Chapter IV: Part 4

The property of preserving areas is, however, a valuable one when the purpose of the map is to exhibit areas. If, for example, it is desired to give an idea of the area and distribution of the various states comprising the British Empire, this is a fairly good projection. Mercator's, which is commonly used in atlases, preserves local shape at the expense of area, and is valueless for the purpose of showing areas.

Many other projections can be and have been devised, which depend for their construction on a purely geometrical relationship between the imaginary model and the plane. Thus projections may be drawn which are derived from cones which touch or cut the sphere, the parallels being formed by the intersection with the cones of planes parallel to the equator, or by lines drawn radially from the centre. It is convenient to describe all projections which are derived from the model by a simple and direct geometrical construction as "geometrical projections." All other projections may be known as "non-geometrical projections." Geometrical projections, which include perspective projections, are generally speaking of small practical value. They have loomed much more largely on the map-maker's horizon than their importance warrants. It is not going too far to say that the expression "map projection" conveys to most well-informed persons the notion of a geometrical projection; and yet by far the greater number of useful projections are non-geometrical. The notion referred to is no doubt due to the very term "projection," which unfortunately appears to indicate an arrangement of the terrestrial parallels and meridians which can be arrived at by direct geometrical construction. Especially has harm been caused by this idea when dealing with the group of conical projections. The most useful conical projections have nothing to do with the secant cones, but are simply projections in which the meridians are straight lines which converge to a point which is the centre of the circular parallels. The number of really useful geometrical projections may be said to be four: the _equal-area cylindrical_ just described, and the following perspective projections--the _central_, the _stereographic_ and _Clarke's external_.

_Perspective Projections._

In perspective drawings of the sphere, the plane on which the representation is actually made may generally be any plane perpendicular to the line joining the centre of the sphere and the point of vision. If V be the point of vision, P any point on the spherical surface, then p, the point in which the straight line VP intersects the plane of the representation, is the projection of P.

_Orthographic Projection._--In this projection the point of vision is at an infinite distance and the rays consequently parallel; in this case the plane of the drawing may be supposed to pass through the centre of the sphere. Let the circle (fig. 3) represent the plane of the equator on which we propose to make an orthographic representation of meridians and parallels. The centre of this circle is clearly the projection of the pole, and the parallels are projected into circles having the pole for a common centre. The diameters aa´, bb´ being at right angles, let the semicircle bab´ be divided into the required number of equal parts; the diameters drawn through these points are the projections of meridians. The distances of c, of d and of e from the diameter aa´ are the radii of the successive circles representing the parallels. It is clear that, when the points of division are very close, the parallels will be very much crowded towards the outside of the map; so much so, that this projection is not much used.

For an orthographic projection of the globe on a meridian plane let
qnrs (fig. 4) be the meridian, ns the axis of rotation, then qr is the
projection of the equator. The parallels will be represented by
straight lines passing through the points of equal division; these
lines are, like the equator, perpendicular to ns. The meridians will
in this case be ellipses described on ns as a common major axis, the
distances of c, of d and of e from ns being the minor semiaxes.

Let us next construct an orthographic projection of the sphere on the
horizon of any place.

Set off the angle aop (fig. 5) from the radius oa, equal to the
latitude. Drop the perpendicular pP on oa, then P is the projection of
the pole. On ao produced take ob = pP, then ob is the minor semiaxis
of the ellipse representing the equator, its major axis being qr at
right angles to ao. The points in which the meridians meet this
elliptic equator are determined by lines drawn parallel to aob through
the points of equal subdivision cdefgh. Take two points, as d and g,
which are 90° apart, and let ik be their projections on the equator;
then i is the pole of the meridian which passes through k. This
meridian is of course an ellipse, and is described with reference to i
exactly as the equator was described with reference to P. Produce io
to l, and make lo equal to half the shortest chord that can be drawn
through i; then lo is the semiaxis of the elliptic meridian, and the
major axis is the diameter perpendicular to iol.

For the parallels: let it be required to describe the parallel whose
co-latitude is u; take pm = pn = u, and let m´n´ be the projections of
m and n on oPa; then m´n´ is the minor axis of the ellipse
representing the parallel. Its centre is of course midway between m´
and n´, and the greater axis is equal to mn. Thus the construction is
obvious. When pm is less than pa the whole of the ellipse is to be
drawn. When pm is greater than pa the ellipse touches the circle in
two points; these points divide the ellipse into two parts, one of
which, being on the other side of the meridian plane aqr, is
invisible. Fig. 6 shows the complete orthographic projection.

_Stereographic Projection._--In this case the point of vision is on the surface, and the projection is made on the plane of the great circle whose pole is V. Let kplV (fig. 7) be a great circle through the point of vision, and ors the trace of the plane of projection. Let c be the centre of a small circle whose radius is cp = cl; the straight line pl represents this small circle in orthographic projection.

