Topology As Fluid Geometry Twodimensional Spaces Volume 2 James W Cannon

Topology As Fluid Geometry Twodimensional Spaces
Volume 2 James W Cannon download
https://ebookbell.com/product/topology-as-fluid-geometry-
twodimensional-spaces-volume-2-james-w-cannon-7014988
Explore and download more ebooks at ebookbell.com

Here are some recommended products that we believe you will be
interested in. You can click the link to download.
Jesus As New Moses In Matthew 89 Jewish Typology In First Century
Greek Literature Michael P Theophilos
https://ebookbell.com/product/jesus-as-new-moses-in-matthew-89-jewish-
typology-in-first-century-greek-literature-michael-p-
theophilos-50340784
Theology As History History As Theology Paul In Ephesus In Acts 19
Reprint 2012 Scott Shauf
https://ebookbell.com/product/theology-as-history-history-as-theology-
paul-in-ephesus-in-acts-19-reprint-2012-scott-shauf-49052974
Theology As Science In Nineteenth Century Germany From Fc Baur To
Ernst Troeltsch Changing Paradigms In Historical And Systematic
Theology Johannes Zachhuber
https://ebookbell.com/product/theology-as-science-in-nineteenth-
century-germany-from-fc-baur-to-ernst-troeltsch-changing-paradigms-in-
historical-and-systematic-theology-johannes-zachhuber-52645120
Theology As Doxology And Dialogue The Essential Writings Of Nikos
Nissiotis Nikos Nissiotis
https://ebookbell.com/product/theology-as-doxology-and-dialogue-the-
essential-writings-of-nikos-nissiotis-nikos-nissiotis-23914840

Theology As Improvisation A Study In The Musical Nature Of Theological
Thinking Nathan Crawford
https://ebookbell.com/product/theology-as-improvisation-a-study-in-
the-musical-nature-of-theological-thinking-nathan-crawford-36361762
Theology As An Empirical Science 1st Edition Douglas Clyde Macintosh
https://ebookbell.com/product/theology-as-an-empirical-science-1st-
edition-douglas-clyde-macintosh-44415608
Theology As Hermeneutics Rudolf Bultmanns Theology Of The History Of
Jesus John Painter
https://ebookbell.com/product/theology-as-hermeneutics-rudolf-
bultmanns-theology-of-the-history-of-jesus-john-painter-50678948
Theology As Discipleship Keith L Johnson
https://ebookbell.com/product/theology-as-discipleship-keith-l-
johnson-19470060
Theology As A Way Of Life On Teaching And Learning The Christian Faith
Adam Neder
https://ebookbell.com/product/theology-as-a-way-of-life-on-teaching-
and-learning-the-christian-faith-adam-neder-48904616

Two-Dimensional Spaces, Volume 2
TOPOLOGY AS FLUID
GEOMETRY
James W. Cannon

TOPOLOGY AS FLUID
GEOMETRY

A M E R I C A N M A T H E M A T I C A L S O C I E T Y
P r o v i d e n c e , R h o d e I s l a n d
TOPOLOGY AS FLUID
GEOMETRY
James W. Cannon

2010 Mathematics Subject Classification. Primary 57-01, 57M20.
For additional information and updates on this book, visit
www.ams.org/bookpages/mbk-109
Library of Congress Cataloging-in-Publication Data
Names: Cannon, James W., author.
Title: Two-dimensional spaces / James W. Cannon.
Description: Providence, Rhode Island : American Mathematical Society, [2017] | Includes bibli-
ographical references.
Identifiers: LCCN 2017024690 | ISBN 9781470437145 (v. 1) | ISBN 9781470437152 (v. 2) | ISBN
9781470437169 (v. 3)
Subjects: LCSH: Geometry. | Geometry, Plane. | Non-Euclidean geometry. | AMS: Geometry –
Instructional exposition (textbooks, tutorial papers, etc.). msc
Classification: LCC QA445 .C27 2017 | DDC 516–dc23
LC record available at https://lccn.loc.gov/2017024690
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting
for them, are permitted to make fair use of the material, such as to copy select pages for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Permissions to reuse
portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink
service. For more information, please visit: http://www.ams.org/rightslink.
Send requests for translation rights and licensed reprints to reprint-permission@ams.org.
Excluded from these provisions is material for which the author holds copyright. In such cases,
requests for permission to reuse or reprint material should be addressed directly to the author(s).
Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the
first page of each article within proceedings volumes.
c
2017 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.

∞ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org/
10 9 8 7 6 5 4 3 2 1 22 21 20 19 18 17

Contents
Preface to the Three Volume Set ix
Preface to Volume 2 xiii
Chapter 1. The Fundamental Theorem of Algebra 1
1.1. Complex Arithmetic 2
1.2. First Proof of the Fundamental Theorem 5
1.3. Second Proof 7
1.4. Exercises 10
Chapter 2. The Brouwer Fixed Point Theorem 11
2.1. Statement of the Theorem 11
2.2. Introducing Extra Structure into a Problem 12
2.3. Two Elementary Problems 12
2.4. Three Advanced Problems 18
2.5. Exercises 27
Chapter 3. Tools 29
3.1. Polyhedral complexes 29
3.2. Urysohn’s Lemma and the Tietze Extension Theorem 31
3.3. Set Convergence 34
3.4. Exercises 36
Chapter 4. Lebesgue Covering Dimension 37
4.1. Deﬁnition of Covering Dimension 37
4.2. Euclidean n-Dimensional Space Rn
Has Covering Dimension n 38
4.3. Construction of Partitions of Unity 40
4.4. Techniques Needed in Higher Dimensions 41
4.5. Exercises 42
Chapter 5. Fat Curves and Peano Curves 45
5.1. The Constructions 45
5.2. The Topological Lemmas 49
5.3. The Analytical Lemmas 51
5.4. Characterization of Peano Curves 52
5.5. Exercises 54
Chapter 6. The Arc, the Simple Closed Curve, and the Cantor Set 57
6.1. Characterizing the Arc and Simple Closed Curve 57
6.2. The Cantor Set and Its Characterization 61
6.3. Interesting Cantor Sets 63
v

vi CONTENTS
6.4. Cantor Sets in the Plane Are Tame 69
6.5. Exercises 73
Chapter 7. Algebraic Topology 75
7.1. Facts Assumed from Algebraic Topology 76
7.2. The Reduced Homology of a Sphere 77
7.3. The Homology of a Ball Complement 77
7.4. The Homology of a Sphere Complement 78
7.5. Proof of the Arc Non-Separation Theorem and the Jordan Curve
Theorem 79
Chapter 8. Characterization of the 2-Sphere 81
8.1. Statement and Proof of the Characterization Theorem 81
8.2. Exercises 89
Chapter 9. 2-Manifolds 91
9.1. Definition and Examples 91
9.2. Exercises 91
Chapter 10. Arcs in S2
Are Tame 95
10.1. Arcs in S2
Are Tame 95
10.2. Disk Isotopies 97
10.3. Exercises 100
Chapter 11. R. L. Moore’s Decomposition Theorem 101
11.1. Examples and Applications 101
11.2. Decomposition Spaces 102
11.3. Proof of the Moore Decomposition Theorem 104
11.4. Exercises 107
Chapter 12. The Open Mapping Theorem 109
12.1. Tools 109
12.2. Two Lemmas 110
12.3. Proof of the Open Mapping Theorem 112
12.4. Exercise 112
Chapter 13. Triangulation of 2-Manifolds 113
13.1. Statement of the Triangulation Theorem 113
13.2. Tools 113
13.3. Proof of the Triangulation Theorem 115
13.4. Exercises 116
Chapter 14. Structure and Classification of 2-Manifolds 117
14.1. Statement of the Structure Theorem 117
14.2. Edge-pairings 118
14.3. Proof of the Structure Theorem 119
14.4. Statement and Proof of the Classification Theorem 123
14.5. Exercises 125
Chapter 15. The Torus 129
15.1. Lines and Arcs in the Plane 129

