranged as to obtain the best results with the least labour. Either he may construct a new map by tracing from a silver print of the photograph whatever topographical details are required for it and omitting the rest, or he may ink in such details at once with black ink on a blue print taken from a transfer of the photograph to stone or zinc, in both cases adding names and writing of appropriate sizes; either result may be reproduced by photography, as the unblackened details of the blue print will disappear in the process. This done, a transfer to stone or zinc may be made from the second photograph for the printing off. Prints from photographic reproductions to full scale exhibit all the blemishes of the hand drawing and somewhat exaggerate them, whereas prints from photo-reductions are freer from blemish, and often as clear and sharp as good hand lithographs. In employing a process of double photography, therefore, the first photo is usually made on a larger scale than that for publication; the lines of the printing and topographical detail are correspondingly exaggerated by the draftsman; and then the second photo is a reduction, which should be sharp, clear, and free from blemish.
IX. Map Printing.
Various processes are employed for the reproduction of maps in large numbers for general issue; some are purely manual, the map being redrawn by hand on copper, stone, or other substance presenting a suitable surface from which prints may be taken, or on paper specially prepared for transfer to such substance; others are carried out with the aid of photography, whereby an exact copy of the original can be obtained either directly upon, or for subsequent transfer to, the surface to be printed from. The former include the processes of copper-plate engraving and lithography, which are the oldest, and still in some respects the best of all, but slow and expensive; the latter include the processes of photo-lithography, photo-zincography, photo-gravure, and photo-collotype. Engraving on stone is much employed on the Continent for map work, being cheaper and quicker than engraving on copper. Electro-metallurgic processes are frequently employed in connexion with copper-plate engraving, either to protect and harden the surface of the plate with a facing of steel or to furnish duplicates to be printed from, instead of the plate itself being used; sometimes the wear of the plate is prevented by transferring a print from it to a lithographic stone or a zinc plate, from which the printing is done in its stead. By the anastatic process an old print of a map may be transferred to a zinc plate to be printed from.
Engraving may be performed on copper, wood, zinc, or stone; see vol. viii. p. 435. As done on copper plate for mapping, it is a combination of ploughing with the burin and etching with an acid, the former being used for the names and topographical outlines, the latter for the features of the ground. The system adopted in the Ordnance Survey of Great Britain—where it has been largely employed and carried to great perfection—is as follows. The 6-inch maps of the survey are engraved on copper plates measuring 36 by 24 inches within the marginal lines and weighing about 35 ft; the 1-inch maps are 18 by 12 for England and 24 by 18 for Scotland. The corners of the maps, the prescribed marginal subdivisions, and the trigonometrical points are first marked on the plate by a scoring machine, in which it is laid, and which is provided with a travelling carriage holding a steel pricker. The carriage is moved along a graduated scale and the pricker along another scale at right angles to the former, and all points of which the rectangular coordinates are known are laid off by vernier-read measurements from the two scales. The plate is then removed from the scoring machine, heated, and given a thin coating of white wax, to form a surface on which the topographical details are plotted before the graving is commenced. This surface is divided into a number of rectangles by fine lines joining marginal subdivisions, the distances between which are usually so regulated as to introduce sixteen of the survey sheets on the 1 / 2500 scale into one sheet on the 6-inch scale. The reductions to this scale are made by photography, and the subsequent reductions to the 1-inch scale either by the pantagraph or by photography. Tracings of the reductions in lamp-black, made to fit into the rectangles, are transferred to the wax ground by nibbing with a steel burnisher. The plate is then ready to be placed in the hands of the engravers, who complete first the outlines, then the printing and writing, and afterwards the ornament, each class of work being usually done by a different person. The figures of latitudes, longitudes, and altitudes, and various conventional symbols, are stamped with steel punches. Parks and sands are ruled with a dotting wheel, and buildings shaded in lines with a ruling machine. When a plate of the 1-inch map is being engraved, all the printing is completed, and line-engraving with the exception of the contour lines, and then an electrotype duplicate of the plate is taken. The contour-lines are engraved on the duplicate, and the hills arc etched on the original plate; thus two editions of the map are obtained, one with contours but without hills, the other with hills but without contours, the topographical details and writing being the same in both. In etching, the surface of the plate is thinly coated with an acid-resisting substance composed of asphalt, Burgundy pitch, and virgin wax, forming an etching ground, on which the outlines of the hill features are traced, and then marked through with a needle which removes the ground where it passes, exposing the surface of the copper. Aquafortis is applied to bite in the finer lines and then poured off; the parts which are bitten sufficiently are painted over with “stopping varnish and acid is again applied. The processes of stopping out and biting in are alternately repeated until all the required tints from the lightest to the darkest are produced. In printing from a copper plate, a much more powerful press has to be used than in printing from stone or zinc, as the ink lies in the furrows that have been ploughed or bitten into the plate and not on its surface; the process of printing is also much slower. In engraving on stone or zinc, the surface is coated with a preparation of gum and lamp-black, and on it the detail is traced with red chalk and afterwards cut in with very fine steel or diamond points so as just to lay bare the surface of the ground without penetrating to any depth, as in copperplate engraving. A little oil having been rubbed over the surface, the gummy composition is washed away and printing-ink applied; the printing is performed almost exactly in the same way as in ordinary lithography, except that the printing-ink is in the first instance spread over the stone or the zinc plate with a dabber instead of a roller.
Electrotyping is employed to conserve work engraved on copper, either by depositing a thin surface of steel over an engraved plate, which enables it to be printed from very much oftener without injury, or by producing a duplicate to be employed in its stead in the printing. In the latter case, a double process is gone through: first, a cast or matrix is produced in relief by the deposition of copper on the surface of the original plate, and then an intaglio of the matrix—which is therefore a duplicate of the original—is formed by depositing copper on the surface of the matrix. For details of these processes, see Electro-Metallurgy, vol. viii. p. 114. In the Ordnance Survey electrotyping was first employed to obtain duplicates on which to make the corrections and additions necessary to show the growth of railroads and towns since the time of the original survey. The alterations are effected more easily when obsolete details are scraped off the electrotype matrix than when they are scooped out of an intaglio; the original plate is also preserved intact. Electrotyping is further serviceable in producing the two editions of the general map, one with contour lines, the other with hill-shading, already mentioned, as well as editions for geological and other details. It is also serviceable in effecting a combination of portions of several plates: matrices of the different portions are riveted together to form a single plate; then an intaglio of this plate is taken, on which any details lost at the junction of the matrices are made good by hand. The dimensions of a full-sized plate are 38½ by 26½ inches; the weight of a matrix is 18 lb, and of the duplicate 38 lb.
There are two essentially distinct processes of lithography,—one in which the map is wholly drawn by hand on the stone, the other, a much quicker but coarser process, in which it is traced with greasy ink on specially prepared paper, which is then laid face downwards on the stone. When lithographs are to be produced by a single printing, all hill features, as well as topographical outlines and names, are drawn with a pen or fine camel-hair brush in ink of one colour. Double printing is necessary when the hills are drawn in chalk, two stones being required, one for the chalk work, the other for the pen-and-ink work; and in chromo-lithography a separate stone is required for the work in each colour. For full details, see Lithography, vol. xiv. p. 699.