We have first to show that the stereographic projection of the small
circle pl is itself a circle; that is to say, a straight line through
V, moving along the circumference of pl, traces a circle on the plane
of projection ors. This line generates an oblique cone standing on a
circular base, its axis being cV (since the angle pVc = angle cVl);
this cone is divided symmetrically by the plane of the great circle
kpl, and also by the plane which passes through the axis Vc,
perpendicular to the plane kpl. Now Vr·Vp, being = Vo sec kVp·Vk cos
kVp = Vo·Vk, is equal to Vs·Vl; therefore the triangles Vrs, Vlp are
similar, and it follows that the section of the cone by the plane rs
is similar to the section by the plane pl. But the latter is a circle,
hence also the projection is a circle; and since the representation of
every infinitely small circle on the surface is itself a circle, it
follows that in this projection the representation of small parts is
strictly similar. Another inference is that the angle in which two
lines on the sphere intersect is represented by the same angle in the
projection. This may otherwise be proved by means of fig. 8, where Vok
is the diameter of the sphere passing through the point of vision, fgh
the plane of projection, kt a great circle, passing of course through
V, and ouv the line of intersection of these two planes. A tangent
plane to the surface at t cuts the plane of projection in the line rvs
perpendicular to ov; tv is a tangent to the circle kt at t, tr and ts
are any two tangents to the surface at t. Now the angle vtu (u being
the projection of t) is 90° - otV = 90° - oVt = ouV = tuv, therefore
tv is equal to uv; and since tvs and uvs are right angles, it follows
that the angles vts and vus are equal. Hence the angle rts also is
equal to its projection rus; that is, any angle formed by two
intersecting lines on the surface is truly represented in the
stereographic projection.

In this projection, therefore, angles are correctly represented and every small triangle is represented by a similar triangle. Projections having this property of similar representation of small parts are called _orthomorphic_, _conform_ or _conformable_. The word orthomorphic, which was introduced by Germain[27] and adopted by Craig,[28] is perhaps the best to use.

Since in orthomorphic projections very small figures are correctly represented, it follows that the scale is the same in all directions round a point in its immediate neighbourhood, and orthomorphic projections may be defined as possessing this property. There are many other orthomorphic projections, of which the best known is Mercator's. These are described below.

We have seen that the stereographic projection of any circle of the sphere is itself a circle. But in the case in which the circle to be projected passes through V, the projection becomes, for a great circle, a line through the centre of the sphere; otherwise, a line anywhere. It follows that meridians and parallels are represented in a projection on the horizon of any place by two systems of orthogonally cutting circles, one system passing through two fixed points, namely, the poles; and the projected meridians as they pass through the poles show the proper differences of longitude.

To construct a stereographic projection of the sphere on the horizon
of a given place. Draw the circle vlkr (fig. 9) with the diameters
kv, lr at right angles; the latter is to represent the central
meridian. Take koP equal to the co-latitude of the given place, say u;
draw the diameter PoP¹, and vP, vP´ cutting lr in pp´: these are the
projections of the poles, through which all the circles representing
meridians have to pass. All their centres then will be in a line smn
which crosses pp´ at right angles through its middle point m. Now to
describe the meridian whose west longitude is [omega], draw pn making
the angle opn = 90° - [omega], then n is the centre of the required
circle, whose direction as it passes through p will make an angle opg
= [omega] with pp´. The lengths of the several lines are

op = tan ½u; op´ = cot ½u; om = cot u; mn = cosec u cot [omega].

Again, for the parallels, take Pb = Pc equal to the co-latitude, say
c, of the parallel to be projected; join vb, vc cutting lr in e, d.
Then ed is the diameter of the circle which is the required
projection; its centre is of course the middle point of ed, and the
lengths of the lines are

od = tan ½(u - c); oe = tan ½(u + c).

The line sn itself is the projection of a parallel, namely, that of
which the co-latitude c = 180° - u, a parallel which passes through
the point of vision.

Notwithstanding the facility of construction, the stereographic projection is not much used in map-making. It is sometimes used for maps of the hemispheres in atlases, and for star charts.

_External Perspective Projection._--We now come to the general case in which the point of vision has any position outside the sphere. Let abcd (fig. 10) be the great circle section of the sphere by a plane passing through c, the central point of the portion of surface to be represented, and V the point of vision. Let pj perpendicular to Vc be the plane of representation, join mV cutting pj in f, then f is the projection of any point m in the circle abc, and ef is the representation of cm.

Let the angle com = u, Ve = k, Vo = h, ef = [rho]; then, since ef: eV
= mg:gV, we have [rho] = k sin u/(h + cos u), which gives the law
connecting a spherical distance u with its rectilinear representation
[rho]. The relative scale at any point in this system of projection is
given by

[sigma] = d[rho]/du, [sigma]´ = [rho]/sin u,
[sigma] = k(1 + h cos u)/(h + cos u)²; [sigma]´ = k/(h + cos u),

the former applying to measurements made in a direction which passes
through the centre of the map, the latter to the transverse direction.
The product [sigma][sigma]´ gives the exaggeration of areas. With
respect to the alteration of angles we have [Sigma] = (h + cos u)/(l +
kcos u), and the greatest alteration of angle is

/h - 1 u \
= sin^(-1) ( ----- tan² --- ).
\h + 1 2 /

This vanishes when h = 1, that is if the projection be stereographic;
or for u = 0, that is at the centre of the map. At a distance of 90°
from the centre, the greatest alteration is 90° - 2 cot^(-1) [root]h.
(See _Phil. Mag._ 1862.)

_Clarke's Projection._--The constants h and k can be determined, so
that the total misrepresentation, viz.:
_
/ [beta]
M = | {([sigma] - 1)² + ([sigma]´ - 1)²} sin u du,
_/ 0

shall be a minimum, [beta] being the greatest value of u, or the
spherical radius of the map. On substituting the expressions for
[sigma] and [sigma]´ the integration is effected without difficulty.
Put

[lambda] = (1 - cos [beta])/(h + cos [beta]); [nu] = (h - 1)[lambda],

H = [nu] - (h + 1) log_e ([lambda] + 1), H´ = [lambda](2 - [nu] + (1/3)[nu]²)/(h + 1).

