_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 l and l´ to represent the
latitudes, a and b the co-latitudes (90-l 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 right-angled triangles. Then we have sin ½ D = sin a sin ½ t,
and since sin a = sin (90° - l) = cos t, it follows that
sin ½ D = cos l sin ½ t.
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
cos t = -------------------
sin a sin b
cos D = cos a cos b + sin a sin b cos t
= sin l sin l´ + cos l cos l´ 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 sea-level, a base-line, or the side of
a primary triangulation, account must be taken of the spheroidal shape
of the earth and of the elevation above the sea-level. 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 base-line 100,000 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 m.
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 coast-line. Thus, the direct distance, as
the crow flies, between Brig and the hospice of the Simplon amounts to
4.42 geogr. m. (slope nearly 9°), 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 devised in 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 tracing-paper. 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.[4]
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
100 sq. m. projected upon a horizontal plane only measures 59 sq. m.,
whilst 100 sq. m. of the snowclad Sentis in Appenzell are reduced to
10 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 l, and the area of the face presented by
the edge of the upper stratum t·h = A1, the slope [alpha] is found to
be tan [alpha] = h·l / (A - A1); hence its superficies, A = A2 sec
[alpha]. 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.[5]
_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 of 1:1,000,000 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 river-courses 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 strata-relief, which facilitates the completion of
the relief so that it shall be a fair representation of nature, which
the strata-relief cannot claim to be. A pantograph armed with
cutting-files[6] which carve the relief out of a block of gypsum, was
employed in 1893-1900 by C. Perron of Geneva, in producing his relief
map of Switzerland on a scale of 1:100,000. 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 papier-mâché, or paste it upon a copy of the relief--a
task of some difficulty. Relief maps are frequently objected to on
account of their cost, bulk and weight, but their great use in teaching
geography is undeniable.
_Globes._[7]--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 0.16%, 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 Dürer it has also engaged the minds of many
mathematicians, foremost among whom was Professor A. G. Kästner of
Göttingen. One of the best instructions for the manufacture of globes we
owe to Altmütter of Vienna.[8]
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 hour-circle,
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 [pi] (3.1416).
_Map Printing._--Maps were first printed in the second half of the 15th
century. Those in the _Rudimentum novitiarum_ published at Lübeck 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
printing-machine, which thus supersedes the slow-working hand-press.[9]
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 copper-engraving), or it is drawn upon the stone by
pen, brush 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 wood-cut. Another
process of producing such blocks is known as cerography (Gr. [Greek:
kêros]), 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.[10] 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
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 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 1 : 75,000, which was completed in seventeen years
(1873-1890). 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 630
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 LITHOGRAPHY;
PROCESS, &c.