Chapter V: Part 5

d²y dy /dy\
sin² z --- + sin z cos z -- - y = z - sin z ( -- )[beta] = 0.
dz² dz \dz/

This method of development is due to Sir George Airy, whose original
paper--the investigation is different in form from the above, which is
due to Colonel Clarke--will be found in the _Philosophical Magazine_
for 1861. The solution of the differential equation leads to this
result--

[rho] = 2 cot ½z log_e sec ½z + C tan ½z,
C = 2 cot² ½[beta] log_e sec ½[beta].

The limiting radius of the map is R = 2C tan ½[beta]. In this system,
called by Sir George Airy _Projection by balance of errors_, the total
misrepresentation is an absolute minimum. For short it may be called
_Airy's Projection_.

Returning to the general case where [rho] is any function of z, let us
consider the local misrepresentation of direction. Take any
indefinitely small line, length = i, making an angle [alpha] with the
meridian in co-latitude z. Its projections on a meridian and parallel
are i cos [alpha], i sin [alpha], which in the map are represented by
i[sigma] cos [alpha], i[sigma]´ sin [alpha]. If then [alpha]´ be the
angle in the map corresponding to [alpha],

tan [alpha]´ = ([sigma]´/[sigma]) tan [alpha].

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.

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.

_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.

Now if [sigma][sigma]´ = 1 the areas are correctly represented, then

h[rho] d[rho] = sin z dz, and integrating ½h[rho]² = C - cos z; (i.)

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, [rho] = 0, and const = 1, we have p = 2 sin ½z/[delta]h (ii.)

Let z1 be the co-latitude of some parallel which is to be correctly
represented, then 2h sin ½z1/[delta]h = sin z1, and h = cos² ½z1;
putting this value of h in equation (ii.) the radius of any parallel

= [rho] = 2 sin ½z sec ½z1 (iii.)

This is Lambert's _conical equal-area projection with one standard
parallel_, the pole being the centre of the parallels.

If we put z1 = [theta], then h = 1, and the meridians are inclined at
their true angles, also the scale at the pole becomes correct, and
equation (iii.) becomes

[rho] = 2 sin ½z; (iv.)

this is the _zenithal equal-area projection_.

Reverting to the general expression for equal-area conical projections

[rho] = [root]{2(C - cos z)/h}, (i.)

we can dispose of C and h so that any two selected parallels shall be
their true lengths; let their co-latitudes be z1 and z2, then

2h(C - cos z1) = sin² z1 (v.)
2h(C - cos z2) = sin² z2 (vi.)

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 x² tan [phi] cosec 1´´ /
2[rho][nu] where x is the length of the perpendicular, [rho] that of
the radius of curvature to the meridian, [nu] that of the normal
terminated by the minor axis, [phi] the latitude of the foot of the
perpendicular. The difference of longitude in seconds is approximately
x sec [rho] cosec 1´´ / [nu]. 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., ½ in. and ¼ 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 [root]2 sin [delta] where [pi] sin [phi] = 2[delta] + sin 2[delta], [phi] being the latitude of the parallel. The length of the central meridian from pole to pole = 2 [root]2, where the radius of the sphere is unity. The length of the equator = 4 [root]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.

_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 general shape and position of the chief land masses; it is better for this purpose than the stereographic, which is commonly employed in atlases.

FIG. 27.--Plane Table Graticule, dimensions in inches, for a scale of 4 in. to 1 m.]

_Projections for Field Sheets._

Field sheets for topographical surveys should be on conical projections with rectified meridians; these projections for small areas and ordinary topographical scales--not less than 1/500,000--are sensibly errorless. But to save labour it is customary to employ for this purpose either form of polyconic projection, in which the errors for such scales are also negligible. In some surveys, to avoid the difficulty of plotting the flat arcs required for the parallels, the arcs are replaced by polygons, each side being the length of the portion of the arc it replaces. This method is especially suitable for scales of 1:125,000 and larger, but it is also sometimes used for smaller scales.

Fig. 27 shows the method of plotting the projection for a field sheet.
Such a projection is usually called a graticule. In this case ABC is
the central meridian; the true meridian lengths of 30´ spaces are
marked on this meridian, and to each of these, such as AB, the figure
(in this case representing a square half degree), such as ABED, is
applied. Thus the point D is the intersection of a circle of radius AD
with a circle of radius BD, these lengths being taken from geodetic
tables. The method has no merit except that of convenience.

_Summary._

The following projections have been briefly described:--

/ 1. Cylindrical equal-area.
| 2. Orthographic.
| 3. Stereographic (which is orthomorphic).
Perspective < 4. General external perspective.
| 5. Minimum error " (Clarke's).
\ 6. Central.

