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MEASUREMENT ON
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MEASUREMENT ON MAPS Measurement of Distance.—The shortest distance between two places on the See also:surface of a globe is represented by the arc of a See also:great circle. If the two places are upon the same See also:meridian or upon the See also:equator the exact distance separating them is to be found by reference to a table giving the lengths of arcs of a meridian and of the equator. In all other cases recourse must be had to a See also:map, a globe or mathematical See also:formula. Measurements made on a topographical map yield the most satisfactory results. Even a See also:general map may be trusted, as See also:long as we keep within ten degrees of its centre. In the See also:case of more considerable distances, however, a globe of suitable See also:size should be consulted, or—and this seems preferable—they should be calculated by the rules of spherical See also:trigonometry. The problem then resolves itself in the See also:solution of a spherical triangle. In the formulae which follow we suppose 1 and 1' to represent the latitudes, a and b the co-latitudes (90°—1 or 90°—l'), and t the difference in See also:longitude between them or the meridian distance, whilst D is the distance required. If both places have the same See also:latitude we have to See also:deal with an isosceles triangle, of which two sides and the included See also:angle are given. This triangle, for the convenience of calculation, we See also:divide into two right-angled triangles. Then we have See also:sin D =sin a sin it, and since sin a=sin (90°—1) = See also:cos 1, it follows that sin iD = cos 1 sin it. If the latitudes differ, we have to solve an oblique-angled spherical triangle, of which two sides and the included angle are given. Thus, cos D — cos a cos b cost = sin asinb cos D = cos a cos b + sin a sin b cos t = sin l sin 1' + cos l cos 1' cos t. In See also:order to adapt this formula to logarithms, we introduce a subsidiary angle p, such that cot p = cot l cos t; we then have cos D = sin l cos(l' — p) / sin p. In the above formulae our See also:earth is assumed to be a See also:sphere, but when calculating and reducing to the See also:sea-level, a See also:base-See also:line, or the See also:side of a See also:primary triangulation, See also:account must be taken of the spheroidal shape of the earth and of the See also:elevation above the sea-level. The See also:error due to the neglect of the former would at most amount to 1%, while a reduction to the mean level of the sea necessitates but a trifling reduction, amounting, in the case of a base-line See also:Ioo,000 metres in length, measured on a See also:plateau of 3700 metres (12,000 ft.) in height, to 57 metres only. These orthodromic distances are of course shorter than those measured along a loxodromic line, which intersects all See also:parallels at the same angle. Thus the distance between New See also:York and See also:Oporto, following the former (great circle sailing), amounts to 3000 m., while following the rhumb, as ,in See also:Mercator sailing, it would amount to 3120 in. These See also:direct distances may of course differ widely with the distance which it is necessary to travel between two places along a road, down a winding See also:river or a sinuous See also:coast-line. Thus, the direct distance, as the See also:crow flies, between Brig and the See also:hospice of the Simplon amounts to 4.42 geogr. In. (slope nearly 90), while the distance by road See also:measures 13.85 geogr. m. (slope nearly 3°) Distances such as these can be measured only on a topographical map of a fairly large See also:scale, for on general maps many of the details needed for that purpose can no longer be represented. Space runners for facilitating these measurements, variously known as chartometers, curvimeters, opisometers, &c., have been devisedin great variety. Nearly all these See also:instruments See also:register the revolution of a small See also:wheel of known circumference, which is run along the line to be measured. The Measurement of Areas is easily effected if the map at our disposal is See also:drawn on an equal See also:area See also:projection. In that case we need simply See also:cover the map with a network of squares—the area of each of which has been determined with reference to the scale of the map—See also:count the squares, and estimate the contents of those only partially enclosed within the boundary, and the result will give the area desired. Instead of See also:drawing these squares upon the map itself, they may be engraved or etched upon See also:glass, or drawn upon transparent celluloid or tracing-See also:paper. Still more expeditious is the use of a planimeter, such as See also:Captain Prytz's " See also:Hatchet Planimeter," which yields fairly accurate results, or G. Coradi's " Polar Planimeter," one of the most trustworthy instruments of the See also:kind.l When dealing with maps not drawn on an equal area projection we substitute quadrilaterals bounded by meridians and parallels, the areas for which are given in the " Smithsonian See also:Geographical Tables " (1894), in See also:Professor H. See also:Wagner's tables in the geographical Jahrbuch, or similar See also:works. It is obvious that the area of a See also:group of mountains projected on a See also:horizontal See also:plane, such as is presented by a map, must differ widely from the area of the superficies or See also:physical surface of those mountains exposed to the See also:air. Thus, a slope of 45° having a surface of too sq. m. projected upon a horizontal plane only measures 59 sq. in., whilst See also:loo sq. m. of the snowclad Sentis in See also:Appenzell are reduced to to sq. m. A hypsographical map affords the readiest solution of this question. Given the area A of the plane between the two horizontal contours, the height h of the upper above the See also:lower See also:contour, the length of the upper contour 1, and the area of the See also:face presented by the edge, of the upper stratum l.h = Ai, the slope a is found to be tan a = h.l / (A — Al) ; hence its superficies, A = See also:A2 sec a. The result is an