![Introduction and overview 3
measurable metric or conformal structures. Indeed, it is from this point of view
that the Beltrami equation initially arose and it is a view which we adopt in the
last few chapters, where we present a selection of topics that represent fairly
recent developments in a different direction from the analytic development in the
first part of the book.
Many of these notions, ideas and results extend to manifolds, and accordingly,
while we do not develop this aspect in full, the reader should note that all the
machinery we set up is ready for these developments. It is a deep result of Sullivan
that all topological n-manifolds (n 54 4) admit quasiconformal structures, and we
discuss this later in our overview. Thus one is able to do analysis on purely
topological objects, relating topological and analytical invariants. For instance,
Donaldson and Sullivan have developed a measurable Yang-Mills theory [78],
there are analogues of the Atiyah-Singer index theory [321], and there is also the
recent work of Connes, Sullivan and Teleman [73] developing the theory of
characteristic classes in this setting.
Quasiconformal mappings provide a class which lies between homeomorph-
isms and diffeomorphisms. Mappings of finite distortion are even more flexible.
Many constructions in analysis, geometry and topology rely on limiting processes.
The compactness properties of families of mappings with finite distortion
make them ideal tools for solving various problems in n-dimensional analysis
and topology. For instance, in studying deformations of elastic bodies and
the related extremals for variational integrals, mappings of finite distortion
are often the natural candidates to consider because they are closed under
uniform convergence, whereas the limit of a diffeomorphism need not be smooth
nor even a homeomorphism. In this book, we present a considerable number
of such compactness results for the class of mappings of finite distortion, see
Chapter 8.
In recent years there has been another well-known theory of mappings
(referred to as deformations) whose ideas have come to the core of geometry
and analysis. This is the non-linear elasticity theory of Antman, Ball and Ciarlet
[12, 21, 22, 66], building on earlier work of Green [118, 119]. The theory was
founded by the eighteenth-century mathematicians Bernoulli and Euler, who
were concerned with the practical problems of mathematics and physics of that
time. Nowadays the theory of elasticity studies mappings (in certain Sobolev
classes) which minimize stored energy integrals. These mappings are not always
quasiregular, but the governing PDEs are the same. It is necessary to study non-
linear equations to observe certain physical phenomena such as bifurcation and
phase transition. In particular, the Jacobian determinant (a highly non-linear
geometric object itself) of these mappings has been subjected to a great deal of
investigation. Its higher integrability properties were already recognized in the
celebrated paper of Gehring in the 1960s, where he discovered the "reverse Holder
inequalities". In this monograph we shall give a comprehensive account of higher
integrability properties of Jacobians and other, more general, non-linear quant-
ities which arise naturally in the L" theory of differential forms.](https://image.slidesharecdn.com/14264-250407175653-0dbbd62a/85/Geometric-Function-Theory-and-Non-linear-Analysis-1st-Edition-Tadeusz-Iwaniec-25-320.jpg)
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almost no restrictions whatsoever. You may copy it, give it away
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included with this ebook or online at www.gutenberg.org. If you
are not located in the United States, you will have to check the
laws of the country where you are located before using this
eBook.
Title: My Arctic journal: a year among ice-fields and Eskimos
Author: Josephine Diebitsch Peary
Contributor: Robert E. Peary
Release date: February 14, 2021 [eBook #64549]
Most recently updated: October 18, 2024
Language: English
Credits: Richard Tonsing and the Online Distributed Proofreading
Team at https://www.pgdp.net (This file was produced
from images generously made available by The
Internet Archive)
*** START OF THE PROJECT GUTENBERG EBOOK MY ARCTIC
JOURNAL: A YEAR AMONG ICE-FIELDS AND ESKIMOS ***](https://image.slidesharecdn.com/14264-250407175653-0dbbd62a/85/Geometric-Function-Theory-and-Non-linear-Analysis-1st-Edition-Tadeusz-Iwaniec-31-320.jpg)
![shade. As we descended a blue mist seemed to hang over that part of
the cliffs that lay in shadow, and the contrast with the white bergs
gleaming in the sapphire waters below was very striking. We
returned to the foot of the cliff after eight o’clock. On Sunday we
made another expedition, to the Blaese Dael, or “windy valley,”
where a beautiful double waterfall comes tumbling through the hard
rock, into which it has graven a deep channel. We gathered more
flowers, and collected some seaweed; the mosquitos, of which we had
had a foretaste the day before, were extremely troublesome, and
recalled to memory the shores of New Jersey. When we reached the
first Eskimo hut, a number of the piccaninnies[1]
came to me and
presented me with bunches of wild flowers. We gave them some
hardtack in return, and they were happy.
1. The Eskimos frequently designate their children as piccaninnies, a word
doubtless introduced by the whalers.
Mr. Peary, Professor Heilprin, myself, and two other members of
our party dined with the inspector in the evening, joining some
members of the Danish community, who had also been invited. The
course consisted of fresh codfish with caper-sauce, roast ptarmigan,
potatoes boiled and then browned; and for dessert, “Rudgrud,” a
“dump,” almonds, and raisins. There was, following European
custom, a varied accompaniment of wines.
After dinner the gentlemen went up-stairs to examine the
geological and oölogical collections of the inspector, while the ladies
preferred the parlor with their coffee. Were it not for the outer
surroundings, it would have been difficult to realize that we were in
the distant Arctic realm, so truly homelike were the scenes of the
little household, and so cheerful the little that was necessary to make
living here not only comfortable, but pleasant. The entire community
numbers barely 120 souls, nine tenths of whom are Eskimos, mainly
half-breeds; the remainder are the Danish officials and their families,
whose recreation lies almost entirely within the little circle which
they themselves constitute.
Toward nine o’clock we visited the storehouse, where a native ball
was in progress. Several of our boys went the rounds with the Eskimo
“belles,” but for me the odor of the interior was too strong to permit
me to say that looking on was an “unalloyed pleasure.” The steps](https://image.slidesharecdn.com/14264-250407175653-0dbbd62a/85/Geometric-Function-Theory-and-Non-linear-Analysis-1st-Edition-Tadeusz-Iwaniec-46-320.jpg)
Geometric Function Theory and Non linear Analysis 1st Edition Tadeusz Iwaniec
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- 5. Geometric Function Theory and Non linear Analysis 1st Edition Tadeusz Iwaniec Digital Instant Download Author(s): Tadeusz Iwaniec, Gaven Martin ISBN(s): 9780198509295, 0198509294 Edition: 1st File Details: PDF, 13.46 MB Year: 2002 Language: english
- 6. D-X-70 RD }I i i1 f 'Z S Jul e'- ra r ,itu'l, caI-, TA DE,USZ .l WAN I E C and GAVE NN Mk.R' TINT OXFOI SC +' N CE PUBUC .IONS
- 7. OXFORD MATHEMATICAL MONOGRAPHS Series Editors J. M. BALL E. M. FRIEDLANDER I. G. MACDONALD L. NIRENBERG R. PENROSE J. T. STUART