/ 7. Conical, with rectified meridians and two
| standard parallels (5 forms).
| 8. Simple conical.
| 9. Simple cylindrical (a special case of 8).
| 10. Modified conical equal-area (Bonne's).
| 11. Sinusoidal " " (Sanson's).
| 12. Werner's conical " "
Conical < 13. Simple polyconic.
| 14. Rectangular polyconic.
| 15. Conical orthomorphic with 2 standard parallels
| (Lambert's, commonly called Gauss's).
| 16. Cylindrical orthomorphic (Mercator's).
| 17. Conical equal-area with one standard parallel.
| 18. " " " " two " parallels.
\ 19. Projection by rectangular spheroidal co-ordinates.

/ 20. Equidistant zenithal.
| 21. Zenithal equal-area.
| 22. Zenithal projection by balance of errors (Airy's).
Zenithal < 23. Elliptical equal-area (Mollweide's).
| 24. Globular (conventional).
\ 25. Field sheet graticule.

Of the above 25 projections, 23 are conical or quasi-conical, if
zenithal and perspective projections be included. The projections may,
if it is preferred, be grouped according to their properties. Thus in
the above list 8 are equal-area, 3 are orthomorphic, 1 balances
errors, 1 represents all great circles by straight lines, and in 5 one
system of great circles is represented correctly.

Among projections which have not been described may be mentioned the
circular orthomorphic (Lagrange's) and the rectilinear equal-area
(Collignon's) and a considerable number of conventional projections,
which latter are for the most part of little value.

The choice of a projection depends on the function which the map is
intended to fulfil. If the map is intended for statistical purposes to
show areas, density of population, incidence of rainfall, of disease,
distribution of wealth, &c., an _equal-area_ projection should be
chosen. In such a case an area scale should be given. At sea,
_Mercator's_ is practically the only projection used except when it is
desired to determine graphically great circle courses in great oceans,
when the _central_ projection must be employed. For conveying good
general ideas of the shape and distribution of the surface features of
continents or of a hemisphere _Clarke's perspective_ projection is the
best. For exhibiting the progress of polar exploration the _polar
equidistant_ projection should be selected. For special maps for
general use on scales of 1/1,000,000 and smaller, and for a series of
which the sheets are to fit together, the _conical, with rectified
meridians and two standard parallels_, is a good projection. For
topographical maps, in which each sheet is plotted independently and
the scale is not smaller than 1/500,000, either form of _polyconic_ is
very convenient.

The following are the projections adopted for some of the principal
official maps of the British Empire:--

_Conical, with Rectified Meridians and Two Standard Parallels._--The
1:1,000,000 Ordnance map of the United Kingdom, special maps of the
topographical section, General Staff, e.g. the 64-mile map of
Afghanistan and Persia. The 1:1,000,000 Survey of India series of
India and adjacent countries.

_Modified Conical, Equal-area (Bonne's)._--The 1 in., ½ in., ¼ in. and
1/10 in. Ordnance maps of Scotland and Ireland. The 1:800,000 map of
the Cape Colony, published by the Surveyor-General.

_Simple Polyconic and Rectangular Polyconic_ maps on scales of
1:1,000,000, 1:500,000, 1:250,000 and 1:125,000 of the topographical
section of the General Staff, including all maps on these scales of
British Africa. A rectilinear approximation to the simple polyconic is
also used for the topographical sheets of the Survey of India. The
simple polyconic is used for the 1 in. maps of the Militia Department
of Canada.

_Zenithal Projection by Balance of Errors (Airy's)._--The 10-mile to 1
in. Ordnance map of England.

_Projection by Rectangular Spheroidal Co-ordinates._--The 1:2500 and
the 6 in. Ordnance sheets of the United Kingdom, and the 1 in., ½ in.
and ¼ in. Ordnance maps of England. The cadastral plans of the Survey
of India, and cadastral plans throughout the empire.

AUTHORITIES.--See _Traité des projections des cartes géographiques_,
by A. Germain (Paris, 1865) and _A Treatise on Projections_, by T.
Craig, United States Coast and Geodetic Survey (Washington, 1882).
Both Germain and Craig (following Germain) make use of the term
_projections by development_, a term which is apt to convey the
impression that the spherical surface is developable. As this is not
the case, and since such projections are conical, it is best to avoid
the use of the term. For the history of the subject see d'Avezac,
"Coup d'oeil historique sur la projection des cartes géographiques,"
_Société de géographie de Paris_ (1863).

