SlideShare a Scribd company logo
Separation of Variables for Partial Differential
Equations An Eigenfunction Approach 1st Edition
George Cain download pdf
https://ebookultra.com/download/separation-of-variables-for-partial-
differential-equations-an-eigenfunction-approach-1st-edition-george-
cain/
Visit ebookultra.com today to download the complete set of
ebook or textbook!
We believe these products will be a great fit for you. Click
the link to download now, or visit ebookultra.com
to discover even more!
Partial differential equations An introduction 2nd Edition
Strauss W.A.
https://ebookultra.com/download/partial-differential-equations-an-
introduction-2nd-edition-strauss-w-a/
Handbook of Differential Equations Stationary Partial
Differential Equations Volume 6 1st Edition Michel Chipot
https://ebookultra.com/download/handbook-of-differential-equations-
stationary-partial-differential-equations-volume-6-1st-edition-michel-
chipot/
Partial Differential Equations A Unified Hilbert Space
Approach 1st Edition Rainer Picard
https://ebookultra.com/download/partial-differential-equations-a-
unified-hilbert-space-approach-1st-edition-rainer-picard/
Applied Partial Differential Equations John Ockendon
https://ebookultra.com/download/applied-partial-differential-
equations-john-ockendon/
Stochastic Partial Differential Equations 2nd Edition Chow
https://ebookultra.com/download/stochastic-partial-differential-
equations-2nd-edition-chow/
Solution techniques for elementary partial differential
equations Third Edition Constanda
https://ebookultra.com/download/solution-techniques-for-elementary-
partial-differential-equations-third-edition-constanda/
Introduction to Partial Differential Equations Rao K.S
https://ebookultra.com/download/introduction-to-partial-differential-
equations-rao-k-s/
Control theory of partial differential equations 1st
Edition Guenter Leugering
https://ebookultra.com/download/control-theory-of-partial-
differential-equations-1st-edition-guenter-leugering/
Partial Differential Equations 2nd Edition Lawrence C.
Evans
https://ebookultra.com/download/partial-differential-equations-2nd-
edition-lawrence-c-evans/
Separation of Variables for Partial Differential Equations An Eigenfunction Approach 1st Edition George Cain
Separation of Variables for Partial Differential Equations
An Eigenfunction Approach 1st Edition George Cain
Digital Instant Download
Author(s): George Cain, Gunter H. Meyer
ISBN(s): 9781584884200, 1584884207
Edition: 1
File Details: PDF, 9.72 MB
Year: 2005
Language: english
Separation
of Variables
for Partial
Differential
Equations
An Eigenfunction
Approach
STUDIES IN ADVANCED MATHEMATICS
Separation
of Variables
for Partial
Differential
Equations
An Eigenfunction
Approach
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
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
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
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.
Separation of Variables for Partial Differential Equations An Eigenfunction Approach 1st Edition George Cain
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
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
PREFACE ix
complication, regardless of the type of equation under consideration. A concise
algorithmic approach results.
Equally important to us is the second reason for a new exposition of the
method of separation of variables. We wish to emphasize the power of the
method by solving a great variety of problems which often go well beyond the
usual textbook examples. Many of the applications ask questions which are
not as easily resolved with numerical methods as with analytic approximate
solutions. Of course, evaluation of these approximate solutions usually relies on
numerical methods to integrate, solve linear systems or nonlinear equations, and
to find values of special functions, but these methods by now may be considered
universally available "black boxes." We are, however, mindful of the gap between
the concept of a solution in principle and a demonstrably computable solution
and try .to convey our experience with how well the eigenfunction approach
actually solves the sample problems.
The method of separation of variables from a spectral expansion view is
presented in nine chapters.
Chapter 1 collects some background information on the three dominant equa-
tions of this text, the potential equation, the heat equation, and the wave equa-
tion. We refer to these results when applying and analyzing the method of
separation of variables.
Chapter 2 contains a discussion of orthogonal projections which are used time
and again to approximate given data functions in a specified finite-dimensional
but parameter-dependent subspace.
Chapter 3 introduces the subspace whose basis consists of the eigenfunctions
of a so-called Sturm-Liouville problem associated with the application under
consideration. These are the eigenfunctions of the title of this text. We cite
results from the Sturm-Liouville theory and provide a table of eigenvalues and
eigenfunctions that arise in the method of separation of variables.
Chapter 4 treats the case in which the eigenfunctions are sine and cosine
functions with a common period. In this case the projection into the subspace is
closely related to the Fourier series representation of the data functions. Precise
information about the convergence of the Fourier series is known. We cite those
results which are helpful later on for the application of separation of variables.
Chapter 5 constitutes the heart of the text. We consider a partial differen-
tial equation in two independent variables with a source term and subject to
boundary and initial conditions. We give the algorithm for approximating such
a problem and for solving it in a finite-dimensional space spanned by eigen-
functions determined by the "spacial part" of the equation and its boundary
conditions. We illustrate in broad outline the application of this approach to
the heat, wave, and potential equations.
Chapter 6 gives an expansive exposition of the algorithm for the one-dimen-
sional heat equation. It contains many worked examples with comments on
the numerical performance of the method, and concludes with a rudimentary
analysis of the error in the approximate solution.
Chapter 7 parallels the previous chapter but treats the wave equation.
x PREF...CE
Chapter 8 deals with the potential equation. It describes how one can pr~
condition the data of problems with smooth solutions in order not to introdi.:ce
artificial discontinuities into the separation of variables solution. We solve P'=-
tential problems with various boundary conditions and conclude with a calcu-
lation of eigenfunctions for the two-dimensional Laplacian.
Chapter 9 uses the eigenfunctions of the preceding chapter to find eigenfunc-
tion expansion solutions of two- and three-dimensional heat, wave, and potenti.a::
equations.
This text is written for advanced undergraduate and graduate students :.::.
science and engineering with previous exposure to a course in engineering math-
ematics, but not necessarily separation of variables. Basic prerequisites beyond
calculus are familiarity with linear algebra, the concept of vector spaces of func-
tions, norms and inner products, the ability to solve linear inhomogeneous first
and second order ordinary differential equations, and some contact with practi-
cal applications of partial differential equations.
The book contains more material than can (and should) be taught in a cour~
on separation of variables. We have introduced the eigenfunction approach to
our own students based on an early version of this text. We covered parts of
Chapters 2-4 to lay the groundwork for an extensive discussion of Chapter 5.
The remainder of the term was filled by working through selected examples
involving the heat, wave, and potential equation. We believe that by term·s
end the students had an appreciation that they could solve realistic problems.
Since we view Chapters 2-5 as suitable for teaching separation of variables,
we have included exercises to help deepen the reader's understanding of the
eigenfunction approach. The examples of Chapters 6-8 and their exercise sets
generally lend themselves for project assignments.
This text will put a bigger burden on the instructor to choose topics and
guide students than more elementary texts on separation of variables that start
with product solutions. The instructor who subscribes to the view put forth
in Chapter 5 should find this text workable. The more advanced applications,
such as interface, inverse, and multidimensional problems, as well as the the
more theoretical topics require more mathematical sophistication and may be
skipped without breaking continuity.
The book is also meant to serve as a reference text for the method of separa-
tion of variables. We hope the many examples will guide the reader in deciding
whether and how to apply the method to any given problem. The examples
should help in interpreting computed solutions, and should give insight into
those cases in which formal answers are useless because of lack of convergence
or unacceptable oscillations. Chapters 1 and 9 are included to support the
reference function. They do not include exercises.
We hasten to add that this text is not a complete reference book. We
do not attempt to characterize the equations and coordinate systems where
a separation of variables is applicable. We do not even mention the various
coordinate systems (beyond cartesian, polar, cylindrical, and spherical) in which
the Laplacian is separable. We _have not scoured the literature for new and
innovative applications of separation of variables. Moreover, the examples we do
PREFACE xi
include are often meant to show structure rather than represent reality because
in general little attention is given to the proper scaling of the equations.
There does not appear to exist any other source that could serve as a prac-
tical reference book for the practicing engineer or scientist. We hope this book
will alert the reader that separation of variables has more to offer than may be
apparent from elementary texts.
Finally, this text does not mention the implementation of our formulas and
calculations on the computer, or do we provide numerical algorithms or pro-
grams. Yet the text, and in particular our numerical examples, could not
have been presented without access to symbolic and numerical packages such as
!viaple, Mathematica, and Matlab. We consider our calculations and the graph-
ical representation of their results routine and well within the competence of
today's students and practitioners of science and engineering.
Separation of Variables for Partial Differential Equations An Eigenfunction Approach 1st Edition George Cain
Contents
Acknowledgments v
Preface vii
1 Potential, Heat, and Wave Equation 1
1.1 Overview . . . . . . . . . . . . . .
1.2 Classification of second order equations 2
1.3 Laplace's and Poisson's equation 3
1.4 The heat equation 11
1.5 The wave equation . 18
2 Basic Approximation Theory 25
2.1 Norms and inner products . . 26
2.2 Projection and best approximation 29
2.3 Important function spaces 34
3 Sturm-Liouville Problems 45
3.1 Sturm-Liouville problems for q/' = µ¢ 45
3.2 Sturm-Liouville problems for £¢ = µ¢ . . . 53
3.3 A Sturm-Liouville problem with an interface 59
4 Fourier Series 67
4.1 Introduction . 67
4.2 Convergence . 68
4.3 Convergence of Fourier series 74
4.4 Cosine and sine series . . . . 77
4.5 Operations on Fourier series . 80
4.6 Partial sums of the Fourier series and the Gibbs phenomenon 84
5 Eigenfunction Expansions for Equations in Two Independent
Variables 95
XiY
6
CONTENTS
One-Dimensional Diffusion Equation
6.1 Applications of the eigenfunction expansion method
Example 6.1 How many terms of the series solution are enough?
Example 6.2 Determination of an unknown diffusivity from mea-
sured data ..
Example 6.3 Thermal waves
Example 6.4 Matching a temperature history
Example 6.5 Phase shift for a thermal wave .
Example 6.6 Dynamic determination of a convective heat transfer
coefficient from measured data . . . . . . . . . . . .
Example 6.7 Radial heat flow in a sphere
Example 6.8 A boundary layer problem .
Example 6.9 The Black-Scholes equation
Example 6.10 Radial heat flow in a disk .
Example 6.11 Heat flow in a composite slab
Example 6.12 Reaction-diffusion with blowup
6.2 Convergence of UN(x, t) to the analytic solution :
6.3 Influence _of the boundary conditions and Duhamel's solution
115
115
115
119
120.
