Mercator’s projection, although indispensable at sea, is of little
value for land maps. For topographical sheets it is obviously
unsuitable; and in cases in which it is required to show large areas
on small scales on an orthomorphic projection, that form should
be chosen which gives two standard parallels (Lambert’s conical
orthomorphic). Mercator’s projection is often used in atlases for
maps of the world. It is not a good projection to select for this
purpose on account of the great exaggeration of scale near the
poles. The misconceptions arising from this exaggeration of scale
may, however, be corrected by the juxtaposition of a map of the
world on an equal-area projection.

It is now necessary to revert to the general consideration of conical projections.

It has been shown that the scales of the projection (fig. 23) as
compared with the sphere are *p*′*q*′ / *pq* = *dp* / *dz* = σ along a meridian,
and *p*′*r* ′ / *pr* ′ = ρ*h*/ sin *z* = σ′ at right angles to a meridian.

Now if σσ′ = 1 the areas are correctly represented, then

*h*ρ

*d*ρ = sin

*zdz*, and integrating 12

*h*ρ

^{2}= C − cos

*z*;

this gives the whole group of *equal-area conical projections*.

As a special case let the pole be the centre of the projected parallels, then when

*z*= 0, ρ = 0, and const = 1, we have

*p*= 2 sin 12

*z*/ δ

*h*

Let *z*_{1} be the co-latitude of some parallel which is to be correctly
represented, then 2*h* sin 12*z*_{1} / δ*h*= sin *z*_{1}, and *h* = cos^{2} 12*z*_{1}; putting
this value of *h* in equation (ii.) the radius of any parallel

*z*sec 12

*z*

_{1}

This is Lambert’s *conical equal-area projection with one standard*
*parallel*, the pole being the centre of the parallels.

If we put *z*_{1}=θ, then *h* = 1, and the meridians are inclined at
their true angles, also the scale at the pole becomes correct, and
equation (iii.) becomes

*z*;

this is the *zenithal equal-area projection*.

Reverting to the general expression for equal-area conical projections

*z*) /

*h*},

we can dispose of C and *h* so that any two selected parallels shall
be their true lengths; let their co-latitudes be *z*_{1} and *z*_{2}, then

*h*(C − cos

*z*

_{1}) = sin

^{2}

*z*

_{1}

*h*(C − cos

*z*

_{2}) = sin

^{2}

*z*

_{2}

from which C and *h* are easily found, and the radii are obtained
from (i.) above. This is H. C. Albers’ *conical equal-area projection*
*with two standard parallels*. The pole is not the centre of
the parallels.

*Projection by Rectangular Spheroidal Co-ordinates.*

If in the simple conical projection the selected parallel is the
equator, this and the other parallels become parallel straight
lines and the meridians are straight lines spaced at equatorial
distances, cutting the parallels at right angles; the parallels are
their true distances apart. This projection is the *simple cylindrical*.
If now we imagine the touching cylinder turned through
a right-angle In such a way as to touch the sphere along any
meridian, a projection is obtained exactly similar to the last,
except that in this case we represent, not parallels and meridians,
but small circles parallel to the given meridian and great circles
at right angles to it. It is clear that the projection is a special
case of conical projection. The position of any point on the
earth’s surface is thus referred, on this projection, to a selected
meridian as one axis, and any great circle at right angles to it as
the other. Or, in other words, any point is fixed by the length
of the perpendicular from it on to the fixed meridian and the
distance of the foot of the perpendicular from some fixed point
on the meridian, these spherical or spheroidal co-ordinates
being plotted as plane rectangular co-ordinates.

The perpendicular is really a plane section of the surface through
the given point at right angles to the chosen meridian, and may be
briefly called a great circle. Such a great circle clearly diverges
from the parallel; the exact difference in latitude and longitude
between the point and the foot of the perpendicular can be at once
obtained by ordinary geodetic formulae, putting the azimuth = 90°.
Approximately the difference of latitude in seconds is *x*^{2} tan φ
cosec 1″ / 2ρν where *x* is the length of the perpendicular, ρ that of
the radius of curvature to the meridian, ν that of the normal terminated
by the minor axis, φ the latitude of the foot of the perpendicular.
The difference of longitude in seconds is approximately *x* sec ρ
cosec 1″ / ν. The resulting error consists principally of an exaggeration
of scale north and south and is approximately equal to sec *x*
(expressing *x* in arc); it is practically independent of the extent in
latitude.

It is on this projection that the 1/2,500 Ordnance maps and the 6-in. Ordnance maps of the United Kingdom are plotted, a meridian being chosen for a group of counties. It is also used for the 1-in., 12 in. and 14 in. Ordnance maps of England, the central meridian chosen being that which passes through a point in Delamere Forest in Cheshire. This projection should not as a rule be used for topographical maps, but is suitable for cadastral plans on account of the convenience of plotting the rectangular co-ordinates of the very numerous trigonometrical or traverse points required in the construction of such plans. As regards the errors involved, a range of about 150 miles each side of the central meridian will give a maximum error in scale in a north and south direction of about 0.1%.

*Elliptical Equal-area Projection.*

In this projection, which is also called Mollweide’s projection the parallels are parallel straight lines and the meridians are ellipses, the central meridian being a straight line at right angles to the equator, which is equally divided. If the whole world is represented on the spherical assumption, the equator is twice the length of the central meridian. Each elliptical meridian has for one axis the central meridian, and for the other the intercepted portion of the equally divided equator. It follows that the meridians 90° east and west of the central meridian form a circle. It is easy to show that to preserve the property of equal areas the distance of any parallel from the equator must be √2 sin δ where π sin φ = 2δ + sin 2δ, φ being the latitude of the parallel. The length of the central meridian from pole to pole = 2 √2, where the radius of the sphere is unity. The length of the equator = 4 √2.

The following equal-area projections may be used to exhibit the entire surface of the globe: Cylindrical equal area, Sinusoidal equal area and Elliptical equal area.

Fig. 26.—Globular Projection. |

*Conventional or Arbitrary Projections.*

These projections are devised for simplicity of drawing and
not for any special properties. The most useful projection of
this class is the *globular projection*. This is a conventional
representation of a hemisphere in which the equator and central
meridian are two equal straight lines at right angles, their intersection
being the centre of the circular boundary. The meridians
divide the equator into equal parts and are arcs of circles passing
through points so determined and the poles. The parallels are
arcs of circles which divide the central and extreme meridians
into equal parts. Thus in fig. 26 NS = EW and each is divided
into equal parts (in this case each division is 10°); the circumference
NESW is also divided into 10° spaces and circular arcs are
drawn through the corresponding points. This is a simple and
effective projection and one well suited for conveying ideas of the