Then the value of M is

M = 4 sin² ½[beta] + 2kH + k²H´.

When this is a minimum,

dM/dh = 0; dM/dk = 0

:. kH´ + H = 0; 2 dH/dh + k dh H´/dh = 0.

Therefore M = 4 sin² ½[beta] - H²/H^1, and h must be determined so as
to make H²:H´ a maximum. In any particular case this maximum can only
be ascertained by trial, that is to say, log H² - log H´ must be
calculated for certain equidistant values of h, and then the
particular value of h which corresponds to the required maximum can
be obtained by interpolation. Thus we find that if it be required to
make the best possible perspective representation of a hemisphere, the
values of h and k are h = 1.47 and k = 2.034; so that in this case

2.034 sin u
[rho] = ------------.
1.47 + cos u

For a map of Africa or South America, the limiting radius [beta] we
may take as 40°; then in this case

2.543 sin u
[rho] = -------------.
1.625 + cos u

For Asia, [beta] = 54, and the distance h of the point of sight in
this case is 1.61. Fig. 11 is a map of Asia having the meridians and
parallels laid down on this system.

Fig. 12 is a perspective representation of more than a hemisphere, the
radius [beta] being 108°, and the distance h of the point of vision,
1.40.

The co-ordinates xy of any point in this perspective may be expressed
in terms of latitude and longitude of the corresponding point on the
sphere in the following manner. The co-ordinates originating at the
centre take the central meridian for the axis of y and a line
perpendicular to it for the axis of x. Let the latitude of the point
G, which is to occupy the centre of the map, be [gamma]; if [phi],
[omega] be the latitude and longitude of any point P (the longitude
being reckoned from the meridian of G), u the distance PG, and [mu]
the azimuth of P at G, then the spherical triangle whose sides are 90°
- [gamma], 90° - [phi], and u gives these relations--

sin u sin [mu] = cos [phi] sin [omega],
sin u cos [mu] = cos [gamma] sin [phi] - sin [gamma] cos [phi] cos [omega],
cos u = sin [gamma] sin [phi] + cos [gamma] cos [phi] cos [omega].

Now x = [rho] sin [mu], y = [rho] cos [mu], that is,

x cos [phi] sin [omega]
--- = -------------------------------------------------------------,
k h + sin [gamma] sin [phi] + cos [gamma] cos [phi] cos [omega]

y cos [gamma] sin [phi] - sin [gamma] cos [phi] cos [omega]
--- = -------------------------------------------------------------,
k h + sin [gamma] sin [phi] + cos [gamma] cos [phi] cos [omega]

by which x and y can be computed for any point of the sphere. If from
these equations we eliminate [omega], we get the equation to the
parallel whose latitude is [phi]; it is an ellipse whose centre is in
the central meridian, and its greater axis perpendicular to the same.
The radius of curvature of this ellipse at its intersection with the
centre meridian is k cos [phi] / (h sin [gamma] + sin [phi]).

The elimination of [phi] between x and y gives the equation of the
meridian whose longitude is [omega], which also is an ellipse whose
centre and axes may be determined.

The following table contains the computed co-ordinates for a map of
Africa, which is included between latitudes 40° north and 40° south
and 40° of longitude east and west of a central meridian.

+-----+------------------------------------------------------------------------------+
| | Values of x and y. |
|[phi]+--------------+---------------+---------------+---------------+---------------+
| | [omega] = 0° | [omega] = 10° | [omega] = 20° | [omega] = 30° | [omega] = 40° |
+-----+--------------+---------------+---------------+---------------+---------------+
| 0° | x = 0.00 | 9.69 | 19.43 | 29.25 | 39.17 |
| | y = 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| | | | | | |
| 10° | x = 0.00 | 9.60 | 19.24 | 28.95 | 38.76 |
| | y = 9.69 | 9.75 | 9.92 | 10.21 | 10.63 |
| | | | | | |
| 20° | x = 0.00 | 9.32 | 18.67 | 28.07 | 37.53 |
| | y = 19.43 | 19.54 | 19.87 | 20.43 | 21.25 |
| | | | | | |
| 30° | x = 0.00 | 8.84 | 17.70 | 26.56 | 35.44 |
| | y = 29.25 | 29.40 | 29.87 | 30.67 | 31.83 |
| | | | | | |
| 40° | x = 0.00 | 8.15 | 16.28 | 24.39 | 32.44 |
| | y = 39.17 | 39.36 | 39.94 | 40.93 | 42.34 |
+-----+--------------+---------------+---------------+---------------+---------------+

_Central or Gnomonic (Perspective) Projection._--In this projection the eye is imagined to be at the centre of the sphere. It is evident that, since the planes of all great circles of the sphere pass through the centre, the representations of all great circles on this projection will be straight lines, and this is the special property of the _central projection_, that any great circle (i.e. shortest line on the spherical surface) is represented by a straight line. The plane of projection may be either parallel to the plane of the equator, in which case the parallels are represented by concentric circles and the meridians by straight lines radiating from the common centre; or the plane of projection may be parallel to the plane of some meridian, in which case the meridians are parallel straight lines and the parallels are hyperbolas; or the plane of projection may be inclined to the axis of the sphere at any angle [lambda].