125
129
131
134
137
139
142
146
148
151
155
7 One-Dimensional Wave Equation 161
7.1 Applicatfons of the eigenfunction expansion method - : 161
Example 7.1 A vibrating string with initial displacement . 161
Example 7.2 A vibrating string with initial velocity . . 166
Example 7.3 A forced wave and resonance . 168
Example 7.4 Wave propagation in a resistive medium. 171
Example 7.5 Oscillations of a hanging chain 175
Example 7.6 Symmetric pressure wave in a sphere 177
Example 7.7 Controlling the shape of a wave . . . 180
Example 7.8 The natural frequencies. of a uniform beam 182
Example 7.9 A system of wave equations . . . . . 185
7.2 Convergence of UN(x, t) to the analytic solution . 188
7.3 Eigenfunction expansions and Duhamel's
principle . 190
8 Potential Problems in the Plane 195
8.1 Applications of the eigenfunction expansion method . 195
Example 8.1 The Dirichlet problem for the Laplacian on a rectanglel95
Example 8.2 Preconditioning for general boundary data . . 201
Example 8.3 Poisson's equation with Neumann boundary data 213
Example 8.4 A discontinuous potential 215
Example 8.5 Lubrication of a plane slider bearing . 218
Example 8.6 Lubrication of a step bearing . . 220
Example 8.7 The Dirichlet problem on an £-shaped domain 221
Example 8.8 Poisson's equation in polar coordinates . . . . 225
Example 8.9 Steady-state heat flow around an insulated pipe I 230
Example 8.10 Steady-state heat flow around an insulated pipe II 232
CONTENTS xv
Example 8.11 Poisson's equation on a triangle . 234
8.2 Eigenvalue problem for the two-dimensional Laplacian 237
Example 8.12 The eigenvalue problem for the Laplacian on a
rectangle . . . . . . . . . . 237
Example 8.13 The Green's function for the Laplacian on a square 239
Example 8.14 The eigenvalue problem for the Laplacian on a disk 243
Example 8.15 The .eigenvalue problem for the Laplacian on the
surface of a sphere . . . . . . . . . 244
8.3 Convergence of UN(x, y) to the analytic solution 247
9 Multidimensional Problems 255
9.1 Applications of the eigenfunction expansion method 255
Example 9.1 A diffusive pulse test . . . . . . . 255
Example 9.2 Standing waves on a circular membrane 258
Example 9.3 The potential inside a charged sphere 260
Example 9.4 Pressure in a porous slider bearing . . . 261
9.2 The eigenvalue problem for the Laplacian in ffi:.3 . . . 265
Example 9.5 An eigenvalue problem for quadrilaterals 265
Example 9.6 An eigenvalue problem for the Laplacian in a cylinder266
Example 9.7 Periodic heat flow in a cylinder 267
Example 9.8 An eigenvalue problem for the Laplacian in a sphere 269
Example 9.9 The eigenvalue problem for Schrodinger's equation
with a spherically symmetric potential well . . . . 271
Bibliography 277
Index 279
Separation of Variables for Partial Differential Equations An Eigenfunction Approach 1st Edition George Cain
Chapter 1
Potential, Heat, and Wave
Equation
This chapter provides a quick look into the vast field of partial differential
equations. The main goal is to extract some qualitative results on the three
dominant equations of mathematical physics, the potential, heat, and wave
equation on which our attention will be focused throughout this text.
1.1 Overview
When processes that change smoothly with two or more independent variables
are modeled mathematically, then partial differential equations arise. Most
common are second order equations of the general form
M M
Lu= L aijUxix, + L biuxi +cu = F, xE DC IRM (1.1)
i,j=l i=l
where the coefficients and the source term may depend on the independent vari-
ables {x1, ... ,xM}, on u, and on its derivatives. D is a given set in !RM (whose
boundary will be denoted by [)D). The equation may reflect conservation and
balance laws, empirical relationships, or may be purely phenomenological. Its
solution is used to explain, predict, and control processes in a bewildering array
of applications ranging from heat, mass, and fluid flow, migration of biological
species, electrostatics, and molecular vibration to mortgage banking.
In (1.1) C. is known as a partial differential operator that maps a smooth
function u to the function C.u. Throughout this text a smooth function denotes
a function with as many continuous derivatives as are necessary to carry out
the operations to which it is subjected. lu = F is the equation to be solved.
Given a partial differential equation and side constraints on its solution,
typically initial and boundary conditions, it becomes a question of mathematical
analysis to establish whether the problem has a solution, whether the solution
2 CHAPTER 1. POTENTIAL, HEAT, AND WAVE EQUATION
is unique, and whether the solution changes continuously with the data of the
problem. If that is the case, then the given problem for (1.1) is said to be well
posed; if not then it is ill posed. We note here that the data of the problem are
the coefficients of£, the source term F, any side conditions imposed on u, and
the shape of D. However, dependence on the coefficients and on the shape of D
will be ignored. Only continuous dependence with respect to the source term
and the side conditions will define well posedness for our purposes.
The technical aspects of in what sense a function u solves the problem and
in what sense it changes with the data of the problem tend to be abstract and
complex and constitute the mathematical theory of partial differential equations
(e.g., [5]). Such theoretical studies are essential to establish that equation (1.1)
and its side conditions are a consistent description of the processes under consid-
eration and to characterize the behavior of its solution. Outside mathematics
the validity of a mathematical model is often taken on faith and its solution
is assumed to exist on "physical grounds." There the emphasis is entirely on
solving the equation, analytically if possible, or approximately and numerically
otherwise. Approximate solutions are the subject of this text.
1.2 Classification of second order equations
The tools for the analysis and solution of (1.1) depend on the structure of the
coefficient matrix
A= {aij}
in (1.1). By assuming that ux,x1 = ux1x, we can always write A in such a way
that it is symmetric. For example, if the equation which arises in modeling a
process is
then it will be rewritten as
so that
A= G~)
We can now introduce three broad classes of differential equations.
Definition The operator £ given by
M M
Lu =L aijUxtx) + L biuXi +cu
i,j=l i::::l
is
i) Elliptic at x= (x1 , ••. , XM) if all eigenvalues of the symmetric matrix A are
nonzero and have the same algebraic sign,
1.3. LAPLACE'S AND POISSON'S EQUATION 3
ii) Hyperbolic at i if all eigenvalues of A are nonzero and one has a different
algebraic sign from all others,
ili) Parabolic at x if A has a zero eigenvalue.
If A depends on u and its derivatives, then £ is elliptic, etc. at a given point
relative to a specific function u. If the operator£ is elliptic at a point then (1.1)
is an elliptic equation at that point. (As mnemonic we note that for M = 2 the
level sets of
AG~),(~~)) =constant,
where (i, if') denotes the dot product of x and y, are elliptic, hyperbolic, and
parabolic under the above conditions on the eigenvalues of A). The lower order
~erms in (1.1) do not affect the type of the equation, but in particular applica-
tions they can dominate the behavior of the solution of (1.1).
Each class of equations has its own admissible side conditions to make (Ll)
well posed, and all solutions of the same class have, broadly speaking, common
characteristics. We shall list some of them for the three dominant equations
of mathematical physics: Laplace's equation, the heat equation, and the wave
equation.
1.3 Laplace's and Poisson's equation
The most extensively studied example of an elliptic equation is Laplace's equa-
~ion
£u =1 ·Ju= 0
which arises in potential problems, steady-state heat conduction, irrotational
flow, minimal surface problems, and myriad other applications. The operator
£u is known as the Laplacian and is generally denoted by
£u = 1 ·Ju= 72 u = 6.u.
The last form is common in the mathematical literature and will be used con-
sistently throughout this text. The Laplacian in cartesian coordinates
assumes the forms
i.i In polar coordinates (r, ())
M
~u. = LUxixi
i=l
ii) In cylindrical coordinates (r, (), z)
Random documents with unrelated
content Scribd suggests to you:
evil282—save insofar as evil may be a beneficent penalty and discipline. At
the same time, while advising the imprisonment or execution of heretics
who did not believe in the Gods, Plato regarded with even greater
detestation the man who taught that they could be persuaded or propitiated
by individual prayer and sacrifice.283 Thus he would have struck alike at
the freethinking few and at the multitude who held by the general religious
beliefs of Greece, dealing damnation on all save his own clique, in a way
that would have made Torquemada blench.284 In the face of such teaching
as this, it may well be said that “Greek philosophy made incomparably
greater advances in the earlier polemic period [of the Ionians] than after its
friendly return to the poetry of Homer and Hesiod”285—that is, to their
polytheistic basis. It is to be said for Plato, finally, that his embitterment at
the downward course of things in Athens is a quite intelligible source for his
own intellectual decadence: a very similar spectacle being seen in the case
of our own great modern Utopist, Sir Thomas More. But Plato’s own
writing bears witness that among the unbelievers against whom he
declaimed there were wise and blameless citizens;286 while in the act of
seeking to lay a religious basis for a good society he admitted the
fundamental immorality of the religious basis of the whole of past Greek
life.
3. Aristotle [384–322], like Sokrates, albeit in a very different way,
rendered rather an indirect than a direct service to Freethought. Where
Sokrates gave the critical or dialectic method or habit, “a process of eternal
value and of universal application,”287 Aristotle supplied the great
inspiration of system, partly correcting the Sokratic dogmatism on the
possibilities of science by endless observation and speculation, though
himself falling into scientific dogmatism only too often. That he was an
unbeliever in the popular and Platonic religion is clear. Apart from the
general rationalistic tenor of his works,288 there was a current
understanding that the Peripatetic school denied the utility of prayer and
sacrifice;289 and though the essentially partisan attempt of the anti-
Macedonian party to impeach him for impiety may have turned largely on
his hyperbolic hymn to his dead friend Hermeias (who was a eunuch, and as
such held peculiarly unworthy of being addressed as on a level with semi-
divine heroes),290 it could hardly have been undertaken at all unless he had
given solider pretexts. The threatened prosecution he avoided by leaving the
city, dying shortly afterwards. Siding as he did with the Macedonian
faction, he had put himself out of touch with the democratic instincts of the
Athenians, and so doubly failed to affect their thinking. But nonetheless the
attack upon him by the democrats was a political stratagem. The
prosecution for blasphemy had now become a recognized weapon in
politics for all who had more piety than principle, and perhaps for some
who had neither. And Aristotle, well aware of the temper of the population
around him, had on the whole been so guarded in his utterance that a
fantastic pretext had to be fastened on for his undoing.
Prof. Bain (Practical Essays, p. 273), citing Grote’s remark on the “cautious prose
compositions of Aristotle,” comments thus: “That is to say, the execution of Sokrates was
always before his eyes; he had to pare his expressions so as not to give offence to Athenian
orthodoxy. We can never know the full bearings of such a disturbing force. The editors of
Aristotle complain of the corruption of his text: a far worse corruption lies behind. In
Greece Sokrates alone had the courage of his opinions. While his views as to a future life,
for example, are plain and frank, the real opinion of Aristotle on the question is an
insoluble problem.” (See, however, the passage in the Metaphysics cited below.)
The opinion of Grote and Bain as to Aristotle’s caution is fully coincided in by Lange, who
writes (Gesch. des Mater. i, 63): “More conservative than Plato and Sokrates, Aristotle
everywhere seeks to attach himself as closely as possible to tradition, to popular notions, to
the ideas embodied in common speech, and his ethical postulates diverge as little as may be
from the customary morals and laws of Greek States. He has therefore been at all times the
favourite philosopher of conservative schools and movements.”