Zincography has of late years largely taken the place of lithography for printing from hand-drawn transfers, though not for hand-drawing on the surface of the zinc, as on stone and copper. Zinc plates are less costly and bulky than lithographic stones, and are much more conveniently handled: thus a plate measuring about 43 by 28½ inches and 3 / 16 of an inch thick weighs 60 ft, is easily carried by one man, and costs 16 shillings; a lithographic stone of the same surface is 4½ inches thick, weighs 450 ft, requires four men to lift it, and costs about £7. Prints from transfers to a zinc plate are as satisfactory as prints from transfers to stone, and there is no liability of the plate being fractured in the press, which not unfrequently happens to the stone. The surface of the plate is prepared by scraping it evenly all over with a razor blade in parallel lines, until all irregularities are removed; the plate is then bent so as to present a slightly convex surface, which is ground with pumice-stone and water, and smoothed with a piece of steatite, and then given a grained surface with sand. It is flattened by being passed through a press, after which it is ready to receive the transfer. The subsequent procedure depends, as in lithography, on the circumstance that greasy substances do not mix with water and are repelled by gummy substances. The greasy ink lines of the transfer are readily absorbed by the surface of the plate; then a preparation of gum and decoction of gall nuts (to which a little phosphoric acid is added), applied to the entire surface of the plate, serves to etch the blank ground without affecting the lines of the transfer; but it prevents the ink from spreading, and also fills up the pores of the blank parts of the plate with a gummy substance, which repels a greasy ink. Printing ink, therefore, applied as usual with a roller to the entire surface of the plate adheres to the inked lines only and can be readily washed off the blank spaces, and then a print taken will show the inked lines only. The tracing for transfer is drawn on paper thinly coated with starch to prevent the graphic writing-ink from soaking into it; the ink is a mixture of Paris black, Castile soap, white wax, tallow or sweet oil, and shellac, which being greasy is readily absorbed by the zinc. The tracing is laid face downwards on the plate and passed several times under the pressure of the roller of the printing press. It is then wetted and peeled off, the ink remaining on the zinc. The surface of the plate is again washed with the etching liquid, which removes stains from the blank spaces and renders them more susceptible of being equably wetted with water, and also—after a few drops of turpentine have been added—removes the unabsorbed writing-ink and helps to fix the lines. The plate is then ready to be printed from. The printing-ink is composed of lamp-black—with a little Prussian blue added—and linseed oil varnish of a thickness depending on the temperature and the subject. Small corrections on the plate can be made by removing the surface with a strong solution of hydrate of potash, and then preparing a new surface to be drawn on by applying dilute nitric acid and afterwards washing off the nitrate of zinc.
Anastatic printing produces facsimiles of any inked print by transfer to a zinc plate, the inked lines on it being absorbed in a greater or less degree by the plate. The print is laid face downwards on blotting-paper, and brushed with a solution of nitric acid diluted with five times its bulk of water until thoroughly and evenly saturated; it is then placed face downwards on a zinc plate with a well-grained surface, and passed under the roller of a powerful copper-plate printing press. The grease of the ink, being set free by the acid, adheres to the surface of the plate; but, as the amount of ink absorbed is much less than in the case of an ordinary transfer, it is strengthened by working up with lithographic ink, oil, and gum water until the surface is sufficiently strong to bear etching with the usual preparation of gum, nut galls, and phosphoric acid. The plate is now ready to be printed from in the usual manner. If the original print is an old one, it must first have its ink softened by immersion in hot water containing half an ounce of caustic strontia for every pint of water, the time of immersion varying, with the condition of the print, from a few minutes to an hour. A print well worked up is often superior to the original.
Photography having already been described in detail (see vol. xviii. p. 821 ), its application to mapping and map-printing need only be noticed here. The action of light can be employed either by placing the map in contact with a sheet of sensitized paper and against a glass plate in a printing frame, when the light will pass through the map and produce a picture of it on the paper, or by using a camera furnished with an object-glass, through which rays of light from the map are transmitted so as to produce a picture on a sensitized glass plate, which can afterwards be printed from. The best known of the processes in which the camera is not used is the “cyanotype”; the paper is sensitized with a mixture containing nearly equal proportions of solutions of ammonio-citrate of iron and the ferrid-cyanide of potassium; the prints give white lines on a dark-blue ground, and are very inexpensive. There are other processes of printing with the salts of iron, uranium, &c., which give an exact transcript of the original drawing with dark lines on a white ground. But they are only suitable for maps drawn in pen and ink not larger than the glass plate of the printing frame; being therefore only serviceable in special cases when few copies are wanted, they are little employed and may be regarded more as curiosities than as ordinary methods of map-printing. Photography is generally effected with the aid of a camera, and employed to obtain a negative of a map on glass, from which prints may be taken either for use per se or for transfer to a flat surface of zinc, stone, or other suitable material to print from. The map is usually attached to a board suspended vertically in an adjust able frame, while the camera is placed on an adjustable stand set at right angles to the map frame on a tramway, along which it can be moved to any desired distance from the map. The camera is furnished with a ground-glass focusing screen, on which is pencilled a rectangle whose dimensions are proportional to those of a corresponding rectangle on the map, in the ratio of the scale of the required photograph to that of the map. The map and the focusing screen are brought into parallelism at such a distance that the image of the rectangle on the map exactly coincides with the rectangle on the focusing screen. A sensitized collodion plate is then substituted for the screen and a negative taken, which is afterwards “fixed” and “intensified” so as to produce the greatest transparence in the lines and an almost opaque density of the ground. Printing from a negative is usually performed by the action of light when only a few copies are wanted, and mechanically when many are wanted; the prints are taken directly from the negative in the one instance, and from a transfer of the negative to the surface of a stone or metal plate in the other. Of the processes of printing directly from the negative, silver printing, the oldest, is as yet unsurpassed for the delicacy of its results, but it is expensive and perishable; the prints are taken on paper coated with albumen containing an alkaline chloride, such as common salt, floated on a bath of nitrate of silver, and allowed to dry in the dark. After exposure to light in a printing frame, the prints are washed, toned with a solution of gold, and then fixed in a bath of hyposulphate of soda, which dissolves all the remaining unaltered chloride of silver. At the Ordnance Survey office platinum printing is now (1887) largely used instead of silver printing for all purposes where only a few copies of a map are required. It is more expensive, but the prints are absolutely permanent and are produced more quickly than silver prints. Their rich velvety black colour and freedom from glaze render them peculiarly suitable. The paper is sensitized with a preparation of platinum and ferric oxalate. After exposure to light, the image is developed almost instantaneously by laying the print on a hot solution of potassic oxalate; it is then washed in successive baths of dilute acid to remove the soluble iron salts, and after that in a few changes of water. Various processes of “collo-chromate” printing are also most usefully employed in mapprinting; they depend on the reaction of the salts of chromium— particularly the alkaline bichromates—on gelatin, gum, albumen, or other colloid substances, which, in proportion to the amount of the action of light upon them, become more or less insoluble in and unabsorbent of water, and acquire the property of taking up greasy ink and not attracting plumbago or other fine dry pigment in powder. When the subject is in line the print is taken on paper that is usually coated with a mixture of gelatin and bichromate of potash, coloured with Indian ink or any other suitable pigment; after a few minutes’ exposure in the copying frame the paper is plunged into tepid water, which dissolves the unaltered gelatin in the blank parts of the print—they have been protected from the light under the dark parts of the negative—leaving a clear image in pigment on a white ground. When the subject is in half-tone, the gelatin film has to be detached from the paper that it may be developed by being washed on the unexposed side, a temporary support being employed to preserve the image from injury during the washing; the most delicate shades in the half-tones are thus perfectly preserved.