In the latter case, which is the most general, if [theta] is the angle
any meridian makes (on paper) with the central meridian, [alpha] the
longitude of any point P with reference to the central meridian, l the
latitude of P, then it is clear that the central meridian is a
straight line at right angles to the equator, which is also a straight
line, also tan [theta] = sin [lambda] tan [alpha], and the distance of
p, the projection of P, from the equator along its meridian is (on
paper) m sec [alpha] sin l / sin (l + x), where tan x = cot [lambda]
cos [alpha], and m is a constant which defines the scale.

The three varieties of the central projection are, as is the case with other perspective projections, known as polar, meridian or horizontal, according to the inclination of the plane of projection.

FIG. 14.--Part of the Atlantic Ocean on a Meridian Central Projection. The shortest path between any two points is shown on this projection by a straight line.]

Fig. 14 is an example of a _meridian central projection_ of part of the Atlantic Ocean. The term "gnomonic" was applied to this projection because the projection of the meridians is a similar problem to that of the graduation of a sun-dial. It is, however, better to use the term "central," which explains itself. The central projection is useful for the study of direct routes by sea and land. The United States Hydrographic Department has published some charts on this projection. False notions of the direction of shortest lines, which are engendered by a study of maps on Mercator's projection, may be corrected by an inspection of maps drawn on the central projection.

There is no projection which accurately possesses the property of showing shortest paths by straight lines when applied to the spheroid; one which very nearly does so is that which results from the intersection of terrestrial normals with a plane.

We have briefly reviewed the most important projections which are derived from the sphere by direct geometrical construction, and we pass to that more important branch of the subject which deals with projections which are not subject to this limitation.

_Conical Projections._

Conical projections are those in which the parallels are represented by concentric circles and the meridians by equally spaced radii. There is no necessary connexion between a conical projection and any touching or secant cone. Projections for instance which are derived by geometrical construction from secant cones are very poor projections, exhibiting large errors, and they will not be discussed. The name conical is given to the group embraced by the above definition, because, as is obvious, a projection so drawn can be bent round to form a cone. The simplest and, at the same time, one of the most useful forms of conical projection is the following:

_Conical Projection with Rectified Meridians and Two Standard Parallels._--In some books this has been, most unfortunately, termed the "secant conical," on account of the fact that there are two parallels of the correct length. The use of this term in the past has caused much confusion. Two selected parallels are represented by concentric circular arcs of their true lengths; the meridians are their radii. The degrees along the meridians are represented by their true lengths; and the other parallels are circular arcs through points so determined and are concentric with the chosen parallels.

Thus in fig. 15 two parallels Gn and G´n´ are represented by their
true lengths on the sphere; all the distances along the meridian PGG´,
pnn´ are the true spherical lengths rectified.

Let [gamma] be the co-latitude of Gn; [gamma]´ that of Gn´; [omega] be
the true difference of longitude of PGG´ and pnn´; h[omega] be the
angle at O; and OP = z, where Pp is the representation of the pole.
Then the true length of parallel Gn on the sphere is [omega] sin
[gamma], and this is equal to the length on the projection, i.e.
[omega] sin [gamma] = h[omega](z + [gamma]); similarly [omega] sin
[gamma]´= h[omega](z + [gamma]´).

The radius of the sphere is assumed to be unity, and z and [gamma] are
expressed in circular measure. Hence h = sin [gamma]/(z + [gamma]) =
sin [gamma]´(z + [gamma]´); from this h and z are easily found.

In the above description it has been assumed that the two errorless parallels have been _selected_. But it is usually desirable to impose some condition which itself will fix the errorless parallels. There are many conditions, any one of which may be imposed. In fig. 15 let Cm and C´m´ represent the extreme parallels of the map, and let the co-latitudes of these parallels be c and c´, then any one of the following conditions may be fulfilled:--

(a) The errors of scale of the extreme parallels may be made equal and may be equated to the error of scale of the parallel of maximum error (which is near the mean parallel).

(b) Or the errors of scale of the extreme parallels may be equated to that of the mean parallel. This is not so good a projection as (a).

(c) Or the absolute errors of the extreme and mean parallels may be equated.

(d) Or in the last the parallel of maximum error may be considered instead of the mean parallel.

(e) Or the mean length of all the parallels may be made correct. This is equivalent to making the total area between the extreme parallels correct, and must be combined with another condition, for example, that the errors of scale on the extreme parallels shall be equal.

We will now discuss (a) above, viz. a conical projection with
rectified meridians and two standard parallels, the scale errors of
the extreme parallels and parallel of maximum error being equated.

Since the scale errors of the extreme parallels are to be equal,

h(z + c) h(z + c´) c´ sin c - c sin c´
-------- - 1 = --------- - 1, whence z = ------------------- (i.)
sin c sin c´ sin c´ - sin c

The error of scale along any parallel (near the centre), of which the
co-latitude is b is

1 - {h(z + b)/sin b}. (ii.)

This is a maximum when

(tan b) - b = z, whence b is found.

Also

h(z + b) h(z + c)
1 - -------- = -------- - 1 whence h is found. (iii.)
sin b sin c

For the errorless parallels of co-latitudes [gamma] and [gamma]´ we
have

h = (z + [gamma])/sin [gamma] = (z + [gamma]´)/sin [gamma]´.

If this is applied to the case of a map of South Africa between the
limits 15° S. and 35° S. (see fig. 16) it will be found that the
parallel of maximum error is 25° 20´; the errorless parallels, to the
nearest degree, are those of 18° and 32°. The greatest scale error in
this case is about 0.7%.

