Put

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

and the error [alpha]´ - [alpha] of representation = [epsilon], then

([Sigma] - 1) tan [alpha]

tan [epsilon] = -------------------------.

1 + [Sigma] tan² [alpha]

Put [Sigma] = cot²[zeta], then [epsilon] is a maximum when [alpha] =

[zeta], and the corresponding value of [epsilon] is

[epsilon] = ½[pi] - 2[zeta].

For simplicity of explanation we have supposed this method of

development so applied as to have the pole in the centre. There is,

however, no necessity for this, and any point on the surface of the

sphere may be taken as the centre. All that is necessary is to calculate

by spherical trigonometry the azimuth and distance, with reference to

the assumed centre, of all the points of intersection of meridians and

parallels within the space which is to be represented in a plane. Then

the azimuth is represented unaltered, and any spherical distance z is

represented by [rho]. Thus we get all the points of intersection

transferred to the representation, and it remains merely to draw

continuous lines through these points, which lines will be the meridians

and parallels in the representation.

Thus treating the earth as a sphere and applying the _Zenithal

Equal-area Projection_ to the case of Africa, the central point selected

being on the equator, we have, if [theta] be the spherical distance of

any point from the centre, [phi], [alpha] the latitude and longitude

(with reference to the centre), of this point, cos [theta] = cos [phi]

cos [alpha]. If A is the azimuth of this point at the centre, tan A =

sin [alpha] cot [phi]. On paper a line from the centre is drawn at an

azimuth A, and the distance [theta] is represented by 2 sin ½[theta].

This makes a very good projection for a single-sheet equal-area map of

Africa. The exaggeration in such systems, it is important to remember,

whether of linear scale, area, or angle, is the same for a given

distance from the centre, whatever be the azimuth; that is, the

exaggeration is a function of the distance from the centre only.

_General Theory of Conical Projections._

Meridians are represented by straight lines drawn through a point, and a

difference of longitude [omega] is represented by an angle h[omega]. The

parallels of latitude are circular arcs, all having as centre the point

of divergence of the meridian lines. It is clear that perspective and

zenithal projections are particular groups of conical projections.

[Illustration: FIG. 24.]

Let z be the co-latitude of a parallel, and [rho], a function of z,

the radius of the circle representing this parallel. Consider the

infinitely small space on the sphere contained by two consecutive

meridians, the difference of whose longitude is d[mu], and two

consecutive parallels whose co-latitudes are z and z + dz. The sides

of this rectangle are pq = dz, pr = sin z d[mu]; in the projection

p´q´r´s´ these become p´q´ = d[rho], and p´r´ = [rho]h d[mu].

The scales of the projection as compared with the sphere are p´q´/pq =

d[rho]/dz = the scale of meridian measurements = [sigma], say, and

p´r´/pr = [rho]h d[mu]/sin z d[mu] = [rho]h/sin z = scale of

measurements perpendicular to the meridian = [sigma]´, say.

Now we may make [sigma] = 1 throughout, then [rho] = z + const. This

gives either the group of _conical projections with rectified

meridians_, or as a particular case the _equidistant zenithal_.

We may make [sigma] = [sigma]´ throughout, which is the same as

requiring that at any point the scale shall be the same in all

directions. This gives a group of _orthomorphic projections_.

In this case d[rho]/dz = [rho]h/sin z, or d[rho]/[rho] = h dz/sin z.

Integrating,

[rho] = k(tan ½z)^h, (i.)

where k is a constant.

Now h is at our disposal and we may give it such a value that two

selected parallels are of the correct lengths. Let z1, z2 be the

co-latitudes of these parallels, then it is easy to show that

log sin z1 - log sin z2

h = ------------------------- (ii.)

log tan ½z1 - log tan ½z2

This projection, given by equations (i.) and (ii.), is Lambert's

orthomorphic projection--commonly called Gauss's projection; its

descriptive name is the _orthomorphic conical projection with two

standard parallels_.

The constant k in (i.) defines the scale and may be used to render the

scale errors along the selected parallels not nil but the same; and

some other parallel, e.g. the central parallel may then be made

errorless.