It is clear, nevertheless, if we can be sure of his writings, that he was a
monotheist, but a monotheist with no practical religion. “Excluding such a
thing as divine interference with Nature, his theology, of course, excludes
the possibility of revelation, inspiration, miracles, and grace.”291 In a
passage in the Metaphysics, after elaborating his monistic conception of
Nature, he dismisses in one or two terse sentences the whole current
religion as a mass of myth framed to persuade the multitude, in the interest
of law and order.292 His influence must thus have been to some extent, at
least, favourable to rational science, though unhappily his own science is
too often a blundering reaction against the surmises of earlier thinkers with
a greater gift of intuition than he, who was rather a methodizer than a
discoverer.293 What was worst in his thinking was its tendency to apriorism,
which made it in a later age so adaptable to the purposes of the Roman
Catholic Church. Thus his doctrines of the absolute levity of fire and of
nature’s abhorrence of a vacuum set up a hypnotizing verbalism, and his
dictum that the earth is the centre of the universe was fatally helpful to
Christian obscurantism. For the rest, while guiltless of Plato’s fanaticism, he
had no scheme of reform whatever, and was as far as any other Greek from
the thought of raising the mass by instruction. His own science, indeed, was
not progressive, save as regards his collation of facts in biology; and his
political ideals were rather reactionary; his clear perception of the nature of
the population problem leaving him in the earlier attitude of Malthus, and
his lack of sympathetic energy making him a defender of slavery when
other men had condemned it.294 He was in some aspects the greatest brain
of the ancient world; and he left it, at the close of the great Grecian period,
without much faith in man, while positing for the modern world its vaguest
conception of Deity. Plato and Aristotle between them had reduced the
ancient God-idea to a thin abstraction. Plato would not have it that God was
the author of evil, thus leaving evil unaccounted for save by sorcery.
Aristotle’s God does nothing at all, existing merely as a potentiality of
thought. And yet upon those positions were to be founded the theisms of the
later world. Plato had not striven, and Aristotle had failed, to create an
adequate basis for thought in real science; and the world gravitated back to
religion.
[In previous editions I remarked that “the lack of fresh science, which was the proximate
cause of the stagnation of Greek thought, has been explained like other things as a result of
race qualities: ‘the Athenians,’ says Mr. Benn (The Greek Philosophers, i, 42), ‘had no
genius for natural science: none of them were ever distinguished as savans.... It was, they
thought, a miserable trifling [and] waste of time.... Pericles, indeed, thought differently....’
On the other hand, Lange decides (i, 6) “that with the freedom and boldness of the Hellenic
spirit was combined ... the talent for scientific deduction. These contrary views,” I
observed, “seem alike arbitrary. If Mr. Benn means that other Hellenes had what the
Athenians lacked, the answer is that only special social conditions could have set up such a
difference, and that it could not be innate, but must be a mere matter of usage.” Mr. Benn
has explained to me that he does not dissent from this view, and that I had not rightly
gathered his from the passage I quoted. In his later work, The Philosophy of Greece
considered in relation to the character and history of its people (1898), he has pointed out
how, in the period of Hippias and Prodikos, “at Athens in particular young men threw
themselves with ardour into the investigation of” problems of cosmography, astronomy,
meteorology, and comparative anatomy (p. 138). The hindering forces were Athenian
bigotry (pp. 113–14, 171) and the mischievous influence of Sokrates (pp. 165, 173).
Speaking broadly, we may say that the Chaldeans were forward in astronomy because their
climate favoured it to begin with, and religion and their superstitions did so later.
Hippokrates of Kos became a great physician because, with natural capacity, he had the
opportunity to compare many practices. The Athenians failed to carry on the sciences, not
because the faculty or the taste was lacking among them, but because their political and
artistic interests, for one thing, preoccupied them—e.g., Sokrates and Plato; and because,
for another, their popular religion, popularly supported, menaced the students of physics.
But the Ionians, who had savans, failed equally to progress after the Alexandrian period;
the explanation being again not stoppage of faculty, but the advent of conditions
unfavourable to the old intellectual life, which in any case, as we saw, had been first set up
by Babylonian contacts. (Compare, on the ethnological theorem of Cousin, G. Bréton,
Essai sur la poésie philos. en Grèce, p. 10.) On the other hand, Lange’s theory of gifts
“innate” in the Hellenic mind in general is the old racial fallacy. Potentialities are “innate”
in all populations, according to their culture stage, and it was their total environment that
specialized the Greeks as a community.]
§ 9
The overthrow of the “free” political life of Athens was followed by a
certain increase in intellectual activity, the result of throwing back the
remaining store of energy on the life of the mind. By this time an almost
open unbelief as to the current tales concerning the Gods would seem to
have become general among educated people, the withdrawal of the old risk
of impeachment by political factions being so far favourable to
outspokenness. It is on record that the historian Ephoros (of Cumæ in
Æolia: fl. 350 B.C.), who was a pupil of Isocrates, openly hinted in his
work at his disbelief in the oracle of Apollo, and in fabulous traditions
generally.295 In other directions there were similar signs of freethought. The
new schools of philosophy founded by Zeno the Stoic (fl. 280: d. 263 or
259) and Epicurus (341–270), whatever their defects, compare not ill with
those of Plato and Aristotle, exhibiting greater ethical sanity and sincerity if
less metaphysical subtlety. Of metaphysics there had been enough for the
age: what it needed was a rational philosophy of life. But the loss of
political freedom, although thus for a time turned to account, was fatal to
continuous progress. The first great thinkers had all been free men in a
politically free environment: the atmosphere of cowed subjection,
especially after the advent of the Romans, could not breed their like; and
originative energy of the higher order soon disappeared. Sane as was the
moral philosophy of Epicurus, and austere as was that of Zeno, they are
alike static or quietist,296 the codes of a society seeking a regulating and
sustaining principle rather than hopeful of new achievement or new truth.
And the universal skepticism of Pyrrho has the same effect of suggesting
that what is wanted is not progress, but balance. It is significant that he,
who carried the Sokratic profession of Nescience to the typical extreme of
doctrinal Nihilism, was made high-priest of his native town of Elis, and had
statues erected in his honour.297
Considered as freethinkers, all three men tell at once of the critical and of
the reactionary work done by the previous age. Pyrrho, the universal
doubter, appears to have taken for granted, with the whole of his followers,
such propositions as that some animals (not insects) are produced by
parthenogenesis, that some live in the fire, and that the legend of the
Phœnix is true.298 Such credences stood for the arrest of biological science
in the Sokratic age, with Aristotle, so often mistakenly, at work; while, on
the other hand, the Sokratic skepticism visibly motives the play of
systematic doubt on the dogmas men had learned to question. Zeno, again,
was substantially a monotheist; Epicurus, adopting but not greatly
developing the science of Demokritos,299 turned the Gods into a far-off
band of glorious spectres, untroubled by human needs, dwelling for ever in
immortal calm, neither ruling nor caring to rule the world of men.300 In
coming to this surprising compromise, Epicurus, indeed, probably did not
carry with him the whole intelligence even of his own school. His friend,
the second Metrodoros of Lampsakos, seems to have been the most
stringent of all the censors of Homer, wholly ignoring his namesake’s
attempts to clear the bard of impiety. “He even advised men not to be
ashamed to confess their utter ignorance of Homer, to the extent of not
knowing whether Hector was a Greek or a Trojan.”301 Such austerity
towards myths can hardly have been compatible with the acceptance of the
residuum of Epicurus. That, however, became the standing creed of the sect,
and a fruitful theme of derision to its opponents. Doubtless the comfort of
avoiding direct conflict with the popular beliefs had a good deal to do with
the acceptance of the doctrine.
This strange retention of the theorem of the existence of anthropomorphic
Gods, with a flat denial that they did anything in the universe, might be
termed the great peculiarity of average ancient rationalism, were it not that
what makes it at all intelligible for us is just the similar practice of modern
non-Christian theists. The Gods of antiquity were non-creative, but strivers
and meddlers and answerers of prayer; and ancient rationalism relieved
them of their striving and meddling, leaving them no active or governing
function whatever, but for the most part cherishing their phantasms. The
God of modern Christendom had been at once a creator and a governor,
ruling, meddling, punishing, rewarding, and hearing prayer; and modern
theism, unable to take the atheistic or agnostic plunge, relieves him of all
interference in things human or cosmic, but retains him as a creative
abstraction who somehow set up “law,” whether or not he made all things
out of nothing. The psychological process in the two cases seems to be the
same—an erection of æsthetic habit into a philosophic dogma, and an
accommodation of phrase to popular prejudice.
Whatever may have been the logical and psychological crudities of
Epicureanism, however, it counted for much as a deliverance of men from
superstitious fears; and nothing is more remarkable in the history of ancient
philosophy than the affectionate reverence paid to the founder’s memory302
on this score through whole centuries. The powerful Lucretius sounds his
highest note of praise in telling how this Greek had first of all men freed
human life from the crashing load of religion, daring to pass the flaming
ramparts of the world, and by his victory putting men on an equality with
heaven.303 The laughter-loving Lucian two hundred years later grows
gravely eloquent on the same theme.304 And for generations the effect of
the Epicurean check on orthodoxy is seen in the whole intellectual life of
the Greek world, already predisposed in that direction.305 The new schools
of the Cynics and the Cyrenaics had alike shown the influence in their
perfect freedom from all religious preoccupation, when they were not flatly
dissenting from the popular beliefs. Antisthenes, the founder of the former
school (fl. 400 B.C.), though a pupil of Sokrates, had been explicitly anti-
polytheistic, and an opponent of anthropomorphism.306 Aristippos of
Cyrene, also a pupil of Socrates, who a little later founded the Hedonic or
Cyrenaic sect, seems to have put theology entirely aside. One of the later
adherents of the school, Theodoros, was like Diagoras labelled “the
Atheist”307 by reason of the directness of his opposition to religion; and in
the Rome of Cicero he and Diagoras are the notorious atheists of history.308
To Theodoros, who had a large following, is attributed an influence over the
thought of Epicurus,309 who, however, took the safer position of a verbal
theism. The atheist is said to have been menaced by Athenian law in the
time of Demetrius Phalereus, who protected him; and there is even a story
that he was condemned to drink hemlock;310 but he was not of the type that
meets martyrdom, though he might go far to provoke it.311 Roaming from
court to court, he seems never to have stooped to flatter any of his
entertainers. “You seem to me,” said the steward of Lysimachos of Thrace
to him on one occasion, “to be the only man who ignores both Gods and
kings.”312
In the same age the same freethinking temper is seen in Stilpo of Megara
(fl. 307), of the school of Euclides, who is said to have been brought before
the Areopagus for the offence of saying that the Pheidian statue of Athênê
was “not a God,” and to have met the charge with the jest that she was in
reality not a God but a Goddess; whereupon he was exiled.313 The stories
told of him make it clear that he was an unbeliever, usually careful not to
betray himself. Euclides, too, with his optimistic pantheism, was clearly a
heretic; though his doctrine that evil is non-ens314 later became the creed of
some Christians. Yet another professed atheist was the witty Bion of
Borysthenes, pupil of Theodoros, of whom it is told, in a fashion familiar to
our own time, that in sickness he grew pious through fear.315 Among his
positions was a protest or rather satire against the doctrine that the Gods
punished children for the crimes of their fathers.316 In the other schools,
Speusippos (fl. 343), the nephew of Plato, leant to monotheism;317 Strato
of Lampsakos, the Peripatetic (fl. 290), called “the Naturalist,” taught sheer
pantheism, anticipating Laplace in declaring that he had no need of the
action of the Gods to account for the making of the world;318 Dikaiarchos
(fl. 326–287), another disciple of Aristotle, denied the existence of separate
souls, and the possibility of foretelling the future;319 and Aristo and
Cleanthes, disciples of Zeno, varied likewise in the direction of
pantheism; the latter’s monotheism, as expressed in his famous hymn, being
one of several doctrines ascribed to him.320
Contemporary with Epicurus and Zeno and Pyrrho, too, was Evêmeros
(Euhemerus), whose peculiar propaganda against Godism seems to imply
theoretic atheism. As an atheist he was vilified in a manner familiar to
modern ears, the Alexandrian poet Callimachus labelling him an “arrogant
old man vomiting impious books.”321 His lost work, of which only a few
extracts remain, undertook to prove that all the Gods had been simply
famous men, deified after death; the proof, however, being by way of a
fiction about old inscriptions found in an imaginary island.322 As above
noted,323 the idea may have been borrowed from skeptical Phoenicians, the
principle having already been monotheistically applied by the Bible-making
Jews,324 though, on the other hand, it had been artistically and to all
appearance uncritically acted on in the Homeric epopees. It may or may not
then have been by way of deliberate or reasoning Evêmerism that certain
early Greek and Roman deities were transformed, as we have seen, into
heroes or hetairai.325 In any case, the principle seems to have had
considerable vogue in the later Hellenistic world; but with the effect rather
of paving the way for new cults than of setting up scientific rationalism in
place of the old ones. Quite a number of writers like Palaiphatos, without
going so far as Evêmeros, sought to reduce myths to natural possibilities
and events, by way of mediating between the credulous and the
incredulous.326 Their method is mostly the naïf one revived by the Abbé
Banier in the eighteenth century of reducing marvels to verbal
misconceptions. Thus for Palaiphatos the myth of Kerberos came from the
facts that the city Trikarenos was commonly spoken of as a beautiful and
great dog; and that Geryon, who lived there, had great dogs called Kerberoi;
Actæon was “devoured by his dogs” in the sense that he neglected his
affairs and wasted his time in hunting; the Amazons were shaved men, clad
as were the women in Thrace, and so on.327 Palaiphatos and the Herakleitos
who also wrote De Incredibilibus agree that Pasiphae’s bull was a man
named Tauros; and the latter writer similarly explains that Scylla was a
beautiful hetaira with avaricious hangers-on, and that the harpies were
ladies of the same profession. If the method seems childish, it is to be
remembered that as regards the explanation of supernatural events it was
adhered to by German theologians of a century ago; and that its credulity in
incredulity is still to be seen in the current view that every narrative in the
sacred books is to be taken as necessarily standing for a fact of some kind.