In the above account the earth has been treated as a sphere. Of course
its real shape is approximately a spheroid of revolution, and the
values of the axes most commonly employed are those of Clarke or of
Bessel. For the spheroid, formulae arrived at by the same principles
but more cumbrous in shape must be used. But it will usually be
sufficient for the selection of the errorless parallels to use the
simple spherical formulae given above; then, having made the selection
of these parallels, the true spheroidal lengths along the meridians
between them can be taken out of the ordinary tables (such as those
published by the Ordnance Survey or by the U.S. Coast and Geodetic
Survey). Thus, if a1, a2, are the lengths of 1° of the errorless
parallels (taken from the tables), d the true rectified length of the
meridian arc between them (taken from the tables),

h = {(a2 - a1)/d}180/[pi],

and the radius on paper of parallel, a1 is a1d/(a2 - a1), and the
radius of any other parallel = radius of a1 ± the true meridian
distance between the parallels.

This class of projection was used for the 1/1,000,000 Ordnance map of
the British Isles. The three maximum scale errors in this case work
out to 0.23%, the range of the projection being from 50° N. to 61° N.,
and the errorless parallels are 59° 31´ and 51°44´.

Where no great refinement is required it will be sufficient to take
the errorless parallels as those distant from the extreme parallels
about one-sixth of the total range in latitude. Thus suppose it is
required to plot a projection for India between latitudes 8° and 40°
N. By this rough rule the errorless parallels should be distant from
the extreme parallels about 32°/6, i.e. 5° 20´; they should therefore,
to the nearest degree, be 13° and 35° N. The maximum scale errors will
be about 2%.

The scale errors vary approximately as the square of the range of
latitude; a rough rule is, largest scale error = L²/50,000, where L is
the range in the latitude in degrees. Thus a country with a range of
7° in latitude (nearly 500 m.) can be plotted on this projection with
a maximum linear scale error (along a parallel) of about 0.1%;[29]
there is no error along any meridian. It is immaterial with this
projection (or with any conical projection) what the extent in
longitude is. It is clear that this class of projection is accurate,
simple and useful.

FIG. 16.--South Africa on a conical projection with rectified
meridians and two standard parallels. Scale 800 m. to 1 in.]

In the projections designated by (c) and (d) above, absolute errors of
length are considered in the place of errors of scale, i.e. between
any two meridians (c) the absolute errors of length of the extreme
parallels are equated to the absolute error of length of the middle
parallel. Using the same notation

h(z + c) - sin c = h(z + c´) - sin c´ = -h(z + ½c + ½c´) - sin ½(c + c´).

L. Euler, in the _Acta Acad. Imp. Petrop._ (1778), first discussed
this projection.

If a map of Asia between parallels 10° N. and 70° N. is constructed on
this system, we have c = 20°, c´ = 80°, whence from the above
equations z = 66.7° and h = .6138. The absolute errors of length along
parallels 10°, 40° and 70° between any two meridians are equal but the
scale errors are respectively 5, 6.7, and 15%.

The modification (d) of this projection was selected for the
1:1,000,000 map of _India and Adjacent Countries_ under publication by
the Survey of India. An account of this is given in a pamphlet
produced by that department in 1903. The limiting parallels are 8° and
40° N., and the parallel of greatest error is 23° 40´ 51´´. The errors
of scale are 1.8, 2.3, and 1.9%.

It is not as a rule desirable to select this form of the projection.
If the surface of the map is everywhere equally valuable it is clear
that an arrangement by which errors of scale are larger towards the
pole than towards the equator is unsound, and it is to be noted that
in the case quoted the great bulk of the land is in the north of the
map. Projection (a) would for the same region have three equal maximum
scale errors of 2%. It may be admitted that the practical difference
between the two forms is in this case insignificant, but linear scale
errors should be reduced as much as possible in maps intended for
general use.

f. In the fifth form of the projection, the total area of the
projection between the extreme parallels and any two meridians is
equated to the area of the portion of the sphere which it represents,
and the errors of scale of the extreme parallels are equated. Then it
is easy to show that

z = (c´ sin c - c sin c´) / (sin c´ - sin c);
h = (cos c - cos c´) / (c´ - c){z + ½(c + c´)}.

It can also be shown that any other zone of the same range in latitude
will have the same scale errors along its limiting parallels. For
instance, a series of projections may be constructed for zones, each
having a range of 10° of latitude, from the equator to the pole.
Treating the earth as a sphere and using the above formulae, the
series will possess the following properties: the meridians will all
be true to scale, the area of each zone will be correct, the scale
errors of the limiting parallels will all be the same, so that the
length of the upper parallel of any zone will be equal to that of the
lower parallel of the zone above it. But the curvatures of these
parallels will be different, and two adjacent zones will not fit but
will be capable of exact rolling contact. Thus a very instructive flat
model of the globe may be constructed which will show by suitably
arranging the points of contact of the zones the paths of great
circles on the sphere. The flat model was devised by Professor J. D.
Everett, F.R.S., who also pointed out that the projection had the
property of the equality of scale errors of the limiting parallels for
zones of the same width. The projection may be termed _Everett's
Projection_.

_Simple Conical Projection._--If in the last group of projections the two selected parallels which are to be errorless approach each other indefinitely closely, we get a projection in which all the meridians are, as before, of the true rectified lengths, in which one parallel is errorless, the curvature of that parallel being clearly that which would result from the unrolling of a cone touching the sphere along the parallel represented. And it was in fact originally by a consideration of the tangent cone that the whole group of conical projections came into being. The quasi-geometrical way of regarding conical projections is legitimate in this instance.

