MEASUREMENT ON MAPS
Measurement of Distance.—The shortest distance between two places on the surface of a globe is represented by the arc of a great circle. If the two places are upon the same meridian or upon the 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 map, a globe or mathematical formula. Measurements made on a topographical map yield the most satisfactory results. Even a general map may be trusted, as long as we keep within ten degrees of its centre. In the case of more considerable distances, however, a globe of suitable size should be consulted, or—and this seems preferable—they should be calculated by the rules of spherical trigonometry. The problem then resolves itself in the solution of a spherical triangle.
In the formulae which follow we suppose 1 and 1' to represent the latitudes, a and b the colatitudes (90°—1 or 90°—l'), and t the difference in longitude between them or the meridian distance, whilst D is the distance required.
If both places have the same latitude we have to deal with an isosceles triangle, of which two sides and the included angle are given. This triangle, for the convenience of calculation, we divide into two rightangled triangles. Then we have sin D =sin a sin it, and since sin a=sin (90°—1) = cos 1, it follows that
sin iD = cos 1 sin it.
If the latitudes differ, we have to solve an obliqueangled 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 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 earth is assumed to be a sphere, but when calculating and reducing to the sealevel, a baseline, or the side of a primary triangulation, account must be taken of the spheroidal shape of the earth and of the elevation above the sealevel. The 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 baseline Ioo,000 metres in length, measured on a 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 parallels at the same angle. Thus the distance between New York and Oporto, following the former (great circle sailing), amounts to 3000 m., while following the rhumb, as ,in Mercator sailing, it would amount to 3120 in.
These 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 river or a sinuous coastline. Thus, the direct distance, as the crow flies, between Brig and the hospice of the Simplon amounts to 4.42 geogr. In. (slope nearly 90), while the distance by road measures 13.85 geogr. m. (slope nearly 3°) Distances such as these can be measured only on a topographical map of a fairly large 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 instruments register the revolution of a small 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 drawn on an equal area projection. In that case we need simply 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—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 drawing these squares upon the map itself, they may be engraved or etched upon glass, or drawn upon transparent celluloid or tracingpaper. Still more expeditious is the use of a planimeter, such as Captain Prytz's " Hatchet Planimeter," which yields fairly accurate results, or G. Coradi's " Polar Planimeter," one of the most trustworthy instruments of the 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 Geographical Tables " (1894), in Professor H. Wagner's tables in the geographical Jahrbuch, or similar works.
It is obvious that the area of a group of mountains projected on a horizontal plane, such as is presented by a map, must differ widely from the area of the superficies or physical surface of those mountains exposed to the air. Thus, a slope of 45° having a surface of too sq. m. projected upon a horizontal plane only measures 59 sq. in., whilst loo sq. m. of the snowclad Sentis in 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 lower contour, the length of the upper contour 1, and the area of the 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 = 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 country, and this height, combined with the area, the determination of volume, or cubic contents, is a simple matter.2
Relief Maps are intended to present a representation of the ground which shall be absolutely true to nature. The object, however, can be fully attained only if the scale of the map is sufficiently large, if the horizontal and 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 ordinary maps, as likewise an exaggeration of the heights. Thus on a relief on a scale oft : t,000,000 a mountain like Ben 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 plan is as follows—draw an outline of the country of which a map is to be produced upon a board; mark all points the altitude of which is known or can be estimated by pins or wires clipped off so as to denote the heights; mark rivercourses and suitable profiles by strips of vellum and finally finish your model with the aid of a good map, in clay or wax. If contoured maps are available it is easy to build up a stratarelief, which facilitates the completion of the relief so that it shall be a fair representation of nature, which the stratarelief cannot claim to be. A pantograph armed with cuttingfiles 3 which carve the relief out of a block of gypsum, was employed in 1893—1000 by C. Perron of Geneva, in producing his relief map of Switzerland on a scale of 1 : 500,000. After copies of such reliefs have been taken in gypsum, cement, statuary pasteboard, fossil dust mixed with vegetable oil, or some other suitable material, they are painted. If a number of copies is required it may be advisable to print a map of the country represented in colours, and either to emboss this map, backed with papiermach or paste it upon a copy of the relief—a task of some difficulty. Relief maps are frequently objected to on
Professor Henrici, Report on Planimeters (64th meeting of the British Association, Oxford, 1894) ; J. Tennant, " The Planimeter (Engineering, xlv. 1903).
2 H. Wagner's Lehrbuch (Hanover, 1908, pp. 241—252) refers to numerous authorities who deal fully with the whole question of measurement.
3 Rienzi of Leoben in 1891 had invented a similar apparatus which he called a Relief Pantograph (Zeitschrift, Vienna Geog. Soc. 1891).
account of their cost, bulk and weight, but their great use in
teaching 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 considerable amount of distortion. On the other hand a map drawn on the surface of a sphere representing a terrestrial globe will prove true to nature, for it possesses, in 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 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 century. A matrix of wood or iron is covered with successive layers of papers, pasted together so as to form pasteboard. The 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 round an axis the ends of which passed through the poles and entered a metal meridian circle. The sphere is then coated with plaster or whiting, and when it has been smoothed on a 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 Albert Durer it has also engaged the minds of many mathematicians, foremost among whom was Professor A. G. Kastner of 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 top of which supports an artificial horizon. The globe itself rotates within a metallic meridian to which its axis is attached. Other accessories are an hourcircle, around the north pole, a 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 world. The size of a globe is usually given in terms of its 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 Printing.—Maps were first printed in the second half of the
15th century. Those in the Rudintentum novitiarum published at Lubeck in 1475 are from woodcuts, while the maps in the first two editions of Ptolemy published in Italy in 1472 are from copper plates. Wood engraving kept its ground for a considerable period, especially in Germany, but copper in the end supplanted it, and owing to the beauty and clearness of the maps produced by a combination of engraving and etching it still maintains its ground. The objection that a copper plate shows signs of wear after a thousand impressions have been taken has been removed, since duplicate plates are readily produced by electrotyping, while transfers of copper engravings, on stone, zinc or aluminium, make it possible to turn out large editions in a printingmachine, which thus supersedes the slowworking handpress.3 These impressions from transfers, however, are liable to be inferior to impressions taken from an original plate or an electrotype. The art of lithography greatly affected the production of maps. The work is either engraved upon the stone (which yields the most satisfactory result at half the cost of copperengraving), or it is drawn upon the stone by pen, brush
1 M. Fiorini, Erd and Himmelsgloben, frei bearbeitet von S. Gunther (Leipzig, 1895).
2 Jahrb. des polytechn. Instituts in Wien, vol. xv.
3 Compare the maps of EUROPE, ASIA, &c., in this work.or chalk (after the stone has been " grained "), or it is transferred from a drawing upon transfer paper in lithographic ink. In chromolithography a stone is required for each 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 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 woodcut. Another process of producing such blocks is known as cerography (Gr. is pbs), wax. A copper plate having been coated with wax, outline and ornament are cut into the wax, the lettering is impressed with type, and the 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 zincplate, or it is impressed upon transferpaper 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 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 pendrawing. The saving in 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 image on a copper plate, was for the first time employed on a large scale for producing a new topographical map of the Austrian Empire in 718 sheets, on a scale of I : 75,000, which was completed in seventeen years (1873189o). 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 average 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 reproduction see LITHO
GRAPHY; PROCESS, &C.
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