One of the inferrible effects of the Evêmerist method was to facilitate for
the time the adoption of the Egyptian and eastern usage of deifying kings. It
has been plausibly argued that this practice stands not so much for
superstition as for skepticism, its opponents being precisely the orthodox
believers, and its promoters those who had learned to doubt the actuality of
the traditional Gods. Evêmerism would clinch such a tendency; and it is
noteworthy that Evêmeros lived at the court of Kassander (319–296 B.C.)
in a period in which every remaining member of the family of the deified
Alexander had perished, mostly by violence; while the contemporary
Ptolemy I of Egypt received the title of Sotêr, “Saviour,” from the people of
Rhodes.328 It is to be observed, however, that while in the next generation
Antiochus I of Syria received the same title, and his successor Antiochus II
that of Theos, “God,” the usage passes away; Ptolemy III being named
merely Evergetês, “the Benefactor” (of the priests), and even Antiochus III
only “the Great.” Superstition was not to be ousted by a political
exploitation of its machinery.329
In Athens the democracy, restored in a subordinate form by Kassander’s
opponent, Demetrius Poliorkêtes (307 B.C.), actually tried to put down the
philosophic schools, all of which, but the Aristotelian in particular, were
anti-democratic, and doubtless also comparatively irreligious. Epicurus and
some of his antagonists were exiled within a year of his opening his school
(306 B.C.); but the law was repealed in the following year.330
Theophrastos, the head of the Aristotelian school, was indicted in the old
fashion for impiety, which seems to have consisted in denouncing animal
sacrifice.331 These repressive attempts, however, failed; and no others
followed at Athens in that era; though in the next century the Epicureans
seem to have been expelled from Lythos in Crete and from Messenê in the
Peloponnesos, nominally for their atheism, in reality probably on political
grounds.332 Thus Zeno was free to publish a treatise in which, besides far
out-going Plato in schemes for dragooning the citizens into an ideal life, he
proposed a State without temples or statues of the Gods or law courts or
gymnasia.333 In the same age there is trace of “an interesting case of
rationalism even in the Delphic oracle.”334 The people of the island of
Astypalaia, plagued by hares or rabbits, solemnly consulted the oracle,
which briefly advised them to keep dogs and take to hunting. About the
same time we find Lachares, temporarily despot at Athens, plundering the
shrine of Pallas of its gold.335 Even in the general public there must have
been a strain of surviving rationalism; for among the fragments of
Menander (fl. 300), who, in general, seems to have leant to a well-bred
orthodoxy,336 there are some speeches savouring of skepticism and
pantheism.337
It was in keeping with this general but mostly placid and non-polemic
latitudinarianism that the New Academy, the second birth, or rather
transformation, of the Platonic school, in the hands of Arkesilaos and the
great Carneades (213–129), and later of the Carthaginian Clitomachos,
should be marked by that species of skepticism thence called Academic—a
skepticism which exposed the doubtfulness of current religious beliefs
without going the Pyrrhonian length of denying that any beliefs could be
proved, or even denying the existence of the Gods.
For the arguments of Carneades against the Stoic doctrine of immortality see Cicero, De
natura Deorum, iii, 12, 17; and for his argument against theism see Sextus Empiricus, Adv.
Math. ix, 172, 183. Mr. Benn pronounces this criticism of theology “the most destructive
that has ever appeared, the armoury whence religious skepticism ever since has been
supplied” (The Philosophy of Greece, etc., p. 258). This seems an over-statement. But it is
just to say, as does Mr. Whittaker (Priests, Philosophers, and Prophets, 1911, p. 60; cp. p.
86), that “there has never been a more drastic attack than that of Carneades, which
furnished Cicero with the materials for his second book, On Divination”; and, as does Prof.
Martha (Études Morales sur l’antiquité, 1889, p. 77), that no philosophic or religious
school has been able to ignore the problems which Carneades raised.
As against the essentially uncritical Stoics, the criticism of Carneades is
sane and sound; and he has been termed by judicious moderns “the greatest
skeptical mind of antiquity”338 and “the Bayle of Antiquity”;339 though he
seems to have written nothing.340 There is such a concurrence of testimony
as to the victorious power of his oratory and the invincible skill of his
dialectic341 that he must be reckoned one of the great intellectual and
rationalizing forces of his day, triumphing as he did in the two diverse
arenas of Greece and Rome. His disciple and successor Clitomachos said of
him, with Cicero’s assent, that he had achieved a labour of Hercules “in
liberating our souls as it were of a fierce monster, credulity, conjecture, rash
belief.”342 He was, in short, a mighty antagonist of thoughtless beliefs,
clearing the ground for a rational life; and the fact that he was chosen with
Diogenes the Peripatetic and Critolaos the Stoic to go to Rome to plead the
cause of ruined Athens, mulcted in an enormous fine, proved that he was
held in high honour at home. Athens, in short, was not at this stage “too
superstitious.” Unreasoning faith was largely discredited by philosophy.
On this basis, in a healthy environment, science and energy might have
reared a constructive rationalism; and for a time astronomy, in the hands of
Aristarchos of Samos (third century B.C.), Eratosthenes of Cyrene, the
second keeper of the great Alexandrian library (2nd cent. B.C.), and above
all of Hipparchos of Nikaia, who did most of his work in the island of
Rhodes, was carried to a height of mastery which could not be maintained,
and was re-attained only in modern times.343 Thus much could be
accomplished by “endowment of research” as practised by the Ptolemies at
Alexandria; and after science had declined with the decline of their polity,
and still further under Roman rule, the new cosmopolitanism of the second
century of the empire reverted to the principle of intelligent evocation,
producing under the Antonines the “Second” School of Alexandria.
But the social conditions remained fundamentally bad; and the earlier
greatness was never recovered. “History records not one astronomer of note
in the three centuries between Hipparchos and Ptolemy”; and Ptolemy (fl.
140 C.E.) not only retrograded into astronomical error, but elaborated on
oriental lines a baseless fabric of astrology.344 Other science mostly
decayed likewise. The Greek world, already led to lower intellectual levels
by the sudden ease and wealth opened up to it through the conquests of
Alexander and the rule of his successors, was cast still lower by the Roman
conquest. Pliny, extolling Hipparchos with little comprehension of his
work, must needs pronounce him to have “dared a thing displeasing to
God” in numbering the stars for posterity.345 In the air of imperialism,
stirred by no other, original thought could not arise; and the mass of the
Greek-speaking populations, rich and poor, gravitated to the level of the
intellectual346 and emotional life of more or less well-fed slaves. In this
society there rapidly multiplied private religious associations—thiasoi,
eranoi, orgeones—in which men and women, denied political life, found
new bonds of union and grounds of division in cultivating worships, mostly
oriental, which stimulated the religious sense and sentiment.347
Such was the soil in which Christianity took root and flourished; while
philosophy, after the freethinking epoch following on the fall of Athenian
power, gradually reverted to one or other form of mystical theism or
theosophy, of which the most successful was the Neo-Platonism of
Alexandria.348 When the theosophic Julian rejoiced that Epicureanism had
disappeared,349 he was exulting in a symptom of the intellectual decline
that made possible the triumph of the faith he most opposed. Christianity
furthered a decadence thus begun under the auspices of pagan imperialism;
and “the fifth century of the Christian era witnessed an almost total
extinction of the sciences in Alexandria”350—an admission which disposes
of the dispute as to the guilt of the Arabs in destroying the great library.
Here and there, through the centuries, the old intellectual flame burns
whitely enough: the noble figure of Epictetus in the first century of the
new era, and that of the brilliant Lucian in the second, in their widely
different ways remind us that the evolved faculty was still there if the
circumstances had been such as to evoke it. Menippos in the first century
B.C. had played a similar part to that of Lucian, in whose freethinking
dialogues he so often figures; but with less of subtlety and intellectuality.
Lucian’s was indeed a mind of the rarest lucidity; and the argumentation of
his dialogue Zeus Tragædos covers every one of the main aspects of the
theistic problem. There is no dubiety as to his atheistic conclusion, which is
smilingly implicit in the reminder he puts in the mouth of Hermes, that,
though a few men may adopt the atheistic view, “there will always be
plenty of others who think the contrary—the majority of the Greeks, the
ignorant many, the populace, and all the barbarians.” But the moral doctrine
of Epictetus is one of endurance and resignation; and the almost unvarying
raillery of Lucian, making mere perpetual sport of the now moribund
Olympian Gods, was hardly better fitted than the all-round skepticism of
the school of Sextus Empiricus to inspire positive and progressive
thinking.