CONTENTS vii
15.2. The Torus as a Euclidean Surface 131
15.3. Curve Straightening 134
15.4. Construction of the Simple Closed Curve with Slope k/ 136
15.5. Exercises 137
Chapter 16. Orientation and Euler Characteristic 139
16.1. Orientation 139
16.2. Euler Characteristic 141
16.3. Exercises 149
Chapter 17. The Riemann-Hurwitz Theorem 151
17.1. Setting 151
17.2. Elementary Facts from Trigonometry 151
17.3. Branched Maps of S2
154
17.4. Statement of the Riemann-Hurwitz Theorem 155
17.5. Proof of the Riemann-Hurwitz Theorem 155
17.6. Rational Maps 156
17.7. Exercises 157
Bibliography 159

Preface to the Three Volume Set
Geometry measures space (geo = earth, metry = measurement). Einstein’s
theory of relativity measures space-time and might be called geochronometry (geo
= space, chrono= time, metry = measurement). The arc of mathematical history
that has led us from the geometry of the plane of Euclid and the Greeks after 2500
years to the physics of space-time of Einstein is an attractive mathematical story.
Geometrical reasoning has proved instrumental in our understanding of the real and
complex numbers, algebra and number theory, the development of calculus with its
elaboration in analysis and diﬀerential equations, our notions of length, area, and
volume, motion, symmetry, topology, and curvature.
These three volumes form a very personal excursion through those parts of the
mathematics of 1- and 2-dimensional geometry that I have found magical. In all
cases, this point of view is the one most meaningful to me. Every section is designed
around results that, as a student, I found interesting in themselves and not just
as preparation for something to come later. Where is the magic? Why are these
things true?
Where is the tension? Every good theorem should have tension
between hypothesis and conclusion. — Dennis Sullivan
Where is the Sullivan tension in the statement and proofs of the theorems?
What are the key ideas? Why is the given proof natural? Are the theorems almost
false? Is there a nice picture? I am not interested in quoting results without proof.
I am not afraid of a little algebra, or calculus, or linear algebra. I do not care
about complete rigor. I want to understand. If every formula in a book cuts the
readership in half, my audience is a small, elite audience. This book is for the
student who likes the magic and wants to understand.
A scientist is someone who is always a child, asking ‘Why?
why? why?’. — Isidor Isaac Rabi, Nobel Prize in Physics
1944
Wir müssen wissen, wir werden wissen. [We must know, we
will know.] — David Hilbert
The three volumes indicate three natural parts into which the material on 2-
dimensional spaces may be divided:
Volume 1: The geometry of the plane, with various historical attempts to
understand lengths and areas: areas by similarity, by cut and paste, by counting,
by slicing. Applications to the understanding of the real numbers, algebra, number
theory, and the development of calculus. Limitations imposed on the measurement
of size given by nonmeasurable sets and the wonderful Hausdorﬀ-Banach-Tarski
paradox.
ix

x PREFACE TO THE THREE VOLUME SET
Volume 2: The topology of the plane, with all of the standard theorems of
1 and 2-dimensional topology, the Fundamental Theorem of Algebra, the Brouwer
Fixed-Point Theorem, space-filling curves, curves of positive area, the Jordan Curve
Theorem, the topological characterization of the plane, the Schoenflies Theorem,
the R. L. Moore Decomposition Theorem, the Open Mapping Theorem, the trian-
gulation of 2-manifolds, the classification of 2-manifolds via orientation and Euler
characteristic, dimension theory.
Volume 3: An introduction to non-Euclidean geometry and curvature. What
is the analogy between the standard trigonometric functions and the hyperbolic
trig functions? Why is non-Euclidean geometry called hyperbolic? What are the
gross intuitive differences between Euclidean and hyperbolic geometry?
The approach to curvature is backwards to that of Gauss, with definitions
that are obviously invariant under bending, with the intent that curvature should
obviously measure the degree to which a surface cannot be flattened into the plane.
Gauss’s Theorema Egregium then comes at the end of the discussion.
Prerequisites: An undergraduate student with a reasonable memory of cal-
culus and linear algebra, but with no fear of proofs, should be able to understand
almost all of the first volume. A student with the rudiments of topology—open and
closed sets, continuous functions, compact sets and uniform continuity—should be
able to understand almost all of the second volume with the exeption of a little
bit of algebraic topology used to prove results that are intuitively reasonable and
can be assumed if necessary. The final volume should be well within the reach of
someone who is comfortable with integration and change of variables. We will make
an attempt in many places to review the tools needed.
Comments on exercises: Most exercises are interlaced with the text in those
places where the development suggests them. They are an essential part of the
text, and the reader should at least make note of their content. Exercise sections
which appear at the end of most chapters refer back to these exercises, sometimes
with hints, occasionally with solutions, and sometimes add additional exercises.
Readers should try as many exercises as attract them, first without looking at hints
or solutions.
Comments on difficulty: Typically, sections and chapters become more diffi-
cult toward the end. Don’t be afraid to quit a chapter when it becomes too difficult.
Digest as much as interests you and move on to the next chapter or section.
Comments on the bibliography: The book was written with very little
direct reference to sources, and many of the proofs may therefore differ from the
standard ones. But there are many wonderful books and wonderful teachers that
we can learn from. I have therefore collected an annotated bibliography that you
may want to explore. I particularly recommend [1, G. H. Hardy, A Mathematician’s
Apology], [2, G. Pólya, How to Solve It], and [3, T. W. Körner, The Pleasure of
Counting], just for fun, light reading. For a bit of hero worship, I also recommend
the biographical references [21, E. T. Bell, Men of Mathematics], [22, C. Henrion,
Women of Mathematics], and [23, W. Dunham, Journey Through Genius]. And
I have to thank my particular heroes: my brother Larry, who taught me about
uncountable sets, space-filling curves, and mathematical induction; Georg Pólya,
who invited me into his home and showed me his mathematical notebooks; my ad-
visor C. E. Burgess, who introduced me to the wonders of Texas-style mathematics;
R. H. Bing, whose Sling, Dogbone Space, Hooked Rug, Baseball Move, epslums and

PREFACE TO THE THREE VOLUME SET xi
deltas, and Crumpled Cubes added color and wonder to the study of topology; and
W. P. Thurston, who often made me feel like Gary Larson’s character of little brain
(“Stop, professor, my brain is full.”) They were all kind and encouraging to me.
And then there are those whom I only know from their writing: especially Euclid,
Archimedes, Gauss, Hilbert, and Poincaré.
Finally, I must thank Bill Floyd and Walter Parry for more than three decades
of mathematical fun. When we would get together, we would work hard every
morning, then talk mathematics for the rest of the day as we hiked the cities, coun-
trysides, mountains, and woods of Utah, Virginia, Michigan, Minnesota, England,
France, and any other place we could manage to get together. And special thanks
to Bill for cleaning up and improving almost all of those ﬁgures in these books
which he had not himself originally drawn.