- 8. OXFORD MATHEMATICAL MONOGRAPHS A. Belleni-Moranti: Applied semigroups and evolution equations A. M. Arthurs: Complementary variational principles 2nd edition M. Rosenblum and J. Rovnyalc Hardy classes and operator theory J. W. P. Hirschfeld: Finite projective spaces of three dimensions A. Pressley and G. Segal: Loop groups D. E. Edmunds and W. D. Evans: Spectral theory and differential operators Wang Jianhua: The theory of games S. Omatu and J. H. Seinfeld: Distributed parameter systems: theory and applications J. Hilgert, K. H. Hofmann, and J. D. Lawson: Lie groups, convex cones, and semi groups S. Dineen: The Schwarz lemma S. K. Donaldson and P. B. Kronheimer: The geometry offour-manifolds D. W. Robinson: Elliptic operators and Lie groups A. G. Werschulz: The computational complexity of differential and integral equations L. Evens: Cohomology of groups G. Effinger and D. R. Hayes: Additive number theory of polynomials J. W. P. Hirschfeld and J. A. Thas: General Galois geometries P. N. Hoffman and J. F. Humphreys: Projective representations of the symmetric groups 1. Gyori and G. Lades: The oscillation theory of delay differential equations J. Heinonen, T. Kilpelainen, and O. Martio: Non-linear potential theory B. Amberg, S. Franciosi, and F. de Giovanni: Products of groups M. E. Gurtin: Thermomechanics of evolving phase boundaries in the plane 1. Ionescu and M. Sofonea: Functional and numerical methods in viscoplasticity N. Woodhouse: Geometric quantization 2nd edition U. Grenander: General pattern theory J. Faraut and A. Koranyi: Analysis on symmetric cones I. G. Macdonald: Symmetric functions and Hall polynomials 2nd edition B. L. R. Shawyer and B. B. Watson: Borel's methods of summability M. Holschneider: Wavelets. an analysis tool Jacques Thhvenaz: G-algebras and modular representation theory Hans-Joachim Baues: Homotopy type and homology P. D. D'Eath: Black holes: gravitational interactions R. Lowen: Approach spaces: the missing link in the topology-uniformity-metric triad Nguyen Dinh Cong: Topological dynamics of random dynamical systems J. W. P. Hirschfeld: Projective geometries over finite fields 2nd edition K. Matsuzald and M. Taniguchi Hyperbolic manifolds and Kleinian groups David E. Evans and Yasuyuld Kawahigashi: Quantum symmetries on operator algebras Norbert Klingen: Arithmetical similarities: prime decomposition andfinite group theory Isabelle Catto, Claude Le Bris, and Pierre-Louis Lions: The mathematical theory of thermodynamic limits: Thomas-Fermi type models D. McDuff and D. Salamon: Introduction to symplectic topology 2nd edition William M. Goldman: Complex hyperbolic geometry Charles J. Colbourn and Alexander Rosa: Triple systems V. A. Kozlov, V. G. Maz'ya and A. B. Movchan: Asymptotic analysis offields in multi-structures Girard A. Maugin: Nonlinear waves in elastic crystals George Dassios and Ralph Kleinman: Low frequency scattering Gerald W. Johnson and Michel L. Lapidus: The Feynman integral and Feynman's operational calculus W. Lay and S. Y. Slavyanov: Special functions: A unified theory based on singularities D. Joyce: Compact manifolds with special holonomy A. Carbone and S. Semmes: A graphic apology for symmetry and implicitness Johann Boos: Classical and modern methods in summability Nigel Higson and John Roe: Analytic K-homology S. Semmes: Some novel types of fractal geometry Tadeusz Iwaniec and Gaven Martin: Geometric function theory and non-linear analysis
- 9. Geometric Function Theory and Non linear Analysis TADEUSZ IWANIEC John Raymond French Distinguished Professor of Mathematics at Syracuse University and GAVEN MARTIN Professor of Mathematics at the University of Auckland and James Cook Fellow of Royal Society (NZ) CLARENDON PRESS OXFORD 2001
- 10. OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Cape Town Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris Si o Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Tadeusz Iwaniec and Gaven Martin, 2001 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2001 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this title is available from the British Library Library of Congress Cataloging in Publication Data Iwaniec, Tadeusz. Geometric function theory and non-linear analysis / Tadeusz Iwaniec and Gaven Martin. (Oxford mathematical monographs) Includes bibliographical references and index. 1. Geometric function theory. 2. Non-linear theories. 3. Numerical analysis. 1. Martin, Gaven. II. Title. III. Series. QA360 .I93 2001 515-dc2l 2001034652 ISBN 0 19 850929 4 10 9 8 7 6 5 4 3 2 1 Typeset by Integra Software Services Pvt. Ltd, Pondicherry, India www.integra-india.com Printed in Great Britain on acid-free paper by T.J. International Ltd., Padstow, Cornwall
- 11. To our families Graiyna & Krystyna and Dianne & Jennifer & Amy
- 12. PREFACE This book is largely about the geometry of mappings - that is, functions or deformations between subsets of the Euclidean n-space R" and more generally between manifolds or other geometric objects. Such mappings may be homeo- morphisms, diffeomorphisms, branched coverings or more abstract cor- respondences such as Sobolev mappings. They may arise as the solutions to differential equations, the minima of certain optimization problems in the calculus of variations, as local coordinates on a manifold or as geometric realizations of abstract isomorphisms between spaces. In each case the regularity and geometric properties of these mappings will tell us something about the problem at hand or the spaces we are investigating. Of course such a general topic intersects many areas of modern mathematics. Thus we will run into aspects of differential geometry, topology, partial differen- tial equations and harmonic analysis, as well as nonlinear analysis, the calculus of variations and so forth. A good deal of this intersection is surveyed in Chapter 1, in which our aim is to give the reader some appreciation of the diversity of applications and directions in which current research is moving, as well as a glimpse of the substantial body of work which we were unable to cover in any detail here. This book is essentially a research monograph. We have tried to present a fairly complete account of the most recent developments in these areas as they pertain to the geometry of mappings, and indeed a significant portion of this book was new or recent at the time of writing. However, we do cover and offer new approaches to many aspects of the classical theory as well as devoting a few chapters to foundational material, and we have pitched the level of the book at the competent graduate student. We wish to express our deep gratitude to the many fellow mathematicians who have contributed in one way or another to this book. In particular, those from the Finnish and Italian Schools with whom we have collaborated and discussed many ideas and whose theorems can be found throughout this book. Also many thanks to Tsukasa Yashiro who created all the pictures for us, and to John Duncan and Volker Mayer who read and commented on a good portion of the manuscript. Both authors would like to acknowledge the support they received from the US National Science Foundation and the NZ Marsden Fund. While this book is dedicated to our families, there are two others we must acknowledge. These are our teachers, Bogdan Bojarski and Fred Gehring, who
- 13. Preface vii pioneered much of the theory presented here. In particular, Fred brought us together in Ann Arbor from either end of the world to do mathematics, and throughout our careers he and his wife Lois have been unfailingly supportive. Thanks! March 2001 T.I. G.M.