This latter school, described by Cicero as dispersed and extinct in his
day,351 appears to have been revived in the first century by Ænesidemos,
who taught at Alexandria.352 It seems to have been through him in
particular that the Pyrrhonic system took the clear-cut form in which it is
presented at the close of the second century by the accomplished Sextus
“Empiricus”—that is, the empirical (i.e., experiential) physician,353 who
lived at Alexandria and Athens (fl. 175–205 C.E.). As a whole, the school
continued to discredit dogmatism without promoting knowledge. Sextus, it
is true, strikes acutely and systematically at ill-founded beliefs, and so
makes for reason;354 but, like the whole Pyrrhonian school, he has no idea
of a method which shall reach sounder conclusions. As the Stoics had
inculcated the control of the passions as such, so the skeptics undertook to
make men rise above the prejudices and presuppositions which swayed
them no less blindly than ever did their passions. But Sextus follows a
purely skeptical method, never rising from the destruction of false beliefs to
the establishment of true. His aim is ataraxia, a philosophic calm of non-
belief in any dogmatic affirmation beyond the positing of phenomena as
such; and while such an attitude is beneficently exclusive of all fanaticism,
it unfortunately never makes any impression on the more intolerant fanatic,
who is shaken only by giving him a measure of critical truth in place of his
error. And as Sextus addressed himself to the students of philosophy, not to
the simple believers in the Gods, he had no wide influence.355 Avowedly
accepting the normal view of moral obligations while rejecting dogmatic
theories of their basis, the doctrine of the strict skeptics had the effect, from
Pyrrho onwards, of giving the same acceptance to the common religion,
merely rejecting the philosophic pretence of justifying it. Taken by
themselves, the arguments against current theism in the third book of the
Hypotyposes356 are unanswerable; but, when bracketed with other
arguments against the ordinary belief in causation, they had the effect of
leaving theism on a par with that belief. Against religious beliefs in
particular, therefore, they had no wide destructive effect.
Lucian, again, thought soundly and sincerely on life; his praise of the men
whose memories he respected, as Epicurus and Demonax (if the Life of
Demonax attributed to him be really his), is grave and heartfelt; and his
ridicule of the discredited Gods was perfectly right so far as it went. It is
certain that the unbelievers and the skeptics alike held their own with the
believers in the matter of right living.357 In the period of declining pagan
belief, the maxim that superstition was a good thing for the people must
have wrought a quantity and a kind of corruption that no amount of ridicule
of religion could ever approach. Polybius (fl. 150 B.C.) agrees with his
complacent Roman masters that their greatness is largely due to the
carefully cultivated superstition of their populace, and charges with
rashness and folly those who would uproot the growth;358 and Strabo,
writing under Tiberius—unless it be a later interpolator of his work—
confidently lays down the same principle of governmental deceit,359 though
in an apparently quite genuine passage he vehemently protests the
incredibility of the traditional tales about Apollo.360 So far had the doctrine
evolved since Plato preached it. But to countervail it there needed more
than a ridicule which after all reached only the class who had already cast
off the beliefs derided, leaving the multitude unenlightened. The lack of the
needed machinery of enlightenment was, of course, part of the general
failure of the Græco-Roman civilization; and no one man’s efforts could
have availed, even if any man of the age could have grasped the whole
situation. Rather the principle of esoteric enlightenment, the ideal of secret
knowledge, took stronger hold as the mass grew more and more
comprehensively superstitious. Even at the beginning of the Christian era
the view that Homer’s deities were allegorical beings was freshly
propounded in the writings of Herakleides and Cornutus (Phornutus); but it
served only as a kind of mystical Gnosis, on all fours with Christian
Gnosticism, and was finally taken up by Neo-Platonists, who were no
nearer rationalism for adopting it.361
So with the rationalism to which we have so many uneasy or hostile
allusions in Plutarch. We find him resenting the scoffs of Epicureans at the
doctrine of Providence, and recoiling from the “abyss of impiety”362
opened up by those who say that “Aphrodite is simply desire, and Hermes
eloquence, and the Muses the arts and sciences, and Athênê wisdom, and
Dionysos merely wine, Hephaistos fire, and Dêmêtêr corn”;363 and in his
essay On Superstition he regretfully recognizes the existence of many
rational atheists, confessing that their state of mind is better than that of the
superstitious who abound around him, with their “impure purifications and
unclean cleansings,” their barbaric rites, and their evil Gods. But the
unbelievers, with their keen contempt for popular folly, availed as little
against it as Plutarch himself, with his doctrine of a just mean. The one
effectual cure would have been widened knowledge; and of such an
evolution the social conditions did not permit.
To return to a state of admiration for the total outcome of Greek thought,
then, it is necessary to pass from the standpoint of simple analysis to that of
comparison. It is in contrast with the relatively slight achievement of the
other ancient civilizations that the Greek, at its height, still stands out for
posterity as a wonderful growth. That which, tried by the test of ideals, is as
a whole only one more tragic chapter in the record of human frustration, yet
contains within it light and leading as well as warning; and for long ages it
was as a lost Paradise to a darkened world. It has been not untruly said that
“the Greek spirit is immortal, because it was free”:364 free not as science
can now conceive freedom, but in contrast with the spiritual bondage of
Jewry and Egypt, the half-barbaric tradition of imperial Babylon, and the
short flight of mental life in Rome. Above all, it was ever in virtue of the
freedom that the high things were accomplished; and it was ever the falling
away from freedom, the tyranny either of common ignorance or of mindless
power, that wrought decadence. There is a danger, too, of injustice in
comparing Athens with later States. When a high authority pronounces that
“the religious views of the Demos were of the narrowest kind,”365 he is not
to be gainsaid; but the further verdict that “hardly any people has sinned
more heavily against the liberty of science” is unduly lenient to Christian
civilization. The heaviest sins of that against science, indeed, lie at the door
of the Catholic Church; but to make that an exoneration of the modern
“peoples” as against the ancient would be to load the scales. And even apart
from the Catholic Church, which practically suppressed all science for a
thousand years, the attitude of Protestant leaders and Protestant peoples,
from Luther down to the second half of the nineteenth century, has been one
of hatred and persecution towards all science that clashed with the sacred
books.366 In the Greek world there was more scientific discussion in the
three hundred years down to Epicurus than took place in the whole of
Christian Europe in thirteen hundred; and the amount of actual violence
used towards innovators in the pagan period, though lamentable enough,
was trifling in comparison with that recorded in Christian history, to say
nothing of the frightful annals of witch-burning, to which there is no
parallel in civilized heathen history. The critic, too, goes on to admit that,
while “Sokrates, Anaxagoras, and Aristotle fell victims in different degrees
to the bigotry of the populace,” “of course their offence was political rather
than religious. They were condemned not as heretics, but as innovators in
the state religion.” And, as we have seen, all three of the men named taught
in freedom for many years till political faction turned popular bigotry
against them. The true measure of Athenian narrowness is not to be reached,
therefore, without keeping in view the long series of modern outrages and
maledictions against the makers and introducers of new machinery, and the
multitude of such episodes as the treatment of Priestley in Christian
Birmingham, little more than a century ago. On a full comparison the
Greeks come out not ill.
It was, in fact, impossible that the Greeks should either stifle or persecute
science or freethought as it was either stifled or persecuted by ancient Jews
(who had almost no science by reason of their theology) or by modern
Christians, simply because the Greeks had no anti-scientific hieratic
literature. It remains profoundly significant for science that the ancient
civilization which on the smallest area evolved the most admirable life,
which most completely transcended all the sources from which it originally
drew, and left a record by which men are still charmed and taught, was a
civilization as nearly as might be without Sacred Books, without an
organized priesthood, and with the largest measure of democratic freedom
that the ancient world ever saw.
1
2
3
4
5
6
7
8
9
10
11
12
Cp. Tiele, Outlines, pp. 205, 207, 212. ↑
Cp. E. Meyer, Geschichte des Alterthums, ii, 533. ↑
Cp. K. O. Müller, Literature of Ancient Greece, ed. 1847, p. 77. ↑
Duncker, Gesch. des Alterth. 2 Aufl. iii, 209–10, 252–54, 319 sq.; E. Meyer, Gesch.
des Alterth. ii, 181, 365, 369, 377, 380, 535 (see also ii, 100, 102, 105, 106, 115 note, etc.);
W. Christ, Gesch. der griech. Lit. 3te Aufl. p. 12; Gruppe, Die griech. Culte und Mythen,
1887, p. 165 sq. ↑
E. Curtius, Griech. Gesch. i, 28, 29, 35, 40, 41, 101, 203, etc.; Meyer, ii, 369. ↑
See the able and learned essay of S. Reinach, Le Mirage Orientate, reprinted from
L’Anthropologie, 1893. I do not find that its arguments affect any of the positions here
taken up. See pp. 40–41. ↑
Meyer, ii. 369; Benn, The Philosophy of Greece, 1898, p. 42. ↑
Cp. Bury, History of Greece, ed. 1906, pp. vi, 10, 27, 32–34, 40, etc.; Burrows, The
Discoveries in Crete, 1907, ch. ix; Maisch, Manual of Greek Antiquities, Eng. tr. §§ 8, 9,
10, 60; H. R. Hall, The Oldest Civilization of Greece, 1901, pp. 31, 32. ↑
Cp. K. O. Müller, Hist. of the Doric Race, Eng. tr. 1830, i, 8–10; Busolt, Griech.
Gesch. 1885, i, 33; Grote, Hist. of Greece, 10-vol. ed. 1888, iii, 3–5, 35–44; Duncker, iii,
136, n.; E. Meyer, Gesch. des Alterthums, i, 299–310 (§§ 250–58); E. Curtius, i, 29;
Schömann, Griech. Alterthümer, as cited, i, 2–3, 89; Burrows, ch. ix. ↑
Cp. Meyer, ii, 97; and his art. “Baal” in Roscher’s Ausführl. Lex. Mythol. i, 2867. ↑
The fallacy of this tradition, as commonly put, was well shown by Renouvier long ago
—Manuel de philosophie ancienne, 1844, i, 3–13. Cp. Ritter, as cited below. ↑
Cp. on one side, Ritter, Hist. of Anc. Philos. Eng. tr. i, 151; Renan, Études d’hist.
religieuse, pp. 47–48; Zeller, Hist. of Greek Philos. Eng. tr. 1881, i, 43–49; and on the
other, Ueberweg, Hist. of Philos. Eng. tr. i, 31, and the weighty criticism of Lange, Gesch.
des Materialismus, i, 126–27 (Eng. tr. i, 9, note 5). ↑
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Cp. Curtius, i, 125; Bury, introd. and ch. i. ↑
Cp. Bury, as cited. ↑
As to the primary mixture of “Pelasgians” and Hellenes, cp. Busolt, i, 27–32; Curtius,
i, 27; Schömann, i, 3–4; Thirlwall, Hist. of Greece, ed. 1839, i, 51–52, 116. K. O. Müller
(Doric Race, Eng. tr. i, 10) and Thirlwall, who follows him (i, 45–47), decide that the
Thracians cannot have been very different from the Hellenes in dialect, else they could not
have influenced the latter as they did. This position is clearly untenable, whatever may
have been the ethnological facts. It would entirely negate the possibility of reaction
between Greeks, Kelts, Egyptians, Semites, Romans, Persians, and Hindus. ↑
Murray, Four Stages of Greek Religion, 1912, p. 59. ↑
Cp. Meyer, Gesch. des Alt. ii, 583. ↑
The question is discussed at some length in the author’s Evolution of States, 1912. ↑
Lit. of Anc. Greece, pp. 41–47. The discussion of the Homeric problem is, of course,
alien to the present inquiry. ↑
Introd. to Scientif. Mythol. Eng. tr. pp. 180, 181, 291. Cp. Curtius, i, 126. ↑
Cp. Curtius, i, 107, as to the absence in Homer of any distinction between Greeks and
barbarians; and Grote, 10-vol. ed. 1888, iii, 37–38, as to the same feature in Archilochos. ↑
Duncker, Gesch. des Alt., as cited, iii. 209–10; pp. 257, 319 sq. Cp. K. O. Müller, as
last cited, pp. 181, 193; Curtius, i, 43–49, 53, 54, 107, 365, 373, 377, etc.; Grote, iii, 39–
41; and Meyer, ii, 104. ↑
Duncker, iii, 214; Curtius, i, 155, 121; Grote, iii, 279–80. ↑
Busolt, Griech. Gesch. 1885, i, 171–72. Cp. pp. 32–34; and Curtius, i, 42. ↑
On the general question cp. Gruppe, Die griechischen Culte und Mythen, pp. 151 ff.,
157, 158 ff., 656 ff., 672 ff. ↑
Preller, Griech. Mythol. 2 Aufl. i, 260; Tiele, Outlines, p. 211; R. Brown, Jr., Semit.
