![Preface
3-=paration of variables is a solution method for partial differential equations.
liile its beginnings date back to work of Daniel Bernoulli (1753), Lagrange
, 1759), and d'Alembert (1763) on wave motion (see [2]), it is commonly asso-
,ciated with the name of Fourier (1822), who developed it for his research on
·:-Jnductive heat transfer. Since Fourier's time it has been an integral part of
e!:gineering mathematics, and in spite of its limited applicability and heavy com-
petition from numerical methods for partial differential equations, it remains a
-;;.·ell-known and widely used technique in applied mathematics.
Separation of variables is commonly considered an analytic solution
::nethod that yields the solution of certain partial differential equations in terms
cf an infinite series such as a Fourier series. While it may be straightforward to
·:uite formally the series solution, the question in what sense it solves the prob-
lem is not readily answered without recourse to abstract mathematical analysis.
A modern treatment focusing in part on the theoretical underpinnings of the
method and employing the language and concepts of Hilbert spaces to analyze
che infinite series may be fonnd in the text of MacCluer [15]. For many problems
•he formal series can be shown to represent an analytic solution of the differ-
ential equation. As a tool of analysis, however, separation of variables with its
:nfirtite series solutions is not needed. Other mathematical methods exist which
guarantee the existence and uniqueness of a solution of the problem under much
::nore general conditions than those required for the applicability of the method
;:,f separation of variables.
In this text we mostly ignore infinite series solutions and their theoretical
and practical complexities. We concentrate instead on the first N terms of the
series which are all that ever are computed in an engineering application. Such
a partial sum of the infinite series is an approximation to the analytic solution
c,f the original problem. Alternatively, it can be viewed as the exact analytic
solution of a new problem that approximates the given problem. This is the
point of view taken in this book.
Specifically, we view the method of separation of variables in the following
context: mathematical analysis applied to the given problem guarantees the
existence and uniqueness of a solution u in some infinite dimensional vector space
o:,f functions X, but in general provides no means to compute it. By modifying
the problem appropriately, however, an approximating problem results which
has a computable closed form solution UN in a subspace M of X. If fl!! is
vii](https://image.slidesharecdn.com/54124-250625132526-2bb02ac5/85/Separation-of-Variables-for-Partial-Differential-Equations-An-Eigenfunction-Approach-1st-Edition-George-Cain-13-320.jpg)
![viii PREFACE
suitably chosen, then UN is a good approximation to the unknown solution u.
As we shall see, M will be defined such that UN is just the partial sum of the
first N terms of the infinite series traditionally associated with the method of
separation of variables.
The reader may recognize this view as identical to the setting of the finite
element, collocation, and spectral methods that have been developed for the
numerical solution of differential equations. All these methods differ in how the
subspace Mis chosen and in what sense the original problem is approximated.
These choices dictate how hard it is to compute the approximate solution UN
and how well it approximates the ai!alytic solution u.
Given the almost universal applicability of numerical methods for the solu-
tion of partial differential equations, the question arises whether separation of
variables with its severe restrictions on the type of equation and the geometry of
the problem is still a viable tool and deserves further exposition. The existence
of this text ·reflects our view that the method of separation of variables still
belongs to the core of applied mathematics. There are a number of reasons.
Closed form (approximate) solutions show structure and exhibit explicitly
the influence of the problem parameters on the solution. We think, for example,
of the decomposition of wave motion into standing waves, of the relationship
between driving frequency and resonance in sound waves, of the influence of
diffusivity on the rate of decay of temperature in a heated bar, or of the gen-
eration of equipotential and stream lines for potential flow. Such structure and
insight are not readily obtained frnm purely numerical solutions of the underly-
ing differential equation. Moreover, optimization, control, and inverse problems
tend to be easier to solve when an analytic representation of the (approximate)
solution is available. In addition, the method is not as limited in its applicability
as one might infer from more elementary texts on separation of variables. Ap-
proximate solutions are readily computable for problems with time-dependent
data, for diffusion with convection and wave motion with dissipation, problems
seldom seen in introductory textbooks. Even domain restrictions can sometimes
be overcome with embedding and domain decomposition techniques. Finally,
there is the class of singularly perturbed and of higher dimensional problems
where numerical methods are not easily applied while separation of variables
still yieltb an analytic approximate solution.
Our rationale for offering a new exposition of separation of variables is then
twofold. First, although quite common in more advanced treatments (such as
[15]), interpreting the separation of variables solution as an eigenfunction expan-
sion is a point of view r.arely taken when introducing the method to students.
Usually the formalism is based on a product solution for the partial differential
equation, and this limits the applicability of the method to homogeneous partial
differential equations. When source terms do appear, then a reformulation of
problems for the heat and wave equation with the help of Duhamel's superposi-
tion principle and an approximation of the source term in the potential equation
with the help of an eigenfunction approximation become necessary. In an expo-
sition based from the beginning on an eigenfunction expansion, the presence of
source terms in the differential equation is only a technical, but not a conceptual](https://image.slidesharecdn.com/54124-250625132526-2bb02ac5/85/Separation-of-Variables-for-Partial-Differential-Equations-An-Eigenfunction-Approach-1st-Edition-George-Cain-14-320.jpg)
![GIRLS OF NUBIA (MAKING POTTERY).
[From St. John’s “Oriental Album.”]
The manners of the communities in the Sahara are imperfectly known; but from
the accounts we have received they appear to be of a far more elevated order than
those of any other part of Africa. It is true that customs prevail which shock our](https://image.slidesharecdn.com/54124-250625132526-2bb02ac5/85/Separation-of-Variables-for-Partial-Differential-Equations-An-Eigenfunction-Approach-1st-Edition-George-Cain-17-320.jpg)
![ideas of decency. A chief, for instance, offered Richardson his two daughters as
wives. It is also true that many women exist who follow the profession of
prostitutes, though we have no distinct account of them. But immorality is usually
among them a secret crime. Their general customs with regard to sexual
intercourse are at least as pure as those of Europe. Among the wandering tribes of
the desert the hardship of their lives, continual occupation, varied scenes of
excitement, and contempt for sensual enjoyments, contribute to preserve chastity
among their virtues; while the Marabouts of the cities are of a generally moral
character. Intoxication never happens among the women. Still, the condition of the
sex is degraded; for they are, with exceptions, regarded only as the materials of a
man’s household, and ministers to the sensual enjoyments of his life[51]. The
Mohammedans of Central Africa, bigoted as to dogmas, are nevertheless more
liberal to women, who enjoy more consideration among them than in the more
important strongholds of that religion[52].
The wandering Arabs of Algeria hold marriage as a business transaction, though
the estimation of the sex is not low. The lover brings to the woman’s home ten
head of cattle, with other presents, which usually form her dowry. The father asks,
“How much does she whom you are going to have for wife cost you?” He replies,
“A prudent and industrious woman can never be too dear.” She is dressed, placed
on a horse, and borne to her new home amid rejoicing. She then drinks the cup of
welcome, and thrusting a stick into the ground, declares, “As this stick will remain
here until some one forces it away, so will I.” She then performs some little office
to show she is ready for the duty of a wife, and the ceremony is ended[53].
Transferring our observations to Abyssinia, we find in its several divisions different
characteristics of manners. In Tajura, on the Red Sea, profligacy is a conspicuous
feature of society. Men live with their wives for a short period, and then sell them,
maintaining thus a succession of favourites in their harems. Parents, also, are
known not only to sell their daughters as wives, but to hire them out as prostitutes.
One chief offered a traveller his daughter either as a temporary or a permanent
companion; he showed another whom he would have sold for 100 dollars. One
woman presented herself, stating, as a recommendation, that she had already lived
with five men. These are nothing but prostitutes, whatever the delicacy of
travellers induces them to term them. Unfortunately the inquiries made into this
system are very slight, affording us no statistics or results of any kind. We are thus
left to judge of morality in Tajura by the fact that syphilis afflicts nearly the whole
population, man and woman, sultan and beggar, priests and their wives included.
In the Christian kingdom of Shoa, the Christian king has one wife, and 500
concubines; seven in the palace, thirteen at different places in the outskirts, and
the rest in various parts of his dominions. He makes a present to the parents of
any women he may desire, and is usually well paid in return for the honour. The
governors of cities and provinces follow this example, keeping establishments of](https://image.slidesharecdn.com/54124-250625132526-2bb02ac5/85/Separation-of-Variables-for-Partial-Differential-Equations-An-Eigenfunction-Approach-1st-Edition-George-Cain-18-320.jpg)
![concubines at different places. Scores of the royal slaves are cast aside, and their
place supplied by others.
In Shoa there are two kinds of marriage; one a mere agreement to cohabitation,
another a holy ceremony; the former is almost universally practised. The men and
women declare before witnesses that they intend to live happily together. The
connection thus easily contracted is easily broken; mutual consent only is
necessary to a divorce. In Shoa a wife is valued according to the amount of her
property. The heiress to a house, a field and a bedstead, is sure to have a
husband. When they quarrel and part, a division of goods takes place. Holy
ceremonies are very rare, and not much relished. A wedded couple, in one sense
of the term, is a phenomenon. Instances of incontinence are frequent; while the
caprice of the men leads them often to increase the number of their concubines.
These are procured as well from the Christians as from the Mohammedans and
Pagans; but the poor girls professing these religions are forced to a blind
profession of Christianity. Favourite slaves and concubines hold the same position
with married women; while illegitimate and legitimate children are treated by the
law with no distinction. Three hundred of the king’s concubines are slaves, taken in
war or purchased from dealers. They are guarded by fifty eunuchs, and live in
seclusion; though this by no means prevents the court from overflowing with
licentiousness. Numerous adulteries take place, and this example is followed by the
people; among whom a chaste married couple is not common.
Women in Abyssinia, which is an agricultural country, mix freely with the men, and
dance in their company; though a few jealous husbands or cautious parents
seclude them. Morality is at an extremely low ebb. At the Christmas saturnalia,
gross and disgusting scenes occur, as well as at other feasts. What else can be
expected in a country where 12,000 priests live devoted, in theory at least, to
celibacy; and where, at the annual baptisms, these priests, with men, women, and
children strip naked, and rush in promiscuous crowds into a stream, where they are
baptised according to the Christian religion! The sacerdotal class of Shoa is
notoriously drunken and profligate. Another cause of corruption is the caprice
which induces men to abandon their concubines after short cohabitation with
them. These women, discarded and neglected, devote themselves to an infamous
profession, and thus immorality is perpetuated through every grade of their
society: in a word, the morals of Shoa are of the lowest description. In the
Mohammedan states in its neighbourhood the condition of the sex is no better. If
there is less general prostitution, it is because every woman is the slave of some
man’s lust, and is imprisoned under his eye. He is jealous only of her person;
scarcely attributing to her a single quality which is not perceptible to his senses[54].