- 15. CONTENTS 1 Introduction and overview 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 The planar theory n-Dimensional quasiconformal mappings The Liouville theorem Higher integrability Stability and rigidity phenomena Quasiconformal structures on manifolds Nevanlinna theory Non-linear potential theory Singular integral operators Removable singularities Quasiconformal groups, semigroups and dynamics Continuum mechanics and non-linear elasticity Mostow rigidity 2 Conformal mappings 2.1 The Cauchy-Riemann system 2.2 The Mobius group 2.3 The Liouville theorem (smooth case) 2.4 Curvature 2.5 Computing the Jacobian 2.6 Conclusions 2.7 Further aspects 3 Stability of the Mobius group 3.1 Mapping classes 3.2 Harnack inequalities 3.3 A stability function 3.4 Passing Harnack inequalities on to Mt 3.5 Local injectivity 4 Sobolev theory and function spaces 4.1 Schwartz distributions 4.2 Definitions of Sobolev spaces 4.3 Mollification 4.4 Lebesgue points 4.5 Pointwise coincidence of Sobolev functions
- 16. x Contents 4.6 Alternative characterizations 4.7 Cross product of gradient fields 4.8 The adjoint differential 4.9 Subharmonic distributions 4.10 Embedding theorems 4.11 Duals and compact embeddings 4.12 Orlicz-Sobolev spaces 4.13 Hardy spaces and BMO 5 The Liouville theorem 5.1 Introduction 5.2 Second-order estimates 5.3 Identities 5.4 Second-order equations 5.5 Continuity of the Jacobian 5.6 A formula for the Jacobian 5.7 Concluding arguments 6 Mappings of finite distortion 6.1 Differentiability 6.2 Integrability of the Jacobian 6.3 Absolute continuity 6.4 Distortion functions 6.5 Examples 6.5.1 Radial stretchings 6.5.2 Winding maps 6.5.3 Cones and cylinders 6.5.4 The Zorich exponential map 6.5.5 A regularity example 6.5.6 Squeezing the Sierpinski sponge 6.5.7 Releasing the sponge 7 Continuity 7.1 Distributional Jacobians 7.2 The Ll integrability of the Jacobian 7.3 Weakly monotone functions 7.4 Oscillation in a ball 7.5 Modulus of continuity 7.6 Exponentially integrable outer distortion 7.7 Holder estimates 7.8 Fundamental LP-inequality for the Jacobian 7.8.1 A class of Orlicz functions 7.8.2 Another proof of Corollary 7.2.1
- 17. Contents xi 8 Compactness 169 8.1 Distributional Jacobians revisited 169 8.2 Weak convergence of Jacobians 172 8.3 Maximal inequalities 175 8.4 Improving the degree of integrability 176 8.5 Weak limits and orientation 181 8.6 L log L integrability 185 8.7 A limit theorem 186 8.8 Polyconvex functions 187 8.8.1 Null Lagrangians 188 8.8.2 Polyconvexity of distortion functions 190 8.9 Biting convergence 191 8.10 Lower semicontinuity of the distortion 193 8.11 The failure of lower semicontinuity 197 8.12 Bounded distortion 200 8.13 Local injectivity revisited 201 8.14 Compactness for exponentially integrable distortion 205 9 Topics from Multilinear Algebra 208 9.1 The 1-covectors 208 9.2 The wedge product 209 9.3 Orientation 211 9.4 The pullback 211 9.5 Matrix representations 212 9.6 Inner products 213 9.7 The volume element 216 9.8 Hodge duality 217 9.9 Hadamard-Schwarz inequality 220 9.10 Submultiplicity of the distortion 221 10 Differential Forms 222 10.1 Differential forms in R" 222 10.2 Pullback of differential forms 228 10.3 Integration by parts 229 10.4 Orlicz-Sobolev spaces of differential forms 232 10.5 The Hodge decomposition 234 10.6 The Hodge decomposition in R" 236 11 Beltrami equations 240 11.1 The Beltrami equation 240 11.2 A fundamental example 244 11.2.1 The construction 245 11.3 Liouville-type theorem 250
- 18. xii Contents 11.4 The principal solution 11.5 Stoilow factorization 11.6 Failure of factorization 11.7 Solutions for integrable distortion 11.8 Distortion in the exponential class 11.8.1 An example 11.8.2 Statement of results 11.9 Distortion in the subexponential class 11.9.1 An example 11.9.2 Statement of results 11.9.3 Further generalities 11.10 Preliminaries 11.10.1 Results from harmonic analysis 11.10.2 Existence for exponentially integrable distortion 11.10.3 Uniqueness 11.10.4 Critical exponents 11.10.5 Existence for subexponentially integrable distortion 11.11 Global solutions 11.12 Holomorphic dependence 11.13 Examples and non-uniqueness 11.14 Compactness 11.15 Removable singularities 11.16 Final comments 12 Riesz transforms 12.1 Singular integral operators 12.2 Fourier multipliers 12.3 Trivial extension of a scalar operator 12.4 Extension to C" 12.5 The real method of rotation 12.6 The complex method of rotation 12.7 Polarization 12.8 The tensor product of Riesz transforms 12.9 Dirac operators and the Hilbert transform on forms 12.10 The LP-norms of the Hilbert transform on forms 12.11 Further estimates 12.12 Interpolation 13 Integral estimates 13.1 Non-linear commutators 13.2 The complex method of interpolation
- 19. Contents xiii 13.3 Jacobians and wedge products revisited 343 13.4 The H'-theory of wedge products 345 13.5 An L log L inequality 347 13.6 Estimates beyond the natural exponent 350 13.7 Proof of the fundamental inequality for Jacobians 352 14 The Gehring lemma 354 14.1 A covering lemma 356 14.2 Calderdn-Zygmund decomposition 357 14.3 Gehring's lemma in Orlicz spaces 359 14.4 Caccioppoli's inequality 363 14.5 The order of zeros 367 15 The governing equations 370 15.1 Equations in the plane 370 15.2 Absolute minima of variational integrals 375 15.3 Conformal mappings 380 15.4 Equations at the level of exterior algebra 386 15.5 Even dimensions 391 15.6 Signature operators 393 15.7 Four dimensions 398 16 Topological properties of mappings of bounded distortion 401 16.1 The energy integrand 402 16.2 The Dirichlet problem 405 16.3 The A-harmonic equation 406 16.4 Caccioppoli inequality 410 16.5 The comparison principle 410 16.6 The polar set 411 16.7 Sets of zero conformal capacity 414 16.8 Qualitative analysis near polar points 416 16.9 Local injectivity of smooth mappings 419 16.10 The Jacobian is non-vanishing 422 16.11 Analytic degree theory 423 16.12 Openness and discreteness for mappings of bounded distortion 426 16.13 Further generalities 427 16.14 An update 428 17 Painlev4's theorem in space 431 17.1 Painleve's theorem in the plane 431 17.2 Hausdorff dimension and capacity 432
- 20. xiv Contents 17.3 Removability of singularities 17.4 Distortion of dimension 18 Even dimensions 18.1 The Beltrami operator 18.2 Integrability theorems in even dimensions 18.3 Mappings with exponentially integrable distortion 18.4 The Lz inverse of I -.a S 18.5 Wl"n-regularity 18.6 Singularities 18.7 An example 19 Picard and Montel theorems in space 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 Picard's theorem in space Serrin's theorem and Harnack functions Estimates in fe(Rn) Harnack inequalities near zeros Collections of Harnack functions Proof of Rickman's theorem Normal families Montel's theorem in space Further generalizations 20 Conformal structures 20.1 The space S(n) 20.2 Conformal structures 20.3 The smallest ball 21 Uniformly quasiregular mappings 21.1 A first uniqueness result 21.2 First examples 21.3 Fatou and Julia sets 21.4 Lattes-type examples 21.5 Invariant conformal structures 22 Quasiconformal groups 22.1 Convergence properties 22.2 The elementary quasiconformal groups 22.3 Non-elementary quasiconformal groups 22.4 The triple space 22.5 Conjugacy results 22.6 Hilbert-Smith conjecture 22.7 Remarks
- 21. Contents xv 23 Analytic continuation for Beltrami systems 528 23.1 Uniqueness 528 23.2 Proof of Theorem 23.1.1 529 23.3 Remarks 530 Bibliography 531 Index 547
- 23. 1 INTRODUCTION AND OVERVIEW The interplay between partial differential equations (PDEs) and the theory of mappings has a long and distinguished history, and that connection underpins this book. Gauss's practical geodesic survey work stimulated him to develop the theory of conformal transformations, for mapping figures from one surface to another. For conformal transformation from plane to plane he used a pair of equations apparently derived by d'Alembert, who first related the derivatives of the real and imaginary part of a complex function in 1746 in his work on hydro- dynamics (311, p. 497. These equations have become known as the Cauchy- Riemann equations. Gauss developed the differential geometry of surfaces around 1827, emphasizing the intrinsic geometry, with Gaussian curvature defined by measurements within the surface. If a surface is deformed conformally (preserving angles), then the Gaussian curvature is unchanged, and hence the intrinsic geometry of the surface is unaffected by such deformations. Gauss also considered geodesic curves within surfaces. In 1829 Lobachevsky constructed a surface (the horosphere) within his non-Euclidean space, such that the intrinsic geometry within that surface is Euclidean. with geodesic curves being called Euclidean lines. For the converse process, he could only suggest tentatively that, within Euclidean space, the intrinsic geometry of a sphere of imaginary radius was Lobachevskian. But imaginary numbers were then regarded with justifiable suspicion, and he did not propose that as an acceptable model of his geometry within Euclidean space. In his most famous work, Beltrami [321 showed that Lobachevsky's geometry is the intrinsic geometry of a surface of constant negative curvature, with geodesic curves being called lines in Lobachevsky's geometry. Beltrami illustrated various surfaces with constant negative curvature, the sim- plest of which is the pseudosphere generated by revolving a tractrix around its axis. Beltrami's paper convinced most mathematicians that the geometries of Euclid and of Lobachevsky are logically equivalent. In that work Beltrami used a differential equation corresponding to Gauss's equation. This has come to be known as Beltrami's equation, and later in this book we shall present the most recent developments in this area, solving Beltrami's equation at the critical point, where uniform ellipticity bounds are lost. This will necessitate the development of some considerable technical machinery to enable us to move away from the classical setting of uniformly elliptic PDEs to the case of degenerate elliptic equations. Beltrami's equation and its solutions, the quasiconformal mappings, have found a home in virtually all aspects of modern complex analysis, from the