approximation, for inequalities of the ground bounded by the two contours have not been considered. The hypsographical map facilitates likewise the determination of the mean height of a See also:country, and this height, combined with the area, the determination of See also:volume, or cubic contents, is a See also:simple See also:matter.2 See also:Relief Maps are intended to See also:present a See also:representation of the ground which shall be absolutely true to nature. The See also:object, however, can be fully attained only if the scale of the map is sufficiently large, if the horizontal and See also:vertical scales are identical, so that there shall be no exaggeration of the heights, and if regard is had, eventually, to the curvature of the earth's surface. Relief maps on a small scale necessitate a generalization of the features of the ground, as in the case of See also:ordinary maps, as like-See also:wise an exaggeration of the heights. Thus on a relief on a scale oft : t,000,000 a See also:mountain like See also:Ben See also:Nevis would only rise to a height of 1.3 mm. The methods of producing reliefs vary according to the scale and the materials available. A simple See also:plan is as follows—draw an outline of the country of which a map is to be produced upon a See also:board; See also:mark all points the See also:altitude of which is known or can be estimated by pins or wires clipped off so as to denote the heights; mark river-courses and suitable profiles by strips of vellum and finally finish your See also:model with the aid of a See also:good map, in See also:clay or See also:wax. If contoured maps are available it is easy to build up a strata-relief, which facilitates the completion of the relief so that it shall be a See also:fair representation of nature, which the strata-relief cannot claim to be. A See also:pantograph armed with cutting-files 3 which carve the relief out of a See also:block of See also:gypsum, was employed in 1893—1000 by C. See also:Perron of See also:Geneva, in producing his relief map of See also:Switzerland on a scale of 1 : 500,000. After copies of such reliefs have been taken in gypsum, See also:cement, statuary pasteboard, fossil dust mixed with See also:vegetable oil, or some other suitable material, they are painted. If a number of copies is required it may be advisable to See also:print a map of the country represented in See also:colours, and either to emboss this map, backed with See also:papier-See also:mach or See also:paste it upon a copy of the relief—a task of some difficulty. Relief maps are frequently objected to on Professor Henrici, See also:Report on Planimeters (64th See also:meeting of the See also:British Association, See also:Oxford, 1894) ; J. See also:Tennant, " The Planimeter (See also:Engineering, xlv. 1903). 2 H. Wagner's Lehrbuch (See also:Hanover, 1908, pp. 241—252) refers to numerous authorities who deal fully with the whole question of measurement. 3 See also:Rienzi of See also:Leoben in 1891 had invented a similar apparatus which he called a Relief Pantograph (Zeitschrift, See also:Vienna Geog. See also:Soc. 1891). account of their cost, bulk and See also:weight, but their great use in teaching See also:geography is undeniable. Globes.'—It is impossible to represent on a plane the whole of the earth's surface, or even a large extent of it, without a consider-able amount of distortion. On the other See also:hand a map drawn on the surface of a sphere representing a terrestrial globe will prove true to nature, for it possesses, in See also:combination, the qualities which the ingenuity of no mathematician has hitherto succeeded in imparting to a projection intended for a map of some extent, namely, equivalence of areas of distances and angles. Nevertheless, it should be observed that our globes take no account of the oblateness of our sphere; but as the difference in length between the circumference of the equator and the perimeter of a meridian See also:ellipse only amounts to o.16%, it could be shown only on a globe of unusual size. The method of manufacturing a globe is much the same as it was at the beginning of the 16th See also:century. A See also:matrix of See also:wood or See also:iron is covered with successive layers of papers, pasted together so as to See also:form pasteboard. The See also:shell thus formed is then cut along the line of the intended equator into two hemispheres, they are then again glued together and made to revolve See also:round an See also:axis the ends of which passed through the poles and entered a See also:metal meridian circle. The sphere is then coated with See also:plaster or See also:whiting, and when it has been smoothed on a See also:lathe and dried, the lines representing meridians and parallels are drawn upon it. Finally the globe is covered with the paper gores upon which the map is drawn. The adaption of these gores to the curvature of the sphere calls for great care. Generally from 12 to 24 gores and two small segments for the polar regions printed on vellum paper are used for each globe. The method of preparing these gores was originally found empirically, but since the days of See also:Albert See also:Durer it has also engaged the minds of many mathematicians, foremost among whom was Professor A. G. Kastner of See also:Gottingen. One of the best instructions for the manufacture of globes we owe to Altmutter of Vienna. 2 Larger globes are usually on a stand the See also:top of which supports an artificial See also:horizon. The globe itself rotates within a metallic meridian to which its axis is attached. Other accessories are an See also:hour-circle, around the See also:north See also:pole, a See also:compass placed beneath the globe, and a flexible quadrant used for finding the distances between places. These accessories are indispensable if it be proposed to solve the problems usually propounded in books on the " use of the globes," but can be dispensed with if the globe is to serve only as a map of the See also:world. The size of a globe is usually given in terms of its See also:diameter. To find its scale divide the mean diameter of the earth (1,273,500 m.) by the diameter of the globe; to find its circumference multiply the diameter by
Ir (3.1416).