In the southern provinces of Kordofan, under the government of Egypt, south of
the Nubian Mountains, immense labour is imposed on the unmarried girls; yet the
sentiment of love is not altogether unknown to them, and men fight duels with](https://image.slidesharecdn.com/54124-250625132526-2bb02ac5/85/Separation-of-Variables-for-Partial-Differential-Equations-An-Eigenfunction-Approach-1st-Edition-George-Cain-19-320.jpg)
![whips of hippopotamus hide on account of a disputed mistress. The wife is
nevertheless a virtual slave, and still more degraded should she prove barren; the
husband, in that case, solaces himself with a concubine, who, if she bears a child,
is elevated to the rank of wife. It is common among the rich for a man to make his
wife a separate allowance after the birth of her second child, when she goes to live
in a separate hut. All their bloom is gone by the time they are twenty-four years
old, and thenceforward they enjoy no estimation from the men. Yet, improvident in
their hearts, the young girls of Kordofan are merry; and, whether at work or idle,
spend the day in songs and laughter; while in the evening they assemble and
dance to the music of the Tarabuka drum. Their demeanour, in general, is modest,
and their lives are chaste. Married women, on the contrary, especially those who
are neglected by their husbands, occupy themselves in gossip, and find solace in
criminal intrigues. In some parts of the country, indeed, men consider it an honour
for their wives to have intercourse with others; and the women are often
forwarded in their advances. Female slaves often have liberty when they bear
children to their proprietors.
Women eat when the men have done, and pretty dancers attend at the feasts to
amuse their employers. These girls, like the Ghawazee of Lower Egypt, are usually
prostitutes, and very skilful in the arts of seduction. Numbers of this class fled from
Egypt into Kordofan, on one occasion, when Mohammed Ali, in one of his affected
fits of morality, endeavoured to suppress their calling altogether.
Marriage, it may be scarcely necessary to say, is concluded without the woman’s
consent. The man bargains for her, pays her price, takes her home, strips off her
virginal girdle, which is the only garment of unmarried girls, and covers her with a
cloth about her loins; a feast and a dance occasionally celebrate the event. When a
wife is ill-treated beyond endurance, she demands a divorce; and, taking her
female offspring, with her dowry, returns home. Trifles often produce these
separations. That her husband has not allowed her sufficient pomatum to anoint
her person with, is not unfrequently the ground of complaint. Few men in Kordofan
have more than two wives; but most have concubines besides, whom the more
opulent protect by a guard of eunuchs.
These remarks apply to the agricultural or fixed population. The Baghaira, or
wandering pastoral tribes of Kordofan, are a modest, moral race—naked, but not
on that account indecent[55].
A chief of the Berbers offered a late traveller the choice of his two daughters for a
bedfellow. They were already both married. Women there, however, as well as in
Dongola, are, many of them, ready to prostitute themselves for a present. A virgin,
whether as wife or concubine, may be purchased for a horse. “Why do you not
marry?” said a traveller to a young Berber. He pointed to a colt and answered
“When that is a horse I shall marry.”[56]](https://image.slidesharecdn.com/54124-250625132526-2bb02ac5/85/Separation-of-Variables-for-Partial-Differential-Equations-An-Eigenfunction-Approach-1st-Edition-George-Cain-20-320.jpg)
![The condition of women and state of manners on the upper borders of the Nile, we
find described in Ferdinand Werne’s account of his recent voyage to discover the
sources of the White Stream. The system in Khartum may be indicated by one
sentence in the traveller’s own language. He speaks of desiring that the pay might
be advanced to prevent starvation from visiting the soldiers’ families, “which, from
the low price of female slaves, were numerous.” It may, without resort to
hyperbole, be said, that the female monkeys peopling the neighbouring woods
occupy a far nobler and more natural position. Among the barbarians on the banks
of the river further up, the state of manners is in a great degree more pure. The
Keks, for example, are described as leading a blameless life. The travellers saw no
marriageable maidens or children, married women alone appearing. The most
singular social economy prevails among them. The women live, during a
considerable part of the year, in villages apart from the men, who possess only
temporary huts. Their wives have regular substantial habitations, which are
common to both sexes during the rainy season. A man dare not approach the
“harem village,” except at the proper period, though some of the women
occasionally creep into their husbands’ village. Polygamy is allowed, but only
practised by the chiefs, since all the wives are bought, which renders the
indulgence costly.
Among some of the tribes on the banks of the White Nile women will sell their
children if they can do so with profit. Everywhere in that region the maidens mingle
naked with the men, but appear by no means immodest. When married they wear
an apron. All exhibit a sense of shame at exhibiting themselves unclothed before
strangers. Beyond the Mountains of the Moon, however, Werne found people,
among whom the unmarried men and women were separated. They were
completely naked, but chaste and decent nevertheless. A heavy price was always
asked for a girl, which prevented common polygamy, though their social code
permitted it[57].
It must be evident that, in an inquiry like the present, a view of the manners and
morals of Africa with regard to the female sex must be incomplete. In the first
place, our information is very limited; in the second, we are confined for space—for
otherwise these sketches could be extended to an indefinite extent. We have,
however, taken observations in Southern, in Western, in Eastern, in Northern, and
Central Africa. Kingdoms and communities, indeed, there are which we have not
included in our description. Of these some wear features so similar to others we
have noticed, that to particularise them is unnecessary in a general view. Of
others, such as Egypt, Nubia, Barca, Tripoli, Algiers, and Morocco, we shall treat in
a future division of the subject, because they are not included, by the character of
their civilization, among the communities of which we have hitherto spoken. The
reader will, we trust, have been enabled to form a fair idea of the average of
morals among the savages and semi-savages of Africa. With modern barbarians, as](https://image.slidesharecdn.com/54124-250625132526-2bb02ac5/85/Separation-of-Variables-for-Partial-Differential-Equations-An-Eigenfunction-Approach-1st-Edition-George-Cain-21-320.jpg)
![children reared. In other places one child to every six women was not an unusual
average. This, however, is not all to be ascribed to infanticide. Many of the females
abandon themselves so recklessly to vice that they lose all their natural powers,
and become incapable of bearing offspring. Eyre found in other parts of Australia
that the average of births was four to every woman. In New South Wales the
proportion of women to men appears to be as two to three; while in the interior,
Sturt calculated that female children outnumbered the male, while with adults the
reverse was true. This indicates an awful spread of the practice of infanticide,
which we cannot refuse to believe when we remember the facts which travellers of
undeniable integrity have made known to us.
To suppose from this that in Australia the natural sentiments of humanity are
unknown, would be extremely rash. On the contrary, we find very much that is
beautiful in the character of its wild people, and are led to believe that civilization
may go far towards elevating them from all their barbarous customs. Women are
known to bear about their necks, as relics sacred to affection, the bones of their
children, whom they have mourned for years with a pure and deep sorrow. Men
have loved and respected their wives; maidens have prized and guarded their
virtue; but it is too true that these are exceptions, and that the character and the
condition of the female sex in Australia is that of debasement and immorality.
With respect to the prostitute class of the colonial towns, to which allusion has
been made, it will be noticed in another part of this inquiry, when we examine into
the manners of English and other settlers abroad.
Of prostitutes as a class among the natives themselves, it is impossible to speak
separately; for prostitution of that kind implies some advance towards the forms of
regular society, and little of this appears yet to be made in that region. From the
sketch we have given, however, a general idea may be gained of the state of
women and the estimation of virtue among a race second only to the lowest tribes
of Africa in barbarity and degradation[58].
Of Prostitution in New Zealand.
In the New Zealand group we find a race considerably elevated above the other
inhabitants of Australasia, with a species of native civilization—a system of art,
industry, and manners. Perhaps the savage of New Holland is one of the most
miserable, and the New Zealander one of the most elevated, barbarians in the
world. By this we do not mean that he has made any progress in refinement, or
been subdued by the amiable amenities of life; but he is quick, intelligent, apt to
learn, swift to imitate, and docile in the school of civilization. The Maories, in their
original state, are low and brutal; but they are easily raised from that condition.
They have exhibited a capacity for the reception of knowledge, and a desire to](https://image.slidesharecdn.com/54124-250625132526-2bb02ac5/85/Separation-of-Variables-for-Partial-Differential-Equations-An-Eigenfunction-Approach-1st-Edition-George-Cain-27-320.jpg)
![countries, release a woman from servitude, but she enjoys a privilege which is
denied to the chief wife—she may marry again after her master’s death.
Formerly the general custom, however, was for a wife to hang, drown, strangle, or
starve herself on the death of her husband. Her relatives often gave her a rope of
flax, with which she retired to a neighbouring thicket and died. It was not a
peremptory obligation, but custom viewed it as almost a sacred duty. Sometimes
three of the wives destroyed themselves, but generally one victim sufficed. Self-
immolation is now, indeed, becoming very rare; but it is still the practice for the
widow, whether she loved her husband or not, to lament him with loud cries, and
lacerate her flesh upon his tomb. Whenever she marries again a priest is consulted
to predict whether she will survive the second husband or not. Occasionally we find
instances of real attachment between man and wife, such as would sanctify any
family hearth; while examples have occurred of women hanging themselves for
sorrow, on the death of a betrothed lover.
These, however, are only indications that humanity is not in New Zealand
universally debased below the brute condition. The general colour of the picture is
dark. Women are degraded; men are profligate; virtue is unknown in its abstract
sense; chastity is rare; and prostitution a characteristic of female society. Fathers,
mothers, and brothers—usually the guardians of a young woman—prostitute her
for gain, and the women themselves delight in this vice. There is, nevertheless,
some amelioration observable in the manners of the people, produced by the
influence of the English colonies. Those colonies themselves, however, are not free
from the stain, as will be shown when we treat of communities of that description
in general[59].
Of Prostitution in the Islands of the Pacific.