- 24. 2 Introduction and overview theory of Riemann surfaces and Teichmiiller and Moduli spaces to more recent developments such as holomorphic dynamics and three-dimensional hyperbolic geometry. We hope the developments presented in this book encourage new applications of quasiconformal mappings in these areas. It has been nearly 200 years since Cauchy created the basic framework of complex function theory, principally during the years 1814-1831 [3061, since when the theory of conformal mappings and analytic functions has expanded in many different directions, far too numerous to relate here. This theory lies at the foundation of virtually all of modern analysis. Moreover, practical applications, such as in fluid flow, hydrodynamics and more modern areas of control theory, robotics and dynamical systems, abound. Geometric function theory (GFT) in higher dimensions is largely concerned with generalizations to R" of aspects of this theory of analytic functions of one complex variable, particularly the geometric and function-theoretic properties. We hope to give the reader a clear picture of these connections. In this sense GFT has been quite a successful theory, with many diverse applications. The category of maps that one usually considers in the higher- dimensional theory are, as in the planar case, the quasiregular mappings, or, if injective, quasiconformal mappings. Both kinds of mappings have the character- istic property of "bounded distortion" and solve PDEs closely analogous to the Cauchy-Riemann and Beltrami equations. Moreover, these mappings preserve the natural Sobolev spaces which arise in consideration of the function theory and PDEs on subdomains of R", or more generally n-manifolds. More recent developments have emphasized the connections between quasi- conformal mappings, harmonic analysis and PDEs. This connection is an import- ant aspect of our book. And that is why we depart from the usual theory of quasiconformal mappings quite early on and develop the theory of mappings with finite distortion. Again, the motivation here is to move into the realm of degen- erate elliptic equations where important applications lie. Usually, however, some control of the distortion functions (or equivalently the ellipticity bounds) will be necessary to achieve concrete results. These often take the form of integral estimates in some Lebesgue or Sobolev space. As mentioned, the governing equations for mappings of finite distortion are non-linear first-order systems of PDEs closely related to the Cauchy-Riemann equations and the complex Beltrami equation. There are also related second-order equations. For example, the components of an analytic function are harmonic, while those of a quasiregular mapping are "A-harmonic". In this way such well- known non-linear differential operators as the p-Laplacian and the associated non- linear potential theory arise naturally. There is also a close analogy between the analytic aspects of the theory of holomorphic functions and higher-dimensional theories of mappings of finite distortion. As we shall see, this analogy is particu- larly pronounced in even dimensions. A fruitful idea when studying quasiregular mappings, or more generally map- pings of finite distortion, is to view them as conformal with respect to certain
- 25. Introduction and overview 3 measurable metric or conformal structures. Indeed, it is from this point of view that the Beltrami equation initially arose and it is a view which we adopt in the last few chapters, where we present a selection of topics that represent fairly recent developments in a different direction from the analytic development in the first part of the book. Many of these notions, ideas and results extend to manifolds, and accordingly, while we do not develop this aspect in full, the reader should note that all the machinery we set up is ready for these developments. It is a deep result of Sullivan that all topological n-manifolds (n 54 4) admit quasiconformal structures, and we discuss this later in our overview. Thus one is able to do analysis on purely topological objects, relating topological and analytical invariants. For instance, Donaldson and Sullivan have developed a measurable Yang-Mills theory [78], there are analogues of the Atiyah-Singer index theory [321], and there is also the recent work of Connes, Sullivan and Teleman [73] developing the theory of characteristic classes in this setting. Quasiconformal mappings provide a class which lies between homeomorph- isms and diffeomorphisms. Mappings of finite distortion are even more flexible. Many constructions in analysis, geometry and topology rely on limiting processes. The compactness properties of families of mappings with finite distortion make them ideal tools for solving various problems in n-dimensional analysis and topology. For instance, in studying deformations of elastic bodies and the related extremals for variational integrals, mappings of finite distortion are often the natural candidates to consider because they are closed under uniform convergence, whereas the limit of a diffeomorphism need not be smooth nor even a homeomorphism. In this book, we present a considerable number of such compactness results for the class of mappings of finite distortion, see Chapter 8. In recent years there has been another well-known theory of mappings (referred to as deformations) whose ideas have come to the core of geometry and analysis. This is the non-linear elasticity theory of Antman, Ball and Ciarlet [12, 21, 22, 66], building on earlier work of Green [118, 119]. The theory was founded by the eighteenth-century mathematicians Bernoulli and Euler, who were concerned with the practical problems of mathematics and physics of that time. Nowadays the theory of elasticity studies mappings (in certain Sobolev classes) which minimize stored energy integrals. These mappings are not always quasiregular, but the governing PDEs are the same. It is necessary to study non- linear equations to observe certain physical phenomena such as bifurcation and phase transition. In particular, the Jacobian determinant (a highly non-linear geometric object itself) of these mappings has been subjected to a great deal of investigation. Its higher integrability properties were already recognized in the celebrated paper of Gehring in the 1960s, where he discovered the "reverse Holder inequalities". In this monograph we shall give a comprehensive account of higher integrability properties of Jacobians and other, more general, non-linear quant- ities which arise naturally in the L" theory of differential forms.
- 26. Discovering Diverse Content Through Random Scribd Documents
- 30. The Project Gutenberg eBook of My Arctic journal: a year among ice-fields and Eskimos
- 31. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: My Arctic journal: a year among ice-fields and Eskimos Author: Josephine Diebitsch Peary Contributor: Robert E. Peary Release date: February 14, 2021 [eBook #64549] Most recently updated: October 18, 2024 Language: English Credits: Richard Tonsing and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive) *** START OF THE PROJECT GUTENBERG EBOOK MY ARCTIC JOURNAL: A YEAR AMONG ICE-FIELDS AND ESKIMOS ***
- 32. Transcriber’s Note: The cover image was created by the transcriber and is placed in the public domain. TAKING ON AN ESKIMO PILOT.
- 33. MY ARCTIC JOURNAL A YEAR AMONG ICE-FIELDS AND ESKIMOS BY JOSEPHINE DIEBITSCH-PEARY WITH AN ACCOUNT OF THE GREAT WHITE JOURNEY ACROSS GREENLAND BY ROBERT E. PEARY CIVIL ENGINEER, U. S. NAVY LONDON LONGMANS, GREEN, AND CO. 1894 All rights reserved
- 34. THE DE VINNE PRESS, NEW YORK, U. S. A.
- 35. INTRODUCTORY NOTE On June 6, 1891, the steam-whaler “Kite,” which was to bear the expedition of the Philadelphia Academy of Natural Sciences northward, set sail from the port of New-York, her destination being Whale Sound, on the northwest coast of Greenland, where it had been determined to pass the winter, preliminary to the long traverse of the inland ice which was to solve the question of the extension of Greenland in the direction of the Pole. The members of the expedition numbered but five besides the commander, Mr. Peary, and his wife. They were Dr. F. A. Cook, Messrs. Langdon Gibson, Eivind Astrup, and John T. Verhoeff, and Mr. Peary’s faithful colored attendant in his surveying labors in Nicaragua, Matthew Henson. This was the smallest number that had ever been banded together for extended explorations in the high Arctic zone. A year and a quarter after their departure, with the aid of a relief expedition conducted by Professor Angelo Heilprin, Mr. Peary’s party, lacking one of its members, the unfortunate Mr. Verhoeff, returned to the American shore. The explorer had traversed northern Greenland from coast to coast, and had added a remarkable chapter to the history of Arctic exploration. The main results of Mr. Peary’s journey were: The determination of the rapid convergence of the shores of Greenland above the 78th parallel of latitude, and consequently the practical demonstration of the insularity of this great land-mass; The discovery of the existence of ice-free land-masses to the northward of Greenland; and The delineation of the northward extension of the great Greenland ice-cap. In the following pages Mrs. Peary recounts her experiences of a full twelvemonth spent on the shores of McCormick Bay, midway
- 36. between the Arctic Circle and the North Pole. The Eskimos with whom she came in contact belong to a little tribe of about three hundred and fifty individuals, completely isolated from the rest of the world. They are separated by hundreds of miles from their nearest neighbors, with whom they have no intercourse whatever. These people had never seen a white woman, and some of them had never beheld a civilized being. The opportunities which Mrs. Peary had of observing their manners and mode of life have enabled her to make a valuable contribution to ethnological learning. THE PUBLISHERS.