In the processes noticed above it is necessary to repeat the operation by exposure to light for every print produced; the rate of printing will therefore be more or less dependent on the sensitiveness of the paper, the strength of the light, and the condition of the atmosphere. In the processes about to be described these disadvantages are obviated by transferring the photographic image to a surface of stone or metal, from which prints may be made mechanically in any numbers independently of light or weather. The photo-mechanical processes are broadly divisible into two classes,— one comprising photo-lithography, photo-zincography, and photo-typography, for the reproduction of subjects in line only; the other, photo-collotype and photo-gravure, for subjects in mezzotint or half-tone as well as line.
Photo-lithography and the analogous photo-zincography are the processes which have hitherto been most extensively employed for map printing. They are the simplest to carry out; they allow the photographs of several sections of a map which may be too large to be reproduced as a whole to be combined; and additions and corrections may be readily made by hand on the stone or zinc plate. The prints for transfer from the negatives are taken on paper coated with a mixture of gelatin and potassium bichromate, as in the pigment printing process, except that the greasy ink or colouring matter is not mixed with the gelatin, but applied evenly over the surface of the prints after exposure to the light. The inked print is immersed for a few minutes in tepid water to soften the gelatin still remaining soluble in the parts not acted upon by light, and is then laid on a sloping plate and washed with a soft sponge until all the unaltered soluble gelatin and the ink overlying it are removed. The lines on which the light has acted remain insoluble and retain the ink, forming a clear image of the subject in a greasy ink. When a map is photographed in several sections, as often happens, each section overlaps well all round to enable the transfers from the different negatives to be neatly joined together without showing lines of juuction; if the whole is too large to be printed on a single sheet of paper, it is cut up into sections for printing separately.
The object of photo-typography is to obtain by photographic agency a surface block which may be set up with type and printed in the same way as a woodcut. The image may be obtained on a zinc plate by transfer in the same way as for photo-zincography, or it may be printed directly from a reversed negative. In the latter case the zinc plate is usually prepared with a thin coating of bitumen, a substance which has the property of becoming insoluble under the influence of light, so that, when after exposure the plate is washed with turpentine or benzole, the image remains on the zinc, while the ground is washed away. In both cases the image is strengthened by careful inking and by the application of powdered resin, which the plate is heated sufficiently to melt. The image is then etched with nitric acid. The operations of inking, applying resin, and biting with acid are repeated several times, until the plate is bitten sufficiently deeply to give clear prints. In another process, which is perhaps preferable for fine work, a mould is obtained by electrotyping a relief in swollen gelatin, the surface of which has been metallized with plumbago or bronze powder. These processes are largely used for producing small maps to illustrate books and newspapers, but not for maps of ordinary size.
The three mechanical processes just noticed are only applicable to maps drawn in line, and to get good prints every line should be of the same blackness, though of different breadth. Attempts have been made to reproduce brush-shaded drawings, exhibiting continuous gradations of shade, by photo-lithography and photo-zincography, but with very partial success, and only by breaking up and destroying the continuity of gradation. The following processes are specially suited for reproducing maps in half-tone.
In photo-collotype, so-called from the printing surface being of gelatin, a plate with a perfectly smooth surface, usually of thick glass, either is coated with a sensitive mixture of gelatin and bichromate of potash, upon which the photographic image is produced by the action of light through a reversed negative, or is employed to support a gelatin film on which the image has been imprinted from an ordinary negative, and which is attached to the plate with suitable cement. The gelatin when properly moistened possesses the valuable property of receiving a greater or less amount of ink in different parts of the image in exact proportion to the intensity of the action of the light on each part; thus it is capable of reproducing the most delicate gradations of shade. The process is admirable for maps of small size, which only require a single plate, but is not suited for making a combination of sections to form a map of ordinary size; nor can additions or corrections be made on the gelatin film, which is, moreover, so tender that it does not readily permit of a large number of prints of uniform quality being taken, and is easily damaged.
The several methods of obtaining an incised image on a copper plate by means of photography are broadly divisible into the two groups of electrotyping and etching processes; one of each will be briefly noticed. (1) A positive pigment print, forming a relief in hardened gelatin, is developed on a silvered copper plate by the ordinary operations of the autotype or pigment printing process; it is then blackleaded and copper is deposited on it to form an electrotype intaglio, from which prints may be taken in the usual way, three to four weeks being required for the deposition of enough copper to produce a plate of sufficient thickness. (2) A negative pigment print is developed on a highly polished copper plate, upon which a very fine grain of powdered resin has been deposited and fixed by heat. The intaglio is obtained directly on the plate by biting in with a solution of perchloride of iron, which penetrates the gelatin film with comparative ease in those parts representing the shades and lines of a map, where there is little or no gelatin, and thus bites the copper to a considerable depth, while in the parts representing the blank spaces and ground of the map, where the gelatin is thicker, it penetrates with more and more difficulty as the thickness of the gelatin increases, and in the highest blanks should leave the copper untouched. The operation of biting takes only a few minutes, and the gravure is remarkable for its delicacy of gradation and richness of effect; there is, however, some difficulty in etching to the proper depth so that the plate may stand much printing without the loss of the finest tint. In both cases the copper plates have to be protected by a facing of steel before they can be printed from. The processes have not yet been used to any great extent for maps with half-tones, but they are very promising. For maps in line the first method gives excellent results, and is largely employed in the Austrian and Italian surveys.
X. Instruments.
The instruments employed in survey operations are broadly divisible into two classes, one for making the requisite linear and angular measurements on the ground, the other for plotting the data thus acquired on paper’ and for measuring from the map, when completed, lengths and areas which it may not be convenient to calculate from the numerical data. As a rule different instruments are employed for the mensuration on the ground and for the plotting on paper; but to this rule there is a notable exception in the plane table, by means of which all bearings may be drawn directly on paper with a sight rule, without previous measurement of any kind, and thus a plot of the ground may be constructed without employing any other instrument.
Field Instruments.— These are of two classes,—linear, for deter-Innung distances directly by actual measurement along the surface of the ground, and angular, for determining the bearings of, or the angles between, any objects. Some instruments are automatic, as the needle, which points to the magnetic north, the plumb-line and the spirit-level, which indicate the direction of gravity, and hypsometers of various kinds, for measuring altitudes; others are entirely controlled by the manipulator. Some require to be rigidly supported on the ground, as measuring bars and theodolites; others are adapted for flexible supports, as reflecting and magnetic instruments, which may be employed either on land or on the oscillating deck of a ship at sea. Some, as magnetic compasses, measure angles in the horizontal plane only; others, as theodolites, in two planes,—one horizontal, the other vertical; others, as reflecting instruments, in all planes; others, as levelling instruments, measure nothing, but simply indicate a plane of reference. And there are certain instruments by which angles are measured in the ordinary way, and direct distances are determined by micrometric measures of the small angles subtended at a distance by objects of known dimensions.
Linear instruments are of two classes,—one for exact measurement of base-lines the lengths of which are required to be known with great precision, the other for ordinary and rough measurements. Among the former may be included the Colby apparatus of compensation bars and microscopes, described in sect. I., § 2 (p. 696 above), Bessel’s apparatus, those of Struve and the United States Coast Survey, and Porro’s (adopted by the Spaniards and the French in Algiers), which have already been described in Earth (Figure of the), vol. vii. pp. 598, 600, and Geodesy, vol. x. pp. 163, 164. For less exact but still essentially accurate measures the instruments most commonly employed are the brass or steel chain of 100 links, the graduated metallic tape, and the offset pole.