Among the innumerable islands which are scattered over the surface of the Pacific,
we discover various phases of manners developed under different influences. In
some of the lonely groups lying out of the usual course of trade or travel,
communities exist whose social habits remain entirely pure—that is, unchanged by
intercourse with foreigners. In others continual communication through a long
period, with white men, has wholly changed the characteristic aspects of the
people—given them a new religion, a new moral code, new ideas of decency and
virtue, new pleasures, and new modes of life. The same process appears likely, at a
future day, to obliterate the ancient system of things. In all the islands of this class,
indeed, the reform of manners is not so thorough as the florid accounts of the
missionaries would induce us to believe; but those pioneers of civilization have
done enough, without assuming more than their due, to deserve the praise of all
Christendom. To have restrained the fiercest passions of human nature among](https://image.slidesharecdn.com/54124-250625132526-2bb02ac5/85/Separation-of-Variables-for-Partial-Differential-Equations-An-Eigenfunction-Approach-1st-Edition-George-Cain-35-320.jpg)
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- 6. Separation of Variables for Partial Differential Equations An Eigenfunction Approach STUDIES IN ADVANCED MATHEMATICS
- 7. Separation of Variables for Partial Differential Equations An Eigenfunction Approach
- 8. Studies in Advanced Mathematics Titles Included in the Series John P. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping John J. Benedetto, Harmonic Analysis·and Applications John J. Benedetto and Michael l¥. Fraz.ier, Wavelets: Mathematics and Applications Albert Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex Keith Bums and Marian Gidea, Differential Geometry and Topology: With a View to Dynamical Systems George Cain and Gunter H. Meyer, Separation of Variables for Partial Differential Equations: An Eigenfunction Approach Goo11g Chen and Jianxi11 Zhou, Vibration and Damping in Distributed Systems Vol. I: Analysis, Estimation, Attenuation, and Design Vol. 2: WKB and Wave Methods, Visualization, and Experimentation Carl C. Cowen and Barbara D. MacCluer, Composition Operators on Spaces of Analytic Functions Jewgeni H. Dshalalow, Real Analysis: An Introduction to the Theory of Real Functions and Integration Dean G. Duffy, Advanced Engineering Mathematics with MATLAB®, 2nd Edition Dean G. Duffy, Green's Functions with Applications Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties of Functions Gerald B. Folland, A Course in Abstract Harmonic Analysis Josi Garcfa-Cuerva, Eugenio Hernd.ndez, Fernando Soria, and Josi-Luis Torrea, Fourier Analysis and Partial Differential Equations Peter 8. Gilkey. Invariance Theory, the Heat Equation,_and the Atiyah·Singer Index Theorem, 2nd Edition Peter B. Gilke.v, John V. Leahy, and Jeonghueong Park, Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture Alfred Gray, Modem Differential Geometry of Curves and Surfaces with Mathematica, 2nd Edition Eugenio Hemd.ndez and Guido Weiss. A First Course on Wavelets Kenneth B. Howell, Principles of Fourier Analysis Steven G. Krantz, The Elements of Advanced Mathematics, Second Edition Steven G. Krantz., Partial Differential Equations and Complex Analysis Steven G. Krantz. Real Analysis and Foundations, Second Edition Kenneth L. Kurt/er, Modern Analysis Michael Pedersen, Functional Analysis in Applied Mathematics and Engineering Ciark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition Jolm Rya11. Clifford Algebras in Analysis and Related Topics Joh11 Sclierk. Algebra: A Computational Introduction Pai·ei Soffn. Karel Segeth, and lvo Doletel, High-Order Finite Element Method Andr<i Unterberger and Harald Upmeier, Pseudodifferential Analysis on Symmetric Cones James S. lfolker, Fast Fourier Transforms, 2nd Edition James S. H'Cilker, A Primer on Wavelets and Their Scientific Applications Gilbert G. U'i?lter and Xiaoping Shen, Wavelets and Other Orthogonal Systems, Second Edition Nik Weaver. Mathematical Quantization Kehe Zhu. An Introduction to Operator Algebras
- 9. Separation of Variables for Partial Differential Equations An Eigenfunction Approach George Cain Georgia Institute of Technology Atlanta, Georgia, USA Gunter H. Meyer Georgia Institute of Technology A.r!anta, Georgia, USA Boc<1 Raton London New York
- 10. Published in 2006 by Chapman & HaIVCRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & HalVCRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10987654321 International Standard Book Number-IO: 1-58488-420-7 (Hardcover) International Standard Book Number-13: 978-1-58488-420-0 (Hardcover) Library of Congress Card Number 2005051950 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and infonnation, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic. mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (hltp://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks. and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Cain, George L. Separation of variables for partial differential equations : an eigenfunction approach I George Cain, Gunter H. Meyer. p. cm. -- (Studies in advanced mathematics) Includes bibliographical references and index. ISBN 1-58488-420-7 (alk. paper) l. Separation of variables. 2. Eigenfunctions. L Meyer, Gunter H. IL Title. III. Series. QA377.C247 2005 5 l5'.353--dc22 informa Taylor & Francis Group is the Academic Division of lnforma plc. 2005051950 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 11. Acknowledgments Ve would like to thank our editor Sunil Nair for welcoming the project and for his willingness to stay with it as it changed its scope and missed promised deadlines. · We also wish to express our gratitude to Ms. Annette Rohrs of the School of Mathematics of Georgia Tech who transformed decidedly low-tech scribbles into a polished manuscript. Without· her talents, and patience, we would·not have completed the book.
- 13. Preface 3-=paration of variables is a solution method for partial differential equations. liile its beginnings date back to work of Daniel Bernoulli (1753), Lagrange , 1759), and d'Alembert (1763) on wave motion (see [2]), it is commonly asso- ,ciated with the name of Fourier (1822), who developed it for his research on ·:-Jnductive heat transfer. Since Fourier's time it has been an integral part of e!:gineering mathematics, and in spite of its limited applicability and heavy com- petition from numerical methods for partial differential equations, it remains a -;;.·ell-known and widely used technique in applied mathematics. Separation of variables is commonly considered an analytic solution ::nethod that yields the solution of certain partial differential equations in terms cf an infinite series such as a Fourier series. While it may be straightforward to ·:uite formally the series solution, the question in what sense it solves the prob- lem is not readily answered without recourse to abstract mathematical analysis. A modern treatment focusing in part on the theoretical underpinnings of the method and employing the language and concepts of Hilbert spaces to analyze che infinite series may be fonnd in the text of MacCluer [15]. For many problems •he formal series can be shown to represent an analytic solution of the differ- ential equation. As a tool of analysis, however, separation of variables with its :nfirtite series solutions is not needed. Other mathematical methods exist which guarantee the existence and uniqueness of a solution of the problem under much ::nore general conditions than those required for the applicability of the method ;:,f separation of variables. In this text we mostly ignore infinite series solutions and their theoretical and practical complexities. We concentrate instead on the first N terms of the series which are all that ever are computed in an engineering application. Such a partial sum of the infinite series is an approximation to the analytic solution c,f the original problem. Alternatively, it can be viewed as the exact analytic solution of a new problem that approximates the given problem. This is the point of view taken in this book. Specifically, we view the method of separation of variables in the following context: mathematical analysis applied to the given problem guarantees the existence and uniqueness of a solution u in some infinite dimensional vector space o:,f functions X, but in general provides no means to compute it. By modifying the problem appropriately, however, an approximating problem results which has a computable closed form solution UN in a subspace M of X. If fl!! is vii
- 14. viii PREFACE suitably chosen, then UN is a good approximation to the unknown solution u. As we shall see, M will be defined such that UN is just the partial sum of the first N terms of the infinite series traditionally associated with the method of separation of variables. The reader may recognize this view as identical to the setting of the finite element, collocation, and spectral methods that have been developed for the numerical solution of differential equations. All these methods differ in how the subspace Mis chosen and in what sense the original problem is approximated. These choices dictate how hard it is to compute the approximate solution UN and how well it approximates the ai!alytic solution u. Given the almost universal applicability of numerical methods for the solu- tion of partial differential equations, the question arises whether separation of variables with its severe restrictions on the type of equation and the geometry of the problem is still a viable tool and deserves further exposition. The existence of this text ·reflects our view that the method of separation of variables still belongs to the core of applied mathematics. There are a number of reasons. Closed form (approximate) solutions show structure and exhibit explicitly the influence of the problem parameters on the solution. We think, for example, of the decomposition of wave motion into standing waves, of the relationship between driving frequency and resonance in sound waves, of the influence of diffusivity on the rate of decay of temperature in a heated bar, or of the gen- eration of equipotential and stream lines for potential flow. Such structure and insight are not readily obtained frnm purely numerical solutions of the underly- ing differential equation. Moreover, optimization, control, and inverse problems tend to be easier to solve when an analytic representation of the (approximate) solution is available. In addition, the method is not as limited in its applicability as one might infer from more elementary texts on separation of variables. Ap- proximate solutions are readily computable for problems with time-dependent data, for diffusion with convection and wave motion with dissipation, problems seldom seen in introductory textbooks. Even domain restrictions can sometimes be overcome with embedding and domain decomposition techniques. Finally, there is the class of singularly perturbed and of higher dimensional problems where numerical methods are not easily applied while separation of variables still yieltb an analytic approximate solution. Our rationale for offering a new exposition of separation of variables is then twofold. First, although quite common in more advanced treatments (such as [15]), interpreting the separation of variables solution as an eigenfunction expan- sion is a point of view r.arely taken when introducing the method to students. Usually the formalism is based on a product solution for the partial differential equation, and this limits the applicability of the method to homogeneous partial differential equations. When source terms do appear, then a reformulation of problems for the heat and wave equation with the help of Duhamel's superposi- tion principle and an approximation of the source term in the potential equation with the help of an eigenfunction approximation become necessary. In an expo- sition based from the beginning on an eigenfunction expansion, the presence of source terms in the differential equation is only a technical, but not a conceptual
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- 16. Among the nomade and stationary tribes of the Sahara, who are not aboriginal to that region, we have a different system of manners. In the Arabian communities you may find women ready to perform indecent actions, and even to prostitute themselves for money; but these are of the low classes. Cases of adultery are rare. The Mohammedans believe that a man cannot have too many wives, or, at least, too many concubines. They declare it assists their devotion; but the feeling is one merely sensual. Pure sentiment is a thing in which they can scarcely believe. Rich men who are accustomed to travel in pursuit of trade, have one family at Ghadames, another, perhaps, at Ghat, and another at Soudan, and live with each of them by turns. These women stand in great fear of their husbands. The rich are veiled, and live in retirement; the poor do not; but all will unveil their faces to a stranger, if it can be done with safety. The white, or respectable women of Ghadames, never descend into the streets, or even into the gardens of their houses. The flat roof of their dwelling is their perpetual promenade, and a suite of two or three rooms their abode. It is said that in these retreats many of the women privately rule their husbands, though no men will confess the fact. Among the Marabouts it is held disgraceful to be unmarried, but shameful also to be under the wife’s control. The negresses and half-castes who may be seen in the streets of the cities of the Sahara, are generally slaves. The women of the Touarik tribes, however, are by no means so. They belong to a fierce and warlike tribe, half vagrant, half stationary, and are bound by few restrictions. Their morals are described as superior to those of the lower class of women in Europe; though exceptions, of course, are found. One Touarik woman offered to prostitute herself to Richardson for a sum of money; or, as it was expressed, to become his wife. Polygamy, though universally allowed in the Sahara, is not carried to an extent at all equal to that prevailing in the savage regions on the east and west. Three wives usually occupy the harem of a rich man. Marriage is, as usual with people of that religion, a civil contract with a shade of sanctity upon it, but celebrated with great feasts and rejoicings. The bridegroom is expected to live in retirement during two or three weeks. He occasionally walks about the town at evening alone, dressed in gay clothes of blue and scarlet, and bearing a fine long stave of brass or polished iron. He never speaks or is spoken to, and vanishes on meeting any one.
