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AVAVA YYAWWWYA
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AVAVA YYAWWWYA , g YAAMY/ AIWA . 146 in which dc and dA represent the errors in the length and See also:azimuth of any See also:side c which have been generated 7 in the course of the triangulation up to • it from the See also:base-See also:line and the azimuth station at the origin . The errors in the See also:latitude and See also:longitude of any station which are due to the triangulation are 3 dX, = [d .AX], and dL, = [d . AL] . Let station i be the origin, and let 2, 3, ... be the succeeding stations taken along a predetermined line of See also:traverse, which may either run from vertex to vertex 2 of the successive triangles, zigzagging between the flanks of the See also:chain, as in fig . 3 (I), or be carried directly along one of the flanks, as in fig . 3 (2) . For the See also:general symbols of the See also:differential equations substitute Wk,,, AL,,, AA,,, c,,, A,,, and B,,, for the side between stations n and n+I of the traverse; and let Sc„ and 8A°be the errors generated between the sides and c,,; then 4 dci bc1 dc2 Sci Sc2 do " Sc c1 c1' c2=ci+ c2' '' = i c ; dA1=SA1; dA2=dBi+SA2; ... dA„=dB„_1+SA,, . Performing the necessary substitutions and summations, we get r Sc2 +A Sc° 1[DA17,- +2 [DA]~2 -{-...An c„ +(I+i[AA cot A] See also:sin i")SA1+(I+2[AA cot A] sin I")SA2 +...+(I+AA. cot A. sin I")SA,, . [Axil '+n[x]b C2 2+...+oa„bn" —['[AX tan A15A1+2[Da tan A]SA2+ ... l +oan tan A.M.] sin rnr 1 Sc2 bc° 1 [ALIs i +2 [AL]c2 + ... +L n c„ +{1[AL cot A]SA1+a[.L cot A]SA2+ ... +AL, cot AnbA„] sin 1" . Thus we have the following expression for any See also:geodetic See also:error:—µ,Sci+ . +gnb n +,1SA1+ ... +,„SA„=E, (8) where g and 4) represent the respective summations which are the coefficients of Sc and SA in each instance but the first, in which I is added to the summation in forming the coefficient of SA . The angular errors x, y and z must now be introduced, in See also:place of Sc and SA, into the general expression, which will then take different forms, according as the route adopted for the line of traverse was the zigzag or the See also: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 See also:common numerical notation may be adopted for both the traverse stations and the See also:collateral triangles; thus the angular errors of every triangle enter the general expression in the See also:form = c~x+cot Y . g'y —cot Z . g'z, in which µ'=g sin i ", and the upper sign of ¢ is taken if the triangle lies to the See also:left, the See also:lower if to the right, of the line of traverse . When the direct traverse is adopted, there are only See also:half as many traverse stations as triangles, and therefore only half the number of g's and 4,'s to determine; but it becomes necessary to adopt different numberings for the stations and the triangles, and the form of the coefficients of the angular errors alternates in successive triangles . Thus, if the pth triangle has no side on the line of the traverse but only an See also:angle at the lth station, the form is +sin .xp+cot Y5.gi •yp—cot Z.gt.z5 . If the qth triangle has a side between the lth and the (l+I)th stations of the traverse, the form is cot Xq(gl — g't+I)xg + + 11'i+1 cot Yq)yq — (4u+1 — g( cot Z0)zq . As each See also:circuit has a right-See also:hand and a left-hand See also:branch, the errors of the angles are finally arranged so as to See also:present equations of the general form[GEODETIC TRIANGULATION u, v and w being the reciprocals of the weights of the observed angles . This necessitates the simultaneous See also:solution of eighty-three equations to obtain as many values of X . The resulting values of the errors of the angles in any, the pth, triangle, are xp= up[apal; y,= vp[b5)]; zp=w5[cpX] . (9) ii . One of the unknown quantities in every triangle, as x, may be eliminated from each of the eleven circuit and base-line equations by substituting its See also:equivalent—(y+z) for it, a similar substitution being made in the minimum . Then the equations take the form [(b—a)y+(c—a)zJ=E, while the minimum becomes [(y+z)2 Y2 zil L u +;v +w Thus we have now to find only eleven values of a 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: rip yp = Zip+vp -[-wp[ (up+wp) [(bp— ap)X] —wp[(cp—ap)r]] • (ro) zp up+vp....wp[ (up+vp)[(cp— ap)xl —:zp[(bp — ap)al ] The second method has invariably been adopted, originally be. cause it was supposed that, the number of the factors A being reduced from the See also:total number of equations to that of the circuit and base-line equations, a See also: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 See also: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 See also: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 See also:Indian triangulation it was decided, after examining the relative magnitudes of the probable errors of the linear and the angular See also:measures and ratios, to