Preface to Volume 2
The first of three volumes in this set was devoted to the measurement of lengths
and areas, and to some of the consequences that study had in number theory,
algebra, and analysis. Euclid was able to solve quadratic equations by geometric
construction. But when mathematicians tried to extend those results to equations
of higher degree and to differential equations, a number of fascinating difficulties
arose, all involving limits and continuity, best modelled by topology.
In this second volume we assume that the reader has had a first course in
topology and is comfortable with open and closed sets, connected sets, compact
sets, limits, and continuity. Two good references are W. S. Massey [25] and J. R.
Munkres [24].
The following discoveries led to the topics of this second volume.
(1) The solution of cubic and quartic equations required serious consideration of
complex numbers, thought at first to be mysterious. But the mystery disappeared
when it was seen that complex numbers simply model the Euclidean plane. Abel
and Galois proved that equations of degrees 5 and higher could not be solved in
the relatively simple manner by formula as had sufficed in equations of degrees 1
through 4. But Gauss, without giving explicit solutions, managed to prove the
Fundamental Theorem of Algebra that ensured that complex numbers sufficed for
their solution. Gauss gave proofs involving the geometry and topology of the plane.
(2) Newton showed that the study of motion could be greatly simplified if, in-
stead of examining standard equations, one examined differential equations. Prov-
ing the existence of solutions to rather general differential equations led to problems
in topology. One of the standard proof techniques involves Brouwer’s Fixed Point
Theorem. This volume proves that theorem in dimension 2 and outlines the proof
in general dimensions.
(3) Descartes demonstrated that mechanical devices other than straight edge
and compass can construct curves of very high degree. Once curves of very general
form are accepted as interesting, further delicate questions of length and area arise:
finite curves of infinite length, finite curves of positive area, space filling curves,
disks whose interiors have smaller areas than their closures, 0-dimensional sets
through which no light rays can penetrate, continuous functions that are nowhere
differentiable, sets of fractional dimension. This volume gives examples of many of
these phenomena.
(4) The study of solutions to equations became more unified when all variables
were considered to be complex variables. Riemann modelled complex curves by
surfaces, which are 2-dimensional manifolds and are called Riemann surfaces. The
analysis of 2-dimensional manifolds led naturally to notions, such as triangulation,
genus, and Euler characteristic. These notions are explained in this volume.
xiii

xiv PREFACE TO VOLUME 2
All of these considerations required the study of limits and continuity, and the
abstract notion that models limits and continuity in their most general settings is
the notion of topology. Henri Poincaré wrote:
As for me, all of the diverse paths which I have successively
followed have led me to topology. I have needed the gifts of this
science to pursue my studies of the curves defined by differential
equations and for the generalization to differential equations of
higher order, and, in particular, to those of the three body
problem. I have needed topology for the study of nonuniform
functions of two variables. I have needed it for the study of
the periods of multiple integrals and for the application of that
study to the expansion of perturbed functions. Finally, I have
glimpsed in topology a means to attack an important problem
in the theory of groups, the search for discrete or finite groups
contained in a given continuous group.

CHAPTER 1
The Fundamental Theorem of Algebra
Our first excursion into the topology of the plane will be in the proof of the
Fundamental Theorem of Algebra:
Theorem 1.1 (Fundamental Theorem of Algebra). If f(x) = xn
+an−1xn−1
+
· · · + a1x + a0 = 0 is a polynomial equation in the unknown x and if the constant
coefficients an−1, . . ., a1, a0 are complex numbers, then there is a complex number
x = α that satisfies the equation: f(α) = 0. (Of course, since the real numbers are
a subset of the complex numbers (the line lies in the plane), these coefficients are
allowed to be real numbers.)
The Greeks solved linear and quadratic equations geometrically. They rejected
solutions that were not real numbers as being of no application or interest. Complex
numbers were first acknowledged as important in the solution of cubic equations.
General solution formulas for the cubic and quartic equations were found only by
great effort and cleverness.
Georg Pólya wrote to me when I was a young Mormon missionary in Austria.
He said that I should solve a hard mathematical problem every week so that I
wouldn’t rust (“Wer rastet, der rostet”). He also gave me a list of books that I
might find in a used bookstore. I learned the following argument from one of them:
Here is a general solution to the cubic equation, f(x) = x3
+ ax2
+ bx + c = 0:
We simplify the equation by translation, setting x = z + k. By subsititution, we
find
f(x) = z3
+ (3k + a)z2
+ (3k2
+ 2ak + b)z + (k3
+ ak2
+ bk + c) = 0.
If we choose k = −a/3, the equation has the form
z3
+ pz + q = 0.
Setting z = u + v and substituting, we find
(u3
+ v3
+ q) + (3uv + p)(u + v) = 0.
If we are able to choose u and v so that
u3
+ v3
+ q = 0 and 3uv + p = 0,
the equation will be satisfied. We can solve this pair of equations for u and v in
the standard way by substitution:
u = −p/3v and, consequently,

−p
3v
3
+ v3
+ q = 0.
1

2 1. THE FUNDAMENTAL THEOREM OF ALGEBRA
Multiplying by v3
,
v6
+ qv3
+

−p
3
3
= 0.
Though this equation has degree 6 in the unknown v, it is only quadratic in the
unknown v3
, so that by the quadratic formula,
v3
= −
q
2
±
1
2

q2 + 4
p3
27
= −
q
2
±

q
2
2
+

p
3
3
.
Assuming that we know how to take cube roots, we obtain v, from which we find
u3
= −v3
− q or u = −p/3v, z = u + v, and x = z − (a/3).
For example, if f(x) = x3
− 15x − 4 (already simplified), we have p = −15,
q = −4, v3
= 2 +
√
4 − 125 = 2 + 11i, and u3
= 2 − 11i. But (2 + i)3
= 2 + 11i
and (2 − i)3
= 2 − 11i. The sum of these cube roots is (2 + i) + (2 − i) = 4,
which is a real root of the original equation. The fact that real roots could be
mediated by formulas involving complex numbers was a huge motivating factor for
the acceptance of complex numbers.
We have ignored some of the obvious difficulties: How do we take cube roots?
Further, if we take the three cube roots of v3
and three of u3
, we would be able
to put together 9 possibilities for z, and there can only be three solutions to the
original equation. To choose those that are in fact roots, we must satisfy the more
restrictive equation, u = −p/3v. In the example,
−p
3(2 + i)
=
15
6 + 3i
·
6 − 3i
6 − 3i
=
15 · (6 − 3i)
36 + 9
= 2 − i.
Mathematicians hoped to find similar solutions for polynomial equations of
higher degree, solutions that only required the arithmetic operations of addition,
subtraction, multiplication, and division together with the extraction of roots. This
hope was dashed by the work of N. H. Abel and E. Galois, who showed that the
four arithmetic operations and extraction of roots were inadequate in general for
expressing the roots of a quintic equation in terms of its coefficients.
I would love to include their proofs here, but Abel’s proof required twenty pages
of work and Galois’s development requires a substantial development of the theory
of fields and finite groups. Instead, we shall only prove the Fundamental Theorem
of Algebra. We first need to review some fundamentals from the arithmetic of
complex numbers.
1.1. Complex Arithmetic
Complex numbers were fully accepted when mathematicians learned to in-
terpret them as the points of the plane, with a + bi represented by the pair
(a, b). Addition is then simply vector addition or parallelogram addition, with
(a + bi) + (c + di) = (a + c) + (b+ d)i corresponding to (a, b) + (c, d) = (a + c, b+ d).
See Figure 1.
Multiplication also has a beautiful geometric interpretation that is based
on the polar representation of a complex number (see Figure 2) and on the sum
formulas for the sin and cos. We review these trigonometric sum formulas and their
proofs here.
The sum formulas for sin and cos are consequences of a simple projection prin-
ciple, that is essentially the definition of sin and cos. See Figure 3. It is helpful

1.1. COMPLEX ARITHMETIC 3
a + bi
c + di
(a + c) + (b + d)i
Figure 1. Addition of complex numbers
(0, 0)
α
r
(r cos α, r sin α) = r(cos α, sin α)
Figure 2. Polar coordinates
to think of the projection principle as determining the eﬀect on lengths when one
length is projected orthogonally onto another.
The projection principle. Consider a right triangle with one of its acute
angles equal to α, and suppose that the length of the hypotenuse is r. Then the
leg adjacent to the angle α has length (cos α) · r, and the leg opposite the angle α
has length (sin α) · r.
To obtain the sum formulas for sin and cos, we apply the projection principle to
the following two diagrams; see Figure 4. Each term of the sum formulas represents
a well-deﬁned geometric segment in the diagrams.