- 17. GIRLS OF NUBIA (MAKING POTTERY). [From St. John’s “Oriental Album.”] The manners of the communities in the Sahara are imperfectly known; but from the accounts we have received they appear to be of a far more elevated order than those of any other part of Africa. It is true that customs prevail which shock our
- 18. ideas of decency. A chief, for instance, offered Richardson his two daughters as wives. It is also true that many women exist who follow the profession of prostitutes, though we have no distinct account of them. But immorality is usually among them a secret crime. Their general customs with regard to sexual intercourse are at least as pure as those of Europe. Among the wandering tribes of the desert the hardship of their lives, continual occupation, varied scenes of excitement, and contempt for sensual enjoyments, contribute to preserve chastity among their virtues; while the Marabouts of the cities are of a generally moral character. Intoxication never happens among the women. Still, the condition of the sex is degraded; for they are, with exceptions, regarded only as the materials of a man’s household, and ministers to the sensual enjoyments of his life[51]. The Mohammedans of Central Africa, bigoted as to dogmas, are nevertheless more liberal to women, who enjoy more consideration among them than in the more important strongholds of that religion[52]. The wandering Arabs of Algeria hold marriage as a business transaction, though the estimation of the sex is not low. The lover brings to the woman’s home ten head of cattle, with other presents, which usually form her dowry. The father asks, “How much does she whom you are going to have for wife cost you?” He replies, “A prudent and industrious woman can never be too dear.” She is dressed, placed on a horse, and borne to her new home amid rejoicing. She then drinks the cup of welcome, and thrusting a stick into the ground, declares, “As this stick will remain here until some one forces it away, so will I.” She then performs some little office to show she is ready for the duty of a wife, and the ceremony is ended[53]. Transferring our observations to Abyssinia, we find in its several divisions different characteristics of manners. In Tajura, on the Red Sea, profligacy is a conspicuous feature of society. Men live with their wives for a short period, and then sell them, maintaining thus a succession of favourites in their harems. Parents, also, are known not only to sell their daughters as wives, but to hire them out as prostitutes. One chief offered a traveller his daughter either as a temporary or a permanent companion; he showed another whom he would have sold for 100 dollars. One woman presented herself, stating, as a recommendation, that she had already lived with five men. These are nothing but prostitutes, whatever the delicacy of travellers induces them to term them. Unfortunately the inquiries made into this system are very slight, affording us no statistics or results of any kind. We are thus left to judge of morality in Tajura by the fact that syphilis afflicts nearly the whole population, man and woman, sultan and beggar, priests and their wives included. In the Christian kingdom of Shoa, the Christian king has one wife, and 500 concubines; seven in the palace, thirteen at different places in the outskirts, and the rest in various parts of his dominions. He makes a present to the parents of any women he may desire, and is usually well paid in return for the honour. The governors of cities and provinces follow this example, keeping establishments of
- 19. concubines at different places. Scores of the royal slaves are cast aside, and their place supplied by others. In Shoa there are two kinds of marriage; one a mere agreement to cohabitation, another a holy ceremony; the former is almost universally practised. The men and women declare before witnesses that they intend to live happily together. The connection thus easily contracted is easily broken; mutual consent only is necessary to a divorce. In Shoa a wife is valued according to the amount of her property. The heiress to a house, a field and a bedstead, is sure to have a husband. When they quarrel and part, a division of goods takes place. Holy ceremonies are very rare, and not much relished. A wedded couple, in one sense of the term, is a phenomenon. Instances of incontinence are frequent; while the caprice of the men leads them often to increase the number of their concubines. These are procured as well from the Christians as from the Mohammedans and Pagans; but the poor girls professing these religions are forced to a blind profession of Christianity. Favourite slaves and concubines hold the same position with married women; while illegitimate and legitimate children are treated by the law with no distinction. Three hundred of the king’s concubines are slaves, taken in war or purchased from dealers. They are guarded by fifty eunuchs, and live in seclusion; though this by no means prevents the court from overflowing with licentiousness. Numerous adulteries take place, and this example is followed by the people; among whom a chaste married couple is not common. Women in Abyssinia, which is an agricultural country, mix freely with the men, and dance in their company; though a few jealous husbands or cautious parents seclude them. Morality is at an extremely low ebb. At the Christmas saturnalia, gross and disgusting scenes occur, as well as at other feasts. What else can be expected in a country where 12,000 priests live devoted, in theory at least, to celibacy; and where, at the annual baptisms, these priests, with men, women, and children strip naked, and rush in promiscuous crowds into a stream, where they are baptised according to the Christian religion! The sacerdotal class of Shoa is notoriously drunken and profligate. Another cause of corruption is the caprice which induces men to abandon their concubines after short cohabitation with them. These women, discarded and neglected, devote themselves to an infamous profession, and thus immorality is perpetuated through every grade of their society: in a word, the morals of Shoa are of the lowest description. In the Mohammedan states in its neighbourhood the condition of the sex is no better. If there is less general prostitution, it is because every woman is the slave of some man’s lust, and is imprisoned under his eye. He is jealous only of her person; scarcely attributing to her a single quality which is not perceptible to his senses[54]. In the southern provinces of Kordofan, under the government of Egypt, south of the Nubian Mountains, immense labour is imposed on the unmarried girls; yet the sentiment of love is not altogether unknown to them, and men fight duels with
- 20. whips of hippopotamus hide on account of a disputed mistress. The wife is nevertheless a virtual slave, and still more degraded should she prove barren; the husband, in that case, solaces himself with a concubine, who, if she bears a child, is elevated to the rank of wife. It is common among the rich for a man to make his wife a separate allowance after the birth of her second child, when she goes to live in a separate hut. All their bloom is gone by the time they are twenty-four years old, and thenceforward they enjoy no estimation from the men. Yet, improvident in their hearts, the young girls of Kordofan are merry; and, whether at work or idle, spend the day in songs and laughter; while in the evening they assemble and dance to the music of the Tarabuka drum. Their demeanour, in general, is modest, and their lives are chaste. Married women, on the contrary, especially those who are neglected by their husbands, occupy themselves in gossip, and find solace in criminal intrigues. In some parts of the country, indeed, men consider it an honour for their wives to have intercourse with others; and the women are often forwarded in their advances. Female slaves often have liberty when they bear children to their proprietors. Women eat when the men have done, and pretty dancers attend at the feasts to amuse their employers. These girls, like the Ghawazee of Lower Egypt, are usually prostitutes, and very skilful in the arts of seduction. Numbers of this class fled from Egypt into Kordofan, on one occasion, when Mohammed Ali, in one of his affected fits of morality, endeavoured to suppress their calling altogether. Marriage, it may be scarcely necessary to say, is concluded without the woman’s consent. The man bargains for her, pays her price, takes her home, strips off her virginal girdle, which is the only garment of unmarried girls, and covers her with a cloth about her loins; a feast and a dance occasionally celebrate the event. When a wife is ill-treated beyond endurance, she demands a divorce; and, taking her female offspring, with her dowry, returns home. Trifles often produce these separations. That her husband has not allowed her sufficient pomatum to anoint her person with, is not unfrequently the ground of complaint. Few men in Kordofan have more than two wives; but most have concubines besides, whom the more opulent protect by a guard of eunuchs. These remarks apply to the agricultural or fixed population. The Baghaira, or wandering pastoral tribes of Kordofan, are a modest, moral race—naked, but not on that account indecent[55]. A chief of the Berbers offered a late traveller the choice of his two daughters for a bedfellow. They were already both married. Women there, however, as well as in Dongola, are, many of them, ready to prostitute themselves for a present. A virgin, whether as wife or concubine, may be purchased for a horse. “Why do you not marry?” said a traveller to a young Berber. He pointed to a colt and answered “When that is a horse I shall marry.”[56]
