68—2 
[547 
547] 
SPHERICAL OR OTHER SURFACE ON A PLANE. 
539 
SUR- 
N. 
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A map would be perfect if each element of the surface and the corresponding 
element of the map were of the same form, and were in a constant ratio as to 
magnitude ; say if it were free from the defects of “ distortion ” and “ inequality ” (of 
scale); the condition as to form, or freedom from distortion, may be otherwise expressed 
by saying that any two contiguous elements of length on the surface and the corre 
sponding two contiguous elements of length on the map must meet at the same angle 
(this at once appears by taking the two elements of area to be each of them a 
triangle). But for a spherical or other non-developable surface, it is not possible to 
construct a map free from the two defects. 
An obvious and usual kind of representation is that by projection: viz. taking af 
fixed point and plane, the line joining any point P of the surface with the fixed 
point meets the fixed plane in a point P' which is taken to be the representation 
of the point P on the surface. 
When the surface is a sphere the projection is called orthographic, gnomonic or 
stereographic according to the positions of the fixed point and plane: the last kind 
is here alone considered; viz. in the stereographic projection the fixed point is on the 
surface of the sphere, and the fixed plane is parallel to the tangent plane at that 
point, and is usually and conveniently taken to pass through the centre of the sphere. 
The stereographic projection is one of those which is free from the defect of dis 
tortion ; it is consequently, and that in a considerable degree, subject to the defect 
of inequality. It possesses in a high degree the important quality of facility of 
construction, viz. any great or small circle on the sphere is represented by a circle in 
the map; and from the general property of the equality of corresponding angles, or 
otherwise, there arise easy rules for the construction of such circles. 
The so-called Mercator’s projection is an instance of a representation which is not 
in the above restricted sense a projection; and which is free from the defect of dis 
tortion: viz. the (equidistant) meridians are here represented by a system of (equidistant) 
parallel lines; and the parallels of latitude by a set of lines at right angles thereto: 
the distance between consecutive parallels in the map being taken in such wise as 
is required to obtain freedom from distortion ; for this purpose the increments of 
latitude and longitude must have in the map the same ratio that they have on the 
sphere, and since in the map the length of a degree of longitude (instead of decreasing 
with the latitude) remains constant, the lengths of the successive degrees of latitude 
in the map must increase with the latitude: the scale of the representation thus 
increases with the latitude, and would for the latitude ± 90° become infinite. 
There is a simple representation of a hemisphere, due to M. Babinet, in which 
the defect of inequality is avoided, viz. the meridians are represented by ellipses having 
their major axes coincident with the diameter through the poles and dividing the 
equator into equal distances, and the parallels by straight lines parallel to the equator.