For reconnaissance and rough measurement, perambulators, with wheels of known periphery and dials to indicate the number of revolutions, are largely used in India. Crinoline wire has been employed with advantage in Australia; it is so light that a length of 1000 feet or more may be easily carried, rolled on a drum, by one man, who pays it out as wanted; he is usually followed by another, who commences rolling it up at the opposite end when an entire length has been laid out on the ground. Air lines are sometimes measured by stretching the wire over the tops of trees in valleys obstructed with forest, also the breadths of rivers by resting the wire on logs anchored at suitable intervals to support it above water.
Angle-measuring instruments are of two classes, direct and reflecting. Both are provided with an aligner, usually a telescope, which is pivoted over the centre of the graduated circle or sector: in one the aligner is pointed in succession to any two objects the angle between which is being measured; in the other it is pointed to one object, while an image of the second is thrown on the first by double reflexion from a pair of mirrors. Reflecting instruments are largely employed in nautical surveys, as they can be held by the hand and do not require a rigid support; but they are very rarely used in land surveys. A description of them will be found under Sextant (vol. xxi. pp. 724-725). They give the angle in the plane in which they are held; and, whenever this plane is sensibly oblique to the horizon, the angle must be reduced by calculation to the plane of the horizon before it can be employed in the work of a land survey. The other instruments give the required horizontal angles, whatever the altitudes of the objects observed.
The circles of angle-measuring instruments are usually divided into 360 equal parts called degrees, and subdivided into spaces ranging downwards from thirty to five minutes of arc, according as the diameter of the circle is increased. Smaller arcs are measured by interpolation between the subdivisions, with the aid of a circle reader which moves with the aligner. All instruments except those of the simplest form are supplied with one or more circle readers and spirit-levels and a telescope; these important adjuncts, which are common to so many instruments, will therefore be first described, and afterwards the more important instruments which are employed in connexion with survey operations.
Circle readers are of two kinds,—the vernier and the microscope. Both are held over the circle at the extremity of a radial arm pivoted over its centre. The vernier moves in contact with the surface of the circle, while the microscope views it a short distance off; the former is usually applied to circles whose diameters do not exceed 12 inches, the latter to those of larger diameter. Both kinds of reader are applicable to linear scales as well as to graduated circles, the microscope being usually employed when most precision is desired.
The vernier is so called after its inventor, a Frenchman. Its principle is very simple. The space between any convenient number, n, of graduations on the circle is set off on the vernier and divided into (n +1) equal parts; then some one division of the vernier will always coincide with a graduation of the circle. On counting the divisions from the index onwards it is found that the coinciding division, say the with, of the vernier is opposite the with graduation of the circle, counting from the last one passed by the index. This indicates that the distance of the index from the last graduation is parts of the space between the graduations; n is invariably taken as an odd number, such that the unit of {n +1 ) may be some convenient aliquot part of the circle, as a minute for a circle divided into degree spaces.
The micrometer microscope presents the combination of object and eye glasses met with in ordinary microscopes, with the addition of a wire-carrying diaphragm, movable by a screw, for micrometric measurements in the plane of the focus of the object-glass. The tube is conical at the object end and cylindrical at the eye end; the box of the micrometer is mounted between these two parts at right angles with the visual axis. The tube is held at the extremity of the arm of an alidade, in a collar in which it may be moved closer to or away from the surface of the circle, or be turned round so as to place the micrometer tangentially to the circle. The distance between the micrometer and the object-glass is usually about four times that between the object-glass and the face of the circle, and thus a correspondingly magnified image of the spaces between the graduations is obtained in the plane of measurement. The objectglass is held in a small tube which can be screwed in or out of the principal tube, to enable the length of the image to be adjusted to an exact integral number of revolutions of the micrometer. The box of the micrometer and the wire diaphragm are rectangular, the latter sliding to the right or left within the former. Slow motion is communicated to the diaphragm by the micrometer screw, which passes into it through a collar in one side of the box, against which the shoulder of the screw is pressed by an internal spiral spring acting against the sides of the diaphragm and the box. The screw is furnished with a circular head divided into a number of equal parts—usually 60, each equivalent to 1" for circular arcs, and 100 for linear scales—and is rotated opposite an index arm fixed on the box; complete revolutions are marked by the teeth of a stationary comb, which is fixed above the wire of the diaphragm and viewed with it through the eye-piece.
The spirit-level consists of a glass tube not quite filled with alcohol, a small quantity of air being left, which rises as a bubble to the highest part of the tube. In small and coarse levels the diameter of the tube is largest in the middle and decreases uniformly towards the ends, which are closed by the blow-pipe; in long and delicate levels the tube is cylindrical, but with a longitudinal portion of the interior surface ground to the curvature of a circle of greater or less radius according as the level is designed to be more or less sensitive, and it is sometimes closed by circular glass stoppers cemented into the ends. When the tube is held horizontally, with the curved surface of the interior uppermost, the middle part is occupied by the air bubble. Lines are etched on the outer surface at equal distances from the central point, to enable the tube to be set with the bubble exactly in the middle, or a scale graduated throughout its entire length is provided, to enable any deviation from centricality to be measured and the corresponding dislevelment to be calculated and allowed for subsequently in the reduction of the observations. The glass tube is commonly fixed in a metal tube, with plaster of Paris for protection; but, as it is then liable, under changes of temperature, to torsion and strain, which may sensibly alter its curvature, it is preferable to place it in a metallic cradle and rest it on cork bearings, with due provision against sliding, the whole being covered with a glass cylinder if need be for further protection. The metallic cradle or tube is attached to any instrument on which the level is to be mounted by adjusting screws, for setting it correctly with reference to the axis of rotation with which it is associated. The value of a division of the scale, in seconds of arc, is usually called the “run,” and is determined by attaching the level with its scale to a (generally) vertical circle, and taking both the circle and the bubble end readings in different positions of the circle. As the length of the bubble is much affected by changes of temperature, and the curvature of the tube may not be identical at all points, values of the run are commonly obtained under widely differing temperatures.
The telescope consists of a tube, carrying an achromatic objectglass and an eye-piece which holds either a pair of lenses for viewing the inverted image transmitted by the object-glass or a combination of four lenses for inverting the image and causing all objects to be viewed naturally. The former is usually employed for observing celestial objects, the latter for observing terrestrial. The field of view being more or less extensive, a central point is established in the tube, usually by the intersection of a pair of fine wires or spider lines—one vertical, the other horizontal—in the plane of the image, and the telescope is directed by bringing this point on any specific object in the field. As the interval between the objectglass and the image varies with the distance of the object, a tube is provided to slide within the telescope tube and carry the objectglass at one end, while the telescope tube carries the diaphragm and eye-piece at the other end, or vice versa. The image and the wires are brought into the same plane by a focusing screw, which acts on the inner through the outer tube. The wires are attached to the surface of an adjustable annular diaphragm, which is held in position by two pairs of antagonizing screws—one pair horizontal, the other vertical—with shoulders working against the exterior of the tube in which the diaphragm is contained, so as to move it to the right or left and up or down, in order to bring the point of intersection of the wires into the visual axis of the telescope. In practice the first adjustment is to set the eye-piece to distinct vision of the wires; the object-glass is then set truly to focus, which is accomplished when no apparent parallax, or movement of the image relatively to the wires, is seen on shifting the position of the eye, for this would indicate that the image is either in front of or behind the plane of the wires. The line joining the point of intersection of the wires with the centre of the object-glass is called the “line of collimation,” and the diaphragm should be so fixed that this line may always be perpendicular to the axis on which the telescope revolves.