- 37. PREFACE This plain and simple narrative of a year spent by a refined woman in the realm of the dreaded Frost King has been written only after persistent and urgent pressure from friends, by one who shrank from publicity, and who reluctantly yielded to the idea that her experiences might be of interest to others besides her immediate friends. I have been requested to write a few words of introduction; and while there may be some to whom it might occur that I was too much interested to perform this task properly, it must nevertheless be admitted that there is probably no one better fitted than myself to do it. Little, indeed, need be said. The feeling that led Mrs. Peary through these experiences was first and foremost a desire to be by my side, coupled with the conviction that she was fitted physically as well as otherwise to share with me a portion at least of the fatigues and hardships of the work. I fully concurred in this feeling, and yet, in spite of my oft-expressed view that the dangers of life and work in the Arctic regions have been greatly exaggerated, I cannot but admire her courage. She has been where no white woman has ever been, and where many a man has hesitated to go; and she has seen phases of the life of the most northerly tribe of human beings on the globe, and in many ways has been enabled to get a closer insight into their ways and customs than had been obtained before. I rarely, if ever, take up the thread of our Arctic experiences without reverting to two pictures: one is the first night that we spent on the Greenland shore after the departure of the “Kite,” when, in a little tent on the rocks—a tent which the furious wind threatened every moment to carry away bodily—she watched by my side as I lay a helpless cripple with a broken leg, our small party the only human
- 38. beings on that shore, and the little “Kite,” from which we had landed, drifted far out among the ice by the storm, and invisible through the rain. Long afterward she told me that every unwonted sound of the wind set her heart beating with the thoughts of some hungry bear roaming along the shore and attracted by the unusual sight of the tent; yet she never gave a sign at the time of her fears, lest it should disturb me. The other picture is that of a scene perhaps a month or two later, when—myself still a cripple, but not entirely helpless—this same woman sat for an hour beside me in the stern of a boat, calmly reloading our empty firearms while a herd of infuriated walrus about us thrust their savage heads with gleaming tusks and bloodshot eyes out of the water close to the muzzles of our rifles, so that she could have touched them with her hand, in their efforts to get their tusks over the gunwale and capsize the boat. I may perhaps be pardoned for saying that I never think of these two experiences without a thrill of pride and admiration for her pluck. In reading the pages of this narrative it should be remembered that within sixty miles of where Kane and his little party endured such untold sufferings, within eighty miles of where Greely’s men one by one starved to death, and within less than fifty miles of where Hayes and his party and one portion of the “Polaris” party underwent their Arctic trials and tribulations, this tenderly nurtured woman lived for a year in safety and comfort: in the summer-time climbed over the lichen-covered rocks, picking flowers and singing familiar home songs, shot deer, ptarmigan, and ducks in the valleys and lakes, and even tried her hand at seal, walrus, and narwhal in the bays; and through the long, dark winter night, with her nimble fingers and ready woman’s insight, was of inestimable assistance in devising and perfecting the details of the costumes which enabled Astrup and myself to make our journey across the great ice-cap in actual comfort. Perhaps no greater or more convincing proof than this could be desired of what great improvements have been made in Arctic methods. That neither Mrs. Peary nor myself regret her Arctic experiences, or consider them ill-advised, may be inferred from the fact that she is once more by my side in my effort to throw more light on the great Arctic mystery.
- 39. R. E. Peary, Civil Engineer, U. S. N. Falcon Harbor, Bowdoin Bay, Greenland, August 20, 1893.
- 41. PAGE Northward Bound 9 In the Melville Bay Pack 18 Establishing Ourselves 31 Hunts and Explorations 41 Boat Journeys and Preparations for Winter 54 Winter Upon Us 65 Eskimo Visitors 74 Arctic Festivities 84 The New Year 101 Sunshine and Storm 112 Sledge Journey into Inglefield Gulf 124 The Sledge Journey—(Continued) 139 Off for the Inland Ice 147 Weary Days of Waiting 156 My Camping Experience in Tooktoo Valley 168 “Oomiaksoak Tigalay!” (The Ship has Come!) 176 Return of the Explorers 182
- 42. Boat Journey into Inglefield Gulf 189 Farewell to Greenland 200 Greenland Revisited 211 The Great White Journey 221
- 43. CHAPTER I NORTHWARD BOUND First Sight of Greenland—Frederikshaab Glacier—Across the Arctic Circle—Perpetual Daylight—Sunlit Disko—The Climb to the Ice- cap—Dinner at Inspector Anderssen’s—A Native Dance—From Disko to Upernavik—Upernavik—The Governor and his Wife— The Duck Islands—Gathering Eggs and Eider-down and Shooting Ducks. Wednesday, June 24. We have sailed and tossed, have broken through the ice-barriers of Belle Isle Straits, and once more ride the rolling swells of the broad Atlantic. Our three days’ jam in the ice has given us a foretaste of Arctic navigation, but the good little “Kite” speeds northward with a confidence which inspires a feeling of security that not even the famed “greyhounds of the ocean” afford. Genial Captain Pike is on the bridge and off the bridge, and his keen eye is casting for the land. When I came on deck to-day I found the bold, wild coast of Greenland on the right. It was a grand sight—the steep, black cliffs, some of them descending almost vertically to the sea, their tops covered with dazzling snow, and the inland ice flowing through the depressions between their summits; at the foot of the cliffs gleamed bergs of various sizes and shapes, some of them a beautiful blue, others white as snow. The feature of the day was the Frederikshaab glacier, which comes down to the sea in latitude 62° 30′. It did not, however, impress me as being very grand, owing perhaps to our being so far from it. Its face is seventeen miles long, and we could see it like a wall of white marble before us. Long after we had passed it, it still appeared to be with us, and it kept us company nearly all day. Just beyond the glacier was disclosed the
- 44. Capt. Richard Pike—“On Duty.” most beautiful mountain scenery imaginable. The weather was deliciously warm, and revealed to us a new aspect of Arctic climate. It seems strange to be sitting on deck in a light coat, not even buttoned, and only a cap on my head, in the most brilliant sunshine, and gazing on snow-covered mountains. Out on the Billowy Sea. The First Fragment of Greenland Ice. Thursday, June 25. We were promised another lovely day, but after noon the weather changed and a cool wind sprang up, which helped to push our little craft along at a good rate. To-night we shall have the midnight sun for the first time, and it will be weeks, even months, before he sets for us again. Everything on deck is dripping from the fog which has gathered about us. Friday, June 26. In spite of the thick fog we have been making good time, and expect to be in Disko, or more properly Godhavn, about noon to-morrow. We saw our first eider-ducks to-day. Numerous bergs again gleam up in the distance, probably the output of the Jakobshavn glacier. Tuesday, June 30. We have been in a constant state of excitement since Saturday morning, when we first set foot on Greenland’s ice-bound shores.