J. H. Lambert (_Beiträge zum Gebrauch der Mathematik, u.s.w._ Berlin,
1772) devised the following projections of the above list: 1, 15, 17,
and 21; his transverse cylindrical orthomorphic and the transverse
cylindrical equal-area have not been described, as they are seldom
used. Among other contributors we mention Mercator, Euler, Gauss, C.
B. Mollweide (1774-1825), Lagrange, Cassini, R. Bonne (1727-1795),
Airy and Colonel A. R. Clarke. (C. F. Cl.; A. R. C.)

FOOTNOTES:

[1] The ancient Greeks called a map _Pinax_, The Romans _Tabula
geographica_. _Mappa mundi_ was the medieval Latin for a map of the
world which the ancients called _Tabula totius orbis descriptionem
continens_.

[2] Close, "The Ideal Topographical Map," _Geog. Journal_, vol. xxv.
(1905).

[3] K. Peucker, _Schattenplastik und Farbenplastik_ (Vienna, 1898);
_Geograph. Zeitschrift_ (1902 and 1908).

[4] Professor Henrici, _Report on Planimeters_ (64th meeting of the
British Association, Oxford, 1894); J. Tennant, "The Planimeter"
(_Engineering_, xlv. 1903).

[5] H. Wagner's _Lehrbuch_ (Hanover, 1908, pp. 241-252) refers to
numerous authorities who deal fully with the whole question of
measurement.

[6] Kienzl of Leoben in 1891 had invented a similar apparatus which
he called a Relief Pantograph (_Zeitschrift_, Vienna Geog. Soc.
1891).

[7] M. Fiorini, _Erd- und Himmelsgloben, frei bearbeitet von S.
Günther_ (Leipzig, 1895).

[8] _Jahrb. des polytechn. Instituts in Wien_, vol. xv.

[9] Compare the maps of EUROPE, ASIA, &c., in this work.

[10] The great majority of the maps in this work are made by this
process.

[11] Lepsius, _Urkundenbuch_, Pl. XXII.

[12] These Colchians certainly were not Egyptians. The maps referred
to may have been Assyrian.

[13] We are indebted to Strabo for nearly all we know about Greek
cartographers anterior to Ptolemy, for none of their maps has been
preserved.

[14] The gnomon was known to the Chinese in the 5th century B.C., and
reached the Greeks (Anaximander) through Babylon. Pytheas, as far as
known, was the first to utilize it for the determination of a
latitude.

[15] If, with W. Dörpfeld, we assume an Attic stadium of 200 steps
(500 ft.) to be equal to 164 metres, a degree of 700 stad. would be
equal to 114,800 metres, its actual length according to modern
measurement being 110,808 metres.

[16] _Climata_ based on the length of the longest day were introduced
by Hippocrates (_c._ 400 B.C.). _Zones_ similar to those already
drawn out for the celestial sphere were first introduced by the
Pythagoreans. Parmenides of Elea (544-430 B.C.) distinguishes five of
these zones, viz. a torrid zone, between the tropics of summer and
winter, which was uninhabitable on account of heat; two frigid zones,
uninhabitable on account of cold, and two intermediate temperate
zones.

[17] Celestial globes were made much earlier than terrestrial ones.
In the museum of Naples there is a celestial globe, 2 metres in
diameter, supported upon the shoulders of an Atlas, which E. Heis,
judging by the constellations engraved upon it (_Atlas coelestis
novus_, Bonn, 1872) judges to date from the 4th century B.C. It may
even be the work of Eudoxus (d. 386 B.C.) the famous astronomer.
Aratus of Soli in Cilicia, in his poetical _Prognostics of Stars and
the World_, refers to a globe in his possession. Archimedes, the
famous mathematician, had a celestial globe of glass, in the centre
of which was a small terrestrial globe. Hero of Alexandria (284-221
B.C.), the ingenious inventor of "Hero's Fountain," is believed to
have possessed a similar apparatus. The celestial globe of Hipparchus
still existed in the Alexandrian library in the time of Ptolemy, who
himself refers to globes in his _Almagest_, as also in the
_Geography_. Leontius, who wrote a book on the manufacture of globes
(first published at Basel in 1539), is identified by Fiorini with a
bishop of Neapolis (Cyprus) of the time of Constantine III.
(642-668).

[18] The oldest MS. of Ptolemy's _Geography_ is found in the Vatopedi
monastery of Mt Athos. It dates from the 12th or 13th century and was
published by Victor Langlois in 1867. For the latest edition we are
indebted to the late Carl Müller (Paris, 1883-1906) to whom we are
likewise indebted for an edition of the _Geographi graeci minores_
(1855-1861).