The surveying compass gives the magnetic bearing of any object, and is the simplest of all instruments for measuring horizontal angles. It consists of a magnetized needle, with an agate centre, poised on the point of an upright pivot in the centre of the bottom of a circular box and carrying a concentric circular card or silver ring, the circumference of which is graduated into 360°, and is sometimes further subdivided. The aligner is constituted by a pair of sight vanes attached to the box at opposite extremities of a diameter, one vane having a narrow slit for the eye to look through, the other with a wider opening bisected by a vertical wire to be set on the observed object. There is no circle reader, the prolongation of the wire on to the graduations being estimated by the eye; aud there is no level, for the circle poises itself horizontally on the supporting pivot.
The prismatic compass is similar to the surveying compass, with the addition of a prism in the eye vane through which the wire of the sight vane and the divisions of the circle are viewed apparently together; the division with which the wire coincides when the needle is at rest indicates the magnetic azimuth of any object bisected by the wire. The sight vane carries a mirror turning on a hinge, to enable objects to be seen by reflexion which may be too high to be seen on the wire; the eye vane is furnished with a pair of dark glasses to be employed when the sun is being observed.
Magnetic instruments are useful for rapid reconnaissance and rough survey, and for filling in the minor details of an exact survey, but they are not to be relied on to give bearings with errors less than ten to fifteen minutes. In plotting, however, bearings are preferable to angles, for, by drawing a number of meridional lines parallel to each other on the paper, each bearing may be plotted from an independent meridian without any accumulation of error, such as arises when a number of angles are plotted in succession with the protractor adjusted on short lines.
The plane table is in its usual form simply a rectangular board mounted horizontally on a stand, on which it may be turned round and set in any required position; it is furnished with a flat sight rule, which usually carries a pair of sight vanes and has a bevelled edge, parallel to the line of sight, to serve as a ruler, also with a magnetic needle. Occasionally the construction is more elaborate, and the board is surrounded by a marginal frame with graduations radiating from the centre as the degrees of a circle, so that it may be used as an instrument for measuring horizontal angles, while the sight rule is furnished with a telescope, which takes the place of the vanes and is mounted on an axle to measure vertical angles. The size is made as great as is consistent with the limits of portability in each instance, so that the sheet of paper to be drawn on may be as large as possible. The standard plane table of the Indian Survey measures 30 inches by 24, and is made of planks of well-seasoned wood 1 inch thick, with transverse edge bars below to prevent warping and buckling. It is set up on a stand, usually a braced tripod, to which it is clamped by a powerful hand screw passing through the head of the stand into a brass socket fixed centrally under the table; the screw when relaxed serves as a pivot, round which the table may be turned in azimuth and set in any required position. The table is then firmly clamped so as to maintain a constant position during all the subsequent laying off of bearings. The sight rule is 30 inches long, 2 wide, and one- third of an inch thick, of ebony, with a brass sight vane at each end, and a fiducial edge parallel to the line of sight; the vanes are about 5 inches high, which gives sufficient elevation and depression for general use. The magnetic needle is about 6 inches long and is held in a rectangular brass box an inch broad, placed on the table whilst it is being set and afterwards removed. Heights may be determined on the spot with the aid of a clinometer, formed of a bar carrying a spirit-level and a pair of sights, one of which has a scale of tangents graduated to radius = the interval between the sights. For the method of employing the table see § 4, p. 709 above.
The theodolite, the most important of all instruments for the purposes of a survey, is a combination of two graduated circles placed at right angles to each other, for the measurement of horizontal and vertical angles, a telescope, which turns on axes mounted centrically to the circles, and an alidade for each circle, which carries two or more readers of the arcs through which the telescope is moved. The whole is supported by a pedestal resting on foot-screws, which are also employed to level the instrument. The size varies from a minimum with circles 3 inches in diameter to a maximum with a 36-inch horizontal and an 18-inch vertical circle, the weight ranging from 4 lb to 1000 lb; the dimensions and magnifying powers of the telescope increase with the diameter of the horizontal circle. The telescope may be connected rigidly with the alidade and move with it while the circle remains stationary, or with the circle and move with it while the alidade remains stationary. The varieties of form as well as of size are numerous: in some the telescope may be completely turned round in altitude as well as azimuth, and pointed to any object celestial or terrestrial; in others the range of movement in altitude is restricted to about 25° above and 25° below the horizon, and a pair of sectors are substituted for the complete vertical circle; in some the telescope and vertical circle are placed between, in others outside of, the pillars which support their common axis; in some the pedestal is a simple tribrach resting on three foot-screws, in others it takes the objectionable form of a ball carrying the vertical axis and a socket holding the ball between two parallel plates, which are antagonized and set firm by two pairs of foot-screws, turning in sockets fixed to the lower plate, while their heads are pressed against the upper plate, to fix it and bring the instrument into level at the same time. There are numerous other specialities of form which have been introduced to meet specific requirements; but these cannot be noticed here.
The transit theodolite is an alt-azimuth instrument with the graduated circles of equal diameter, usually 6 to 8 inches. The telescope is mounted between a pair of conical arms which taper outwards and end in cylindrical pivots, constituting what is called the transit axis of the instrument. The pivots rest on Y’s or in semicircular collars, on the heads of a pair of pillars, which are made of sufficient height to enable the telescope to revolve between them and be pointed to stars in the zenith. These pillars stand on a circular plate, which serves as the alidade of the horizontal circle and is usually constructed to revolve round a vertical axis fixed in the centre of the plate of the horizontal circle; this axis passes downwards into a socket in the centre of a tribrach, which forms the pedestal of the instrument and rests on three mill-headed foot-screws by which the instrument is levelled. The vertical circle is mounted centrically on one of the cones of the transit axis, near the pivot end; its alidade, usually a rectangular plate carrying a pair of verniers, is fitted centrically over that axis, in contact with the circle but nearer the shoulder of the pivot, and, while the telescope and the circle revolve together, it is held stationary by an adjustable arm the end of which is pinched between a pair of antagonizing screws mounted on the nearest pillar. The alidade of the horizontal circle carries two or three equidistant verniers, because any error in centring an alidade over a circle is eliminated in the mean of the readings whenever two or more verniers, placed at equal distances apart round the circle, are read. A clamp, with a tangent screw for communicating slow motion, is attached to the nearest pillar, to act on the vertical circle and the telescope; another is attached to the plate of the horizontal circle, to act on the alidade of that circle and so also on the telescope for azimuthal motion; and a third to the pedestal, to act on the plate of the horizontal circle. The first two are employed in measuring the vertical and azimuthal angles, the third in setting the zero-diameter of the horizontal circle in any specific direction, with a view to the repetition of the measurements of the azimuthal angles at different parts of the circle. For levelling the instrument, two levels are fixed at right angles to each other on the plate of the alidade of the horizontal circle; a third is attached to the telescope, or, preferably, to the alidade of the vertical circle; a fourth is mounted on the transit axis when levelling for astronomical observations. A magnetic compass or needle is added, and also a plummet for centring the instrument over the station mark.