- 45. The pilot, a half-breed Eskimo, came on board and took us into the harbor of Godhavn shortly after nine o’clock. Mr. Peary, Captain Pike, Professor Heilprin, and myself went ashore and paid our respects to Inspector Anderssen and his family. They were very attentive to us, and invited “Mr. and Mistress Peary” to stay with them during their stop in Godhavn—a pleasure they were, however, compelled to forego. In the afternoon a party of us from the “Kite” set out on our first Arctic tramp, our objective point being the summit of the lofty basalt cliffs that tower above the harbor. My outfit consisted of a red blanket combination suit reaching to the knee, long knit stockings, a short eider-down flannel skirt reaching to the ankles, and the “kamiks,” or long-legged moccasins, which I had purchased in Sidney. The day was exceptionally fine and sunny, and we started off in the best of spirits. Never had I seen so many different wild flowers in bloom at once. I could not put my foot down without crushing two or three different varieties. Mr. Gibson, while chasing a butterfly, slipped and strained the cords of his left foot so that he was obliged to return to the ship. Never had I stepped on moss so soft and beautiful, all shades of green and red, some beds of it covered so thickly with tiny pink flowers that you could not put the head of a pin down between them. We gathered and pressed as many flowers as we could conveniently carry—anemones, yellow poppies, mountain pinks, various Ericaceæ, etc. Sometimes our path was across snow- drifts, and sometimes we were ankle-deep in flowers and moss. Mountain streams came tumbling down in every little gully, and their water was so delicious that it seemed impossible to cross one of these streams without stooping to drink. Our advance was very slow, as we could not resist the temptation of constantly stopping to look back and feast upon the beauties of the view. Disko Bay, blue as sapphire, thickly studded with icebergs of all sizes and beautifully colored by the sun’s rays, lay at our feet, with the little settlement of Godhavn on one side and the brown cliffs towering over it. As far as the eye could reach, the sea was dotted with icebergs, which looked like a fleet of sail-boats. The scene was simply indescribable. We reached the summit, at an elevation of 2400 feet, and built a cairn, in which we placed a tin box containing a piece of paper with our names written upon it, and some American coins. From the summit of these cliffs we stepped upon the ice-cap, which seemed to roll right down to their tops. The temperature was 91° F. in the sun, and 56° in the
- 46. shade. As we descended a blue mist seemed to hang over that part of the cliffs that lay in shadow, and the contrast with the white bergs gleaming in the sapphire waters below was very striking. We returned to the foot of the cliff after eight o’clock. On Sunday we made another expedition, to the Blaese Dael, or “windy valley,” where a beautiful double waterfall comes tumbling through the hard rock, into which it has graven a deep channel. We gathered more flowers, and collected some seaweed; the mosquitos, of which we had had a foretaste the day before, were extremely troublesome, and recalled to memory the shores of New Jersey. When we reached the first Eskimo hut, a number of the piccaninnies[1] came to me and presented me with bunches of wild flowers. We gave them some hardtack in return, and they were happy. 1. The Eskimos frequently designate their children as piccaninnies, a word doubtless introduced by the whalers. Mr. Peary, Professor Heilprin, myself, and two other members of our party dined with the inspector in the evening, joining some members of the Danish community, who had also been invited. The course consisted of fresh codfish with caper-sauce, roast ptarmigan, potatoes boiled and then browned; and for dessert, “Rudgrud,” a “dump,” almonds, and raisins. There was, following European custom, a varied accompaniment of wines. After dinner the gentlemen went up-stairs to examine the geological and oölogical collections of the inspector, while the ladies preferred the parlor with their coffee. Were it not for the outer surroundings, it would have been difficult to realize that we were in the distant Arctic realm, so truly homelike were the scenes of the little household, and so cheerful the little that was necessary to make living here not only comfortable, but pleasant. The entire community numbers barely 120 souls, nine tenths of whom are Eskimos, mainly half-breeds; the remainder are the Danish officials and their families, whose recreation lies almost entirely within the little circle which they themselves constitute. Toward nine o’clock we visited the storehouse, where a native ball was in progress. Several of our boys went the rounds with the Eskimo “belles,” but for me the odor of the interior was too strong to permit me to say that looking on was an “unalloyed pleasure.” The steps
- 47. were made to the music of stringed instruments, over which the resident half-breeds have acquired a fair mastery. The participants and onlookers were all in a lively frame of mind, but not uproarious; and at the appointed time of closing—ten o’clock—all traces of hilarity had virtually been banished. The Most Northern Outpost of Civilization on the Globe—Upernavik. We had hoped to leave early on the following morning, but it was not until near two o’clock that the fog began to lift, and that a departure was made possible. Firing the official salute, and dipping our colors, we gave three hearty cheers in honor of our first Greenland hosts, and sailed out of the rock-bound harbor. It soon cleared up, and we were able to make our normal seven knots an hour. This morning it was foggy for a while, but it cleared up beautifully, and now we are just skimming along, and expect to reach Upernavik, the most northern of the Danish settlements in Greenland, about nine o’clock in the evening. Thursday, July 2. We did not reach Upernavik until 2.30 yesterday morning, owing to a very strong current which was running against us all the way from Godhavn. We remained up all night, and at 1.30 A. M. were enjoying the dazzling brightness of the sunshine. Mr. Peary took a number of photographs between midnight and morning. Upernavik is a very different-looking place from Godhavn. There are four frame-houses and a little church. The natives live in turf huts, very miserable-looking habitations, built right down in the mud. As soon as our ship steamed into the harbor, in which two Danish
- 48. vessels were at anchor, the governor, Herr Beyer, came on board with his lieutenant-governor, a young fellow who had arrived only three days before. We returned the visit at noon, and were pleasantly received by the governor and his wife, a charming woman of about thirty years, who had been married but a year, and whose fondness for home decoration had expressed itself in the pictures, bric-à-brac, fancy embroideries, and flowering plants which were everywhere scattered about, and helped to make up an extremely cozy home. As in all other houses in the country, the guests were treated to wine immediately on entering, and with a delicate politeness the governor presented me with a corsage bouquet of the flowers of Upernavik, neatly tied up with the colors of Denmark. Our visit was fruitful in the receipt of presents, among which were Eskimo carvings, a dozen bottles of native Greenland beer, and a box of “goodies,” addressed to “Miss Peary,” and to be opened, as a reminder, on Christmas eve. The hospitality shown to us could not have been more marked had our friendship extended over many years. THE SUNSET GLOW—BERG OFF SVARTENHOEK. Our visit was a brief one, as we were to weigh anchor early in the afternoon. We steamed away from Upernavik and headed north. The fog had cleared away and disclosed a giant mountain towering above us in the harbor. The sun shone brightly, and the sea was smooth as glass and blue as turquoise. The night promised to be a beautiful one,
- 49. but I resisted the temptation to stay up, having been up the entire night before, and the greater part of the one before that. At 4 A. M. Captain Pike knocked at our door and informed us that in half an hour we would be at the Duck Islands. Here we were to land and all hands shoot eider-ducks and gather their eggs for our winter supply. We were soon on shore, and then began a day’s sport such as I had often read about, but never expected to see. The ducks flew in thick flocks all about us, and on every side were nests as large as a large hen-nest, made of eider-down and containing from three to six eggs. The nests were not hidden, but right out on the rocks in full sight. Alas! we were too late; the ducks were breeding, and out of 960 eggs we did not get over 150 good ones. As I had not taken my gun, I spent the time in gathering down, and collected forty-three pounds in five hours. After returning to the “Kite” for breakfast, we visited a second island, and there I bagged a bird, much to my satisfaction. Altogether ninety-six ducks were shot.