4 1. THE FUNDAMENTAL THEOREM OF ALGEBRA
α
(cos α) · r
r
(sin α) · r
Figure 3. The projection principle
(0, 0) (0, 0)
c(α)c(β) c(α)c(β)
α
β
α
β
1 1
c(α) c(α)
s(α)
s(α)
c(α)s(β) c(α)s(β)
s(α)c(β)
s(α)c(β)
s(α)s(β) s(α)s(β)
Figure 4. The addition formulas
We use c and s as shorthand for cos(α + β) and sin(α + β). We use c(α) and
s(α) as shorthand for cos(α) and sin(α), and similarly for cos(β) and sin(β). We
have drawn two right triangles in each of the diagrams, one with angle α, the other
with angle β. We have scaled the triangles so that the larger one has hypotenuse
1 so that the vertex emphasized by the large dot has coordinates (c, s). All of the
other entries are consequences of the projection principle. From the ﬁgures, it is
clear that
c = c(α)c(β) − s(α)s(α), or
cos(α + β) = cos(α) cos(β) − sin(α) sin(β), and similarly
sin(α + β) = cos(α) sin(β) + sin(α) cos(β).
These are the sum formulas for sin and cos.
Admittedly, the diagrams deal only with positive angles α and β whose sum is
≤ π, but all other angles can readily be reduced to these cases.

Exploring the Variety of Random
Documents with Different Content

breeding female or immature—it matters not; kill all you can to-day and
leave the morrow to itself. True, there are game-laws and close-seasons, but
none observe them.[4]
Types of Spanish Bird-Life
DARTFORD WARBLER (Sylvia undata)
Resident. Frequents deep furze-coverts, seldom seen (as we are
constrained to represent it) in separate outline.
We have selected these examples because we know and can speak with
absolute authority. Presumption and analogy will naturally suggest that the
same intelligence, the same blind improvidence will apply equally in other
and far more important matters. Not one of our Spanish friends with whom
we have discussed these subjects time and again but agrees to the letter with
the above conclusions and most bitterly regrets them.

FANTAIL WARBLER
(Cisticola cursitans)
Resident: builds a deep purse-
like nest supported on long
grass or rushes.
CHAPTER II
UNEXPLORED SPAIN (Continued)
ON TRAVELAND OTHER THINGS
TRAVEL in all the wilder regions of Spain implies the saddle. Our Spain
begins, as premised, where roads end. For us railways exist merely to help us
one degree nearer to the final plunge into the unknown; and not railways
only, but roads and bridges soon “petter out” into trackless waste, and leave
the explorer face to face with open wilds—despoblados, that is, uninhabited
regions—with a route-map in his pocket that is quite unreliable, and a trusty
local guide who is just the reverse.
Riding light, with the “irreducible
minimum” stowed in the saddle-bags, one may
traverse Spain from end to end. But it is only a
hasty and superficial view that is thus
obtainable, and except for those who love
roughing it for roughness’ sake, even the
freedom of the saddle presents grave
drawbacks in a land where none live in the
country and none travel off stated tracks. In the
campo, nothing—neither food for man nor
beast—can be obtained, and no provision exists
for travellers where travellers never come. The
little rural hostelry of northern lands has no
place; there is instead a venta or posada which
may too often be likened to a stable for beasts
with an extra stall for their riders. It is a
characteristic of pastoral countries everywhere
that their rude inhabitants discriminate little
between the needs of man and beast.

Types of Spanish Bird-Life ROCK-
THRUSH (Petrocincla saxatilis) A
beautiful spring-migrant to the highest
sierras. Colours of male: opal, orange,
and black, with a white “mirror” in
centre of back. Female, yellow-brown
barred with black.
But even towns of quite considerable size—when far removed from the
track—are totally devoid of inns in our sense. Inns are not needed. The few
Spanish travellers who, greatly daring, venture so far afield, usually bespeak
beforehand the hospitality of some local friend or acquaintance.
Incidentally it may be added that a
visit to one of these out-of-the-world
cities—asleep most of them for the
last few centuries—is a pleasing and
restful change amidst the racket of
exploration. One breathes a mediæval
atmosphere and marvels at the
revelation, enjoying prehistoric peeps
in lost cities replete for the antiquary
with historic memorial and long-
forgotten lore. No one cares.
Yet in those bygone days of Spain’s
world-power these somnolent spots
produced the right stuff,—a minority,
no doubt, belonged to the type
satirised by Cervantes,—but many
more strong in mind as in muscle, who
went forth, knights-errant, Paladins
and Crusaders, to conquer and to
shape the course of history. Is the old
spirit extinct? Our own impression is
that the material is there all right ready
to spring to life like the stones of
Deucalion, so soon as Spain shall have
shaken off her incubus of lethargy and the tyranny that clogs the wheels of
progress. Nor need the interval be long.
That sound human material continues to exist in rural Spain we have had
recent evidence during the calling-out of levies of young troops ordered
abroad to serve their country in Morocco. None could witness the
entrainment at some remote station of a detachment of these fine lads
without being struck by their bearing, their set purpose, and above all their
patriotism. With such material, with a well cared-for, contented, and loyal

army and a broadening of view, wisely graduated but equally resolute, Spain
moves forward. Alfonso XIII. is a soldier first—No! Above that he is a king
by nature, but his care for his army and its well-being has already borne
fruits that are making and will make for the honour, safety, and advancement
of his country.
To resume our interrupted note on travel: whether you are riding across
bush-clad hills, over far-spread prairie, or through the defiles of the sierra, as
shadows lengthen the problem of a night’s lodging obtrudes. There is a
variety of solutions. At a pinch—as when belated or benighted—one may, in
desperate resort, seek shelter in a choza. Now a choza is the reed-thatched
hut which forms the rural peasant’s lonely home. Assuredly you will be
made welcome, and that with a grace and a courtesy—aye, a courtliness—
that characterises even the humblest in Spain. The best there is will be at
your disposal; yet—if permissible to say so in face of such splendid
hospitality (and in the hope that these good leather-clad friends of ours may
not read this book)—the open air is preferable. There exists in a choza
absolutely no accommodation—not a separate room; a low settee running
round the interior, or a withy frame, forms the bed; those kindly folk live all
together, along with their domestic animals—and pigs are reckoned such in
Spain. Let us gratefully pay this due tribute to our peasant friends—but let us
sleep outside.
At each village will usually be found a posada. These differ in degree,
mostly from bad downwards. The lowlier sort—little better than the choza—
is but a long, low, one-storeyed barn which you share with fellow-wayfarers,
and your own and their beasts, or any others that may come in, barely
separated by a thatched partition that is neither noise-proof nor scent-proof.
We can call instances to mind when even that small luxury was lacking, and
all, human and other, shared alike. There are no windows—merely wooden
hatches. If shut, both light and air are excluded; if open, hens, dogs, and cats
will enter with the dawn—the former to finish what remains of supper. The
cats will at least disperse the regiment of rats which, during the night, have
scurried across your sleeping form.
Here we relate, as a specific example, a night we spent this last spring in
northern Estremadura:—

A VILLAGE POSADA
Owing to a miscalculation of distance, it was an hour after sundown ere
we reached our destination, a lonely hamlet among the hills. Our good little
Galician ponies were dead-beat, for we had been in the saddle since 5 A.M.,
and it was past eight ere we toiled up that last steep, rock-terraced slope. We
were a party of three, with a local guide and our own Sancho Panza—
faithful companion, friend, and servant of many years’ standing. At a
dilapidated hovel, the last in the village and perched on a crag, we drew rein,
and after repeated knocks the door was opened by a girl—she had set down a
five-year-old child among the donkeys while she drew the bolt, the ground-
floor being (as usual) a stable. To our inquiry as to food—and the hunger of
the lost was upon us—our hostess merely shrugged her shoulders, and with
an expressive gesture of open hands, answered “Nada”—nothing! Sancho,
however, was equal to the occasion. Within two minutes, while we yet stood
disconsolate, he returned with a cackling cockerel in his arms. “Stew him
quick before he crows,” he adjured the girl, and turned to unload the ponies.
What an age a cockerel takes to cook! It was midnight ere he smoked on
the board and, hunger satisfied, we could turn in. In an upper den were two
alcoves with beds, or rather stone ledges, ordinarily used by the family, and
which were assigned to us, the luckless No. 3 by lot having to make shift (in
preference to sleeping on a filthy floor) with three cranky tables of varying
heights, and whose united lengths proved a foot too short at either end!
Oh, the joy of the morning’s dawn and delicious freshness of the
mountain air, as we turned out at five o’clock for yet another ten-league spell
to our next destination. Two nights later we slept in the gilded luxury of
Madrid! But how we abused our previous neglect in not having brought a
camp-outfit.