- 21. The condition of women and state of manners on the upper borders of the Nile, we find described in Ferdinand Werne’s account of his recent voyage to discover the sources of the White Stream. The system in Khartum may be indicated by one sentence in the traveller’s own language. He speaks of desiring that the pay might be advanced to prevent starvation from visiting the soldiers’ families, “which, from the low price of female slaves, were numerous.” It may, without resort to hyperbole, be said, that the female monkeys peopling the neighbouring woods occupy a far nobler and more natural position. Among the barbarians on the banks of the river further up, the state of manners is in a great degree more pure. The Keks, for example, are described as leading a blameless life. The travellers saw no marriageable maidens or children, married women alone appearing. The most singular social economy prevails among them. The women live, during a considerable part of the year, in villages apart from the men, who possess only temporary huts. Their wives have regular substantial habitations, which are common to both sexes during the rainy season. A man dare not approach the “harem village,” except at the proper period, though some of the women occasionally creep into their husbands’ village. Polygamy is allowed, but only practised by the chiefs, since all the wives are bought, which renders the indulgence costly. Among some of the tribes on the banks of the White Nile women will sell their children if they can do so with profit. Everywhere in that region the maidens mingle naked with the men, but appear by no means immodest. When married they wear an apron. All exhibit a sense of shame at exhibiting themselves unclothed before strangers. Beyond the Mountains of the Moon, however, Werne found people, among whom the unmarried men and women were separated. They were completely naked, but chaste and decent nevertheless. A heavy price was always asked for a girl, which prevented common polygamy, though their social code permitted it[57]. It must be evident that, in an inquiry like the present, a view of the manners and morals of Africa with regard to the female sex must be incomplete. In the first place, our information is very limited; in the second, we are confined for space—for otherwise these sketches could be extended to an indefinite extent. We have, however, taken observations in Southern, in Western, in Eastern, in Northern, and Central Africa. Kingdoms and communities, indeed, there are which we have not included in our description. Of these some wear features so similar to others we have noticed, that to particularise them is unnecessary in a general view. Of others, such as Egypt, Nubia, Barca, Tripoli, Algiers, and Morocco, we shall treat in a future division of the subject, because they are not included, by the character of their civilization, among the communities of which we have hitherto spoken. The reader will, we trust, have been enabled to form a fair idea of the average of morals among the savages and semi-savages of Africa. With modern barbarians, as
- 22. with ancient states, tabular statistics are impossible: but from a description in general terms, we cannot always refuse to ground a confident opinion. Women in Australia. In Australia we have a family of the human race still more uneducated, though not more barbarous, than that which inhabits the woods of the African continent. There is among them less approach to the arts of civilization, less ingenuity, less intelligence, but there is more simplicity. Their customs are not so brutal as those prevailing on the banks of the Joliba or the Senegal. Nevertheless they are true savages, and the condition of their women is consistent with all the other features of their irreclaimed state. Of the Australians, however, as of all races imperfectly known, there obtains in this country a vulgar idea drawn from the old accounts, which are little better than caricatures. They have been represented as a hideous race, scarcely elevated above the brute, blood-thirsty, destitute of human feeling, without any redeeming characteristics, and, moreover, incapable of civilization. Such a description is calculated only to mislead. The aborigines of Australia are certainly a low, barbarous, and even a brutal race, but the true picture of their manners, which form the expression of their character, is not without encouraging traits. Considering the great extent of New Holland, it is surprising to find such an uniformity of character and customs, as we actually discover among its nations. The language, varied by dialects, the habits, social laws, and ideas of the people, are extremely similar, whether we visit them in that province called the Happy or in the districts around Port Essington. Consequently, though it occupy a large space on the map, this region will not require any very extended notice. An idea of the condition and morality of its women may be afforded by one general view, with reference to the various local peculiarities noticed by travellers. The native inhabitants of Australia are generally nomadic. They dwell in temporary villages scattered over vast surfaces of country, and move from place to place, as the supply of provisions, spontaneously provided by the earth, is more or less abundant. Separated as they are into small isolated communities—rarely numbering more than eighty members—they resort to the borders of lakes and streams, which dry up at certain seasons, and force them to seek elsewhere a home. A rude copy of the patriarchal form of government prevails among them— old men being the rulers of the tribe. The condition of women among these primitive savages is extremely low. They are servants of the stronger sex. In some of their dialects wife and slave are synonymous. All the labour devolves on her, and, as no form of agriculture is practised, this consists principally in the search for the means of life. She collects
- 23. the daily food, she prepares the camp or the hut at night, she piles fire-wood, draws water, weaves baskets, carries all burdens, and bears the children on her back, and the return for all this willing devotion is frequently the grossest ill-usage. There is no form of marriage ceremony observed. A man gets a wife in various ways. Sometimes she is betrothed to him while an infant—even before her birth, and sometimes she devolves to him with other property. The eldest surviving brother, or next male relative, inherits the women of a whole family. Thus many households are supplied. Others steal their wives from hostile tribes, and frequent wars arise from such proceedings. Polygamy is universally allowed, but not by any means generally practised; for there are few parts of Australia where the female sex is not outnumbered by the male. Plurality of wives consequently implies wealth and distinction—each additional one being regarded as a new slave, an increase of property. Nor are the women jealous of polygamy. When a man has many wives, they subdivide the labour, which otherwise would devolve on one, thus lightening each others’ burdens, and procuring companionship. There can indeed be little jealous feeling where affection on the part of the husband to the wife is almost a thing unknown. The Australian wife when past the prime of life is usually a wretched object. She is often deformed and crippled by excessive toil—her body bent, her legs crooked, her ankles swollen, her face wearing an aspect of sullen apathy, produced by long hardship. When young, however, they are frequently lively and happy, not being cursed with keen feelings, and caring for little beyond the present hour. Should a young woman, nevertheless, be distinguished by peculiar beauty, she leads, while her attractions last, a miserable course of existence. Betrothed at an early age, she is perpetually watched by the future husband, and upon the least suspicion of infidelity is subjected to the most brutal treatment. To thrust a spear through her thigh or the calf of her leg is the common mode of punishment. She may, in spite of all precautions, be snatched away: whether consenting or not, she must endure the same penalty. If she be chaste, the man who has attempted to seduce her may strike her with a club, stun her, and bear her to a wood, where she is violated by force. Still she is punished, and it is, says Sir George Grey, no common sight to see a woman of superior elegance or beauty who has not some scars disfiguring various parts of her person. This period, however, is soon over, for the bloom of an Australian woman is very short-lived. When the seducer is found, he is punished in a similar manner, and if he have committed adultery with a married woman, suffers death. The jealousy of the married men is excessive, and would be ridiculous were it not that their vigilance is absolutely called for. A careless husband would speedily suffer for his neglect. Accordingly we find the Australian savages practising in their woods or open plains restrictions not dissimilar to those adopted in the seraglios of the East. When an encampment is formed for the night every man overlooks his wives while they build one or more temporary huts, over which he then places himself as
- 24. a guard. The young children and the unmarried girls occupy this portion of the village. Boys above ten years of age and all single men are forced to sleep in a separate encampment, constructed for them by their mothers, and are not allowed to visit the bivouacs of the married men. Under no circumstances is a strange native allowed to approach one of the family huts. Each of these little dwellings is placed far from the rest, so that when their inmates desire to hold converse they sing to each other from a distance. When the young men collect to dance, the maidens and wives are allowed to be spectators, but only on a few occasions to join. They have dances of their own, at which the youth of the other sex are not permitted to be present. In spite of this excessive jealousy the idea of a husband’s affection for his wife appears strange to them. Men return from journeys without exchanging a greeting with the mothers of their children, but those children they salute with many endearing terms, falling on their necks and shedding tears with every demonstration of love. A man has been known, when his wife was grievously sick, to leave her to die in the wilderness, rather than be troubled with her on his journey. Yet the influence of women is not by any means small. In some of the tribes they obtain a position of moderate equality with the husband, are well-fed, clothed, and treated as rational beings. Everywhere the men, young and old, strive to deserve their praise; and exhibitions of vanity take place, perfectly ludicrous to those European travellers who forget that the silly dandyism of the Australian savage, with his paint and opossum skin, is only peculiar in its form of expression. Women are often present on the field of battle, to inspire their husbands by exhortations, to rouse them by clamours of revenge or appeals to their valour; and among the chief punishments of cowardice is their contempt. The man failing in any great duty of a warrior is so disgraced. Thus, if he neglect to avenge the death of his nearest relation, his wives may quit him; the unmarried girls shun him with scorn, and he is driven by their reproaches to perform his bloody and dangerous task. Where polygamy exists it is seldom the woman’s consent is required before her union with a suitor. In Australia it is never required or expected. The transaction is entirely between her father and the man who desires her for a wife, or, rather, for a concubine. She is ordered, perhaps, to take up her household bag, and go to a certain man’s hut, and this may be the first notice she has of the marriage. There she is in the position of a slave to her master. If she be obedient, toil without torture is her mitigated lot; but if she rebel, the club is employed to enforce submission. She is her husband’s absolute property. He may give her away, exchange her, or lend her as he pleases. Indeed, old men will sometimes offer their wives to friends, or as a mark of respect to strangers; and the offer is not uncommonly accepted.