Theodolites are designed to measure horizontal angles with greater accuracy than vertical, because it is on the former that the most important work of a survey depends, and they are measurable with greatest accuracy; measures of vertical angles are liable to be much impaired by variations in the refractive condition of the lower strata of the atmosphere, more particularly on long lines, so that when heights have to be determined with much accuracy the theodolite must be discarded for a levelling instrument, to be set up repeatedly with staves at short distances. When truly adjusted the theodolite measures the horizontal angle between any two objects, however much they may differ in altitude, as the pole star and any terrestrial object; but, as adjustments are not always made with accuracy nor permanently maintained, it is desirable always to take the observations in pairs, with the face of the vertical circle alternately to the right and left of the observer, for this eliminates collimation error from the horizontal angles and index error in the setting of the spirit-level from the vertical angles.
When a horizontal angle is measured several times for greater accuracy, one of two methods of procedure is adopted. (1) The angle is measured once or oftener in the usual way, the horizontal circle remaining clamped and the telescope and alidade moving over it; then the position of the horizontal circle is shifted as often as may be desired, and after each shifting the angle is again measured as formerly; thus a separate numerical result is obtained for each operation. Or (2), the first object A having been observed and the telescope set on the second object B, the horizontal circle is undamped and turned round until the telescope is brought back on A, when it is again clamped; then the alidade is undamped and the telescope again moved over the horizontal circle to be set on B. The operation is repeated as often as may be desired. The vernier readings are only taken for the first telescope pointing to A and the last to B; their difference +360° for every complete revolution of the circle, divided by the number of repetitions, gives the angle. This method is objectionable when a round of several angles has to be measured, but it enables the value of a single angle—more particularly a small one, as between objects in the same field of the telescope—to be determined accurately with much greater rapidity than the first method.
An auxiliary telescope is sometimes fixed below the plate of the horizontal circle of a theodolite, to be pointed to a referring mark while the upper telescope is being moved about, and thus to serve as a check on the general stability of the instrument and on the permanence of the initial setting of the circle during the measurement of a round of angles. When a theodolite is set up on a lofty scaffolding which is liable to be swayed by the wind, or on a stand which cannot be readily isolated from the observer, horizontal angles may be measured accurately by employing a second observer to keep the auxiliary telescope truly pointed to a referring mark while the observing telescope is being pointed.
The subtense transit theodolite differs from the ordinary transit theodolite merely in having a pair of wire-carrying micrometers mounted in the telescope tube, in order that the small angle subtended by a distant object of known dimensions, or by two objects sufficiently near each other to be seen in the same field of the telescope, may be measured with greater facility and precision than on the graduated circles in the usual way. The micrometers are held in a rectangular box, one on the right hand, the other on the left, with the wires brought as closely as possible into the plane of the fixed wires in the ordinary diaphragm; the box can be turned on the telescope tube through an angle of rather more than 90°, to enable the micrometer wires to be set parallel to either the horizontal or the vertical wire of the diaphragm, or to be placed at any desired angle of inclination. The subtense object usually employed in survey work is a pole of known length; if held perpendicularly to the line of sight of the telescope, its direct distance may be determined from the angle measured by the micrometers with a sufficiently small percentage of error to make this method preferable to chaining over rough ground. The instrument has been advantageously employed in carrying traverses of considerable length over ground which was impracticable for direct linear measurements. The micrometers are also serviceable in astronomical observations for time and longitude, for they give additional wires on which to observe the passage of a star, at distances from the fixed wire which may be varied with the speed of the star; and for determining the longitude they permit numerous measures of the distance between the edge of the moon and a star to be taken, immediately before and after occultation.
Eckhold’s omnimeter is a theodolite furnished with a microscope of considerable magnitude facing a graduated linear scale; the tube of the. microscope is rigidly attached to the telescope tube, either at right angles or parallel to it, so that the two always move together. The scale is fixed either parallel or perpendicular to the alidade plate of the horizontal circle; thus, when the telescope is moved through vertical arcs within the range of the scale, the tangents of the arcs are measured by the microscope on the scale. The latest and best form of the instrument is shown in fig. 8, which represents a transit theodolite converted into an omnimeter by the application of a microscope AB to the telescope at right angles to it, and of a scale C to the plate of the alidade of the horizontal circle in a plane parallel to that of the vertical circle. The microscope is furnished with a diagonal eyepiece, through which the observer looks down on the scale. The scale is divided into 100 equal parts, and is movable in its bedplate through the length of one of these divisions by one rotation of a micrometer screw, with a large head, D , the circumference of which is divided into 100 equal parts, each divisible into fifths by a vernier. The microscope has a fixed wire in a diaphragm at its eye end, A, and, when the telescope is set on an object anil the wire is seen between a pair of divisions on the scale, the scale is moved by the micrometer screw until the nearest division is brought under the wire; the scale reading corresponding to the horizontal position of the telescope being known, the difference between it and the reading when the telescope is pointing above or below the horizontal plane is the tangent of the arc of elevation or depression, to radius = the perpendicular from the axis of rotation of the telescope to the scale. Thus both the distance and the height of any point over which a staff of known length has been set up vertically may be readily determined with fair accuracy. Let O (fig.
9) be the position of the transit axis of the telescope, OA the direction of the telescope when horizontal, and Oa the corresponding direction of the microscope at right angles to the scale amn ; let be a distant point over which the staff MN has been set up vertically, and let m and n be the graduations under the microscope when the telescope is pointing to the bottom and top of the staff; then, since MN and Oa are known, the horizontal distance OA and the height AM are determined from the proportions OA: oa AM: am }:: MN: mn
It is essential that the focusing tube of the microscope should always move parallel to the visual axis when different divisions of the scale are being brought into focus, otherwise errors materially exceeding the quantities appreciable by the micrometer may be caused. The linear results thus obtained are satisfactory when the subtense staff is set up at a moderate distance; the instrument has often been used with advantage in localities where measuring chains could not be conveniently employed. As an angular instrument it is identical with the ordinary transit theodolite, as will be seen from the figure, which may be referred to as illustrating the description of that instrument; the foot-screws are represented as resting on a plate such as is usually fixed on the head of a folding tripod stand, their lower extremities, as well as the grooves in which they are placed on the plate, being concealed from view by a capping upper plate, which is clamped over their shoulders to prevent the instrument from falling off the stand.
In any theodolite with a telescope of the ordinary form the height of the pillars must necessarily be somewhat greater than half the length of the telescope if stars in the zenith are to be observed or if the telescope is to be completely rotated on its transit axis; the higher the pillars the higher the centre of gravity, the less perfect the stability of the instrument when set up for observation, and the greater its weight and cumbersomeness for transport. In Germany and Russia theodolites and transit instruments are sometimes em ployed in which the eye end of the telescope tube is removed—a counterpoise to the object end being substituted in its place; and a prism is inserted at the intersection of the visual axis with the transit axis, so that the rays of light from the object-glass may be reflected through one of the tubes of the transit axis to an eye-piece in the pivot of this tube. In this case the pillars need only be high enough for the counterpoise to pass freely over the plate of the horizontal circle; but the observer has always to place himself at right angles to the direction of the object he is observing.