Influ. in Hellenic Mythol. 1898, p. 130; Murray, Hist. of Anc. Greek Lit. p. 35; H. R. Hall,
Oldest Civilization of Greece, 1901, p. 290. ↑
See Tiele, Outlines, pp. 210, 212. Cp., again, Curtius, Griech. Gesch. i, 95, as to the
probability that the “twelve Gods” were adjusted to the confederations of twelve cities; and
again p. 126. ↑
28
29
30
31
32
33
34
35
36
37
38
39
40
“Even the title ‘king’ (Αναξ) seems to have been borrowed by the Greek from
Phrygian.... It is expressly recorded that τύραννος is a Lydian word. Βασιλεύς (‘king’)
resists all attempts to explain it as a purely Greek formation, and the termination
assimilates it to certain Phrygian words.” (Prof. Ramsay, in Encyc. Brit. art. Phrygia). In
this connection note the number of names containing Anax (Anaximenes, Anaximandros,
Anaxagoras, etc.) among the Ionian Greeks. ↑
iv, 561 sq. ↑
It is now agreed that this is merely a guess. The document, further, has been redacted
and interpolated. ↑
Prehist. Antiq. of the Aryan Peoples, Eng. tr. p. 423. Wilamowitz holds that the verses
Od. xi, 566–631, are interpolations made later than 600 B.C. ↑
Tiele, Outlines, p. 209; Preller, p. 263. ↑
Meyer says on the contrary (Gesch. des Alt. ii, 103, Anm.) that “Kronos is certainly a
Greek figure”; but he cannot be supposed to dispute that the Greek Kronos cult is grafted
on a Semitic one. ↑
Sayce, Hibbert Lectures, pp. 54, 181. Cp. Cox, Mythol. of the Aryan Nations, p. 260,
note. It has not, however, been noted in the discussions on Semelê that Semlje is the Slavic
name for the Earth as Goddess. Ranke, History of Servia, Eng. tr. p. 43. ↑
Iliad, xiv, 201, 302. ↑
Sayce, Hibbert Lectures, p. 367 sq.; Ancient Empires, p. 158. Note p. 387 in the
Lectures as to the Assyrian influence, and p. 391 as to the Homeric notion in particular. Cp.
W. Christ, Gesch. der griech. Literatur, § 68. ↑
It is unnecessary to examine here the view of Herodotos that many of the Greek cults
were borrowed from Egypt. Herodotos reasoned from analogies, with no exact historical
knowledge. But cp. Renouvier, Manuel, i, 67, as to probable Egyptian influence. ↑
Cp. Meyer, ii, §§ 453–60, as to the eastern initiative of Orphic theology. ↑
It is noteworthy that the traditional doctrine associated with the name of Orpheus
included a similar materialistic theory of the beginning of things. Athenagoras, Apol. c. 19.
Cp. Renouvier, Manuel de philos. anc. i, 69–72; and Meyer, ii, 743. ↑
Cp. Meyer, ii, 726. As to the oriental elements in Hesiod see further Gruppe, Die
griechischen Culte und Mythen, 1887, pp. 577, 587, 589, 593. ↑
41
42
43
44
45
46
47
48
49
50
51
52
53
54
Cp. however, Bury (Hist. of Greece, pp. 6, 65), who assumes that the Greeks brought
the hexameter with them to Hellas. Contrast Murray, Four Stages, p. 61. ↑
Mahaffy, History of Classical Greek Literature, 1880, i, 15. ↑
Id. p. 16. Cp. W. Christ, as cited, p. 79. ↑
Mahaffy, pp. 16–17. ↑
Od. xviii, 352. ↑
Od. vi, 240; Il. v, 185. ↑
Od. xxii, 39. ↑
In Od. xiv, 18, αντίθεοι means not “opposed to the Gods,” but “God-like,” in the
ordinary Homeric sense of noble-looking or richly attired, as men in the presence of the
Gods. Cp. vi, 241. Yet a Scholiast on a former passage took it in the sense of God-
opposing. Clarke’s ed. in loc. Liddell and Scott give no use of ἄθεος, in the sense of
denying the Gods, before Plato (Apol. 26 C. etc.), or in the sense of ungodly before Pindar
(P. iv, 288) and Æschylus (Eumen. 151). For Sophocles it has the force of “God-
forsaken”—Oedip. Tyr. 254 (245), 661 (640), 1360 (1326). Cp. Electra, 1181 (1162). But
already before Plato we find the terms ἄπιστος and ἄθεος, “faithless” or “infidel” and
“atheist,” used as terms of moral aspersion, quite in the Christian manner (Euripides,
Helena, 1147), where there is no question of incredulity. ↑
Cp. Lang, Myth, Ritual, and Religion, 2nd ed. i, 14–15. and cit. there from Professor
Jebb. ↑
Cp. Meyer, Gesch. des Alterthums, ii, 724–27; Grote, as cited, i, 279–81. ↑
Meyer, ii, 724, 727. ↑
The tradition is confused. Stesichoros is said first to have aspersed Helen, whereupon
she, as Goddess, struck him with blindness: thereafter he published a retractation, in which
he declared that she had never been at Troy, an eidolon or phantasm taking her name; and
on this his sight was restored. We can but divine through the legend the probable reality,
the documents being lost. See Grote, as cited, for the details. For the eulogies of
Stesichoros by ancient writers, see Girard, Sentiment religieux en Grèce, 1869, pp. 175–
79. ↑
Cp. Meyer (1901), iii. § 244. ↑
Ol. i, 42–57, 80–85. ↑
55
56
57
58
59
60
61
62
63
64
65
66
67
68
Ol. ix, 54–61. ↑
He dedicated statues to Zeus, Apollo, and Hermes. Pausanias, ix, 16, 17. ↑
Herodot. ii. 53. ↑
A ruler of Libyan stock, and so led by old Libyan connections to make friends with
Greeks. He reigned over fifty years, and the Greek connection grew very close. Curtius, i,
344–45. Cp. Grote, i, 144–55. ↑
Grote, 10-vol. ed. 1888, i, 307, 326, 329, 413. Cp. i, 27–30; ii, 52; iii, 39–41, etc. ↑
K. O. Müller, Introd. to Mythology, p. 192. ↑
“Then one [of the Persians] who before had in nowise believed in [or, recognized the
existence of] the Gods, offered prayer and supplication, doing obeisance to Earth and
Heaven” (Persae, 497–99). ↑
Agamemnon, 370–372. This is commonly supposed to be a reference to Diagoras the
Melian (below, p. 159). ↑
Agam. 170–72 (160–62). ↑
So Whittaker, Priests, Philosophers, and Prophets, 1911, pp. 42–43. ↑
So Buckley, in Bohn trans. of Æschylus, p. 100. He characterizes as a “skeptical
formula” the phrase “Zeus, whoever he may be”; but goes on to show that such formulas
were grounded on the Semitic notion that the true name of God was concealed from man. ↑
Grote, ed. 1888, vii, 8–21. See the whole exposition of the exceptionally interesting
67th chapter. ↑
Cp. Meyer, ii, 431; K. O. Müller, Introd. to Mythol. pp. 189–92; Duncker, p. 340;
Curtius, i, 384; Thirlwall, i, 200–203; Burckhardt, Griech. Culturgesch. 1898, ii. 19. As to
the ancient beginnings of a priestly organization, see Curtius, i, 92–94, 97. As to the effects
of its absence, see Heeren, Polit. Hist. of Anc. Greece, Eng. tr. 1829, pp. 59–63;
Burckhardt, as cited, ii, 31–32; Meyer, as last cited; Zeller, Philos. der Griechen, 3te Aufl.
i, 44 sq. Lange’s criticism of Zeller’s statement (Gesch. des Materialismus, 3te Aufl. i,
124–26, note 2) practically concedes the proposition. The influence of a few powerful
priestly families is not denied. The point is that they remained isolated. ↑
Cp. K. O. MÜller, Introd. to Mythol. p. 195; Curtius, i, 387, 389, 392; Duncker, iii,
519–21, 563; Thirlwall, i, 204; Barthélemy St. Hilaire, préf. to tr. of Metaphys. of Aristotle,
p. 14. Professor Gilbert Murray, noting that Homer and Hesiod treated the Gods as
69
70
71
72
73
74
75
76
77
78
79
elements of romance, or as facts to be catalogued, asks: “Where is the literature of religion:
the literature which treated the Gods as Gods? It must,” he adds, “have existed”; and he
holds that we “can see that the religious writings were both early and multitudinous” (Hist.
of Anc. Greek Lit. p. 62; cp. Meyer and Mahaffy as cited above, pp. 125–26. “Writings” is
not here to be taken literally; the early hymns were unwritten). The priestly hymns and
oracles and mystery-rituals in question were never collected; but perhaps we may form
some idea of their nature from the “Homeridian” and Orphic hymns to the Gods, and those
of the Alexandrian antiquary Callimachus. It is further to be inferred that they enter into the
Hesiodic Theogony. (Decharme, p. 3, citing Bergk.) ↑
Meyer, ii, 426; Curtius, i, 390–91, 417; Thirlwall, i, 204; Grote, i, 48–49. ↑
Meyer, ii, 410–14. ↑
Cp. Curtius, i, 392–400, 416; Duncker, iii, 529. ↑
Curtius, i, 112; Meyer, ii, 366. ↑
Curtius, i, 201, 204, 205, 381; Grote, iii, 5; Lange, Gesch. des Materialismus, 3te
Aufl. i, 23 (Eng. tr. i, 23). ↑
Herodotos, i, 170; Diogenes Laërtius, Thales, ch. i. ↑
On the essentially anti-religious rationalism of the whole Ionian movement, cp.