The simple conical projection is therefore arrived at in this way:
imagine a cone to touch the sphere along any selected parallel, the
radius of this parallel on paper (Pp, fig. 17) will be r cot [phi],
where r is the radius of the sphere and [phi] is the latitude; or if
the spheroidal shape is taken into account, the radius of the parallel
on paper will be [nu] cot [phi] where [nu] is the normal terminated by
the minor axis (the value [nu] can be found from ordinary geodetic
tables). The meridians are generators of the cone and every parallel
such as HH´ is a circle, concentric with the selected parallel Pp and
distant from it the true rectified length of the meridian arc between
them.

This projection has no merits as compared with the group just
described. The errors of scale along the parallels increase rapidly as
the selected parallel is departed from, the parallels on paper being
always too large. As an example we may take the case of a map of South
Africa of the same range as that of the example given in (a) above,
viz. from 15° S. to 35° S. Let the selected parallel be 25° S.; the
radius of this parallel on paper (taking the radius of the sphere as
unity) is cot 25°; the radius of parallel 35° S. = radius of 25° -
meridian distance between 25° and 35° = cot 25° - 10[pi]/180 = 1·970.
Also h = sin of selected latitude = sin 25°, and length on paper along
parallel 35° of [omega]° = [omega]h × 1.970 = [omega] × 1.970 × sin
25°,

but length on sphere of [omega] = [omega] cos 35°,

1.970 sin 25°
hence scale error = ------------- - 1 = 1.6%,
cos 35°

an error which is more than twice as great as that obtained by method
(a).

_Bonne's Projection._--This projection, which is also called the "modified conical projection," is derived from the simple conical, just described, in the following way: a central meridian is chosen and drawn as a straight line; degrees of latitude spaced at the true rectified distances are marked along this line; the parallels are concentric circular arcs drawn through the proper points on the central meridian, the centre of the arcs being fixed by describing one chosen parallel with a radius of [nu] cot [phi] as before; the meridians on each side of the central meridian are drawn as follows: along _each_ parallel distances are marked equal to the true lengths along the parallels on sphere or spheroid, and the curve through corresponding points so fixed are the meridians (fig. 18).

This system is that which was adopted in 1803 by the "Dépôt de la Guerre" for the map of France, and is there known by the title of _Projection de Bonne_. It is that on which the ordnance survey map of Scotland on the scale of 1 in. to a mile is constructed, and it is frequently met with in ordinary atlases. It is ill-adapted for countries having great extent in longitude, as the intersections of the meridians and parallels become very oblique--as will be seen on examining the map of Asia in most atlases.

If [phi]0 be taken as the latitude of the centre parallel, and
co-ordinates be measured from the intersection of this parallel with
the central meridian, then, if [rho] be the radius of the parallel of
latitude [phi], we have [rho] = cot [phi]0 + [phi]0 - [phi]. Also, if
S be a point on this parallel whose co-ordinates are x, y, so that VS
= [rho], and [theta] be the angle VS makes with the central meridian,
then [rho][theta] = [omega] cos [phi]; and x = [rho] sin [theta], y =
cot [phi]0 - [rho] cos [theta].

The projection has the property of equal areas, since each small element bounded by two infinitely close parallels is equal in length and width to the corresponding element on the sphere or spheroid. Also all the meridians cross the chosen parallel (but no other) at right angles, since in the immediate neighbourhood of that parallel the projection is identical with the simple conical projection. Where an equal-area projection is required for a country having no great extent in longitude, such as France, Scotland or Madagascar, this projection is a good one to select.

_Sinusoidal Equal-area Projection._--This projection, which is sometimes known as Sanson's, and is also sometimes incorrectly called Flamsteed's, is a particular case of Bonne's in which the selected parallel is the equator. The equator is a straight line at right angles to the central meridian which is also a straight line. Along the central meridian the latitudes are marked off at the true rectified distances, and from points so found the parallels are drawn as straight lines parallel to the equator, and therefore at right angles to the central meridian. True rectified lengths are marked along the parallels and through corresponding points the meridians are drawn. If the earth is treated as a sphere the meridians are clearly sine curves, and for this reason d'Avezac has given the projection the name sinusoidal. But it is equally easy to plot the spheroidal lengths. It is a very suitable projection for an equal-area map of Africa.

_Werner's Projection._--This is another limiting case of Bonne's equal-area projection in which the selected parallel is the pole. The parallels on paper then become incomplete circular arcs of which the pole is the centre. The central meridian is still a straight line which is cut by the parallels at true distances. The projection (after Johann Werner, 1468-1528), though interesting, is practically useless.

_Polyconic Projections._

These pseudo-conical projections are valuable not so much for their intrinsic merits as for the fact that they lend themselves to tabulation. There are two forms, the _simple_ or _equidistant polyconic_, and the _rectangular polyconic_.

_The Simple Polyconic._--If a cone touches the sphere or spheroid along a parallel of latitude [phi] and is then unrolled, the parallel will on paper have a radius of [nu] cot [phi], where [nu] is the normal terminated by the minor axis. If we imagine a series of cones, each of which touches one of a selected series of parallels, the apex of each cone will lie on the prolonged axis of the spheroid; the generators of each cone lie in meridian planes, and if each cone is unrolled and the generators in any one plane are superposed to form a straight central meridian, we obtain a projection in which the central meridian is a straight line and the parallels are circular arcs each of which has a different centre which lies on the prolongation of the central meridian, the radius of any parallel being [nu] cot [phi].