The above, however, presents the gloomier side of the picture, and there
is a reverse, even in posadas. We cannot better describe the latter side than
in our own words from Wild Spain:—
A Night at a Posada (Andalucia)
The wayfarer has been travelling all day across the scrub-clad wastes,
fragrant with rosemary and wild thyme, without perhaps seeing a human
being beyond a stray shepherd or a band of nomad gypsies encamped amidst
the green palmettos. Towards night he reaches some small village where he
seeks the rude posada. He sees his horse provided with a good feed of barley
and as much broken straw as he can eat. He is himself regaled with one dish
—probably the olla or a guiso (stew) of kid, either of them, as a rule, of a
rich red-brick hue, from the colour of the red pepper or capsicum in the
chorizo or sausage, which is an important (and potent) component of most
Spanish dishes. The steaming olla will presently be set on a table before the
large wood-fire, and with the best of crisp white bread and wine, the
traveller enjoys his meal in company with any other guest that may have
arrived at the time—be he muleteer or hidalgo. What a fund of information
may be picked up during that promiscuous supper! There will be the
housewife, the barber, and the padre of the village, perhaps a goatherd come
down from the mountains, a muleteer, and a charcoal-burner or two, each
ready to tell his own tale, or to enter into friendly discussion with the
“Ingles.” Then, as you light your breva, a note or two struck on the guitar
falls on ears predisposed to be pleased.
How well one knows those first few opening notes: no occasion to ask
that it may go on: it will all come in time, and one knows there is a merry
evening in prospect. One by one the villagers drop in, and an ever-widening
circle is formed around the open hearth, rows of children collect, even the
dogs draw around to look on. The player and the company gradually warm
up till couplet after couplet of pathetic malagueñas follow in quick
succession. These songs are generally topical, and almost always extempore;
and as most Spaniards can—or rather are anxious to—sing, one enjoys many
verses that are very prettily as well as wittily conceived.
But girls must dance, and find no difficulty in getting partners to join
them. The malagueñas cease, and one or perhaps two couples stand up, and
a pretty sight they afford! Seldom does one see girl-faces so full of fun and

so supremely happy as they adjust the castanets, and one damsel steps aside
to whisper something sly to a sister or friend. And now the dance begins;
observe there is no slurring or attempt to save themselves in any movement.
Each step and figure is carefully executed, but with easy, spontaneous grace
and precision both by the girl and her partner.
Though two or more pairs may be dancing at once, each is quite
independent of the others, and only dance to themselves; nor do the partners
ever touch each other.[5] The steps are difficult and somewhat intricate, and
there is plenty of scope for individual skill, though grace of movement and
supple pliancy of limb and body are almost universal, and are strong points
in dancing both the fandango and minuet. Presently the climax of the dance
approaches. The notes of the guitar grow faster and faster; the man—a
stalwart shepherd-lad—leaps and bounds around his pirouetting partner, and
the steps, though still well ordered and in time, grow so fast that one can
hardly follow their movements.
Now others rise and take the places of the first dancers, and so the
evening passes; perhaps a few glasses of aguardiente are handed round—
certainly much tobacco is smoked—the older folks keep time to the music
with hand-clapping, and all is good nature and merriment.
What is it that makes the recollection of such evenings so pleasant? Is it
merely the fascinating simplicity and freedom of the dance, or the spectacle
of those weird, picturesque groups, bronze-visaged men and dark-eyed
maidens, all lit up by the blaze of the great wood-fire on the hearth, and low-
burning oil-lamps suspended from the rafters? Perhaps it is only the
remembrance of many happy evenings spent among these people since our
boyhood. This we can truly say, that when at last you turn in to sleep you
feel happy and secure among a peasantry with whom politeness and
sympathy are the only passports required to secure to you both friendship
and protection if required. Nor is there a pleasanter means of forming
acquaintance with Spanish country life and customs than a few evenings
spent thus at a farm-house or village inn in any retired district of laughter-
loving Andalucia.
For rough living we are of course prepared, and accept the necessity
without demur or second thought while travelling. But when more serious
objects are in hand—say big-game or the study of nature, objects which
demand more leisurely progress, or actually encamping for a week or more

SERIN (Serinus hortulanus)
A true European canary, but its
song is harsh and hissing.
at selected points—then we prefer to assure complete independence of all
local assistance and shelter.
An expedition on this scale involves an
amount of care and forethought that only
those who have experienced it would credit.
For in Spain it is an unknown undertaking,
and to engineer something new is always
difficult. Quite an extensive camping-trip
can be organised in Africa, where the system
is understood, with less than a hundredth part
of the care needed for a comparatively short
trip in Spain where it is not. The necessary
bulk of camp-outfit and equipment requires a
considerable cavalcade, and this mule-
transport (since no provender is obtainable in
the country) involves carrying along all the
food for the animals—the heaviest item of
all. Naturally the cost of such expeditions
works out to nearly double that of simple
riding.
But, after all, it is worth it! Compare some of the miseries we have above
but lightly touched upon—the dirt and squalor, the nameless horrors of
choza or posada—with the sense of joyous exhilaration felt when encamped
by the banks of some babbling trout-stream or in the glorious freedom of the
open hill. Casting back in mental reverie over a lengthening vista of years,
we certainly count as among the happiest days of life those spent thus under
canvas—whether on the sierras and marismas of Spain, on high field or dark
forest in Scandinavia, or on Afric’s blazing veld.
Should some remarks (here or elsewhere in this book) appear self-
contradictory the reason will be found rather in our inadequate expression
than in any confusion of idea. We love Spain primarily because she is wild
and waste; but, loving her, are naturally desirous that she should advance to
that position among nations that is her due. Such material development,
nevertheless, need not—and will not—imply the total destruction of her wild
beauties. Development on those lines would not consist with the peculiar
genius of the Spanish race, and, while we trust the development will come,
we fear no such collateral results. Take, for instance, the corn-lands. There

the great bustard is alike the index and the price of vast, unwieldy farms
unfenced and but half tilled, remote from rail, road, or market. That
condition we neither expect nor hope to see exchanged for smug fields with
a network of railways. For “three acres and a cow” is not the line of Spanish
regeneration; it is rather a claptrap catch-word of politicians—a murrain on
the lot of them!
True, the plan seems to answer in Denmark, and if the Danes are
satisfied, well and good—that is no business of ours. But no such
mathematical and Procrustean restriction of vital energies and ambitions will
subserve our British race, nor the Spanish. In Spanish sierra may the howl of
the wolf at dawn never be replaced by blast from factory siren, nor the
curling blue smoke of the charcoal-burner in primeval forest be abolished in
favour of black clouds belching from bristling chimneys that pierce a murky
sky. Either in such circumstance would be misplaced.
Similarly, when the engineer shall have been turned loose in the Spanish
marismas, he can, beyond all doubt, destroy them for ever. His straight lines
and intersecting canals, hideous in utilitarian rectitude, would right soon
demolish that glory of lonely desolation—those leagues of marshland,
samphire, and glittering lucio. And all for nothing! Since the desecration will
not “pay” financially—the reason we give in detail elsewhere—and you
sacrifice for a shadow some of the grandest bits of wild nature that yet
survive—the finest length and breadth of utter abandonment that still enrich
a humdrum Europe. Should “progress” only advance on these lines no scrap
of that continent will be left to wanderer in the wilds—no spot where
clanging skeins of wild-geese serry the skies, and the swish of ten thousand
wigeon be heard overhead; or that marvellous iridescence—as of triple flame
—the passing of a flight of flamingoes, be enjoyed.[6]
That national progress and development may come, for Spain’s sake, we
earnestly pray. But does there exist inherent reason why progress, in itself,
should always come to ruin natural and racial beauties? Progress seems
nowadays to be misunderstood as a synonym for uniformity—and
uniformity to a single type. Disciples of the cult of insensate haste, of self-
assertion and advertisement, have pretty well conquered the civilised world;
but in Spain they find no foothold, and we glory to think they never will.
Spain will never be “dragooned” into a servile uniformity. There remain
many, among whom we count our humble selves, who bow no knee to the
modern Baal, and who (while conceding to the “hustling” crowd not one iota