Meyer, ii, 753–57. ↑
The First Philosophers of Greece, by A. Fairbanks, 1898, pp. 2, 3, 6. This compilation
usefully supplies a revised text of the ancient philosophic fragments, with a translation of
these and of the passages on the early thinkers by the later, and by the epitomists. A good
conspectus of the remains of the early Greek thinkers is supplied also in Grote’s Plato and
the other Companions of Sokrates, ch. i; and a valuable critical analysis of the sources in
Prof. J. Burnet’s Early Greek Philosophy. ↑
Cp. Lange, Gesch. des Mat. i, 126 (Eng. tr. i, 8, n.). Mr. Benn (The Greek
Philosophers, i, 8) and Prof. Decharme (p. 39) seem to read this as a profession of belief in
deities in the ordinary sense. But cp. R. W. Mackay, The Progress of the Intellect, 1850, i,
338. Burnet (ch. i, § 11) doubts the authenticity of this saying, but thinks it “extremely
probable that Thales did say that the magnet and amber had souls.” ↑
Mackay, as cited, p. 331. ↑
Fairbanks, p. 4. ↑
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
Diogenes Laërtius, Thales, ch. 9. ↑
Fairbanks, pp. 3, 7. ↑
Herodotos, i, 74. ↑
Cp. Burnet, Early Greek Philos. 2nd. ed. introd. § 3. To Thales is ascribed by the
Greeks the “discovery” of the constellation Ursus Major. Diog. ch. 2. As it was called
“Phoenike” by the Greeks, his knowledge would be of Phoenician derivation. Cp.
Humboldt, Kosmos, Bohn tr. iii, 160. ↑
Diog. Laërt. ch. 3. On this cp. Burnet, introd. § 6. ↑
Herod. i, 170. Cp. Diog. Laërt. ch. 3. ↑
Diog. Laërt. ch. 9. ↑
Cp. Burnet, p. 57. ↑
Fairbanks, pp. 9–10. Mr. Benn (Greek Philosophers, i, 9) decides that the early
philosophers, while realizing that ex nihilo nihil fit, had not grasped the complementary
truth that nothing can be annihilated. But even if the teaching ascribed to Anaximandros be
set aside as contradictory (since he spoke of generation and destruction within the infinite),
we have the statement of Diogenes Laërtius (bk. ix, ch. 9, § 57) that Diogenes of
Apollonia, pupil of Anaximenes, gave the full Lucretian formula. ↑
Diogenes Laërtius, however (ii, 2), makes him agree with Thales. ↑
Fairbanks, pp. 9–16. Diogenes makes him the inventor of the gnomon and of the first
map and globe, as well as a maker of clocks. Cp. Grote, i, 330, note. ↑
See below, p. 158, as to Demokritos’ statement concerning the Eastern currency of
scientific views which, when put by Anaxagoras, scandalized the Greeks. ↑
Fairbanks, pp. 17–22. ↑
See Windelband, Hist. of Anc. Philos. Eng. tr. 1900, p. 25, citing Diels and
Wilamowitz-Möllendorf. Cp. Burnet, introd. § 14. ↑
It will be observed that Mr. Cornford’s book, though somewhat loosely speculative is
very freshly suggestive. It is well worth study, alongside of the work of Prof. Burnet, by
those interested in the scientific presentation of the evolution of thought. ↑
Diog. Laërt. ix, 19; Fairbanks, p. 76. ↑
Herodotos, i, 163–67; Grote, iii, 421; Meyer, ii, § 438. ↑
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
Cp. Guillaume Bréton, Essai sur la poésie philosophique en Grèce, 1882, pp. 23–25.
The life period of Xenophanes is still uncertain. Meyer (ii, § 466) and Windelband (Hist. of
Anc. Philos. Eng. tr. p. 47) still adhere to the chronology which puts him in the century
570–470, making him a young man at the foundation of Elea. ↑
Cousin, developed by G. Bréton, work cited, p. 31 sq., traces Xenophanes’s doctrine
of the unity of things to the school of Pythagoras. It clearly had antecedents. But
Xenophanes is recorded to have argued against Pythagoras as well as Thales and
Epimenides (Diog. Laërt. ix, 2, §§ 18, 20). ↑
Metaphysics, i, 5; cp. Fairbanks, pp. 79–80. ↑
One of several so entitled in that age. Cp. Burnet, introd. § 7. ↑
Metaph., as cited; Plato, Soph. 242 D. ↑
Long fragment in Athenæus, xi, 7; Burnet, p. 130. ↑
Burnet, p. 141. ↑
Cp. Burnet, p. 131. ↑
Fairbanks, p. 67, Fr. 5, 6; Clem. Alex. Stromata, bk. v, Wilson’s tr. ii, 285–86. Cp. bk.
vii, c. 4. ↑
Fairbanks, Fr. 7. ↑
Cicero, De divinatione, i, 3, 5; Aetius, De placitis reliquiæ, in Fairbanks, p. 85. ↑
Aristotle, Rhetoric, ii, 23, § 27. A similar saying is attributed to Herakleitos, on slight
authority (Fairbanks, p. 54). ↑
Cicero, Academica, ii, 39; Lactantius, Div. Inst. iii, 23. Anaxagoras and Demokritos
held the same view. Diog. Laërt, bk. ii, ch. iii, iv (§ 8); Pseudo-Plutarch, De placitis
philosoph. ii, 25. ↑
Cp. Mackay, Progress of the Intellect, i, 340. ↑
Diog. Laërt. in life of Pyrrho, bk. ix, ch. xi, 8 (§ 72). The passage, however, is
uncertain. See Fairbanks, p. 70. ↑
Fairbanks. Fr. 1. Fairbanks translates with Zeller: “The whole [of God].” Grote: “The
whole Kosmos, or the whole God.” It should be noted that the original in Sextus Empiricus
(Adv. Math. ix, 144) is given without the name of Xenophanes, and the ascription is
modern. ↑
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
Grote, as last cited, p. 18. ↑
Fairbanks, Fr. 19. In Athenæus, x, 413. ↑
Polybius, iv, 40; Sextus Empiricus, Adversus Mathematicos, viii, 126; Fairbanks, pp.
25, 27; Frag. 4, 14. Cp. 92, 111, 113. ↑
Diog. Laërt. ix, i, 2. ↑
Fairbanks, Fr. 134. ↑
Id. Frag. 36, 67. ↑
Id. Frag. 43, 44, 46, 62. ↑
Diog. Laërt. last cited. This saying is by some ascribed to the later Herakleides (see
Fairbanks, Fr. 119 and note); but it does not seem to be in his vein, which is wholly pro-
Homeric. ↑
Clem. Alex. Protrept. ch. 2, Wilson’s tr. p. 41. The passage is obscure, but Mr.
Fairbanks’s translation (Fr. 127) is excessively so. ↑
Clemens, as cited, p.32; Fairbanks, Fr. 124, 125, 130. Cp. Burnet, p. 139. ↑
Fairbanks, Fr. 21. ↑
Cp. Burnet, pp. 175–90. ↑
Theaetetus, 180 D. See good estimates of Parmenides in Benn’s Greek Philosophers,
i, 17–19, and Philosophy of Greece in Relation to the Character of its People, pp. 83–95;
in J. A. Symonds’s Studies of the Greek Poets, 3rd ed. 1893, vol. i, ch. 6; and in Zeller, i,
580 sq. ↑
Plutarch, Perikles, ch. 26. ↑
Mr. Benn finally gives very high praise to Melissos (Philos. of Greece, pp. 91–92); as
does Prof. Burnet (Early Gr. Philos. p. 378). He held strongly by the Ionian conception of
the eternity of matter. Fairbanks, p. 125. ↑
Diog. Laërt. bk. ix, ch. iv, 3 (§ 24). ↑
Diog. Laërt. ix, 3 (§ 21). ↑
As to this see Windelband, Hist. Anc. Philos. pp. 91–92. ↑
Cp. Mackay, Progress of the Intellect, i. 340. ↑
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
“The difference between the Ionians and Eleatæ was this: the former endeavoured to
trace an idea among phenomena by aid of observation; the latter evaded the difficulty by
dogmatically asserting the objective existence of an idea” (Mackay, as last cited). ↑
Cp. Mackay, i, 352–53, as to the survival of veneration of the heavenly bodies in the
various schools. ↑
Grote, i, 350. ↑
Meyer, ii, 9, 759 (§§ 5, 465). ↑
Id. §§ 6, 466. ↑
Jevons, Hist. of Greek Lit. 1886, p. 210. ↑
Compare Meyer, ii, § 502, as to the close resemblances between Pythagoreanism and
Orphicism. ↑
Meyer, i, 186; ii, 635. ↑
Fairbanks, pp. 145, 151, 155, etc. ↑
Id. p. 143. ↑
Id. p. 154. ↑
Prof. Burnet insists (introd. p. 30) that “the” Greeks must be reckoned good observers
because their later sculptors were so. As well say that artists make the best men of
science. ↑
Metaph. i, 5; Fairbanks, p. 136. “It is quite safe to attribute the substance of the First
Book of Euclid to Pythagoras.” Burnet, Early Greek Philos. 2nd ed. p. 117. ↑
Diog. Laërt. Philolaos (bk. viii, ch. 7). ↑
L. U. K. Hist. of Astron. p. 20; A. Berry’s Short Hist. of Astron. 1898, p. 25; Narrien’s
Histor. Acc. of the Orig. and Prog. of Astron. 1850, p. 163. ↑
See Benn, Greek Philosophers, i, 11. ↑
Diog. Laërt. in life of Philolaos; Cicero, Academica, ii, 39. Cicero, following
Theophrastus, is explicit as to the teaching of Hiketas. ↑
Hippolytos, Ref. of all Heresies, i, 13. Cp. Renouvier, Manuel de la philos. anc. i, 201,
205, 238–39. ↑
Pseudo-Plutarch, De Placitis Philosoph. iii, 13, 14. ↑
Welcome to our website – the ideal destination for book lovers and
knowledge seekers. With a mission to inspire endlessly, we offer a
vast collection of books, ranging from classic literary works to
specialized publications, self-development books, and children's
literature. Each book is a new journey of discovery, expanding
knowledge and enriching the soul of the reade
Our website is not just a platform for buying books, but a bridge
connecting readers to the timeless values of culture and wisdom. With
an elegant, user-friendly interface and an intelligent search system,
we are committed to providing a quick and convenient shopping
experience. Additionally, our special promotions and home delivery
services ensure that you save time and fully enjoy the joy of reading.
Let us accompany you on the journey of exploring knowledge and
personal growth!
ebookultra.com

More Related Content

PDF
Separation of Variables for Partial Differential Equations An Eigenfunction A...
PDF
Separation of Variables for Partial Differential Equations An Eigenfunction A...
PDF
Separation of Variables for Partial Differential Equations An Eigenfunction A...
PDF
Wavelets and other orthogonal systems Second Edition Shen
PDF
Wavelets And Other Orthogonal Systems Second Edition Shen Xiaoping Walter
PDF
Willmore Energy and Willmore Conjecture 1st Edition Magdalena D. Toda
PDF
Difference Equations Theory Applications and Advanced Topics 3rd Edition Rona...