assume the base-lines to be errorless . The chains of triangles being largely composed of polygons or other networks, and not merely of single triangles, as has been assumed for simplicity in the illustration, the geometrical See also:harmony to be maintained involved the introduction of a large number of " side," " central and " toto-partial " equations of See also: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—I, the number of unknown quantities contained in the equations being that of the whole of the observed angles' the method of See also:procedure, if rigorous, would be precisely similar to that already indicated for " harmonizing the angles of trigonometrical figures," of which it is merely an expansion from single figures to great See also:groups . The rigorous treatment would, however, have involved the simultaneous solution of about 4000 equations between 9230 unknown quantities, which was impracticable . The triangulation was therefore divided into sections for See also:separate reduction, of which the most important were the five between the meridians of 67° and 92° (see fig . I), consisting of four See also:quadrilateral figures and a trigon, each comprising several chains of triangles and some base-lines . This arrangement had the See also:advantage of enabling the final reductions to be taken in hand as soon as convenient after the completion of any See also: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 See also:part of the section first reduced, it was obviously desirable to run no See also:risk of impairing the best chains by forcing them into See also:adjustment with others of inferior quality . It happened that both the See also:north-See also:east and the See also:south-See also:west quadrilaterals contained several of the older chains; their reduction was therefore made to follow that of the collateral sections containing the See also: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 See also:cognizance had to be taken of a dB„= dX,,+, _ rl L +, = [ax+by+cz].—[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 error, which will satisfy these equations, and be the most probable of any See also:system of values that will do so, and at the same See also:time will not disturb the existing harmony of the angles in each of the seventy-two triangles . Harmony is maintained by introducing the See also:equation of condition x+y+z=o for every triangle . The most probable results are obtained by the method of minimum squares, which may be applied in two ways . i . A See also:factor X may be obtained for each of the eighty-three equa- tions under the condition that [u2+ v +w] is made a minimum, number of " side " and other geometrical equations of condition, which entered irregularly and caused great entanglement . Equations 9 and ro of the illustration are of a See also:simple form because they have a single geometrical condition to maintain, the triangular, which is not only expressed by the simple and symmetrical equation x+y+z=o, but—what is of much greater importance—recurs in a See also:regular See also:order of sequence that materially facilitates the general solution . Thus, though the calculations must in all cases be very numerous and laborious, rules can be formulated under which they can be well controlled at every See also:stage and eventually brought to a successful issue . The other geometrical conditions of networks are expressed by equations which are not merely of a more complex form but have no regular order of sequence, for the networks pre-sent a variety of forms; thus their introduction would cause much entanglement and complication, and greatly increase the labour of the calculations and the chances of failure . Wherever, therefore, any See also: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 See also:independent figural adjustments, been made to exert their full See also:influence on the included angles . The figural adjustments had, how-ever, introduced new relations between the angles of different figures, causing their weights to increase caeteris paribus with the number of geometrical conditions satisfied in each instance . Thus, suppose w to be the See also:average weight of the t observed angles of any figure, and n the number of geometrical conditions presented for See also:satisfaction; then the average weight of the angles after adjustment may be taken as w. t the factor thus being 1.5 for a triangle, 1.8 for a hexagon, 2 for a quadrilateral, 2.5 for the network around the Sironj base-line, &c . In framing the normal equations between the indeterminate factors X for the final simultaneous reduction, it would have greatly added to the labour of the subsequent calculations if a separate weight had been given to each angle, as was done in the See also:primary figural reductions; this was obviously unnecessary, for theoretical requirements would now be amply satisfied by giving equal weights to all the angles of each independent figure . The mean weight that