of their pretensions to fuller efficiency in any shape or form) are proud to
find fascination in simplicity, a solace in honest purpose and in old-world
styles of life—right down (if you will) to its inertia.
Yes, may progress come, yet leave unchanged the innate courtesy, the
dignity and independence of rural Spain—unspoilt her sierras and glorious
heaths aromatic of myrtle and mimosa, alternating with natural woods of ilex
and cork-oak—self-sown and park-like, carpeted between in spring-time
with wondrous wealth of wild flowers. There is nothing incongruous in such
aspiration. Incongruity rather comes in with misappreciation of the fitness of
things, as when a coal-mine is planked down in the midst of sylvan beauties,
to save some hypothetic penny-a-ton (as per Prospectus); where pellucid
streams are polluted with chemical filth and vegetation blasted by noisome
fumes; or where God’s fairest landscapes are ruined by forests of hideous
smoke-stacks.
If vandalisms such as these be progress then we prefer Spain as she is.
A Note on the Spanish Fauna
After all, it is less with the human element that this book is concerned
than with the wild Fauna of Spain; a brief introductory notice thereof cannot,
therefore, be omitted.
BONELLI’S EAGLE (Aquila bonellii)
A pair disturbed at their eyrie.

As head of the list must stand the Spanish Ibex (Capra hispánica), a
game-animal of quite first rank, peculiar to the Iberian Peninsula, and whose
nearest relative—the Bharal (Capra cylindricornis)—lives 2500 miles away
in the far Caucasus. In Spain the ibex inhabits six great mountain-ranges,
each covering a vast area but all widely separated. After a crisis that five
years ago threatened extermination, this grand species is now happily
increasing under a measure of protection and the ægis of King Alfonso. Next
—a notable neighbour of the ibex (and practically extinct in central Europe)
—we place the lone and lordly Lammergeyer. A memorable spectacle it is to
watch the huge Gypaëtus sweeping through space o’er glens and corries of
the sierra in striking similitude to some weird flying dragon of Miocene age
—a vision of blood-red irides set on a cruel head with bristly black beard, of
hoary grey plumage and golden breast. Watch him for half an hour—for half
a day—yet never will you discern a sign of force exerted by those 3-yard
pinions. With slightly reflexed wings he sinks 1000 feet; then, shifting
course, rises 2000, 3000 feet till lost to sight over some appalling skyline.
You have seen the long cuneate tail deflected ever so slightly—more gently
than a well-handled helm—but the wide lavender wings remain rigid, not an
effort that indicates force have you descried. Yet the power (so defined as
“horse-power”) required to raise a deadweight of 20 lbs. through such
altitudes can be calculated by engineers to a nicety—how is it exerted? That
the power is there is conspicuous enough, and at least it serves to explain
fabled traditions of giant lammergeyers hurling ibex-hunter from perilous
hand-hold on the crag, to feast on the remains below; or, in idler moment,
bearing off untended babes to their eyries—alas! that the duty of nature-
students involves dissipating all such romance.

BLACK VULTURE (Vultur monachus)
Nests in the mountain-forests of Central Spain, and winters in
Andalucia. Sketched in Cote Doñana—“Getting under way.”
Spain, as geologically designed, being, as to one-half of her superficies,
either a desert wilderness or a mountain solitude, naturally lends congenial
conditions of life to the predatory forms that rely on hooked bill, on tooth
and claw, fang and talon, to ravage their more gentle neighbours. Savage
raptores, furred and feathered, characterise her wilder scenes. Wherever one
may travel, a day’s ride will surely reveal huge vultures and eagles circling
aloft, intent on blood. Throughout the wooded plains the majestic Imperial
Eagle is overlord—you know him afar in sable uniform, offset by snow-
white epaulets. Among the sierras a like condominium is shared by the
Golden and Bonelli’s Eagles—and they have half-a-dozen rivals, to say
nothing of lynxes and fierce wolves (we give a photo of one, the gape of
whose jaws exceeds by one-half that of an African hyaena). Then there
patrol the wastes a horde of savage night-rovers, denominated in Spanish
Alimañas, to which a special chapter is devoted.

WHITE-FACED DUCK (Erismatura leucocephala)
Bill much dilated, waxy-blue in colour. Wings extremely short; a sheeny
grebe-like plumage, and long stiff tail, often carried erect.
In Estremadura, where man is a negligible quantity, and along the wild
wooded valley of the Tagus, roams the Fallow-deer in aboriginal purity of
blood—whether any other European country can so claim it, the authors
have been unable to ascertain. In Cantabria and the Pyrenees the Chamois
abounds.
Of the big game (the list includes red, roe, and fallow-deer, wild-boar,
ibex, chamois, brown bear, etc.), we treat in full detail hereafter.
As regards winged game, this south-western corner of Europe, is
singularly weak. There exists but a single resident species of true game-bird
—the redleg. Compare this with northern Europe, where, in a Scandinavian
elk-forest, we have shot five kinds of grouse within five miles; while
southwards, in Africa, francolins and guinea-fowl are counted in dozens of
species. True, there are ptarmigan in the Pyrenees, capercaillie, hazel-grouse,
and grey partridge in Cantabria, but all these are confined to the Biscayan
area. Nor are we overlooking the grandest game-bird of all, the Great
Bustard, chiefest ornament of Spanish steppe, and there are others—the
lesser bustard, quail, sand-grouse, etc.—but these hardly fall within our
definition. As for the teeming hosts of wildfowl and waterfowl that throng
the Spanish marismas (some coming from Africa in spring, the bulk fleeing
hither from the Arctic winter), all these are so fully treated elsewhere as to
need no further notice here.

Spain boasts several distinct species peculiar to her limits. Among such
(besides the ibex) are that curious amphibian, the Pyrenean musk-rat
(Myogale pyrenaica), not again to be met with nearer than the eastern
confines of Europe. Birds afford an even more striking instance. The Spanish
azure-winged magpie (Cyanopica cooki) abounds in Castile, Estremadura,
and the Sierra Moréna, but its like is seen nowhere else on earth till you
reach China and Japan!

CHAPTER III
THE COTO DOÑANA: OUR HISTORIC HUNTING-
GROUND
A Foreword by Sir Maurice de Bunsen, G.C.M.G., British Ambassador at
Madrid.
Among my recollections of Spain none will be more vivid and delightful than
those of my visits to the Coto Doñana. From beginning to end, climate,
scenery, sport, and hospitable entertainment combine, in that happy region,
to make the hours all too short for the joys they bring. Equipped with
Paradox-gun or rifle, and some variety of ammunition, to suit the shifting
requirements of deer and boar, lynx, partridge, wild-geese and ducks, snipe,
rabbit and hare, nay, perhaps a chance shot at flamingo, vulture, or eagle, the
favoured visitor steps from the Bonanza pier into the broad wherry waiting
to carry him across the Guadalquivir, a few miles only from its outflow into
the Atlantic. In its hold the first of many enticing bocadillos is spread before
him. Table utensils are superfluous luxuries, but, armed with hunting blade
and a formidable appetite, he plays havoc with the red mullet, tortilla, and
carne de membrillo, washed down with a tumbler of sherry which has
ripened through many a year in a not far distant bodega.
In half an hour he is in the saddle. Distances and sandy soil prohibit much
walking in the Coto Doñana.
Sand Waste in Coto Doñana.