[19] Facsimiles of it have been published by Desjardins(1869-1871),
by K. Miller (1886), who ascribes it to Castorius, A.D. 366, and by
others.

[20] R. Gough, _British Topography_ (London 1768). His "Histories"
are published in _Rerum brit. scriptores_ XL. and LVII. 1866-1869.

[21] M. Bittner, _Die topogr. Capital des ind. Seespiegels_ (Vienna,
1897).

[22] E. G. Ravenstein, _Martin Behaim, his Life and his Globe_
(London, 1908). On the original only equator, ecliptics, tropics,
polar circles and one meridian 80° to the west of Lisbon are laid
down.

[23] See fig. 23, Catalan Map of the World (1375).

[24] J. G. Kohl published facsimiles of the American section of the
maps (Weimar, 1860).

[25] Facsimiles of the maps of 1507 and 1517 were published by J.
Fischer and F. M. von Wieser (Innsbruck, 1903).

[26] See "The Survey in British Africa": the _Annual Report_ of the
Colonial Survey Commission.

[27] A. Germain, _Traité des Projections_ (Paris, 1865).

[28] T. Craig, _A Treatise on Projections_ (U.S. Coast and Geodetic
Survey, Washington, 1882).

[29] This error is much less than that which may be expected from
contraction and expansion of the paper upon which the projection is
drawn or printed.

MAPLE, SIR JOHN BLUNDELL, BART. (1845-1903), English business magnate, was born on the 1st of March 1845. His father, John Maple (d. 1900), had a small furniture shop in Tottenham Court Road, London, and his business began to develop about the time that his son entered it. The practical management soon devolved on the younger Maple, under whom it attained colossal dimensions. The firm became a limited liability company, with a capital of two millions, in 1890, with Mr Maple as chairman. He entered parliament as Conservative member for Dulwich in 1887, was knighted in 1892, and was made a baronet in 1897. He was the owner of a large stud of race-horses, and from 1885 onwards won many important races, appearing at first under the name of "Mr Childwick." His public benefactions included a hospital and a recreation ground to the city of St Albans, near which his residence, Childwickbury, was situated, and the rebuilding, at a cost of more than £150,000, of University College Hospital, London. He died on the 24th of November 1903. His only surviving daughter married in 1896 Baron von Eckhardstein, of the German Embassy.

MAPLE, in botany. The maple (O.E. _mapel-tréow, mapulder_) and sycamore trees are species of _Acer_, of the order _Acerineae_. The genus includes about sixty species, natives of Europe, North America and Asia, especially the Himalayas, China and Japan. Maples are for the most part trees with opposite, long-stalked, palmately lobed leaves. The flowers are in fascicles, appearing before the leaves as in the Norway maple, or in racemes or panicles appearing with, or later than, the leaves as in sycamore. Some of the flowers are often imperfect, the stamens or pistil being more or less aborted. The fruit is a two-winged "samara." The genus was represented in the Tertiary flora of Europe, when it extended into the polar regions; nineteen species have been recorded from the Miocene strata of Oeningen in Switzerland. The common maple, _A. campestre_, is the only species indigenous to Great Britain. This and the sycamore were described by Gerard in 1597 (_Herball_, p. 1299), the latter being "a stranger to England." Many species have been introduced, especially from Japan, for ornamental purposes. The following are more especially worthy of notice.

_Acer campestre_, the common maple, is common in hedgerows, but less
often seen as a tree, when it is seldom more than 20 ft. high, though
in sheltered situations 30 ft. or more is attained. The leaves are
generally less than 2 in. across, and the five main lobes are blunter
than in the sycamore. The clusters of green flowers terminate the
young shoots and are erect; the two wings of the fruit spread almost
horizontally, and are smaller than in the sycamore. It occurs in
northern Europe, the Caucasus, and northern Asia. The wood is
excellent fuel, and makes the best charcoal. It is compact, of a fine
grain, sometimes beautifully veined, and takes a high polish. Hence it
has been celebrated from antiquity for tables, &c. The wood of the
roots is frequently knotted, and valuable for small objects of cabinet
work. The young shoots, being flexible and tough, are employed in
France as whips.