Map See also:Printing.—Maps were first printed in the second See also:half of the
15th century. Those in the Rudintentum novitiarum published at See also:Lubeck in 1475 are from woodcuts, while the maps in the first two See also:editions of See also:Ptolemy published in See also:Italy in 1472 are from See also:copper plates. Wood See also:engraving kept its ground for a consider-able See also:period, especially in See also:Germany, but copper in the end sup-planted it, and owing to the beauty and clearness of the maps produced by a combination of engraving and See also:etching it still maintains its ground. The objection that a copper See also:plate shows signs of See also:wear after a thousand impressions have been taken has been removed, since duplicate plates are readily produced by See also:electrotyping, while transfers of copper engravings, on See also: See also:Gunther (See also:Leipzig, 1895). 2 Jahrb. See also:des polytechn. Instituts in Wien, vol. xv. 3 Compare the maps of See also:EUROPE, See also:ASIA, &c., in this work.or See also:chalk (after the stone has been " grained "), or it is transferred from a drawing upon See also:transfer paper in lithographic See also:ink. In chromolithography a stone is required for each See also:colour. Owing to the great weight of stones, their cost and their liability of being fractured in the press, zinc plates, and more recently aluminium plates, have largely taken the See also:place of stone. The processes of zincography and of algraphy (aluminium printing) are essentially the same as lithography. Zincographs are generally used for producing surface blocks or plates which may be printed in the same way as a wood-cut. Another See also:process of producing such blocks is known as See also:cerography (Gr. is See also:pbs), wax. A copper plate having been coated with wax, outline and See also:ornament are cut into the wax, the lettering is impressed with type, and the See also:intaglio thus produced is electrotyped.' Movable types are utilized in several other ways in the production of maps. Thus the lettering of the map, having been set up in type, is inked in and transferred to a stone or a zinc-plate, or it is impressed upon transfer-paper and transferred to the stone. Photographic processes have been utilized not only in reducing maps to a smaller scale, but also for producing stones and plates from which they may be printed. The See also:manuscript maps intended to be produced by photographic processes upon stone, zinc or aluminium, are drawn on a scale somewhat larger than the scale on which they are to be printed, thus eliminating all those imperfections which are inherent in a pen-drawing. The saving in See also:time and cost by adopting this process is considerable, for a plan, the engraving of which takes two years, can now be produced in two days. Another process, photo- or heliogravure, for obtaining an engraved See also:image on a copper plate, was for the first time employed on a large scale for producing a new topographical map of the See also:Austrian See also:Empire in 718 sheets, on a scale of I : 75,000, which was completed in seventeen years (1873-189o). The original drawings for this map had to be done with exceptional neatness, the draughtsman spending twelve months on that which he would have completed in four months had it been intended to engrave the map on copper; yet. an See also:average See also:chart, measuring 530 by 63o mm., which would have taken two years and nine months for drawing and engraving, was completed in less than fifteen months—fifty days of which were spent in " retouching " the copper plate. It only cost £169 as compared with £360 had the old method been pursued. For details of the various methods of See also:reproduction see LITHO- GRAPHY; PROCESS, &C. Additional information and CommentsThere are no comments yet for this article.
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