The value h = 1/3, as suggested by Sir John Herschel, is admirably

suited for a map of the world. The representation is fan-shaped, with

remarkably little distortion (fig. 24).

If any parallel of co-latitude z is true to scale hk(tan ½z1)^h = sin

z, if this parallel is the equator, so that z1 = 90°, kh = 1, then

equation (i.) becomes [rho] = (tan ½z)^h/h, and the radius of the

equator = 1/h. The distance r of any parallel from the equator is 1/h

- (tan ½z)^h/h = (1/h){1 - (tan ½z)^h}.

If, instead of taking the radius of the earth as unity we call it a, r

= (a/h){1 - (tan ½z)^h}. When h is very small, the angles between the

meridian lines in the representation are very small; and proceeding to

the limit, when h is zero the meridians are parallel--that is, the

vertex of the cone has removed to infinity. And at the limit when h is

zero we have r = a log_e cot ½z, which is the characteristic equation

of Mercator's projection.

[Illustration: FIG. 25.--Elliptical equal-area Projection, showing the

whole surface of the globe.]

_Mercator's Projection._--From the manner in which we have arrived at

this projection it is clear that it retains the characteristic property

of orthomorphic projections--namely, similarity of representation of

small parts of the surface. In Mercator's chart the equator is

represented by a straight line, which is crossed at right angles by a

system of parallel and equidistant straight lines representing the

meridians. The parallels are straight lines parallel to the equator, and

the distance of the parallel of latitude [phi] from the equator is, as

we have seen above, r = a log_e tan (45° + ½[phi]). In the vicinity of

the equator, or indeed within 30° of latitude of the equator, the

representation is very accurate, but as we proceed northwards or

southwards the exaggeration of area becomes larger, and eventually

excessive--the poles being at infinity. This distance of the parallels

may be expressed in the form r = a (sin [phi] + 1/3 sin ^3[phi] + 1/5

sin ^5[phi] + ...), showing that near the equator r is nearly

proportional to the latitude. As a consequence of the similar

representation of small parts, a curve drawn on the sphere cutting all

meridians at the same angle--the loxodromic curve--is projected into a

straight line, and it is this property which renders Mercator's chart so

valuable to seamen. For instance: join by a straight line on the chart

Land's End and Bermuda, and measure the angle of intersection of this

line with the meridian. We get thus the bearing which a ship has to

retain during its course between these ports. This is not great-circle

sailing, and the ship so navigated does not take the shortest path. The

projection of a great circle (being neither a meridian nor the equator)

is a curve which cannot be represented by a simple algebraic equation.

If the true spheroidal shape of the earth is considered, the semiaxes

being a and b, putting e = [root] (a² - b²)/a, and using common

logarithms, the distance of any parallel from the equator can be shown

to be

(a/M) {log tan (45° + ½[phi]) - e² sin [phi] - 1/3 e^4 sin ^3[phi] ...}

where M, the modulus of common logarithms, = 0.434294. Of course

Mercator's projection was not originally arrived at in the manner

above described; the description has been given to show that

Mercator's projection is a particular case of the conical orthomorphic

group. The introduction of the projection is due to the fact that for

navigation it is very desirable to possess charts which shall give

correct local outlines (i.e. in modern phraseology shall be

orthomorphic) and shall at the same time show as a straight line any

line which cuts the meridians at a constant angle. The latter

condition clearly necessitates parallel meridians, and the former a

continuous increase of scale as the equator is departed from, i.e. the

scale at any point must be equal to the scale at the equator × sec.

latitude. In early days the calculations were made by assuming that

for a small increase of latitude, say 1´, the scale was constant, then

summing up the small lengths so obtained. Nowadays (for simplicity the

earth will be taken as a sphere) we should say that a small length of

meridian ad[phi] is represented in this projection by a sec [phi]

d[phi], and the length of the meridian in the projection between the

equator and latitude [phi],

/[phi]

/ a sec [phi] d[phi] = a log_e tan (45° + ½[phi]),

\/ 0

which is the direct way of arriving at the law of the construction of

this very important projection.

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 = [sigma] along a

meridian, and p´r´/pr´ = [rho]h / sin z = [sigma]´ at right angles to

a meridian.