_A. pseudo-platanus_, the sycamore or great maple, is a handsome tree
of quick growth, with a smooth bark. The leaves are large, with finely
acute and serrated lobes, affording abundant shade. The flowers are
borne in long pendulous racemes, and the two wings of the fruit are
ascending. It lives from 140 to 200 years. It is found wild chiefly in
wooded mountainous situations in central Europe. The wood when young
is white, but old heartwood is yellow or brownish. Like the common
maple it is hard and takes a high polish. It is much prized by
wheelwrights, cabinet-makers, sculptors, &c., on the Continent; while
knotted roots are used for inlaying. Sugar has been obtained from the
sap of this as from other species, the most being one ounce from a
quart of sap. The latter has also been made into wine in the Highlands
of Scotland. It withstands the sea and mountain breezes better than
most other timber trees, and is often planted near farm-houses and
cottages in exposed localities for the sake of its dense foliage. Its
wood is valued in turnery for cups, bowls and pattern blocks. It
produces abundance of seeds, and is easily raised, but it requires
good and tolerably dry soil; it will not thrive on stiff clays nor on
dry sands or chalks. There are many varieties, the variegated and
cut-leaved being the most noticeable. The lobed shape of its leaf and
its dense foliage caused it to be confused with the true
sycamore--_Ficus sycamorus_--of scripture.

_A. platanoides_, the Norway maple, is met with from Norway to Italy,
Greece, and central and south Russia. It was introduced into Britain
in 1683. It is a lofty tree (from 40 to 70 ft.), resembling the
sycamore, but with yellow flowers, appearing before the leaves, and
more spreading wings to the fruit. There are several varieties. The
wood is used for the same purposes as that of the sycamore. Sugar has
been made from the sap in Norway and Sweden.

Many varieties of _A. palmatum_, generally known as _polymorphum_,
with variously laciniated and more or less coloured foliage, have been
introduced from Japan as ornamental shrubs. The branches and corolla
are purple, the fruit woolly. The foliage of the typical form is
bright green with very pointed lobes. It occurs in the central
mountains of Nippon and near Nagasaki. Beautiful varieties have been
introduced under the varietal names, _ampelopsifolium_,
_atropurpureum_, _dissectum_, &c. They are remarkable for the coppery
purple tint that pervades the leaves and young growths of some of the
varieties. Other Japanese species are _A. japonicum_, the varieties of
which are among the most handsome of small deciduous shrubs; _A.
rufinerve_, with the habit of the sycamore; _A. distylum_, bearing
leaves without lobes; _A. diabolicum_, with large plane-like leaves;
and _A. carpinifolium_, with foliage resembling that of the hornbeam.

_A. saccharinum_, a North American species, the sugar, rock, or
bird's-eye maple, was introduced in 1735. It sometimes attains to 70
or even over 100 ft., more commonly 50 to 60 ft. It is remarkable for
the whiteness of the bark. The wood is white, but acquires a rosy
tinge after exposure to light. The grain is fine and close, and when
polished has a silky lustre. The timber is used instead of oak where
the latter is scarce, and is employed for axle-trees and spokes, as
well as for Windsor chairs, &c. It exhibits two accidental forms in
the arrangement of the fibres, an undulated one like those of the
curled maple (_A. rubrum_), and one of spots, which gives the name
bird's-eye to the wood of this species. Like the curled maple, it is
used for inlaying mahogany. It is much prized for bedsteads,
writing-desks, shoe-lasts, &c. The wood forms excellent fuel and
charcoal, while the ashes are rich in alkaline principles, furnishing
a large proportion of the potash exported from Boston and New York.
Sugar is principally extracted from this species, the sap being boiled
and the syrup when reduced to a proper consistence runs into moulds to
form cakes. Trees growing in low and moist situations afford the most
sap but least sugar. A cold north-west wind, with frosty nights and
sunny days in alternation, tends to incite the flow, which is more
abundant during the day than the night. A thawing night is said to
promote the flow, and it ceases during a south-west wind and at the
approach of a storm; and so sensitive are the trees to aspect and
climatic variations that the flow of sap on the south and east side
has been noticed to be earlier than on the north and west side of the
same tree. The average quantity of sap per tree is from 12 to 24
gallons in a season.

_A. rubrum_, the red-flowering or scarlet maple, is a middle-sized
tree, and was introduced in 1656. The bright scarlet or dull red
flowers appear before the leaves in March and April. The wood, like
that of other species, is applicable to many purposes--as for the
seats of Windsor chairs, turnery, &c. The grain in very old trees is
sometimes undulated, which suggested the name of curled maple, and
gives beautiful effects of light and shade on polished surfaces. The
most constant use of curled maple is for the stocks of fowling-pieces
and rifles, as it affords toughness and strength combined with
lightness and elegance. The inner bark is dusky red. On boiling, it
yields a purple colour which with sulphate of iron affords a black
dye. The wood is inferior to that of the preceding species in strength
and as fuel. Sugar was made from the sap by the French Canadians, but
the production is only half as great as that from the sugar maple. In
Britain it is cultivated as an ornamental tree, as being conspicuous
for its flowers in spring, and for its red fruit and foliage in
autumn.