was finally adopted for the angles of each See also:group was therefore taken as t w.ot n' p being the modulus . The second of the two processes for applying the method of minimum squares having been adopted, the values of the errors y and z of the angles appertaining to any, the pth, triangle were finally expressed by the following equations, which are derived from (1o) by substituting u for the reciprocal final mean weight as above determined: yn = 3 [(2bv — av — c5)X] zo = 3 [(2ca — a5 — b5)\] The following table gives the number of equations of condition and unknown quantities—the angular errors—in the five great sections of the triangulation, which were respectively included in the simultaneous general reductions and relegated to the subsequent adjustments of each figure per se: Simultaneous . See also:External Figural . Equations . Equations . Section . Tlc t ot:jz °~ era e Z sl F` m F E'w Side . u ' 1 . N.W . Quad . . 23 550 165o 267 104 152 6 761 110 2 . S.E . Quad . 15 277 831 164 64 92 2 476 68 3 . N.E . Quad . . 49 573 1719 112 56 69 0 341 50 4 . Trigon . . . 22 303 909 192 79 101 2 547 77 5 . S . W . Quad . 24 172 516 83 32 52 1 237 40 The corrections to the angles were generally See also:minute, rarely exceeding the theoretical probable errors of the angles, and therefore applicable without taking any liberties with the facts of observation . Azimuth observations in connexion with the See also:principal triangulation were determined by measuring the See also:horizontal angle between a referring See also:mark and a circumpolar See also:star, shortly before and after See also:elongation, and usually at both elongations in order to eliminate the error of the star's place . Systematic changes of " See also:face " and of the zero settings of the azimuthal circle were made as in the measurement of the principal angles; but the repetitions on each zero were more numerous; the azimuthal levels were read and corrections applied to the star observations for dislevelinent . 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 See also: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 See also:differences (observed at the origin—computed from a distance) between the primary and the geodetic azimuths; the differences were assumed to be mainly due to the local deflexions of the plumb-line and only partially to error in the triangulation, and each was multiplied by the factor tangent of latitude of origin, =tangent of latitude of comparing station in order that the effect of the local attraction on the azimuth observed at the distant station—which varies with the latitude and is =the deflexion in the See also:prime See also:vertical X the tangent of the latitude —might be converted to what it would have been had the station been situated in the same latitude as the origin . Each See also: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] (12); [w] the value of x thus obtained was —PI" . The formulae employed in the reduction of the azimuth observations were as follows . In the spherical triangle PZS, in which P is the See also:pole, Z the See also:zenith and S the star, the co-latitude PZ and the polar distance PS are known, and, as the angle at S is a right angle at the elongation, the See also:hour angle and the azimuth at that time are found from the equations cosP = tanPScotPZ, cosZ = cosPSsinP . The See also:interval, 6P, between the time of any observation and that of the elongation being known, the corresponding azimuthal angle, SZ, between the two positions of the star at the times of observation and elongation is given rigorously by the following expression —tan EZ 2sin'2SP cotPSsinPZsinP{1+tan2PScosSP+sec2PScotPsinbP} (13)' which is expressed as follows for logarithmic computation SZ = — m tan Z See also:cos' PS 1 — n +1 where m = 2 sine- cosec i', n = 2 sin2PS sin' , and 2 2 l=cot P sin SP; 1, m, and n are tabulated . Let A and B (fig . 4) be any two points the normals at which meet at C, cutting the See also:sea-level at p and q; take Dq=Ap, then BD is the difference of height; draw Height and ¢ .8 the tangents Aa and Bb at See also:Refraction . A and B, then aAB is the depression of B at A and bBA that of A at B; join AD, then BD is determined from the triangle ABD . The triangulation gives the distance between A and B at the sea-level, whence pq=c; thus, putting Ap, the height of A above the sea-level, =H, and pC= r, AD = c (t H c2 ) + r—24,.2 Putting Da and Db for the actual depressions at A and B, S for the angle at A, usually called the " subtended angle," and h for BD S = a(Db—Da) (15), and h = AD sin s (16) cos Db Azimuth Observations . (i4) . The angle at C being=Db-f-Da, S may FiG' 4' be expressed in terms of a single vertical angle and C when observations have been taken at only one of the two points . P C, the "contained arc," =c—cosec 1" in seconds . Putting D'a PvY and D's for the observed vertical angles, and Oa, ¢b for the amounts by which they are affected by refraction, Da=D'a+4a and Ds=D'b-i-0b; and 4'b may differ in amount, but as they cannot be separately ascertained they are always assumed to be equal; the See also:hypothesis is sufficiently exact for See also:practical purposes when both verticals have been measured under similar atmospheric conditions . The retractions being taken equal, the observed verticals are substituted for the true in (15) to find S, and the difference of height is calculated by (16); the third See also:term within the brackets of (14) is usually omitted . The mean value of the refraction is deduced from the See also:formula ct.