- 50. CHAPTER II IN THE MELVILLE BAY PACK Melville Bay—On the Edge of the Dreaded Ice-pack—Fourth of July—Butting the Ice—Accident to the Leader of the Expedition— Gloom on the “Kite”—Blasting the “Kite” out of a Nip—A Real Bear and a Bear Hunt—A Chase on the Ice—A Phantom Ship— Free of the Pack and in the North Water at Last—The Greenland Shore to Barden Bay—First Sight of the Arctic Highlanders. Thursday, July 2. We are opposite the “Devil’s Thumb,” latitude 74° 20′, and now, at 8 P. M., are slowly making our way through the ice which marks the entrance into the Melville Bay “pack.” Friday, July 3. At midnight the engine was stopped, the ice being too thick for the “Kite” to make any headway. At 6.30 A. M. we started again, and rammed our way along for an hour, but were again forced to stop. At eleven o’clock we tried it once more, but after a couple of hours came to a standstill. We remained in this condition until after five o’clock, when the engine was again started. For two hours we made fairly good progress, and we thought that we should soon be in open water, but a small neck of very heavy ice stopped us. While we were on deck, the mate in the “crow’s-nest,” which was hoisted to- day, sang out, “A bear! A bear!” Off in the distance we could see an object floating, or rather swimming, in the water, and in a minute the boys were climbing helter-skelter over the sides of the “Kite,” all with guns, although some soon discovered that theirs were not loaded; but the bear turned out to be a seal, and not one of about thirty shots hit him. It is now nearly 11 P. M. The sun is shining beautifully, and it is perfectly calm. I have worn only a gray spring jacket, which I have found sufficient for the balmy temperature. At midnight the cannon
- 51. “A Bear! A Bear!” was fired, the flags were run up and dipped, and the boys fired their rifles and gave three cheers for the Fourth of July. The thermometer marked 31°. Saturday, July 4. The ice remains stubborn, and we are fast bound. All around the eye sees nothing but the immovable pack, here smooth as a table, at other places tossed up into long hummock-ridges which define the individual ice-cakes. Occasional lanes of water appear and disappear, and their presence gives us the one hope of an early disentanglement. The event of the day has been a dinner to Captain Pike, in which most of the members of our party participated. After dinner hunting- parties scoured the ice after seals, with the result of bringing in two specimens, one weighing twenty-six pounds, and the other thirty-three pounds. Sunday, July 5. All night we steamed along slowly, but at 8 A. M. we were forced once more to stop. The day has been very disagreeable, foggy, rainy, and even snowy. We have done nothing but eat and sleep. A lazily hovering ivory-gull, which ventured within near gunshot, has been added to our collections. Tuesday, July 7. The weather yesterday was dreary and disagreeable, but to-day it seems warmer. The snow has ceased falling, although the sky is still overcast, and the fog prevents us from seeing the horizon. At noon the sun came through the clouds for a few moments, and the fog lifted sufficiently for the captain to make an observation and find that our position was latitude 74° 51′. During the afternoon the wind died down, and an attempt was made to get through the ice; but after boring and ramming the immovable pack for nearly an hour, and gaining only a ship’s length, we concluded that we were burning coal for nothing. Mr. Peary, with Gibson, Astrup, Cook, and Matt, has been busy all the afternoon sawing, marking, and fitting the lumber for our Whale Sound cottage. The curing of a large number of drake-
- 52. Sailing Through the Pack. skins, intended to be made up into undershirts for winter wear, was a part of the day’s work. Thursday, July 9. Yesterday and to-day the fog lifted sufficiently at times to permit us to see the land, about forty miles distant. A good observation places us in latitude 74° 51′, and longitude about 60° W. Mr. Peary fixed the points with his pocket sextant and the ship’s compass, and then made a sketch of the headlands. The ice looks rotten, but yet it holds together too firmly to permit us to force a passage. We measured some of the floes, and found the thickest to be two and a half feet. It has seemed very raw to-day, owing largely to a slight northwest wind; and for the first time the average temperature has been below the freezing-point, being 31½° F. Friday, July 10. This morning the rigging was covered with hoar-frost, making the “Kite” look like a “phantom ship.” The fog hung heavily about us, shutting out the land completely. In the forenoon a sounding was made, but no bottom was found at 343 fathoms. While we were at dinner, without any warning the “Kite” began to move, steam was immediately gotten up, and for an hour and a half we cut our way through the ice, which had become very rotten, large floes splitting into several pieces as soon as they were struck by the “Kite.” We made about three knots, when we were again obliged to halt on account of a lowering fog. Our little move was made just in time to keep up the courage of some of the West Greenland party, who were beginning to believe that we should be nipped and kept here for the winter. Although we realized that we were still ice-bound in the great and much-dreaded Melville Bay pack, we could not but enjoy at times the peculiar features of our forced imprisonment. Efforts to escape, with full promise of success, followed by a condition of impotency and absolute relaxation, would alternately elevate and depress our spirits to the extent of casting joy and gloom into the little household. The novelty of the situation, however, helped greatly to keep up a good feeling, and all despondency was immediately dispelled by the sound
- 53. of the order to “fire up,” and the dull rumbling of the bell-metal propeller. We never tired of watching our little craft cut her way through the unbroken pans of ice. The great masses of ice were thrust aside very readily; sometimes a piece was split from a large floe and wedged under a still larger one, pushing this out of the way, the commotion causing the ice in the immediate vicinity fairly to boil. Then we would run against an unusually hard, solid floe that would not move when the “Kite” struck it, but let her ride right up on it and then allow her gradually to slide off and along the edge until she struck a weak place, when the floe would be shivered just as a sheet of glass is shivered when struck a sharp, hard blow. The pieces were hurled against and on top of other pieces, crashing and splashing about until it seemed as though the ice must be as thick again as it was before the break-up; but the good old “Kite” pushed them aside, leaving them in the distance groaning and creaking at having been disturbed. The day has been pleasant, in spite of an average temperature of 27½°. Tuesday, July 14. How different everything looks to us since I last wrote in this journal! Saturday the weather was, as usual, cold and foggy; and when, at 5.30 P. M., we found ourselves suddenly moving, every one was elated, hoping we would be able to get into the clear water ahead, which the mate said could be seen from the crow’s-nest. Mr. Peary was particularly pleased, as he said we should then reach Whale Sound by July 15, the limit he had set for getting there. After supper he and I bundled up and went on deck, and watched the “Kite” cut through the rotten ice like butter. We had been on the bridge for some time, when Mr. Peary left me to warm his feet in the cabin. Coming on deck again, he stepped for a moment behind the wheel-house, and immediately after, I saw the wheel torn from the grasp of the two helmsmen, whirling around so rapidly that the spokes could not be seen. One of the men was thrown completely over it, but on recovering himself he stepped quickly behind the house, and I instantly realized that something must have happened to my husband. How I got to him I do not know, but I reached him before any one else, and found him standing on one foot looking pale as death. “Don’t be frightened, dearest; I have hurt my leg,” was all he said. Mr. Gibson and Dr. Sharp helped, or rather carried, him down into the cabin and laid him on the table. He was ice-cold, and while I covered him with blankets, our physicians gave him whisky,
- 54. cut off his boot, and cut open his trousers. They found that both bones of the right leg had been fractured between the knee and the ankle. The leg was put into a box and padded with cotton. The fracture being what the doctors pronounced a “good one,” it was not necessary to have the bones pulled into place. Poor Bert suffered agonies in spite of the fact that the doctors handled him as tenderly as they could. We found it impossible to get him into our state-room, so a bed was improvised across the upper end of the cabin, and there my poor sufferer lies. He is as good and patient as it is possible to be under the circumstances. The accident happened in this way. The “Kite” had been for some time pounding, or, as the whalers say, “butting,” a passage through the ice, slowly but steadily forging a way through the spongy sheets which had already for upward of a week imprisoned her. To gain strength for every assault it was necessary constantly to reverse, and it was during one of these evolutions, when going astern, that a detached cake of ice struck the rudder, crowding the iron tiller against the wheel-house where Mr. Peary was standing, and against his leg, which it held pinned long enough for him to hear it snap. Wednesday, July 15. Mr. Peary passed a fairly comfortable night, and had a good sleep without morphine to-day, consequently he feels better. As for myself, I could not keep up any longer, and at 11 A. M., after Dr. Cook had dressed the leg and made an additional splint, I lay down, and neither moved nor heard a sound until after five o’clock. This was the first sleep I have had since Friday night. Dr. Cook, who has been more than attentive, has made a pair of crutches for the poor sufferer, but he will not be able to use them for a month. We find to-day that our latitude is 75° 1′, and our longitude 60° 9′; consequently our headway has been very slow. It seems as if when the ice is loose the fog is too thick for us to travel in safety, and when the fog lifts the ice closes in around us. Once to-day the ice suddenly opened and a crack which visibly widened allowed us to make nearly four miles in one stretch. Throughout much of the night and day we steamed back and forth and hither and thither, trying to get through or around the ice, and to prevent the “Kite” from getting nipped between two floes. A little after supper the fog suddenly closed in upon us, and before we could complete the passage of a narrow and tortuous lead, through which we were seeking escape from the