_A. macrophyllum_, a north-western American species, is a valuable
timber tree.

For a good account of the North American species see C. S. Sargent's
_Silva of North America_, vol. ii. See also under SUGAR.

MAPU, ABRAHAM (1808-1867), Hebrew novelist. His works are chiefly historical romances in Hebrew. His most famous books were _The Love of Zion_ and the _Transgression of Samaria_. Besides their intrinsic merits, these novels stand high among the works which produced the romantic movement in modern Hebrew literature. Mapu's plots were somewhat sensational, incident being more prominent than characterization. But underlying all was a criticism of contemporary life. His novels made a deep impression and became instantly popular. Mapu's Hebrew style is simple and classical. An English translation of the _Love of Zion_ bears the title _Amnon, Prince and Peasant_, by F. Jaffe (1887). Mapu's stories have been often translated into other languages.

See N. Slouschz, _The Renascence of Hebrew Literature_ (1909), ch. v.
(I. A.)

MAQQARI, or Makkari [Abu-l-'Abbas Ahmad ibn Mahommed ul-Maqqari] (c. 1591-1632), Arabian historian, was born at Tlemcen in Algeria and studied at Fez and Marrakesh, where he remained engaged in literary work until he made the pilgrimage to Mecca in 1618. In the following year he settled in Cairo. In 1620 he visited Jerusalem and Damascus, and during the next six years made the pilgrimage five times. In 1628 he was again in Damascus, where he gave a course of lectures on Bukhari's collection of _Traditions_, spoke much of the glories of Moslem Spain, and received the impulse to write his work on this subject later. In the same year he returned to Cairo, where he spent a year in writing his history. He was just making preparations to settle definitely in Damascus when he died in 1632.

His great work, _The Breath of Perfume from the Branch of Green
Andalusia and Memorials of its Vizier Lisan ud-Din ibn ul-Khat[-i]b_,
consists of two parts. The first is a compilation from many authors on
the description and history of Moslem Spain; it was published by
Wright, Krehl, Dozy and Dugat as _Analectes sur l'histoire et la
littérature des Arabes d'Espagne_ (Leiden, 1855-1861), and in an
abridged English translation by P. de Gayangos (London, 1840-1843).
The whole work has been published at Bulaq (1863) and Cairo (1885).

For other works of Maqqari see C. Brockelmann's _Gesch. der arabischen
Litteratur_ (Berlin, 1902), ii. 297. (G. W. T.)

MAQRIZI, or MAKRIZI [Taqi ud-Din Ahmad ibn 'Ali] (1364-1442), Arabian historian, known as al-Maqrizi because of his ancestral connexion with Maqriz, a suburb of Baalbek, was born at Cairo and spent most of his life in Egypt, where he was trained in the Hanifite school of law, though later he became a Shafi'ite with an inclination to Zahirite views. In 1385 he made the pilgrimage. For some time he was secretary in a government office, and in 1399 became inspector of markets for Cairo and northern Egypt. This post he soon gave up to become preacher at the mosque of 'Amr, president of the mosque ul-Hakim, and a lecturer on tradition. In 1408 he went to Damascus to become inspector of the Qalanisiyya and lecturer. Later he retired into private life at Cairo. In 1430 he made the pilgrimage with his family and travelled for some five years. His learning was great, his observation accurate and his judgment good, but his books are largely compilations, and he does not always acknowledge the sources to which he is indebted. Most of his works are concerned with Egypt. The most important is the _Mawa'iz wal-I'tibar fi dhikr ul-Hitat wal-Aihar_ (2 vols., Bulaq, 1854), translated into French by U. Bouriant as _Description topographique et historique de l'Égypte_ (Paris, 1895-1900; cf. A. R. Guest, "A List of Writers, Books and other Authorities mentioned by El Maqrizi in his _Khitat_," in _Journal of the Royal Asiatic Society_, 1902, pp. 103-125). Of his _History of the Fatimites_ an extract was published by J. G. L. Kosegarten in his _Chrestomathia_ (Leipzig, 1828), pp. 115-123; the _History of the Ayyubit and Mameluke Rulers_ has been translated into French by E. Quatremère (2 vols., Paris, 1837-1845). Maqrizi began a large work called the _Muqaffa_, a cyclopaedia of Egyptian biography in alphabetic order. It was intended to be in 80 volumes, but only 16 were written. Three autograph volumes exist in MS. in Leiden, and one in Paris.