Landscape in Coto Doñana, with Marisma in background.
FROM PHOTOGRAPHS BY H.R.H. PHILIPPE, DUKE OF ORLEANS.
Marshalled by our host, the soul of the party, the cavalcade canters lightly
up the sandy beach of the river. Thence it strikes to the left into the pine-
coverts, leading in five hours more to the friendly roof of the “Palacio.” A
picturesque group it is with Vazquez, Caraballo, and other well-known
figures in the van, packhorses loaded with luggage and implements of the
chase, and lean, hungry podencos hunting hither and thither for a stray rabbit
on the way. The views are not to be forgotten, the distant Ronda mountains
seen through a framework of stone-pines, across seventy miles of sandy
dunes, marismas, and intervening plains. After a couple of hours we skirt the
famous sandhills, innocent of the slightest dash of green, which for some
inscrutable reason attract, morning after morning, at the first tinge of dawn,
countless greylag geese to their barren expanse and on which, si Dios quiere,
toll shall be levied ere long. The marismas and long lagoons are covered
here and there with black patches crawling with myriads of waterfowl, to be
described after supper by the careful Vazquez as muy pocos, un salpicon—a
mere sprinkling. Their names and habits, are they not written, with the most
competent of pens, in this very volume? We stop, perhaps, for a first deer-
drive on our line of march. How thrilling that sudden rustle in the
brushwood! Stag is it, or hind, or grisly porker? As we approach the
“Palacio” we see the spreading oak on which perched, contemptuous and
unsuspecting, the imperial eagle, honoured this year by a bullet from King
Alfonso’s unerring rifle. As we ride through the scrub the whirr of the red-
legged partridge sends an involuntary hand to the gun. They may await
another day. At dusk we ride into the whitewashed patio, just in time to sally
forth and get a flighting woodcock between gun and lingering glow of the
setting sun.

SPANISH IMPERIAL EAGLE
For no precious hours are wasted in the Coto Doñana. Next day at early
dawn, maybe, if the lagoon be our destination, or at any rate after a timely
breakfast, off starts again the eager cavalcade, be it in quest of red deer or
less noble quarry. Then all day in the saddle, from drive to drive,
dismounting only to lie in wait for a stag, or trudge through the sage-bushes
after partridge, or flounder through the boggy soto, beloved of snipe, with
intervening oases for the unforgotten bocadillo.
If Vazquez be kind, he will take you one day to crouch with him behind
his well-trained stalking-horse, drawing craftily nearer and nearer to where
the duck sit thickest, till, straightening your aching back, you have leave to
put in your two barrels, as Vazquez lays low some twenty couples with one
booming shot from his four-bore, into the brown.

Egret-Heronry at Santolalla, Coto Doñana.
(THE FOREGROUND IS SAND.)
FROM PHOTOGRAPHS BY H. R. H. PHILIPPE, DUKE OF ORLEANS.
But one morning surely a visit must be paid to the sandhills. Caraballo
will call you at 4 A.M., and soon after you will be jogging over the six or
eight miles which separate the “Palacio” from that morning rendezvous of
the greylag. The stars still shine brightly as you dismount at the foot of the
long stretch of dunes. A few minutes’ trudge will deposit you in a round hole
dug deep in the dazzling white expanse the day before; for a hole too freshly
dug will expose the damp brown sand from below, staining the spotless
surface with a warning blotch, and causing the wary geese to swerve beyond
the range of your No. 1 shot. It is still dark as you drop into your hole.
Gradually the sky grows greyer and lighter, till the sun rises from the round
yellow rim of the blue morning sky. Who shall describe the magic thrill of
the first hoarse notes falling on your straining ear? The temptation to peep
out is strong, but crouching deep down, you wait till the mighty pinions beat
above you, and the first wedge of eight or ten sails grandly away in the
morning sun. You judge them out of shot. But surely this second batch is
lower down? Are they not close upon you? Why then no response to your
two barrels? Was the emotion too great, or have you misjudged the speed of
that easy flight or its distance through the crystal air? All the keener is the
joy when, with heavy thump, your first goose is landed on the sand amid the

tin decoys. When three or four lie there, Vazquez will send his fleet two-
legged “water-dog” to set them up with twigs supporting their bills, to
beguile more of their kind into line with the barrels. If the day be propitious,
the sky will be dotted at times with geese in all directions. Now and again
they will give you a shot, the expert taking surely three or four to the tyro’s
one. It is half-past eight, and you have sat in your hole close on two hours
before Vazquez comes to gather the slain, to which he will add two or three
more, marked down afar, and picked up as dead as the rest. Never have two
of your waking hours passed so quickly. What would you not give to live
them over again and undo some of those inexplicable misses? But one goose
alone would amply repay that early start. Even four or five are all you can
carry, and the twenty or thirty that our expert [who must be nameless] would
have shot, will live to stock the world afresh.
SPANISH LYNX
Among the fauna of the Coto Doñana, a word must be given to the lynx.
Never can I forget sitting one afternoon, Paradox in hand, on the fringe of a
covert. I was waiting for stag, rather drowsily, for the beat was a long one
and the sun hot, when my eyes suddenly rested on a lynx standing broadside
among the bushes, beyond a bare belt of sand, some fifty yards off. Fain
would I have changed my bullet for slugs, but those sharp ears would have
detected the slightest click; so I loosed my bullet for what it was worth.
The lynx was gone. When the beat came at last to an end, I thought I
would just have a look at his tracks. He lay stone-dead behind a bush, shot
through the heart.

The eventful days are all too soon over. But the recollection remains of
happy companionship and varying adventure, of easy intercourse between
Spaniard and Englishman, with the echo of many a sporting tale, mingled
with sage discourse from qualified lips on the habits of bird and beast. Who
can tell you more about them than that group of true sportsmen and lovers of
nature whose names, Garvey, Buck, Gonzalez, and Chapman, are
indissolubly linked with the more modern history of the famous Coto
Doñana?
Maurice de Bunsen.
British Embassy, Madrid,
July 1910.
GREENSHANK (Totanus canescens)

CHAPTER IV
THE COTO DOÑANA
NOTES ON ITS PHYSICAL FORMATION, FAUNA, AND RED DEER
THE great river Guadalquivir, dividing in its oblique course seawards into
double channels and finally swerving, as though reluctant to lose all identity
in the infinite Atlantic, practically cuts off from the Spanish mainland a
triangular region, some forty miles of waste and wilderness, an isolated
desert, singular as it is beautiful, which we now endeavour to describe. This,
from our having for many years held the rights of chase, we can at least
undertake with knowledge and affection.