-= 2(C—D'.+D'b)) (17) . An approximate value is thus obtained from the observations between the pairs of reciprocating stations in each See also:district, and the corresponding mean " coefficient of refraction," 4=C, is computed for the district, and is employed when heights have to be deter-See also:mined from observations at a single station only . When either of the vertical angles is an See also:elevation—E must be substituted for D in the above expressions.' 2 . LEVELLING Levelling is the See also:art of determining the relative heights of points on the See also:surface of the ground as referred to a hypothetical surface which cuts the direction of gravity everywhere at right angles . When a line of instrumental levels is begun at the sea-level, a See also:series of heights is determined corresponding to what would be found by perpendicular measurements upwards from the surface of See also: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 See also:earth's curvature . The trigonometrical determination of the relative heights of points at known distances apart, by the measurements of their mutual vertical angles—is a method of levelling . But the method to which the term " levelling " is always applied is that of the direct determination of the differences of height from the readings of the lines at which graduated staves, held vertically over the points, are cut by,the horizontal See also:plane which passes through the See also:eye of the observer . Each method has its own advantages . The former is less accurate, but best suited for the requirements of a general See also:geographical survey, to obtain the heights of all the more prominent See also: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 See also:short intervals apart; thus it is very generally undertaken pari passu with geographical surveys to furnish lines of level for ready reference as a check on the accuracy of the trigonometrical heights
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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 See also: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
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Thus, either a correction for curvature must be applied to every See also:staff See also:reading, or the See also: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
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Levelling has to be repeated frequently in executing a See also:long line of levels—say seven times on an average in every mile—and must be conducted with precaution against various errors
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Instrumental errors arise when the visual See also:axis of the See also:telescope is not perpendicular to the axis of rotation, and when the focusing See also:tube does not move truly parallel to the visual axis on a See also:change of See also:focus
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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
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' In topographical and levelling operations it is sometimes convenient to apply small corrections to observations of the height for curvature and refraction simultaneously
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Putting d for the distance, r for the earth's See also:radius, and ic for the coefficient of refraction, and expressing the distance and radius in See also:miles and the correction to height in feet, then correction for curvature = d2; correction for refraction = —ticd2; correction for both
3Errors of staff readings should be guarded against by having the staves graduated on both faces, but differently figured, so that the observer may not be biased to repeat an error of the first reading in the second