Among smaller works published are the _Mahommedan Coinage_ (ed. O. G.
Tychsen, Rostock, 1797; French translation by S. de Sacy, Paris,
1797); _Arab Weights and Measures_ (ed. Tychsen, Rostock, 1800); the
_Arabian Tribes that migrated to Egypt_ (ed. F. Wüstenfeld, Göttingen,
1847); the _Account of Hadhramaut_ (ed. P. B. Noskowyj, Bonn, 1866);
the _Strife between the Bani Umayya and the Bani Hashim_ (ed G. Vos,
Leiden, 1888), and the _Moslems in Abyssinia_ (ed. F. T. Rink, Leiden,
1790). For Maqrizi's life see the quotations from contemporary
biographies in S. de Sacy's _Chrestomathie arabe_ (2nd ed., Paris,
1826), ii. 112 seq., and for other works still in MS. C. Brockelmann,
_Gesch. der arabischen Litteratur_ (Berlin, 1902), ii. 38-41.
(G. W. T.)

MAR, EARLDOM OF. Mar, one of the ancient divisions or provinces of Scotland, comprised the larger portion of Aberdeenshire, extending from north of the Don southward to the Mounth. Like other such districts, it was in Celtic times under the rule of a _mormaer_. In the 12th century his place was taken by an earl, but no definite succession of earls appears till the 13th century, nor is any connexion established between them and the _mormaers_. From the middle of the 13th century the earls were recognized as among "the seven earls of Scotland" and held a great position. Earl Gratney (fl. c. 1300) married a sister of (King) Robert Bruce, who brought him the lordship of Garioch and castle of Kildrummy, which she held against the earl of Athole, an ally of the English (1335). Their son Donald was made regent in July 1332, but was disastrously defeated and slain at Dupplin next month. His daughter and eventual heir, Margaret, brought the earldom to her husband, William, earl of Douglas, and on the accession of her daughter Isabél a troublous time followed.

While she was living as a widow at her castle of Kildrummy, it was stormed by Alexander Stewart, a bastard, who forced her to execute a charter (August 12, 1404) settling the reversion to the earldom on himself and his heirs. This act she revoked by a charter of the 19th of September 1404, which cannot now be found; but on marrying him, on the 9th of December 1404, she granted him the earldom for life, the king confirming this on the 21st of June 1405. After her death in 1408 the earl played a great part, commanding the royal forces at the battle of Harlaw, when the Lord of the Isles was defeated in 1411, and afterwards acting as warden of the Marches. In 1426 he resigned the earldom to the Crown, the king granting it by a fresh creation to him and certain heirs, with reversion to the Crown. On the earl's death in 1435 the earldom was claimed by Robert, Lord Erskine, as heir of Gratney, earl of Mar, through a daughter; but the Crown claimed as reversionary under the creation of 1426. A long struggle followed, till in 1457 James II. obtained from a justiciary court at Aberdeen a recognition of the Crown's right to the earldom and its lands, and shortly after bestowed them on his son John as earl of Mar and Garioch. He died unmarried in 1479, and in 1483 his elder brother Alexander duke of Albany received the earldom, but was soon forfeited. James III. created his son John earl of Mar and Garioch in 1486, and after his death unmarried in 1503, James IV. alienated to Lord Elphinstone (1507-1510) many of the Mar lands, including Kildrummy. The title was not revived till 1562, when James Stewart, earl of Murray, held it for a few months.

In 1565 John, Lord Erskine, succeeded in getting returned heir to the earldom, and shortly after (June 23, 1565) Queen Margaret restored the charter to him and his heirs "all and hail the said earldom of Mar." As earl he took part against the queen in 1567, and in 1571 was made regent of Scotland, which post he retained till his death (1572). His son, earl John (c. 1558-1634), played a great part in the history of the family. His great achievement was the recovery of the Mar estates, alienated by the Crown during the long period that his family had been out of possession, including Kildrummy, the "head" of the earldom. It was in his time that the precedence of the earldom (see below) was settled. John, the next earl (c. 1585-1654) was a Royalist, as was his son John (d. 1668), much to the injury of the family fortune, which was further impaired by the attachment of the family, after the Revolution, to the Stuarts. His son Charles (1650-1689) was arrested by the government just before his death (1689), and the next earl, John (1675-1732), a prominent Jacobite (see below), was attainted, the earldom remaining under forfeiture for 108 years; by the Old Pretender he was created duke of Mar.