- 55. Bruin at Rest. advancing floes in our rear, we were caught fast between two large pans. The ice was only about fourteen inches thick, and there was but little danger of the “Kite” being crushed; still, Captain Pike, with the memories of former disasters fresh in his mind, did not relish the situation, and blasted our way out with gunpowder at 8.15 P. M. This is our first “nip.” An hour later the captain called down to me to come up at once, as a bear was advancing toward the ship. The boys had been watching and longing for a bear ever since we left New- York, and many false alarms had been given; but here was a real live polar coming straight for the “Kite.” A very, very pretty sight he was, with black snout, black eyes, and black toes. Against the white snow and ice, he seemed to be of a cream color. His head was thrown up as he loped along toward us, and when, within a short distance of the “Kite,” a gull flew over his head, he made a playful jump at it, all unconscious of the doom which awaited him. Eleven men with guns were stooping down on the quarter-deck waiting for the captain to give the word to fire. A bullet disabled one of the fore legs, while another struck the animal in the head, instantly dyeing it crimson; the bear stopped short, wheeled round, fell over on his head, and then got up. By this time it was simply raining bullets about the poor beast; still he staggered on toward the water. Gibson, who had jumped on the ice as soon as he fired, was now close to him, and, just as he started to swim away, put a ball in his neck, which stopped him short. A boat was lowered, and he was brought alongside the “Kite.” He measured seven feet one inch in length, and we estimated his weight at from eight to ten hundred pounds. Friday, July 17. Last night was the worst night my poor husband has had. His leg pained him more than it had done so far, and he begged me to give him a sedative, which, with the doctor’s consent, I did; but even then his sleep was disturbed to such an extent that it amounted to delirium. He would plead with me to do something for his leg. After doing everything I could think of, I said, “Can’t you tell me where it hurts you most, and what you think might help you?” His answer was, “Oh, my dear, pack it in ice until some one can shoot it!” In this way he spent the night, and this morning he was
- 56. thoroughly exhausted. Dr. Cook has succeeded in making his leg more comfortable, and now he sleeps. It seems very hard that I cannot take him away to some place where he can rest in peace. Tuesday, July 21. Since last writing in my journal, four days ago, we have been steadily nearing Cape York, and we hope soon to clear the ice of Melville Bay, and pass into the open North Water beyond. Our hopes have, however, so often been disappointed that day by day, even when in full view of the land, we become less and less confident of ever being able to disengage ourselves from our confinement. Huge grounded bergs still hold the ice together, and until they show signs of moving there is little prospect of a general break-up. On Saturday a bear with two cubs was seen on the ice ahead of us, and immediately every man was over the side of the vessel making for the animals. The mother, with a tender affection for her young, guided an immediate retreat, herself taking the rear, and alternately inciting the one cub and then the other to more rapid movement. Our boys were wholly unacquainted with the art of rapid traveling on the rough and hummocky ice, and before long the race was admitted to be a very unequal one; they were all quickly distanced. One of the men, in the excitement of the moment, joined in the chase without his gun, and, even without this implement, when he returned to the “Kite” he was so out of breath that he had to be hauled up the sides of the vessel like a dead seal. He lay sprawling and breathless on the deck for at least five minutes, much to the merriment of the crew and the more fortunate members of the party. A round weight of over two hundred pounds was responsible for his discomfiture. Monday morning about two o’clock the fog suddenly lifted, and we found ourselves almost upon the land. The visible shore extended from Cape York to Wolstenholme Island, and we could clearly distinguish Capes Dudley Diggs and Atholl. I held a looking-glass over the open skylight in such a way that Mr. Peary could see something of the outline of the coast. Poor fellow! he wanted to go on deck so badly, thinking that if he were strapped to a board he could be moved in safety, but the doctor persuaded him to give up the thought. As the doctors have all agreed that in six months his leg will be as good as it ever was, he refuses to consider the idea of returning on the “Kite”; as for myself, now that we have started, I want to keep on too. The air
- 57. is almost black with flocks of the little auk, and a party on the ice to- day brought in sixteen birds in a very short time. Wednesday, July 22. Drs. Hughes and Sharp brought in sixty-four birds as the result of an all-night catch. We are still in the ice, with no signs of our getting out, although the captain says we have drifted twenty miles to the northward since Monday morning. We are now abreast of Conical Rock. Second Mate Dunphy has just reported seeing from the crow’s-nest a steamer off Cape York, but it is not visible to the naked eye, and we are in doubt as to what it is. Friday, July 24. The steamer did not materialize; either the mate was mistaken or the vessel drifted away from us. The ice parted early yesterday morning, much to everybody’s relief, and we have since been pushing steadily on our course. The long line of table-topped bergs off Cape York, some of which measured not less than two hundred to three hundred feet in height, and perhaps considerably over a mile in length, is visibly moving over to the American waters, and to this disrupting force we are doubtless largely indebted for our liberation. The scenery of this portion of the Greenland coast is surpassingly fine. The steep red-brown cliffs are frequently interrupted by small glaciers reaching down to the water’s edge. The entrance to Wolstenholme Sound, guarded as it was by huge bergs, was particularly beautiful. Saunders Island in the distance, and Dalrymple Rock immediately in the foreground, stood up like great black giants, contrasting with the snow-white bergs surrounding them and the red cliffs of the mainland on either side. Whenever anything particularly striking or beautiful appears, I am called on deck, and with my hand-glass placed at the open transom over Mr. Peary’s head, manage to give him a faint glimpse of our surroundings. At nine o’clock this evening we rounded Cape Parry, and about ten o’clock stopped at the little Eskimo village of Netchiolumy in Barden Bay, where we hoped to obtain a native house, sledge, kayak, and various native utensils and implements for the World’s Columbian Exposition. Our search-party found only three houses in the settlement, and the lonely inhabitants numbered six adults and five children; five dogs added life to the solitude. These people had quantities of sealskins and narwhal tusks, many of which were obtained in exchange for knives, saws, files, and tools in general. Wood of any kind, to be used in the construction of sledges,
- 58. kayak frames, and spear- and harpoon-shafts, was especially in demand; they cared nothing for our woven clothing, nor for articles of simple show and finery. We stopped this morning at Herbert Island, where we had hoped to visit a native graveyard, but no graves were found.
- 59. CHAPTER III ESTABLISHING OURSELVES Arrival at McCormick Bay—Selecting the Site for the House— Temporary Quarters—Hurrying the Erection of the House— White Whales—Departure of the “Kite”—Alone on the Arctic Shore—A Summer Storm—Arctic Picnicking—The First Birthday and the First Deer—Birthday-dinner Menu—Departure of the Boat Party for Hakluyt and Northumberland Islands after Birds and Eskimos—Occupations during their Absence—Return of the Party with an Eskimo Family. Sunday, July 26. Mr. Peary is getting along nicely. His nights are fairly comfortable, and consequently he feels much better by day; his back now troubles him more than his leg. Yesterday morning at three o’clock he was awakened and told that the ice prevented our getting to Cape Acland, and that we were just abreast of McCormick Bay, and could not proceed further into the sound. He accordingly decided to put up our quarters on the shores of this bay. It was now a question as to which side of the bay would be most favorable for a home. At 9 A. M., together with several members of our party, I rowed over to the southeast shore, and walked along the coast for about three miles, prospecting for a site, and made a provisional choice of what seemed a desirable knoll. We returned to the “Kite” about noon. After dinner Professor Heilprin, Dr. Cook, Astrup, and three others went over to the other shore, and toward evening they returned with the report that the place was perfectly desolate and not at all suitable for a camp. After supper we returned to the southeast shore to see if we could improve on the location selected in the morning, but after tramping for miles came back to the old site. While it cannot in truth
- 60. be said that the spot is a specially attractive one, it would be equally untrue to describe it as being entirely devoid of charm or attraction. Flowers bloom in abundance on all sides, and their varied colors,— white, pink, and yellow,—scattered through a somewhat somber base of green, picture a carpet of almost surpassing beauty. Rugged cliffs of sandstone, some sixteen hundred to eighteen hundred feet high, in which the volcanic forces have built up long black walls of basalt, rise steeply behind us, and over their tops the eternal ice-cap is plainly visible. Only a few paces from the base of the knoll are the silent and still partially ice-covered waters of the bay, which extends five miles or more over to the opposite shore, and perhaps three times that distance eastward to its termination. A number of lazy icebergs still stand guard between us and the open waters of the western horizon, where the gray and ice-flecked bluffs of Northumberland and Hakluyt Islands disappear from sight. ON THE BEACH OF McCORMICK BAY. This morning the members of our party went ashore with pickaxes and shovels, and they are now digging the foundations of our “cottage by the sea.” They are also putting up a tent for our disabled commander, whence he can superintend the erection of the structure. The men are working in their undershirts and trousers, and it is quite warm enough for me to stay on deck without a wrap, even when I am not exercising; yet, if we had this temperature at
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