So far the construction is the same for both forms of polyconic. In the _simple polyconic_ the meridians are obtained by measuring outwards from the central meridian along each parallel the true lengths of the degrees of longitude. Through corresponding points so found the meridian curves are drawn. The resulting projection is accurate near the central meridian, but as this is departed from the parallels increasingly separate from each other, and the parallels and meridians (except along the equator) intersect at angles which increasingly differ from a right angle. The real merit of the projection is that each particular parallel has for every map the same absolute radius, and it is thus easy to construct tables which shall be of universal use. This is especially valuable for the projection of single sheets on comparatively large scales. A sheet of a degree square on a scale of 1:250,000 projected in this manner differs inappreciably from the same sheet projected on a better system, e.g. an orthomorphic conical projection or the conical with rectified meridians and two standard parallels; there is thus the advantage that the simple polyconic when used for single sheets and large scales is a sufficiently close approximation to the better forms of conical projection. The simple polyconic is used by the topographical section of the general staff, by the United States coast and geodetic survey and by the topographical division of the U.S. geological survey. Useful tables, based on Clarke's spheroid of 1866, have been published by the war office and by the U.S. coast and geodetic survey.

_Rectangular Polyconic._--In this the central meridian and the parallels are drawn as in the simple polyconic, but the meridians are curves which cut the parallels at right angles.

In this case, let P (fig. 20) be the north pole, CPU the central
meridian, U, U´ points in that meridian whose co-latitudes are z and
z+dz, so that UU´ = dz. Make PU = z, UC = tan z, U´C´ = tan (z + dz);
and with CC´ as centres describe the arcs UQ, U´Q´, which represent
the parallels of co-latitude z and z+dz. Let PQQ´ be part of a
meridian curve cutting the parallels at right angles. Join CQ, C´Q´;
these being perpendicular to the circles will be tangents to the
curve. Let UCQ = 2[alpha], UC´Q´ = 2([alpha] + d[alpha]), then the
small angle CQC´, or the angle between the tangents at QQ´, will =
2d[alpha]. Now

CC´ = C´U´- CU - UU´ = tan (z + dz) - tan z - dz = tan² z dz.

The tangents CQ, C´Q´ will intersect at q, and in the triangle CC´q
the perpendicular from C on C´q is (omitting small quantities of the
second order) equal to either side of the equation

tan² z dz sin 2[alpha] = -2 tan zd [alpha].
-tan z dz = 2d[alpha] / sin 2[alpha],

which is the differential equation of the meridian: the integral is
tan [alpha] = [omega] cos z, where [omega], a constant, determines a
particular meridian curve. The distance of Q from the central
meridian, tan z sin 2[alpha], is equal to

2 tan z tan [alpha] 2[omega] sin z
------------------- = -------------------------
1 + tan² [alpha] 1 + [omega]² cos² [alpha]

At the equator this becomes simply 2[omega]. Let any equatorial point
whose actual longitude is 2[omega] be represented by a point on the
developed equator at the distance 2[omega] from the central meridian,
then we have the following very simple construction (due to O'Farrell
of the ordnance survey). Let P (fig. 21) be the pole, U any point in
the central meridian, QUQ´ the represented parallel whose radius CU =
tan z. Draw SUS´ perpendicular to the meridian through U; then to
determine the point Q, whose longitude is, say, 3°, lay off US equal
to half the true length of the arc of parallel on the sphere, i.e. 1°
30´ to radius sin z, and with the centre S and radius SU describe a
circular arc, which will intersect the parallel in the required point
Q. For if we suppose 2[omega] to be the longitude of the required
point Q, US is by construction = [omega] sin z, and the angle
subtended by SU at C is

/[omega] sin z\
tan^(-1) ( ------------- ) = tan^(-1) ([omega] cos z) = [alpha],
\ tan z /

and therefore UCQ = 2[alpha] as it should be. The advantages of this
method are that with a remarkably simple and convenient mode of
construction we have a map in which the parallels and meridians
intersect at right angles.

Fig. 22 is a representation of this system of the continents of Europe and Africa, for which it is well suited. For Asia this system would not do, as in the northern latitudes, say along the parallel of 70°, the representation is much cramped.

With regard to the distortion in the map of Africa as thus constructed, consider a small square in latitude 40° and in 40° longitude east or west of the central meridian, the square being so placed as to be transformed into a rectangle. The sides, originally unity, became 0.95 and 1.13, and the area 1.08, the diagonals intersecting at 90° ± 9° 56´. In Clarke's perspective projection a square of unit side occupying the same position, when transformed to a rectangle, has its sides 1.02 and 1.15, its area 1.17, and its diagonals intersect at 90° ± 7° 6´. The latter projection is therefore the best in point of "similarity," but the former represents areas best. This applies, however, only to a particular part of the map; along the equator towards 30° or 40° longitude, the polyconic is certainly inferior, while along the meridian it is better than the perspective--except, of course, near the centre. Upon the whole the more even distribution of distortion gives the advantage to the perspective system. For single sheets on large scales there is nothing to choose between this projection and the simple polyconic. Both are sensibly perfect representations. The rectangular polyconic is occasionally used by the topographical section of the general staff.

_Zenithal Projections._

Some point on the earth is selected as the central point of the map; great circles radiating from this point are represented by straight lines which are inclined at their true angles at the point of intersection. Distances along the radiating lines vary according to any law outwards from the centre. It follows (on the spherical assumption), that circles of which the selected point is the centre are also circles on the projection. It is obvious that all perspective projections are zenithal.