Alloa and other Erskine estates of the attainted earl were repurchased for the family, and descended to John Francis Erskine (1741-1825), his heir-male, who was also his heir of line through his daughter. To him, in his eighty-third year, as grandson and lineal representative of the attainted earl, the earldom was restored by act of parliament in 1824. His grandson, who succeeded him in 1828, inherited the earldom of Kellie (1619) and other Erskine dignities by decision of 1835. At his death in 1866, his earldom of Mar was the subject of rival claims, and the right to the succession was not determined till 1875. His estates passed to his cousin and heir-male, who succeeded to his earldom of Kellie and claimed "the honour and dignity of earl of Mar." But the latter was also claimed by a Mr Goodeve, whose father had married the late earl's eldest sister, and who assumed the title. It was not suggested that the late earl had more than one earldom of Mar, but Lord Kellie claimed it as descendible to heirs-male under a creation by Queen Mary, and Mr Goodeve as descendible to heirs of line under an earlier creation. The House of Lords decided (Feb. 25, 1875) that Lord Kellie was entitled to the earldom as having been created by Queen Mary in 1565, with a limitation which must be presumed to be to heirs-male of the body. This decision gave great dissatisfaction, but was described as "final, right or wrong, and not to be questioned" by Lord Selborne and the lord chancellor in 1877, and Lord Kellie was thenceforth recognized as holding the earldom on the Union Roll, the only one known, though Mr Goodeve continued to assume the title. The Lords' decision could not be reversed, but in 1885, after much agitation, a means was found of evading it in practice by the "Earldom of Mar Restitution Act." By "an equivocation on the facts of the case," it was recited that "doubts may exist whether the said ancient honour, dignity, and title of peerage of earl of Mar ... was or was not ... by any lawful means surrendered or merged in the Crown" before 1565, and that the House of Lords had decided that Queen Mary's known charter of 1565 applied only to lands and "did not operate or extend to restore" the peerage dignity, and enacted that "John Francis Erskine Goodeve Erskine" (which last name the claimant had added) should be "restored to" the ancient earldom. His previous assumption of the title was thus rejected as invalid, but from the passing of the act two earldoms of Mar were in existence, that of Lord Kellie being confirmed and allowed the precedence of 1565, while the restored earldom was allowed that of the dignity on the Union Roll, the only one known till then. This precedence had been assigned to it by the Decreet of Ranking (1606), and assigns to it an origin in 1404 (or, as some say, 1395). It is frequently, but absurdly, stated to have been "created before 1014," and wrongly spoken of as the Premier Scottish Earldom (see EARL). A barony of Garioch is also wrongly said to be annexed to it, but the title is used by the earl's eldest son in default of any other.

BIBLIOGRAPHY.--_Minutes of Evidence_, 1875 and 1885; Riddell's
_Peerage and Consistorial Law_; Skene, _Celtic Scotland_; Lord
Crawford's _Earldom of Mar in Sunshine and Shade_; articles by G.
Burnett (Lyon), Sir H. Barkly, Cornelius Hallen, W. A. Lindsay and J.
H. Round in _Genealogist_ (N.S.), vols. 3, 4, 9; Lord Redesdale's _The
Earldom of Mar, a Letter to the Lord Clerk Register_ (reply to Lord
Crawford) (1883); J. H. Round's "Are there two Earls of Mar?" in
Foster's _Collectanea genealogica_, and "The later Earldom of Mar" in
Walford's _Antiquarian Magazine_, vol. ii.; also his _Studies in
Peerage and Family History_. (J. H. R.)

MAR, JOHN ERSKINE, 1ST OR 6TH EARL OF (d. 1572), regent of Scotland, was a son of John, 5th Lord Erskine (d. 1552), who was guardian of King James V., and afterwards of Mary Queen of Scots. The younger John, who succeeded his father as 6th Lord Erskine in 1552, joined the religious reformers, but he was never very ardent in the cause, although he subscribed the letter asking Knox to return to Scotland in 1557. The custody of Edinburgh Castle was in his hands, and during the struggle between the regent, Mary of Lorraine, and the lords of the Congregation he appears to have acted consistently in the interests of peace. When Mary Stuart returned to Scotland in 1561 Lord Erskine was a member of her council, he favoured her marriage with Lord Darnley, and his wife, Annabella Murray, called by Knox a "verray Jesabell," was a frequent companion of the queen. In 1565 Erskine was granted the earldom of Mar (see above). As guardian of James, afterwards King James VI., he prevented the young prince from falling into the hands of Bothwell, and when the Scottish nobles rose against Mary and Both well, Mar was one of their leaders; he took part in the government of Scotland during Mary's imprisonment at Lochleven, and also after her subsequent abdication. In September 1571 he was chosen regent of Scotland, but he was overshadowed and perhaps slighted by the earl of Morton, and he died at Stirling on the 29th of October 1572.