- 25. Though we have mentioned three ways of obtaining a wife, the system of betrothal is the most general. Almost every female child is so disposed of a few days after its birth. From that moment the parents have no control whatever over her future settlement; she is in fact a bought slave. Should her betrothed die she becomes the property of his heir. Whatever her age she may be taken into the hut; cohabitation often commencing while the girl is not twelve years old, and her husband only a boy. Three days after her first husband’s death the widow goes to the hut of the second. Some restrictions, however, are imposed on the intercourse of the sexes. Thus all children take the family name of their mother, and a man may not marry a woman of his own family name. Relations nearer than cousins are not allowed to marry, and an alliance even within this degree is very rare. The Australians have, indeed, a horror of all connections with the least stigma of incest upon them, and adjudge the punishment of death to such an offence. Their laws, which are matters not of enactment but of custom, are extremely severe upon this and all other points connected with their women. Chastity, nevertheless, is neither highly appreciated nor often practised. It is far from being prized by the women as a jewel of value; on the contrary, they plot for opportunities to yield it illicitly, and can scarcely be said to know the idea. Profligacy is all but universal among them; it is a characteristic even of the children. When some schools were formed at Perth, for the education of the natives, it was found absolutely necessary to separate children of tender years, in order to prevent scenes of vile debauch from being enacted. It should be said, however, that though indiscriminate prostitution among the women, and depraved sensuality among the men, exist in the most savage communities, disease and vice are far less characteristic of them than of those tribes which have come in contact with Europeans. In all the colonial towns there is a class of native women following the calling of prostitutes, and there the venereal disease and syphilis are most deadly and widely prevalent. The former appears to have been brought from Europe, and makes terrible havoc among them. The latter, ascribed by their traditions to the East, has been found among tribes which had apparently never held intercourse with the whites; in such cases, however, it is in a milder form. Several causes contribute to the corruption of manners among these savage tribes. One of the principal is, the monopoly of women claimed by the old men. The patriarchs of the tribe, contrive to secure all the young girls, leaving to their more youthful brethren only common prostitutes, prisoners of war, and such women as they can ravish from a neighbouring community, or seduce from their husbands’ dwellings. They also abandon to them their own wives when 30 or 40 years old, obtaining in exchange the little girls belonging to the young man’s family. The youthful warrior, therefore, with a number of sisters, can usually succeed in obtaining a few wives by barter. That their personal attractions are faded is not of any high importance; since they are needed chiefly to render him independent of
- 26. labour. His sensual appetites he is content to gratify, until he becomes a patriarch, by illicit intrigues with other women of the tribe. Of these there are generally some ready to sell or give away their favours. The wives, especially of the very old chiefs, look anxiously forward to the death of their husbands, when they hope, in the usual course of inheritance, to be transferred to the hut of a younger man; for, among nations in this debased state, it is not the woman that is prized, but a woman. Personal attachment is rare. The husband whose wife has been ravished away by a warrior from a neighbouring tribe may be pacified by being presented with another companion. Even in Australia Felix, which is peopled by the most intelligent, industrious, and manly of the Australian race, the young man disappointed of a wife in his own tribe sets off to another, waylays some woman, asks her to elope with him, and, on her refusal, stuns her with his club, and drags her away in triumph. Marriage, indeed, appears too dignified a term to apply to this system of concubinage and servitude which in Australia goes under that name. Travellers have found in the far interior happy families of man and wife, roaming together, with common interests, and united by affection; but such instances are rare. A large proportion of the young men in Australia can by no means obtain wives. This arises from the numerical disparity between the sexes, which is almost universal in that region, and is chiefly attributable to the practice of infanticide. Child-killing is indeed among the social institutions of that poor and barbarous race. Women have been known to kill and eat their offspring, and men to swing them by the legs and dash out their brains against a tree. The custom is becoming rare among those tribes in constant intercourse with Europeans, but that intercourse itself has caused much of the evil. Half-castes, or the offspring of native women by European fathers, are almost invariably sacrificed. They are held in dread by the people, who fear the growth of a mixed race which may one day conquer or destroy them. Females, also, are killed in great numbers. This class of infanticide is regulated by various circumstances in different communities. Among some tribes all the girls are destroyed until a boy is born; in others, the firstborn is exposed; in others, all above a certain number perish; but everywhere the custom prevails. One of two twins—a rare birth—is almost always killed. It may be ascribed to the miserably poor condition of the people, and the degraded state of the female sex; for in a region where the aborigines have not yet learned to till the soil, and where the means of life are scanty, there will always be an inducement to check the growth of numbers by infanticide; and where women have to perform all the labour, and follow their husbands in long marches or campaigns, ministering to every want they may experience, the trouble of nursing an infant is often saved at the cost of the infant’s life. Neglect also effects the same purpose. The population, under these circumstances, has always been thin, and is apparently decreasing. Among 421 persons belonging to various tribes in Australia Felix, Eyre remarked that there were in the course of two years and a half only ten
- 27. children reared. In other places one child to every six women was not an unusual average. This, however, is not all to be ascribed to infanticide. Many of the females abandon themselves so recklessly to vice that they lose all their natural powers, and become incapable of bearing offspring. Eyre found in other parts of Australia that the average of births was four to every woman. In New South Wales the proportion of women to men appears to be as two to three; while in the interior, Sturt calculated that female children outnumbered the male, while with adults the reverse was true. This indicates an awful spread of the practice of infanticide, which we cannot refuse to believe when we remember the facts which travellers of undeniable integrity have made known to us. To suppose from this that in Australia the natural sentiments of humanity are unknown, would be extremely rash. On the contrary, we find very much that is beautiful in the character of its wild people, and are led to believe that civilization may go far towards elevating them from all their barbarous customs. Women are known to bear about their necks, as relics sacred to affection, the bones of their children, whom they have mourned for years with a pure and deep sorrow. Men have loved and respected their wives; maidens have prized and guarded their virtue; but it is too true that these are exceptions, and that the character and the condition of the female sex in Australia is that of debasement and immorality. With respect to the prostitute class of the colonial towns, to which allusion has been made, it will be noticed in another part of this inquiry, when we examine into the manners of English and other settlers abroad. Of prostitutes as a class among the natives themselves, it is impossible to speak separately; for prostitution of that kind implies some advance towards the forms of regular society, and little of this appears yet to be made in that region. From the sketch we have given, however, a general idea may be gained of the state of women and the estimation of virtue among a race second only to the lowest tribes of Africa in barbarity and degradation[58]. Of Prostitution in New Zealand. In the New Zealand group we find a race considerably elevated above the other inhabitants of Australasia, with a species of native civilization—a system of art, industry, and manners. Perhaps the savage of New Holland is one of the most miserable, and the New Zealander one of the most elevated, barbarians in the world. By this we do not mean that he has made any progress in refinement, or been subdued by the amiable amenities of life; but he is quick, intelligent, apt to learn, swift to imitate, and docile in the school of civilization. The Maories, in their original state, are low and brutal; but they are easily raised from that condition. They have exhibited a capacity for the reception of knowledge, and a desire to
- 28. adopt what they are taught to admire—which encourage strong hopes of their reclamation. Among them, however, vice was, until recently, almost universal, and at the present day it is so, with the exception of a few tribes brought directly under the influence of educated and moral European communities. The only class which has discarded the most systematic immorality is that which has reconciled itself to the Christian religion, or been persuaded to follow the manners of the white men. The unreclaimed tribes present a spectacle of licentiousness which distinguishes them even among barbarous nations. They show, indeed, an advance in profligacy. Their immorality is upon a plan, and recognised in that unwritten social law which among barbarians remedies the want of a written code. It is not the beastly lust of the savage, who appears merely obedient to an animal instinct, against which there is no principle of morals or sentiment of decency to contend;—it is the appetite of the sensualist, deliberately gratified, and by means similar, in many respects, to those adopted among the lowest classes in Europe. We may, indeed, compare the Maori village, unsubjected to missionary influence, with some of the hamlets in our rural provinces, where moral education of every kind is equally an exile. The New Zealanders have been divided into the descendants of two races, the one inferior to the other; and the Malay has been taken as the superior. Ethnologists may prove a difference between them, and trace it through their manners; but these distinctions of race are not sufficiently marked to require separate investigations. The social institutions of the islanders are very generally the same, with some unimportant variations among the several tribes. We are placed in this peculiar difficulty when inquiring into the manners of New Zealand—that they appear to have undergone considerable modification since, and in consequence of, the arrival of Europeans. The natives refer to this change themselves, and in some cases charge the whites with introducing various evils into their country. Undoubtedly this is as true of New Zealand as of every other portion of the globe whither men have carried from Christendom the vices as well as the advantages of civilization. But in speaking of European settlers, a broad distinction must be borne in mind. White is not more contrasted with black, than are the regular orderly colonies established under the authority of Great Britain with the irregular scattered settlements planted by whalers, runaway or released convicts, land speculators, and other adventurers before the formal hoisting of our flag. The influence of the one has been to enlighten and to elevate, of the other to debase and demoralize, the native population. Gambling, drinking, and prostitution were encouraged or introduced by the one, Christianity, order, and morality are spreading through the exertions of the other; and it is, therefore, unjust to confound them in one general panegyric or condemnation. Nor shall we include all the unrecognised settlements in this description. Many of the hardy whalers and others have taken to themselves Maori wives, who, sober, thrifty, and industrious, submit without complaining to rough usage and hard work, and are animated by a
- 29. deep affection for their husbands. Contented with a calico gown and blanket, an occasional pipe of tobacco, and a very frugal life, they cost little to support, and appear for the most part not only willing but cheerful. The female sex throughout New Zealand is not in such complete subjection to the male as in New Holland. With the right they have acquired the power to resist any unnatural encroachment upon their liberties, though still in a state of comparative bondage. They are influential in society, and whenever this is the case they enjoy, more or less, remission of oppression. We find them declaiming at public meetings of the people, and fiercely denouncing the warriors who may be dishonourably averse to war, or have behaved ignominiously in the field. By influencing their friends and relatives they often secure to themselves revenge for an injury, and thus security against the same in future. In various other ways their position is defended against utter abasement. They are not regarded merely as subservient to the lust and indolence of the male sex. When dead they are buried with ceremony according to the husband’s rank, and formal rites of mourning are observed for them. In public and in domestic affairs their opinions are consulted, and often their hands are obtained in marriage by the most humble supplication, or the most difficult course of persuasion, by the lover. All this is evidence of a higher state than that which is occupied by females either in Africa or New Holland. Polygamy is permitted and practised by those who can afford it. In reality, however, the man has but one wife and a number of concubines, for though the second and third may be ceremoniously wedded to him, they are in subjection to the first, and his intercourse with them is frequently checked by her. She is paramount and all but supreme, though a man of determination will sometimes divorce his first wife to punish her contumelious behaviour to his second. It is customary for a man to marry two or more sisters, the eldest being recognised as the chief or head of the family. They all eat with the men, accompanying them, as well as their lovers and relations, before marriage, on their war expeditions or to their feasts. Betrothal takes place at a very early age—often conditionally before birth. Thus two brothers or two friends will agree that if their first children prove respectively a boy and a girl, they shall be married. When it is not settled so early, it is arranged during infancy, or at least childhood—for a girl of sixteen without an accepted lover is regarded as having outlived her attractions and all chance of an alliance. The betrothal is usually the occasion of a great feast, where wishes for the good success and welfare of the young couple are proclaimed by a company of friends. Three varieties of marriage formality are observed—differing as the girl is wanted to fill the place of first, second, third, or fourth wife. The first is a regular ceremony, the second less formal, and the last, which is merely conventional, is when a slave is raised from servitude to the marital embrace. The highest is that in which the priest pronounces a benediction, and a hope, not a prayer, for the prosperity of the married couple. The rest, which is the most approved and common, is for the man to conduct his betrothed to his hut, and she is