_Equidistant Zenithal Projection._--In this projection, which is commonly called the "equidistant projection," any point on the sphere being taken as the centre of the map, great circles through this point are represented by straight lines of the true rectified lengths, and intersect each other at the true angles.

In the general case--

if z1 is the co-latitude of the centre of the map, z the co-latitude
of any other point, [alpha] the difference of longitude of the two
points, A the azimuth of the line joining them, and c the spherical
length of the line joining them, then the position of the intersection
of any meridian with any parallel is given (on the spherical
assumption) by the solution of a simple spherical triangle.

Thus--

let tan [theta] = tan z cos [alpha], then cos c = cos z sec [theta]
cos (z - [theta]), and sin A = sin z sin [alpha] cosec c.

The most useful case is that in which the central point is the pole; the meridians are straight lines inclined to each other at the true angular differences of longitude, and the parallels are equidistant circles with the pole as centre. This is the best projection to use for maps exhibiting the progress of polar discovery, and is called the _polar equidistant projection_. The errors are smaller than might be supposed. There are no scale errors along the meridians, and along the parallels the scale error is (z / sin x) - 1, where z is the co-latitude of the parallel. On a parallel 10° distant from the pole the error of scale is only 0.5%.

_General Theory of Zenithal Projections._--For the sake of simplicity
it will be at first assumed that the pole is the centre of the map,
and that the earth is a sphere. According to what has been said above,
the meridians are now straight lines diverging from the pole, dividing
the 360° into equal angles; and the parallels are represented by
circles having the pole as centre, the radius of the parallel whose
co-latitude is z being [rho], a certain function of z. The particular
function selected determines the nature of the projection.

Let Ppq, Prs (fig. 23) be two contiguous meridians crossed by
parallels rp, sq, and Op´q´, Or´s´ the straight lines representing
these meridians. If the angle at P is d[mu], this also is the value of
the angle at O. Let the co-latitude

Pp = z, Pq = z + dz; Op´ = [rho], Oq´ = [rho] + d[rho],

the circular arcs p´r´, q´s´ representing the parallels pr, qs. If the
radius of the sphere be unity,

p´q´ = d[rho]; p´r´ = [rho] d[mu],
pq = dz; pr = sin z d[mu].

Put

[sigma] = d[rho]/dz; [sigma]´ = [rho]/sin z,

then p´q´ = [sigma]pq and p´r´ = [sigma]´pr. That is to say, [sigma],
[sigma]´ may be regarded as the relative scales, at co-latitude z, of
the representation, [sigma] applying to meridional measurements,
[sigma]´ to measurements perpendicular to the meridian. A small square
situated in co-latitude z, having one side in the direction of the
meridian--the length of its side being i--is represented by a
rectangle whose sides are i[sigma] and i[sigma]´; its area
consequently is i²[sigma][sigma]´.

If it were possible to make a perfect representation, then we should have [sigma] = 1, [sigma]´ = 1 throughout. This, however, is impossible. We may make [sigma] = 1 throughout by taking [rho] = z. This is the _Equidistant Projection_ just described, a very simple and effective method of representation.

Or we may make [sigma]´= 1 throughout. This gives [rho] = sin z, a perspective projection, namely, the _Orthographic_.

Or we may require that areas be strictly represented in the development. This will be effected by making [sigma][sigma]´ = 1, or [rho]d[rho] = sin z dz, the integral of which is [rho] = 2 sin ½z, which is the _Zenithal Equal-area Projection_ of Lambert, sometimes, though wrongly referred to as _Lorgna's Projection_ after Antonio Lorgna (b. 1736). In this system there is misrepresentation of form, but no misrepresentation of areas.

Or we may require a projection in which all small parts are to be represented in their true forms i.e. an orthomorphic projection. For instance, a small square on the spherical surface is to be represented as a small square in the development. This condition will be attained by making [sigma] = [sigma]´, or d[rho]/[rho] = dz/sin z, the integral of which is, c being an arbitrary constant, [rho] = c tan ½z. This, again, is a perspective projection, namely, the _Stereographic_. In this, though all small parts of the surface are represented in their correct shapes, yet, the scale varying from one part of the map to another, the _whole_ is not a similar representation of the original. The scale, [sigma] = ½c sec² ½z, at any point, applies to all directions round that point.

These two last projections are, as it were, at the extremes of the
scale; each, perfect in its own way, is in other respects
objectionable. We may avoid both extremes by the following
considerations. Although we cannot make [sigma] = 1 and [sigma]´ = 1,
so as to have a perfect picture of the spherical surface, yet
considering [sigma] - 1 and [sigma]´ - 1 as the local errors of the
representation, we may make ([sigma] - 1)² + ([sigma]´ - 1)² a minimum
over the whole surface to be represented. To effect this we must
multiply this expression by the element of surface to which it
applies, viz. sin zd zd [mu], and then integrate from the centre to
the (circular) limits of the map. Let [beta] be the spherical radius
of the segment to be represented, then the total misrepresentation is
to be taken as
_ _ _
/ [beta] | /d[rho] \² /[rho] \² |
| | ( ------ - 1 ) + ( ----- - 1 ) | sin z dz,
_/ 0 |_ \ dz / \sin z / _|

which is to be made a minimum. Putting [rho] = z + y, and giving to y
only a variation subject to the condition [delta]y = 0 when z = 0, the
equations of solution--using the ordinary notation of the calculus of
variations--are

d(P)
N - ---- = 0; P[beta] = 0,
dz

P[beta] being the value of 2p sin z when z = [beta]. This gives