Its precise geological formation ‘twere beyond
our power, unskilled in that science, to diagnose.
But even to untaught eye, the existence of the
whole area is obviously due to an age-long conflict
waged between two Powers—the great river from
within, the greater ocean without. The
Guadalquivir, draining the distant mountains of
Moréna and full 200 miles of intervening plain,
rolls down a tawny flood charged with yellow mud
till its colour resembles café au lait. Thus proceeds
a ceaseless deposit of sediment upon the sea-bed; but the external Power
forcibly opposes such infringement of its area. Here the elemental battle is
joined. The river has so far prevailed as to have grabbed from the sea many
hundred square miles of alluvial plain, that known as the marisma; but at
this precise epoch, the Sea-Power appears to have called checkmate by
interposing a vast barrier of sand along the whole battle-front. The net result
remains that to-day there is tacked on to the southernmost confines of
Europe a singular exotic patch of African desert.
This sand-barrier, known as the Coto Doñana, occupies, together with its
adjoining dunes on the west, upwards of forty miles of the Spanish coast-
line, its maximum breadth reaching in places to eight or ten miles. The Coto
Doñana is cut off from the mainland of Spain not only by the great river,
but by the marisma—a watery wilderness wide enough to provide a home
for wandering herds of wild camels. (See rough sketch-map above.)
Sand and sand alone constitutes the soil-substance of Doñana, overlying,
presumably, the buried alluvia beneath. Yet a wondrous beauty and variety
of landscape this desolate region affords. From the river’s mouth forests of
stone-pine extend unbroken league beyond league, hill and hollow glorious
in deep-green foliage, while the forest-floor revels in wealth of aromatic
shrubbery all lit up by chequered rays of dappled sunlight. Westward,
beyond the pine-limit, stretch regions of Saharan barrenness where miles of
glistening sand-wastes devoid of any vestige of vegetation dazzle one’s
sight—a glory of magnificent desolation, the splendour of sterility. To
home-naturalists the scene may recall St. John’s classic sandhills of Moray,
but magnified out of recognition by the vastly greater scale, as befits their
respective creators—in the one case the 100-league North Sea, here the
1000-league Atlantic. Rather would we compare these marram-tufted,

wind-sculptured sand-wastes with the Red Sea litoral and the Egyptian
Soudan, where Osman Digna led British troops memorable dances in the
‘nineties—alike both in their physical aspect and in their climate, red-hot by
day, yet apt to be deadly chilly after sundown. Resonant with the weird cry
of the stone-curlew and the rhythmic roar of the Atlantic beyond, these
seaward dunes are everywhere traced with infinite spoor of wild beasts, and
dotted by the conical pitfalls dug by ant-lions (Myrmeleon).
In Doñana.
Between these extremes of deep forest and barren dune are interposed
intermediate regions partaking of the character of both. Here the intrusive
pine projects forest-strips, called Corrales, as it were long oases of verdure,
into the heart of the desert, hidden away between impending dunes which
rear themselves as a mural menace on either hand, and towering above the

summits of the tallest trees. Nor is the menace wholly hypothetic; for not
seldom has the unstable element shifted bodily onwards to engulf in
molecular ruin whole stretches of these isolated and enclosed corrales.
Noble pines, already half submerged, struggle in death-grips with the
treacherous foe; of others, already dead, naught save the topmost summits,
sere and shrunk, protrude above that devouring smiling surface, beneath
which, one assumes, there lie the skeletons of buried forests of a bygone
age.
All along these lonely dunes there stand at regular intervals the grim old
watch-towers of the Moors, reminiscent of half-forgotten times and of a
vanished race. Arab telegraphy was neither wireless nor fireless when
beacon-lights blazing out from tower to tower spread instant alarm from sea
to sierra, seventy miles away.
In contrast with the scenery of both these zones, shows up the landscape
of a third region, on the west—that of scrub. Here, one day later in
geological sense, the eye roams over endless horizons of rolling grey-green
brushwood, the chief component of which is cistus (Helianthemum), but
interspersed in its moister dells with denser jungle of arbutus and lentisk,
genista, tree-heath, and giant-heather, with wondrous variety of other
shrubs; the whole studded and ornamented by groves of stately cork-oaks or
single scattered trees. All these, with the ilex, being evergreen, one misses
those ever-changing autumnal tints that glorify the “fall” in northern climes.
Here only a sporadic splash of sere or yellow relieves the uniform verdure.
Obviously regions of such physical character can ill subserve any human
purpose. As designed by nature, they afford but a home for wild beasts,
fowls of the air, and other ferae which abound in striking and charming
variety. For centuries the Coto Doñana formed, as the name imports, the
hunting-ground of its lords, the Dukes of Medina Sidonia, and to not a few
of the Spanish kings—from Phillip IV. in the early part of the seventeenth
century (as recorded by the contemporary chronicler, Pedro Espinosa) to
Alfonso XII. in 1882, and quite recently to H.M. Don Alfonso XIII. For
five-and-twenty years the authors have been co-tenants, previously under
the aforesaid ducal house; latterly under our old friend, the present owner.
The sparse population of Doñana includes a few herdsmen (vaqueros)
who tend the wild-bred cattle and horses that in semi-feral condition wander
both in the regions of scrub and out in the open marisma. Nomadic

MARSH-HARRIER (Circus aeruginosus)
charcoal-burners squat in the forests, shifting their reed-built wigwams
(chozas) as the exigencies of work require; while the gathering of pine-
cones yields a precarious living to a handful of piñoneros. Lastly, but most
important to us, there are the guardas or keepers, keen-eyed, leather-clad,
and sun-bronzed to the hue of Red Indians. There are a dozen of these wild
men distributed at salient points of the Coto, most of them belonging to
families which have held these posts, sons succeeding fathers, for
generations. Of three such cycles we have ourselves already been witnesses.
Briefly to summarise a rich and heterogeneous fauna is not easy; a
volume might be devoted to this region alone. Elsewhere in this book some
few subjects are treated in detail. Here we merely attempt an outline sketch.
Throughout the winter (excepting
only the wildfowl) there exists no
such conspicuous ornithic display as
appeals to casual eye or ear—those,
say, of the average traveller. Ride far
and wide through these wild
landscapes in December or January,
and you may wonder if their oft-
boasted wealth of bird-life be not
exaggerated. You see, perhaps, little
beyond the ubiquitous birds-of-prey. These are ever the first feature to strike
a stranger. Great eagles, soaring in eccentric circles, hunt the cistus-clad
plain; the wild scream of the kite rings out above the pines, and shapely
buzzards adorn some dead tree. Over rush-girt bogs soar weird marsh-
harriers—three flaps and a drift as, with piercing sight, they scan each tuft
and miss not so much as a frog or a wounded wigeon. All these and others
of their race are naturally conspicuous. But, though unseen, there lurk all
around other forms of equal beauty and interest, abundant enough, but
secretive and apt to be overlooked save by closest scrutiny. That, however,
is a characteristic of winter in all temperate lands. Birds at that season are
apt to be silent and elusive, but their absence is apparent rather than real.
All around you, in fact, forest and jungle, scrub, sallow, and bramble-
brake abound with minor bird-forms—with our British summer visitors,
here settled down in their winter quarters; with charming exotic warblers
and silent songsters—all off work for the season. Where nodding bulrush
fringes quaking bog, or miles of tasselled cane-brakes border the marsh,

“SILENT SONGSTERS”
there is the home of infinite feathered
amphibians, crakes and rails, of reed-climbers
and bush-skulkers, all for the nonce silent,
shy, reclusive.
BLACKSTART (Ruticilla titys)
Abundant in winter; retires to the sierra to nest.
Their portraits, roughly caught during hours of patient waiting, may be
found (some of them) scattered through these chapters. But the present is
not the place for detail.
The land-birds in winter you hardly see, for they “take cover.”
Diametrically different—in cause and effect—is the case of wildfowl.
These, by the essence of their natures and by their economic necessities, are
always conspicuous, for they inhabit solely the open spaces of earth—the

Welcome to our website – the perfect destination for book lovers and
knowledge seekers. We believe that every book holds a new world,
offering opportunities for learning, discovery, and personal growth.
That’s why we are dedicated to bringing you a diverse collection of
books, ranging from classic literature and specialized publications to
self-development guides and children's books.
More than just a book-buying platform, we strive to be a bridge
connecting you with timeless cultural and intellectual values. With an
elegant, user-friendly interface and a smart search system, you can
quickly find the books that best suit your interests. Additionally,
our special promotions and home delivery services help you save time
and fully enjoy the joy of reading.
Join us on a journey of knowledge exploration, passion nurturing, and
personal growth every day!
ebookbell.com

Topology As Fluid Geometry Twodimensional Spaces Volume 2 James W Cannon

More Related Content

Similar to Topology As Fluid Geometry Twodimensional Spaces Volume 2 James W Cannon (20)

Recently uploaded (20)

Topology As Fluid Geometry Twodimensional Spaces Volume 2 James W Cannon