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The staves of the Indian survey have one face painted See also: 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 See also:sun's See also:light will often fall endwise on the bubble of the level, See also:illuminating the See also:outer edge of the rim at the nearer end and the inner edge at the farther end, and so biasing the observer to take scale readings of edges which are not equidistant from the centre of the bubble; this introduces a tendency to raise the south or depress the north ends of lines of level in the See also: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 See also:main roads of the See also:country, a number of " See also:bench marks " bring fixed for future reference . In the See also:ordnance survey of Great See also:Britain lines have been carried across from See also:coast to coast in such a manner that the level of any common See also:crossing point may be found by several independent lines . Of these points there are 166 in See also:England, See also:Scotland and See also:Wales; the discrepancies met with at them were adjusted simultaneously by the method of minimum squares . The sea-level is the natural datum plane for levelling operations, more particularly in countries bordering on the ocean . The earliest surveys of coasts were made for the use sea-,See also:eve, of navigators and, as it was considered very important that the charts should everywhere show the minimum See also:depth of water which a See also:vessel would meet with, See also:low water of See also:spring-tides was adopted as the datum . But this does not See also:answer the requirements of a See also: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 See also:uniform all the See also:world over, is much more nearly so than low water . Tidal observations have been taken at nearly fifty points on the coasts of Great Britain, which were connected by levelling operations; the local levels of mean sea were found to differ by larger magnitudes than could fairly be attributed to errors in the lines of level, having a range of 12 to 15 in. above or below the mean of all at points on the open coast, and more in tidal See also: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 See also:period components of the tides, and ignores the See also:annual and semi-annual components, which occasionally attain considerable magnitudes . The mean sea-levels at See also:Port Said in the Mediterranean and at See also:Suez in the Red Sea have been found to be identical, and a similar identity is said to exist in the levels of the See also:Atlantic and the Pacific oceans on the opposite coasts of the See also:Isthmus of See also:Panama . This is in favour of a uniform level all the world over; but, on the other hand, lines of level carried across the See also:continent of See also:Europe make the mean sea-level of the Mediterranean at See also:Marseilles and See also:Trieste from 2 to 5 ft. below that of the North Sea and the Atlantic at See also:Amsterdam and See also:Brest—a result which 2In tidal estuaries and rivers the mean water-level rises above the mean sea-level as the distance from the open coast-line increases; for instance, in the Hooghly See also:river, passing See also:Calcutta, there is a rise of to in. in 42 M. between See also:Sagar (See also:Saugor) See also:Island at the mouth of the river and See also:Diamond See also:Harbour, and a further rise of 20 in. in 43 m. between Diamond Harbour and Kidderpur . it is not easy to explain on See also:mechanical principles . In See also: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 See also:probability to error generated in levelling over lines of great length than to actual differences of sea-level in different localities . The sea-level, however, may not coincide everywhere with the geometrical figure which most closely represents the earth's Qeoldor surface, but may be raised or lowered, here and there, Deformed under the influence of local and abnormal attrac- tions, presenting an equipotential surface—an See also:ellipsoid or See also:spheroid of revolution slightly deformed by bumps and hollows—which H . Bruns calls a " See also:geoid." See also:Archdeacon See also:Pratt has shown that, under the combined influence of the See also:positive attraction of the Himalayan Mountains and the negative attraction of the Indian Ocean, the sea-level may be some 56o ft. higher at See also:Karachi than at Cape See also:Comorin; but, on the other hand, the Indian pendulum operations have shown that there is a deficiency of See also:density under the Himalayas and an increase under the See also:bed of the ocean, which may wholly compensate for the excess of the See also:mountain masses and deficiency of the ocean, and leave the surface undisturbed . If any bumps and hollows exist, they cannot be measured, instrumentally; for the instrumental levels will be affected by the local attractions precisely as the sea-level is, and will thus invariably show level surfaces even should there be considerable deviations from the geometrical figure . . 