The levelling instrument consists of a telescope which carries a long spirit-level parallel to itself and is mounted on a horizontal plate, which is fixed rigidly either on the head of a vertical axis revolving within a socket in the centre of the pedestal or on that of a hollow cone revolving round a vertical axis which projects upwards from the pedestal. There are various forms of the instrument; in the Y-level the telescope rests on a pair of Y’s, in which it can be both rotated and turned end for end; in the dumpy level the telescope is rigidly attached to its supports, and its tube is made shorter and of greater diameter, to carry an object-glass of shorter focal length and larger aperture. A magnetic compass is attached to the instrument to enable the bearings of the levelling staves to be taken whenever desired. Levelling staves are of a variety of patterns and are graduated in various ways, best on both faces and dissimilarly, for a check on accidental errors of reading, as indicated in sect. III.
Reflecting levels are portable instruments which may be held by the hand for rough and rapid survey work. They are of two forms: in one an image of the eye of the observer, in the other an image of the bubble of a spirit-level, is seen by reflexion on a level with the observed object. The first consists of a square of common looking-glass, which is set in a frame suspended from a ring on the line of prolongation of one of the diagonals in such a manner as to swing freely but not turn round on its axis of suspension; the frame is weighted by a metal plate behind, to which it is so adjusted that, when suspended, the plane of the surface of the mirror will be vertical. A small portion of the glass at one end of the horizontal diagonal is either cut away or unsilvered. When the image of the observer’s eye is seen on the diagonal, all objects bisected by the diagonal, whether viewed through the opening in the mirror or by reflexion, are on the level of the eye. The second consists of a tube open at the object end and closed at the eye end by a disk which is perforated with a sight hole; a mirror filling up half the section is fixed in the tube, facing the eye end at an angle of 45° with the axis; and an all-round transparent spirit-level is mounted over an opening above the mirror, and its bubble is seen by reflexion in the axis of the tube. Abney’s level is of the latter construction, but with the spirit-level attached to the alidade of a graduated arc fixed to one side of the (rectangular) tube; thus vertical angles as well as levels may be determined with it.
The optical square is a reflecting instrument indicating a right angle, and is of great use in laying off perpendiculars for the measurement of offsets from a line of survey. It consists of two glass plates, one wholly the other partially silvered, which are fixed permanently in a shallow circular box at an angle of 45°, so that any two objects seen together through a sight hole in the box-one directly through the transparent portion, the other by reflexion in the mirror of the partially silvered glass plate — subtend an angle of 90° at the point where the observer is standing.
Blotting and Plot-measuring Instruments.— These comprise linear scales, common compasses, and angular protractors for laying off distances and angles measured on the ground, proportional compasses and pantagraphs for reproducing a finished plot on some other scale, and opisometers and planimeter for measuring plotted lines and areas.
Scales are divided, cither decimally or fractionally, into equal parts, each of which is a portion of a fixed unit of length, as a foot or an inch; some are subdivided more or less minutely throughout their entire length between a pair of parallel lines; others are subdivided at their extremities only. Diagonal scales are formed by eleven equidistant parallel lines, the outer ones of which are divided primarily and subdivided into tenths at their extremities. The primary divisions are joined by cross lines perpendicular to the eleven parallel lines; the end subdivisions are joined diagonally, the first on the lower line with the second on the upper, and so on, each diagonal cutting every horizontal line in a point a tenth of a subdivision beyond the cutting point on the parallel line below, as measured from any one of the perpendicular lines; and each of these tenths is further divisible into tenths by measuring from the perpendicular at intervals of tenths between the parallel lines; thus great precision of measurement is obtained.
The Marquois scale and triangle consist of a scale divided throughout into equal parts more or less minutely and a right-angled triangle of which the hypothenuse is three times the shortest ∣ side. An arrow is drawn perpendicular to the hypothenuse to 1 serve as a pointer to the divisions of the scale. The third side has a bevelled edge for ruling. When the triangle is placed with its hypothenuse against the scale and is moved along it, all lines drawn along the bevelled edge are parallel to each other, their distances apart being one-third of the distances travelled by the arrow along the scale.
Compasses usually take the form of a pair of legs movable about a joint, so that their extremities, which are of steel, finely pointed, may be set at any required distance apart; the legs may be knee-jointed, and one is usually adapted to hold either a pencil, a ruling pen, or a steel pointer, as may be desired. A beam compass is employed when long lengths are laid off; it consists of a light tubular metal bar, or a rectangular deal rod, fitted with a pair of boxes, which slide along it and carry either pen, pencil, or pointer, and may be set and clamped at any desired distance apart.
Proportional compasses consist of two parts so exactly similar that when held in contact throughout they appear as one; each is pointed at both ends, flat and grooved through one-half its length, and tapering to a point in the other half. The two are coupled together by a pair of similar sliders, one for each groove, turning on a common axle which carries a disk at one end and a clamping screw at the other; by shifting the position of the sliders in the grooves the distances between the points at the opposite ends can be brought into any desired proportion. The settings for different proportions are effected by bringing a line on the slider opposite the lines of a fractional scale engraved on one side of the groove.
Protractors are of two forms circular (or semicircular) and rectangular; the circumferences of the former are divided into 360° or 180°; the latter are divided on three sides of their periphery by lines drawn from the centre of the fourth side to the degree points on the circumference of a semicircle of which that side is the diameter. The protractor being set with its centre on a given point and its zero line on a given line passing through the point, any angle with this line at the point can be readily laid off. Protractors for plotting traverses are commonly annular, that they may be centred over the station of origin with the zero diameter on the initial meridian; their bearings at any other station may be laid off without moving the protractor by drawing lines parallel to the same bearings at the origin. Rectangular protractors sometimes have parallel lines engraved on their faces at equal distances, for setting over paper ruled with parallel lines at unequal distances, and their backs engraved with scales of rhumbs, sines, secants, and tangents and common scales of equal parts.
The station pointer enables the position of any station at which angles between three fixed points have been measured to be plotted on paper. It consists of three arms: the centre arm carries a graduated circle fixed over an axis at one end; the other two are movable round this axis, and each cames a vernier for reading the circle. Each arm has a straight edge bevelled as a ruler, and the lines on the prolongations of these edges meet in the centre of the axis, where there is a small opening through which a point may be pricked on the paper. The arms having been set to the observed angles, the instrument is moved about until each edge is over one of the fixed points on the paper, when its centre will be exactly over the position of the station if none of the angles are very acute. The instrument is much used in nautical surveying, for laying down the position of a vessel at sea by angles measured to fixed objects on shore.
The triangular compass is serviceable in reproducing plans to full scale; it is formed by jointing a third leg to the centre pin of the joint of an ordinary pair of compasses, so as to be movable in any direction.
The pantagraph is employed in reproducing a map on a different —generally a smaller—scale. It consists of two long arms, AB and AC, jointed together at A, and two short arms, FD and FE, jointed together at F and with the long arms at D and E ; FD is made exactly equal to AE and FE to AD, so that ADFE is a true parallelogram whatever the angle at A. The instrument is supported parallel to the paper on ivory castors, on which it moves freely. A tube is usually fixed vertically at c, near the extremity of the long arm AC, and similar tubes are mounted on plates which slide along the short arms BD and FD ; they are intended to hold either the axle pin on a weighted fulcrum round which the instrument turns, or a steel pointer, or a pencil, interchangeably. When the centres of the tubes are exactly in a straight line, as on the dotted line bfc, the small triangle bfD will always be similar to the large triangle bcA; and then, if the fulcrum is placed under b, the pencil at f , and the pointer at c, when the instrument is moved round the fulcrum as a pivot, the pencil and the pointer will move parallel to each other through distances which will be respectively in the proportion of bf to be ; thus the pencil at f draws a reduced copy of the map under the pointer at c ; if the pencil and the pointer were interchanged an enlarged copy would be drawn; if the fulcrum and pencil were interchanged, and the sliders set for f to bisect bc, the map would be copied exactly. Lines are engraved on the arms BD and FD, to indicate the positions to which the sliders must be set for the ratios ½, 1 / 3 ,. . ., which are commonly required.