- 30. thenceforward mistress of the place. Unless she be divorced, no one can take away her power, and no inferior wife can divide it. When they have entered the dwelling a party of friends surround it, make an attack, force their way, strip the newly- married pair nearly naked, plunder all they can find, and retire. By taking a woman to his house a man makes her his wife, or virtually, except in the case of the first, his concubine. When he merely desires to cohabit with one, without being formally united to her, he visits her habitation. Though polygamy or concubinage has been practised in New Zealand from immemorial time, jealousy still burns among the wives as fiercely as in any Christian country where the institution is forbidden by the social law. It is the cause of bitter domestic feuds. The household, with a plurality of women, is rarely at peace. It is universally known to what an extent the jealousy of the Dutch women in Batavia carried them when their husbands indulged in the practice—common in Dutch settlements—of keeping female slaves. They watched their opportunity, and when it occurred would carry a poor girl into the woods, strip her entirely naked, smear her person all over with honey, and leave her to be tortured by the attacks of insects and vermin. A similar spirit of ferocious jealousy is characteristic of the women in New Zealand. The inferior wives consequently lead a miserable life, subjected to the severest tyranny from the chief, who makes them her handmaids, and sometimes terrifies her husband from marital intercourse with them. She exposes them to perpetual danger by endeavouring to insinuate into his mind suspicions of their fidelity, and thus the household is rendered miserable. When a man takes a journey he is usually accompanied by one of his wives, or, if he goes alone, will bring one back with him. Hence arise bitter heart-burnings and quarrels. Occasionally they lead to the death of one among the disputants, and frequently to infanticide. So furious are the passions of the women when their jealousy is excited against their younger rivals, that many of the chiefs in New Zealand fear to enjoy the privilege allowed them by their social law. When they resolve upon it, they often proceed with a caution very amusing to contemplate. More than one anecdote in illustration of this is related in the works of recent travellers. A man having a first wife of bad temper and faded beauty, whom he fears, nevertheless, to offend altogether, is attracted by some young girl of superior charms, and offers to take her home; she accepts, and the husband prepares to execute his design. It is often long before he acquires courage to inform his wife, and only by the most skilful mixture of persuasion, management, and threats, that she is ever brought to consent. Women captured in battle, however, may be made slaves, or taken at once to their captor’s bed. Thus raised from actual slavery, their condition is little improved. The tyranny of the chief wife is exercised to oppress, insult, and irritate them. Should one of them prove pregnant, her mistress—especially if herself barren—will often exert the most abominable arts to ensure her miscarriage, that
- 31. the husband may be disappointed of his child, and the concubine of his favour which would thence accrue to her. Divorces, according to the testimony of most writers, are not unfrequent in New Zealand. Among the ordinary causes are, mere decline of conjugal affection, barrenness in the wife, and a multiplication of concubines. A stepmother ill-treating the children, or a mother wantonly killing one of them, is liable to divorce. The latter is not an useless precaution, for jealous wives have been known in cold blood to murder an infant, merely to revenge themselves upon their husbands, or irritate them into divorce. A woman extravagantly squandering the common property, idling her time, playing the coquette, becoming suspected of infidelity, or refusing to admit a new wife into the house, is sometimes put away. This is effected by expelling her from the house. When it is she who seeks it, she flies to her relatives or friends. Should the husband be content with his loss, both are at liberty to marry; but if he desire to regain her, he seeks to coax her back, and, failing in that, employs force. She is compelled to submit unless her parents are powerful enough to defend her—for in New Zealand arms are the arbiters of law. When the desire to separate is mutual, it is effected by agreement, which is a complete release to both. If the husband insist on taking away the children, he may, but he is forbidden, on pain of severe punishment, from annoying his former wife any further. There is among the New Zealanders a rite known as Tapu, and the person performing it is sacred against the touch of another. While in this condition no contact is allowed with any person or thing. There are, however, comparative forms of Tapu. Thus a woman, in the matter of sexual intercourse, is tapu to all but her husband, and adultery is severely punished. Formerly the irrevocable remedy was death, and this may still be inflicted; but jealousy is seldom strong in the New Zealand husband, who often contents himself with receiving a heavy fine from his enemy. The crime is always infamous, but not inexpiable. The husband occasionally, when his wife has been guilty, takes her out of the house, strips her, and exposes her entirely naked, then receiving her back with forgiveness. The paramour usually attempts to fly. If he be not put to death, he also is sometimes subjected to a similar disgrace. When a wife discovers any girl carrying on a secret and illicit connection with her husband, a favourite mode of revenge is, to strip and expose her in this manner. For, in New Zealand, libidinous as the conduct of the people may be, their outward behaviour is, on the whole, decorous. They indulge in few indecencies before a third person. The exposure of the person is one of the most terrible punishments which can be inflicted. A woman has hanged herself on its being said that she has been seen naked. One girl at Karawanga, on the river Thames, charged with this offence, was hung up by the heels and ignominiously flogged before all the tribe. Shame drove her mad, and she shot herself. They are otherwise obscene, and the children are adepts in indecency and immorality. One strong characteristic of their rude attempts at art is the obscenity in their paintings
- 32. and carvings. In those singular specimens which crowd the rocks of Depuch Island, on the coast of New Holland, not a trace of this grossness is visible. One of the most melancholy features in the manners of this barbarous race, is the prevalence of infanticide. The Christian converts, as well as some of the natives who hold frequent intercourse with the more respectable Europeans, have abandoned it, as well as polygamy; but, with these exceptions, it is general throughout the thinly-scattered population of New Zealand. It almost always takes place immediately after birth, before the sentiment of maternal affection grows strong in the mother’s breast. After keeping a child a little while they seldom, except under the influence of frenzy, destroy it. As they have said to travellers, they do not look on them, lest they should love them. The weakly or deformed are always slain. The victim is sometimes buried alive, sometimes killed by violent compression of its head. This practice has contributed greatly to keep the population down. It is openly and unblushingly pursued, the principal victims being the females. The chief reasons for it are usually—revenge in the woman against her husband’s neglect, poverty, dread of shame, and superstition. One of the most common causes is the wife’s belief that her husband cares no longer for his offspring. The priests, whose low cunning is as characteristic of the class in those islands as elsewhere, frequently demand a victim for an oblation of blood to the spirit of evil, and never fail to extort the sacrifice from some poor ignorant mother. Another injurious and unnatural practice is, that of checking or neutralizing the operations of nature by procuring abortion. Tyrone Power, in his observations on the immorality prevalent in New Zealand, remarks that some of the young girls, betrothed from an early age, are tapu, and thus preserved chaste. He regrets that this superstition is not more influential, since it would check the system of almost universal and indiscriminate prostitution, which prevails among those not subject to this rite. Except when the woman is tapu, her profligacy is neither punished nor censured. Fathers, mothers, and brothers will, without a blush, give, sell, or lend on hire, the persons of their female relatives. The women themselves willingly acknowledge the bargain, and Mr. Power declares the most modest of them will succumb to a liberal offer of money. Nor is anything else to be expected, in any general degree. The children are educated to obscenity and vice. Their intercourse is scarcely restrained, and the early age at which it takes place has proved physically injurious to the race. Even those who are betrothed in infancy and rendered tapu to each other, commence cohabitation before they have emerged, according to English ideas, from childhood. Except in the case of those couples thus pledged before they can make a choice of their own, the laws which in New Zealand regulate the intercourse of the sexes with regard to preparations for marriage, approach in spirit to our own. A man desiring to take as wife a woman who is bound by no betrothment has to court her, and sometimes does so with supplication. The girls exhibit great coyness of manner, and are particular in hiding their faces from the stranger’s eye. When they bathe it
- 33. is in a secluded spot; but they exercise all the arts which attract the opposite sex. When one or two suitors woo an independent woman, the choice is naturally given to the wealthiest; but should she decline to fix her preference on either, a desperate feud occurs, and she is won by force of arms. Sometimes a young girl is seized by two rivals, who pull on either side until her arms are loosened in the sockets, and one gives way. Perhaps, under these circumstances, the system of betrothal is productive of useful results, since it prevents the feuds and conflicts which might otherwise spring from the rivalry of suitors. The girl thus bound must submit to marriage with the man, whatever may be her indifference or aversion to him. Occasionally, indeed, some more youthful, or otherwise attractive, lover gains her consent to an elopement. If caught, however, both of the culprits are severely whipped. Should the young suitor be of poor and mean condition, he runs the chance of being robbed and murdered for his audacity. When, on the contrary, a powerful chief is desirous of obtaining a maiden who is betrothed, he has little difficulty in effecting his object, for in New Zealand the liberty of the individual is proportionate to his strength. It is a feudal system, where the strong may evade the regulations of the social law, and the weak must submit. Justice, however, to the missionaries in those islands requires us to add, that in the districts where their influence is strong, a beneficial change in this, as in other respects, has been produced upon the people. They acknowledge more readily the supremacy of law; they prefer a judicial tribunal to the trial of arms; they restrain their animal passions in obedience to the moral code which has been exhibited to them; and many old polygamists have put away all their wives but one, contented to live faithfully with her. Among the heathen population chastity is not viewed in the same light as with us. It not so much required from the woman as from the wife, from the young girl as from the betrothed maiden. In fact, it signifies little more than faithful conduct in marriage, not for the sake of honour or virtue, but for that of the husband. With such a social theory, we can expect no general refinement in morality. Indeed, the term is not translatable into the language of New Zealand. Modesty is a fashion, not a sentiment, with them. The woman who would retire from the stranger’s gaze may, previous to marriage or betrothal, intrigue with any man without incurring an infamous reputation. Prostitution is not only a common but a recognised thing. Men care little to receive virgins into their huts as wives. Husbands have boasted that their wives had been the concubines of Europeans; and one declared to Polack that he was married to a woman who had regularly followed the calling of a prostitute among the crews of ships in the harbour. This he mentioned with no inconsiderable pride, as a proof of the beauty of the prize he had carried away. Formerly many of the chiefs dwelling on the coast were known to derive a part of their revenue from the prostitution of young females. It was, indeed, converted into a regular trade, and to a great extent with the European ships visiting the group. The handsomest and plumpest women in the villages were chosen, and