3 . TOPOGRAPHICAL SURVEYS The See also:skeleton framework of a survey over a large See also:area should be triangulation, although it is frequently combined with traversing . The method of filling in the details is necessarily influenced to some extent by the nature of the framework, but it depends mainly on the magnitude of the scale and the requisite degree of minutiae . In all instances the principal triangles and circuit traverses have to be broken down into smaller ones to furnish a sufficient number of fixed points and lines for the subsequent operations . The filling in may be performed wholly by linear measurements or wholly by direction intersections, but is most frequently effected by both linear and angular measures, the former taken with chains and tapes and offset poles, the latter with small theodolites, sextants, See also:optical squares or other reflecting See also: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 See also: ~When a considerable area has to be treated by traverses it is divided into a number of blocks of convenient See also:size, bounded by roads, rivers or See also:parish boundaries, and a " traverse on the See also:meridian of the origin " is carried round the periphery of each See also:block . Be-ginning at a trigonometrical station, the theodolite is set to circle reading o° o' with the telescope pointing to the north, and at every " forward " station of the traverse the circle is set to the same reading when the telescope is pointed at the " back " station as was obtained at the back station when the telescope was pointing to the forward one . When the circuit is completed and the theodolite again put up at the origin and set on the last back station with the appropriate circle reading, the circle reading, with the telescope again pointed to the first forward station, will be the same as at first, if no error has been committed . This system establishes a convenient check on the accuracy of the operations and enables the angles to be readily protracted on a system of lines parallel to the meridian of the origin . As a further check the traverse is connected with all contiguous trigonometrical stations by measured angles and distances . Traverses are frequently carried between the points already fixed on the sides of the minor triangles; the initial side is then adopted, instead of the meridian, as the axis of co-ordinates for the plotting, the telescope being pointed with circle reading o° o' to either of the trigonometrical stations at the extremities of the side . The plotting is done from the field-books of the surveyors by a separate agency . Its accuracy is tested by examination on the ground, when all necessary addenda are made . The examiner —who should be surveyor, plotter and draughtsman—verifies the accuracy of the detail by intersections and productions and occasional direct measurements, and generally endeavours to cause the details under examination to prove the accuracy of each other rather than to obtain direct See also:proof by remeasurement . He fixes conspicuous trees and delineates the woods, footpaths, rocks, precipices, steep slopes, embankments, &c., and supplies the requisite See also:information regarding minor objects to enable a draughtsman to make a perfect See also:representation according to the scale of the map . In examining a coast-line he delineates the See also:foreshore and sketches the strike and See also:dip of the stratified rocks . In tidal rivers he ascertains and marks the highest points to which the See also:ordinary tides flow . The examiner on the 25.344 in. scale (=-2510) is required to give all necessary information regarding the parcels of ground of different See also:character—whether arable, pasture, See also:wood, See also:moor, See also:moss, sandy—defining the limits of each on a separate tracing if necessary . He has also to distinguish between See also: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 See also:charge of the levelling to have the levels and See also:contour lines inserted, and finally to the hill sketchers, whose See also:duty it is to make an See also:artistic representation of the features of the ground . In the Indian survey all filling in is done by plane-tabling on a basis of points previously fixed; the methods differ simply in the extent to which linear measures are introduced to supplement the direction rays of the plane-table . When the scale of the survey is small, direct measurements of distance are rarely made and the filling is usually done wholly by direction intersections, which See also: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 See also:sheet of See also:drawing paper is mounted on See also:cloth over the face of the plane-table; the points, previously fixed by triangulation or otherwise, are projected on it—the collateral meridians and See also:parallels, or the rectangular co-ordinates, when these are more convenient for employment than the spherical, having first been drawn; the plane-table is then ready for use . Operations are begun at a fixed point by aligning with the sight rule on another fixed point, which brings the meridian line of the table on that of the station . The magnetic See also: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 See also: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 See also: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, See also:interpolation is best performed by rays drawn from near points, and checked by rays drawn to distant points, as the latter show most strongly the magnitude