The square pantagraph of Adrian Gavard consists of two graduated arms which are pivoted on a plain bar and connected by a graduated bar sliding between them throughout their entire length, to be set at any required distance from the plain bar; a sliding plate carrying a vertical tube, to hold either the axle of the fulcrum, the pencil, or the pointer, is mounted on one of the arms and on a prolongation of the plain bar beyond the other arm, and also on the graduated connecting bar; and an additional arm is provided by means of which reductions below or enlargements above the scales given on the instrument can be readily effected.
The eidograph is designed to supersede the pantagraph, which is somewhat unsteady, having several supports and joints. It is composed of three graduated bars, one of which is held over a fulcrum and carries the others, which are lighter, one at each extremity. The three bars are movable from end to end in box-sockets, each having an index and a vernier in contact with the graduated scale. The box-socket of the principal bar turns round the vertical axle of the fulcrum; that of each side bar is attached to a vertical axle, which also carries a grooved wheel of large diameter and turns in a collar at either end of the principal bar. The two wheels are of exactly the same diameter and are connected by a steel band fitting tightly into the grooves, so that they always turn together through identical arcs; thus the side bars over which they are respectively mounted, when once set parallel, turn with them and always remain parallel. A pointer is held at the end of one of the side bars and a pencil at the diagonally opposite end of the other. The bars may be readily set by their graduated scales to positions in which the distances of the pencil and the pointer from the fulcrum will always be in the ratio of the given and the required map scales.
The opisometer is intended to measure the lengths of roads, rivers, and other lines on a map. It consists simply of a milled wheel mounted in a forked handle on a steel screw with a very fine thread. The wheel, being turned up to one end of the screw, is put down on the map with the handle held vertically over the point at which the measurement is to commence, and is run over the road or line until the point is reached at which the measurement is to stop; it is then lifted off the paper, placed on the scale of the map, and run backwards to the initial end of the screw, over a length of the scale which corresponds to the length run over on the map.
The polar planimeter was invented by Professor Amsler of Schaffhausen for the measurement of areas on maps and plans. It consists essentially of two arms jointed together and a roller, carried at right angles to one of the arms and moving in touch with the paper, which by its revolutions records the area of a figure whose perimeter is traced by a point on that arm, while the instrument is turned bodily on a point on the other arm as a fixed centre. There are two forms of the instrument: in one the position of the roller is fixed and the arms are jointed on a common pinion; in the other the roller and a pinion, to which the holding arm is attached, are both carried by a slider, which is movable along the tracing arm and can be set at any required distance from the tracing point. The first form gives areas in a single unit of measure only, the second in various units. The annexed figure represents the first form, showing the joint A, the tracing point P, the fixed point O, and the roller with its graduated dial and vernier, for indicating the lengths of line rolled over while the tracer moves round the perimeter of the area under measurement.
The following explanation of the theory of the instrument is due to Professor Greenhill. Let OA, AP be the two arms jointed at A, with the fixed point at O and the tracer at P, and suppose the wheel to be fixed at R on the prolongation of the arm PA. Let OA=a, AP=b, AR= c, and the radius of the roller = ?·; and let the direction of a positive rotation of the roller, as marked by the graduations, be that of rotation on a right-handed screw on the axle of R which would give motion in the direction AR. Drop the perpendicular ΟI from O on AR, and first suppose the joint A to be clamped. Then, if I is in AR produced, a rotation of the instrument about O with angular velocity ω will give to R the component velocities OI.ω in the direction IR and IR. ω in the direction perpendicular to IR, and will therefore compel the roller to turn with the angular velocity RI / r ω; but, if I is on the other side of R, the angular velocity of the roller will be - RI / r ω. Therefore, keeping A clamped, the roller will turn through an angle RI / r θ or - RI / r θ, according as I is or is not on the same side of R as A, when the instrument is rotated through an angle θ about O ; but, when I coincides with R, the roller will not turn, and then P describes a circle, called the “zero circle,” represented by the middle dotted circular line, of radius
√(OR 2 + RP 2 ) = √{ a 2 - c 2 + (b + c) 2 } = √(a 2 + b 2 + 2 bc ).
Next unclamp the joint A and clamp the arm OA ; then the roller will turn through an angle - c / r φ, while AP turns through an angle φ.
Now suppose P to travel round the finite circuit PP 1 P 2 P 3 by a combination of the preceding motions in the following order. (1) Clamp the joint, and move P to P 1 and A to A 1 on arcs of circles of centre O ; then the roller will turn through an angle RI / r θ, θ being =AOA 1 =POP 1 . (2) Unclamp the joint and clamp the arm, and move the pointer from P 1 to P 2 on the arc of a circle of centre A 1 ; then the roller will turn through an angle = - c / r φ,φ being = P 1 A 1 P 2 , (3) Unclamp the arm and clamp the joint, and move the pointer from P 2 backwards to P 3 and A 1 to A, on arcs of circles of centre 0, through an angle 0 ; then the roller will turn through an angle RI / r θ, if OI 1 is the perpendicular from O on P 3 A.
(4) Unclamp the joint and clamp the arm, and move the pointer from P 3 to P on the arc of a circle of centre A, and consequently through an angle φ; the roller will turn through an angle φ, which cancels the angle due to motion (2). Thus in completing the finite circuit PP 1 P 3 P 3 the roller will have turned through an angle (RI-RI 1 ) θ / r =(AI-AH) θ / r .
But the area PP 1 P 2 P 3 = area PP 1 Q 1 Q, = sector OPP 1 - sector OQQ 1 = 1 / 2 (OP 2 - ΟP 3 2 )θ, = 1 / 2 {OA 2 + AP 2 + 2AI.AP-(OA 2 + AP 2 +2AI 1 .AP)} θ, = (AI-AI 1 ) bθ, = br times the angle turned through by the roller.
The area PP 1 P 2 P 3 is therefore b times the travel of the circumference of the roller.
Any irregular area, supposed to be built up of infinitesimal elements found in the same manner as PP 1 P 2 P 3 , will be accurately measured by the roller when the point P completes a circuit of the perimeter, the arm AP being free to turn on the joint at A and the arm OA on a fixed point O. If, however, O is inside the area, the area of the zero circle must be added to the area deduced from the readings of the roller. When the roller is fixed permanently, this area is constant, and is usually engraved on the arm in units of the adopted length b ; when the roller is held on a slider which also carries the pinion of the arm OA, the length b may be so adjusted that the areas described will be expressed in any desired unit of measure.
Literature and Authorities consulted.—Accounts of the operations of the Great Trigonometrical Survey of India; Manual of Survey for India; Col. A. R. Clarke, Geodesy; Methods and Processes of the ordnance Survey; Col. Waterhouse, on the Application of Photography to Maps and Plans ; and Professional Papers of the Royal Engineers.