- 34. bartered for certain sums of money or articles of merchandise, some for a longer, some for a shorter period. The practice is now, if not abolished, at least held in great reprobation, as the following anecdote will show. It exhibits the depraved manners of the people in a striking light, and is an illustration of that want of affection between married people which has been remarked as a characteristic of the New Zealanders. A chief from Wallatani, in the Bay of Plenty, went on an excursion to the Bay of Islands, and was accompanied by his wife and her sister. There he met a chief of the neighbourhood, who possessed some merchandise which he coveted. He at once offered to barter the chastity of his wife for the goods, and the proposal was accepted. The woman told her sister of the transaction, and she divulged the secret. So much reproach was brought upon the chief among his people, that he shot his wife’s sister to punish her incontinent tongue. Jerningham Wakefield describes the arrival of the whalers in port. He mentions as one of the most important transactions following this event, the providing of the company with “wives for the season.” Some had their regular helpmates, but others were forced to hire women. Bargains were formally struck, and when a woman failed to give satisfaction, she was exchanged for another. She was at once the slave and the companion of her master. This is neither more nor less than a regular system of prostitution; but it is gradually going out of fashion, and is only carried on in a clandestine manner in the colonies properly so called. Indeed this is, unfortunately, one of the chief products of imperfect civilization—that vice, which before was open, is driven into the dark; it is not extirpated, but is concealed. A man offered his wife to the traveller Earl, and the woman was by no means loth to prostitute herself for a donation. Barbarians readily acquire the modes of vice practised by Europeans. In the criminal calendar of Wellington for 1846, we find one native convicted and punished for keeping a house of ill-fame. Extraordinary as it may appear, prostitution in New Zealand has tended to cure one great evil. It has largely checked the practice of infanticide. For, as the female children were usually destroyed, it was on the supposition that, instead of being valuable, they would be burdensome to their parents. This continued to be the case until the discovery was made that by prostituting the young girls considerable profits might be made. It is to Europeans that the introduction of this idea is chiefly owing. The females were then, in many cases, carefully reared, and brought up to this dishonourable calling without reluctance. No difficulty was ever experienced from their resistance, as they would probably have become prostitutes of their own free will, had they not been directed to the occupation. Slavery, which has from the earliest time existed in New Zealand, has supplied the materials of prostitution, female servants being consigned to it. When possessed of any attractions they are almost invariably debauched by their masters, and frequently suffer nameless punishments from the jealous head wife. Concubinage does not, as in some other
- 35. countries, release a woman from servitude, but she enjoys a privilege which is denied to the chief wife—she may marry again after her master’s death. Formerly the general custom, however, was for a wife to hang, drown, strangle, or starve herself on the death of her husband. Her relatives often gave her a rope of flax, with which she retired to a neighbouring thicket and died. It was not a peremptory obligation, but custom viewed it as almost a sacred duty. Sometimes three of the wives destroyed themselves, but generally one victim sufficed. Self- immolation is now, indeed, becoming very rare; but it is still the practice for the widow, whether she loved her husband or not, to lament him with loud cries, and lacerate her flesh upon his tomb. Whenever she marries again a priest is consulted to predict whether she will survive the second husband or not. Occasionally we find instances of real attachment between man and wife, such as would sanctify any family hearth; while examples have occurred of women hanging themselves for sorrow, on the death of a betrothed lover. These, however, are only indications that humanity is not in New Zealand universally debased below the brute condition. The general colour of the picture is dark. Women are degraded; men are profligate; virtue is unknown in its abstract sense; chastity is rare; and prostitution a characteristic of female society. Fathers, mothers, and brothers—usually the guardians of a young woman—prostitute her for gain, and the women themselves delight in this vice. There is, nevertheless, some amelioration observable in the manners of the people, produced by the influence of the English colonies. Those colonies themselves, however, are not free from the stain, as will be shown when we treat of communities of that description in general[59]. Of Prostitution in the Islands of the Pacific. Among the innumerable islands which are scattered over the surface of the Pacific, we discover various phases of manners developed under different influences. In some of the lonely groups lying out of the usual course of trade or travel, communities exist whose social habits remain entirely pure—that is, unchanged by intercourse with foreigners. In others continual communication through a long period, with white men, has wholly changed the characteristic aspects of the people—given them a new religion, a new moral code, new ideas of decency and virtue, new pleasures, and new modes of life. The same process appears likely, at a future day, to obliterate the ancient system of things. In all the islands of this class, indeed, the reform of manners is not so thorough as the florid accounts of the missionaries would induce us to believe; but those pioneers of civilization have done enough, without assuming more than their due, to deserve the praise of all Christendom. To have restrained the fiercest passions of human nature among
- 36. ignorant and wilful savages; to have converted base libidinous heathens into decent Christians; to have checked the practice of polygamy; and in many places to have extinguished the crime of infanticide;—these are achievements which entitle the missionaries to the applause and respect of Europe; but it is no disparagement of their labours to show, where it is true, that immense things yet remain to be performed before the islanders of the Pacific are raised to the ordinary level of civilized humanity. The main family of the Pacific—the Society, the Friendly, the Sandwich, the Navigators’, and the Marquesas Islands—present a state of society interesting and curious. Inhabiting one of the most beautiful regions on the face of the earth, with every natural advantage, the inhabitants of those groups were originally among the most degraded of mankind. Superior to the savage hordes of Africa and the wandering tribes of Australia, they are in physical and intellectual qualities inferior to the natives of New Zealand, though excelling them in simplicity and willingness to learn. Tahiti may be considered the capital of Polynesia, as it is the head of its politics, trade, and general civilization. Before the settlement of the missionaries and the introduction of a new social scheme, its manners were barbarous and disgusting. The condition of the female sex corresponded to this order of things. It was humiliated to the last degree. Most of the men, by a sacred rite, were rendered too holy for any intercourse with the women except such as was pleasant to their own lusts. It was similar to the tapu of the New Zealanders, but was not, as among them, common to all. It was an exclusive privilege of the males. In consequence of this, women lived in a condition of exile from all the pleasures of life. They never sat at meals with their husbands, dared not eat the flesh of pigs, of fowls, of certain fish, or touch the utensils used by the men. They never entered the houses of their “tabooed” lords, dwelling in separate habitations, which these might enter when they chose. Those of the royal blood, however, were excepted from the action of this law. They might mingle with the other sex, might inherit the throne, and enjoy the advantages of society. With almost all others, beggary, toil, and degradation was the universal lot. Marriage under such circumstances could not be looked upon as a sacred tie, or even a dignified state. It was held to serve only the purposes of nature and the pleasures of the men. With all, indeed, except the rich, it was a mere unceremonious bargain, in which the woman was purchased, though the parents usually made a present to their son-in-law. Among the nobler orders of society there was a little more parade, though an equal absence of sanctity. A person with a beautiful daughter brought her to some chief, saying, “Here is a wife for you.” If she pleased him he took her from her father’s hands, placed her under the care of a confidential servant, and had her fattened, until old and plump enough for marriage. All her friends assembled with his at the temple, and proceeded to the altar. The bride, with a rope hanging about her neck, was accompanied by a man
- 37. bearing a bunch of the fragrant fern. Prayers were muttered, and blessing invoked upon the union. Then the names of their ancestors were whispered, and at each one of the leaves was torn. The nearest kinsman of the woman next loosened the rope from about her neck, and delivered her over to the bridegroom, bidding him take her home. Presents of various kinds were made to the newly-married pair, but, with all this ceremony, the tie was merely one of convenience. Within a month the man might tire of his partner and wish to be rid of her. All he had to do was to desire her departure, saying, “It is enough—go away.” She immediately left him, and almost invariably became a prostitute. This process might be repeated as often as he pleased. The caprice of the male sex thus threw numbers of the females into a necessity of supporting themselves by the public hire of their persons. For, although polygamy existed, it was practised only by the rich, since the facility of divorce rendered it more convenient to take one wife, dwell with her a short time, and abandon her for another, than to be troubled or burdened with several at the same time. The wealthy, however, took numerous concubines—indulging in this luxury more than any of the other islanders. In all their customs and national characteristics, if we desire to view them in their original form, we must contemplate the people of those islands as they were twenty years ago. A great change is now apparent among them. The accounts, therefore, published at that period, though improved by later inquiries, afford us the information we are in search of. We are not surprised to find an indolent licentious people, as they were, when under no restraint, addicted to the most odious forms of vice. One natural result of their manner of life was infanticide. It was practised to a frightful extent, and was encouraged by a variety of causes. In the first place, poverty and idleness often induced parents to destroy their children—choosing to suffer that short pang of natural sorrow than the long struggles with starvation which awaited the indigent—even in those prolific islands. Next the common licentiousness produced innumerable bastards, which were generally killed. Thirdly, the social institutions of the country, with the division of classes, contributed to increase the prevalence of the custom—for the fruit of all unequal matches was cast aside. Superstition also aided it, for the priests demanded for their gods frequent oblations of infant blood. The missionary Williams was informed that, from the constant occurrence of wars, women, being abandoned by their husbands, slew their children, whom they knew not how to support. When a man married a girl of inferior rank, two, four, or six of her children were sacrificed before she could claim equality with him, and should she bear any more they were spared. Vanity, too, exercised its influence, for, as nursing impaired the beauty of the women, they sought to preserve their attractions by sparing themselves the labour. Perhaps, however, we should not lay it to the charge of vanity. The miserable women of these islands found in the flower of their persons the only chance of attachment or respect from their husbands. When this had faded, nothing could save them from neglect.
- 38. Whatever the cause, the extent of the practice was fearful. Three-fourths of the children were destroyed, and sometimes in the most atrocious manner. A wet cloth placed on the infant’s mouth, the hands clenched round its throat, or the earth heaped over it while alive in a grave, were among the most humane. Others broke the infant’s joints, one by one, until it expired. This was usually the plan of the professional child-killers, of whom there was a class—male and female—though the parents often performed the office themselves. Before the establishment of Christianity, Williams declares he never conversed with a woman who had not destroyed one or two of her offspring. Many confessed to him, as well as to Wilmer, that they had killed, some three, some five, some nine, and one seventeen. Connected with infanticide was one of the most extraordinary institutions ever established in a savage or a civilized country. This was the Areoi Society. It was at once the source of their greatest amusements and their greatest sorrow, and was strictly confined to the Society group, though indications of a similar thing have been discovered in the Ladrones. The delicacy of the missionary writers—in many instances extremely absurd—has induced them to neglect informing us in detail of the practices and regulations adopted by this society; but enough is known from them, and from less timid narrators, to allow of a tolerably full sketch. From the traditions of the people it appears that the society was of very ancient date: they said there had been Areois as long as there had been men. Its origin is traced to two heroes—brothers, who, in consequence of some adventures with the gods, were deified, and made kings of the Areoi, which included all who would adhere to them as their lords in heaven. Living in celibacy themselves, they did not enjoin the same on their followers; but required that they should leave no descendants. Thus the great law of the Areois was that all their children should be slain. What the real origin of the institution was it is impossible to discover. This legend, however, indicates a part of its nature. The Areois formed a body of privileged libertines, who spent their days travelling from province to province, from island to island, exhibiting a kind of licentious dramatic spectacle to the people, and everywhere indulging the grossest of their passions. The company located itself in a particular spot as its head-quarters, and at certain seasons departed on an excursion through the group. Great parade was made on the occasion of their setting out. They bore with them portable temples for the worship of their tutelar gods, and, wherever they halted, performed their pantomimes for the amusement of the people. The priests and others—all classes and things—were ridiculed by them in their speeches, with entire impunity, and they were entertained by the chiefs with sumptuous feasts. There were, however, seven classes of the Areois, of which the first was select and small, while the seventh performed the lower and more laborious parts in their entertainments. Numbers of servants followed them to prepare their food and their dresses, and
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