of any error of the primary magnetic setting . In this way, and by self-verificatory traverses " on the back ray " between fixed points, plane-table stations are established over the ground at appropriate intervals, depending on the scale of the survey; and from these stations all surrounding objects which the scale permits of being shown are laid down on the table, sometimes by rays only, sometimes by a single ray and a measured distance . The general configuration- of the ground is delineated simultaneously . In checking and examination various methods are followed . For large scale work in plains it is customary to run arbitrary lines across it and make an independent survey of the See also:belt of ground to a distance of a few chains on either side for comparison with the See also:original survey; the smaller scale hill See also:topography is checked by examination from commanding points, and also by traverses run across the finished work on the table . 4 . GEOGRAPHICAL SURVEYING The introduction by mechanical means of See also:superior See also:graduation in instruments of the smaller class has enabled surveyors to effect Base See also:good results more rapidly, and with less See also:expenditure Measure- on equipment and on the staff necessary for transport meats. in the field, than was formerly possible . The 12-in. theodolite of the present See also:day, with See also:micrometer adjustments to assist in the reading of minute subdivisions of angular graduation, is found to be equal to the old 24-in. or even 36-in. instruments . New Methods for the measurement of bases have largely superseded the laborious process of measurement by the alignment of " See also:compensation " bars, though not entirely independent of them . The Jaderin apparatus, which consists of a See also:wire 25 metres in length stretched along a series of cradles or supports, is the simplest means of measuring a base yet devised; and experiments with it at the Pulkova See also:observatory show it to be capable of producing most accurate results . But there is' a measurable defect in the apparatus, owing to the liability of the wires to change in length under variable conditions of temperature . It is therefore considered necessary, where base measurements for geodetic purposes are to be made with scientific exactness, that the Jaderin wires should be compared before and after use with a See also:standard measurement, and this standard is best attained by the use of the See also:Brunner, or See also:Colby, bars . The direct process of measurement is not extended to such lengths as formerly, but from the ends of a shorter line, the length of which has been exactly determined, the base is extended by a process of triangulation . There are vast areas in which, while it is impossible to apply the elaborate processes of first-class or " geodetic " triangulation, secondary it is nevertheless desirable that we should rapidly Triangula- acquire such geographical knowledge as will enable tien. us to See also:lay down See also:political boundaries, to project roads and railways, and to attain such exact knowledge of See also:special localities as will further military ends . Such surveys are called by various names—military surveys, first surveys, geographical surveys, &c.; but, inasmuch as they are all undertaken with the same end in view, i.e. the acquisition of a See also:sound topographical map on various scales, and as that end serves See also:civil purposes as much as military, it seems appropriate to designate them geographical surveys only . The governing principles of geographical surveys are rapidity and See also:economy . Accuracy is, of course, a recognized See also:necessity, but Principles the term must admit of a certain See also:elasticity in See also:gee-which graphical work which is inadmissible in geodetic govern Geo- or cadastral functions . It is obviously foolish to graphical expend as much See also:money over the elaboration of toposurveys, graphy in the unpeopled See also:sand wastes which border the See also:Nile valley, for instance (albeit those deserts may be full of triangulation, or at least by some process analogous surveds of 3' Survey . to triangulation, which will furnish the necessary skeleton on which to adjust the topography so as to ensure a See also:complete and homogeneous map . This base may be found in a variety of ways . If geodetic triangulation exists in the country, that triangulation should of course include a wide extent of secondary determina- TieBase. tions, the fixing of peaks and points in the landscape far away to either flank, which will either give the data for further See also:extension of geographical triangulation, or which may even serve the purposes of the map-maker without any such extension at all . In this manner the See also:Indus valley series of the triangulation of India has furnished the basis for surveys across See also:Afghanistan and See also:Baluchistan to the See also:Oxus and See also: |