PDF
Measure and Integral An Introduction to Real Analysis 2nd Edition Richard L. ...
Separation of Variables for Partial Differential Equations An Eigenfunction A...
Separation of Variables for Partial Differential Equations An Eigenfunction A...
Separation of Variables for Partial Differential Equations An Eigenfunction A...
Wavelets and other orthogonal systems Second Edition Shen
Wavelets And Other Orthogonal Systems Second Edition Shen Xiaoping Walter
Willmore Energy and Willmore Conjecture 1st Edition Magdalena D. Toda
Difference Equations Theory Applications and Advanced Topics 3rd Edition Rona...
Measure and Integral An Introduction to Real Analysis 2nd Edition Richard L. ...

Similar to Separation of Variables for Partial Differential Equations An Eigenfunction Approach 1st Edition George Cain (20)

PDF
Higherorder Finite Element Methods Pavel Solin Karel Segeth
PDF
Spectral And Scattering Theory For Second Order Partial Differential Operator...
PDF
Real Analysis And Foundations 2nd Edition Steven G Krantz
PDF
Method Of Averaging For Differential Equations On An Infinite Interval Theory...
PDF
Computational Functional Analysis Second Edition Ramon E Moore
PDF
Advanced Differential Quadrature Methods 1st Edition Zhi Zong
PDF
Applied Analysis By The Hilbert Space Method An Introduction With Application...
PDF
Differential forms on singular varieties De Rham and Hodge theory simplified ...
PDF
Classes of modules 1st Edition John Dauns
PDF
Fourier analysis and Hausdorff dimension 1st Edition Mattila
PDF
Control theory of partial differential equations 1st Edition Guenter Leugering
PDF
Measure theory and fine properties of functions Evans
PDF
Mathematical Models and Methods for Real World Systems 1st Edition K.M. Furati
PDF
Fundamental Number Theory With Applications 2nd Edition Richard A Mollin
PDF
Classes of modules 1st Edition John Dauns
PDF
Geometric Function Theory and Non linear Analysis 1st Edition Tadeusz Iwaniec
PDF
Hermitian Analysis From Fourier Series To Cauchyriemann Geometry 1st Edition ...
PDF
Differential forms on singular varieties De Rham and Hodge theory simplified ...
PDF
Geometric Function Theory and Non linear Analysis 1st Edition Tadeusz Iwaniec
PDF
Diffusions Superdiffusions And Pdes Dynkin Eb
Higherorder Finite Element Methods Pavel Solin Karel Segeth
Spectral And Scattering Theory For Second Order Partial Differential Operator...
Real Analysis And Foundations 2nd Edition Steven G Krantz
Method Of Averaging For Differential Equations On An Infinite Interval Theory...
Computational Functional Analysis Second Edition Ramon E Moore
Advanced Differential Quadrature Methods 1st Edition Zhi Zong
Applied Analysis By The Hilbert Space Method An Introduction With Application...
Differential forms on singular varieties De Rham and Hodge theory simplified ...
Classes of modules 1st Edition John Dauns
Fourier analysis and Hausdorff dimension 1st Edition Mattila
Control theory of partial differential equations 1st Edition Guenter Leugering
Measure theory and fine properties of functions Evans
Mathematical Models and Methods for Real World Systems 1st Edition K.M. Furati
Fundamental Number Theory With Applications 2nd Edition Richard A Mollin
Classes of modules 1st Edition John Dauns
Geometric Function Theory and Non linear Analysis 1st Edition Tadeusz Iwaniec
Hermitian Analysis From Fourier Series To Cauchyriemann Geometry 1st Edition ...
Differential forms on singular varieties De Rham and Hodge theory simplified ...
Geometric Function Theory and Non linear Analysis 1st Edition Tadeusz Iwaniec
Diffusions Superdiffusions And Pdes Dynkin Eb
Ad

Recently uploaded (20)

PDF
01-Introduction-to-Information-Management.pdf
PDF
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
PDF
Anesthesia in Laparoscopic Surgery in India
PPTX
school management -TNTEU- B.Ed., Semester II Unit 1.pptx
PDF
Supply Chain Operations Speaking Notes -ICLT Program
PDF
TR - Agricultural Crops Production NC III.pdf
PDF
Abdominal Access Techniques with Prof. Dr. R K Mishra
PDF
Classroom Observation Tools for Teachers
PDF
VCE English Exam - Section C Student Revision Booklet
PPTX
human mycosis Human fungal infections are called human mycosis..pptx
PDF
Sports Quiz easy sports quiz sports quiz
PPTX
Pharma ospi slides which help in ospi learning
PDF
O5-L3 Freight Transport Ops (International) V1.pdf
PDF
Computing-Curriculum for Schools in Ghana
PDF
STATICS OF THE RIGID BODIES Hibbelers.pdf
PDF
Physiotherapy_for_Respiratory_and_Cardiac_Problems WEBBER.pdf
PPTX
Pharmacology of Heart Failure /Pharmacotherapy of CHF
PDF
O7-L3 Supply Chain Operations - ICLT Program
PPTX
Microbial diseases, their pathogenesis and prophylaxis
PPTX
1st Inaugural Professorial Lecture held on 19th February 2020 (Governance and...
01-Introduction-to-Information-Management.pdf
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
Anesthesia in Laparoscopic Surgery in India
school management -TNTEU- B.Ed., Semester II Unit 1.pptx
Supply Chain Operations Speaking Notes -ICLT Program
TR - Agricultural Crops Production NC III.pdf
Abdominal Access Techniques with Prof. Dr. R K Mishra
Classroom Observation Tools for Teachers
VCE English Exam - Section C Student Revision Booklet
human mycosis Human fungal infections are called human mycosis..pptx
Sports Quiz easy sports quiz sports quiz
Pharma ospi slides which help in ospi learning
O5-L3 Freight Transport Ops (International) V1.pdf
Computing-Curriculum for Schools in Ghana
STATICS OF THE RIGID BODIES Hibbelers.pdf
Physiotherapy_for_Respiratory_and_Cardiac_Problems WEBBER.pdf
Pharmacology of Heart Failure /Pharmacotherapy of CHF
O7-L3 Supply Chain Operations - ICLT Program
Microbial diseases, their pathogenesis and prophylaxis
1st Inaugural Professorial Lecture held on 19th February 2020 (Governance and...
Ad

Separation of Variables for Partial Differential Equations An Eigenfunction Approach 1st Edition George Cain

  • 1. Separation of Variables for Partial Differential Equations An Eigenfunction Approach 1st Edition George Cain download pdf https://ebookultra.com/download/separation-of-variables-for-partial- differential-equations-an-eigenfunction-approach-1st-edition-george- cain/ Visit ebookultra.com today to download the complete set of ebook or textbook!
  • 2. We believe these products will be a great fit for you. Click the link to download now, or visit ebookultra.com to discover even more! Partial differential equations An introduction 2nd Edition Strauss W.A. https://ebookultra.com/download/partial-differential-equations-an- introduction-2nd-edition-strauss-w-a/ Handbook of Differential Equations Stationary Partial Differential Equations Volume 6 1st Edition Michel Chipot https://ebookultra.com/download/handbook-of-differential-equations- stationary-partial-differential-equations-volume-6-1st-edition-michel- chipot/ Partial Differential Equations A Unified Hilbert Space Approach 1st Edition Rainer Picard https://ebookultra.com/download/partial-differential-equations-a- unified-hilbert-space-approach-1st-edition-rainer-picard/ Applied Partial Differential Equations John Ockendon https://ebookultra.com/download/applied-partial-differential- equations-john-ockendon/
  • 3. Stochastic Partial Differential Equations 2nd Edition Chow https://ebookultra.com/download/stochastic-partial-differential- equations-2nd-edition-chow/ Solution techniques for elementary partial differential equations Third Edition Constanda https://ebookultra.com/download/solution-techniques-for-elementary- partial-differential-equations-third-edition-constanda/ Introduction to Partial Differential Equations Rao K.S https://ebookultra.com/download/introduction-to-partial-differential- equations-rao-k-s/ Control theory of partial differential equations 1st Edition Guenter Leugering https://ebookultra.com/download/control-theory-of-partial- differential-equations-1st-edition-guenter-leugering/ Partial Differential Equations 2nd Edition Lawrence C. Evans https://ebookultra.com/download/partial-differential-equations-2nd- edition-lawrence-c-evans/
  • 5. Separation of Variables for Partial Differential Equations An Eigenfunction Approach 1st Edition George Cain Digital Instant Download Author(s): George Cain, Gunter H. Meyer ISBN(s): 9781584884200, 1584884207 Edition: 1 File Details: PDF, 9.72 MB Year: 2005 Language: english
  • 6. Separation of Variables for Partial Differential Equations An Eigenfunction Approach STUDIES IN ADVANCED MATHEMATICS
  • 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
  • 15. PREFACE ix complication, regardless of the type of equation under consideration. A concise algorithmic approach results. Equally important to us is the second reason for a new exposition of the method of separation of variables. We wish to emphasize the power of the method by solving a great variety of problems which often go well beyond the usual textbook examples. Many of the applications ask questions which are not as easily resolved with numerical methods as with analytic approximate solutions. Of course, evaluation of these approximate solutions usually relies on numerical methods to integrate, solve linear systems or nonlinear equations, and to find values of special functions, but these methods by now may be considered universally available "black boxes." We are, however, mindful of the gap between the concept of a solution in principle and a demonstrably computable solution and try .to convey our experience with how well the eigenfunction approach actually solves the sample problems. The method of separation of variables from a spectral expansion view is presented in nine chapters. Chapter 1 collects some background information on the three dominant equa- tions of this text, the potential equation, the heat equation, and the wave equa- tion. We refer to these results when applying and analyzing the method of separation of variables. Chapter 2 contains a discussion of orthogonal projections which are used time and again to approximate given data functions in a specified finite-dimensional but parameter-dependent subspace. Chapter 3 introduces the subspace whose basis consists of the eigenfunctions of a so-called Sturm-Liouville problem associated with the application under consideration. These are the eigenfunctions of the title of this text. We cite results from the Sturm-Liouville theory and provide a table of eigenvalues and eigenfunctions that arise in the method of separation of variables. Chapter 4 treats the case in which the eigenfunctions are sine and cosine functions with a common period. In this case the projection into the subspace is closely related to the Fourier series representation of the data functions. Precise information about the convergence of the Fourier series is known. We cite those results which are helpful later on for the application of separation of variables. Chapter 5 constitutes the heart of the text. We consider a partial differen- tial equation in two independent variables with a source term and subject to boundary and initial conditions. We give the algorithm for approximating such a problem and for solving it in a finite-dimensional space spanned by eigen- functions determined by the "spacial part" of the equation and its boundary conditions. We illustrate in broad outline the application of this approach to the heat, wave, and potential equations. Chapter 6 gives an expansive exposition of the algorithm for the one-dimen- sional heat equation. It contains many worked examples with comments on the numerical performance of the method, and concludes with a rudimentary analysis of the error in the approximate solution. Chapter 7 parallels the previous chapter but treats the wave equation.
  • 16. x PREF...CE Chapter 8 deals with the potential equation. It describes how one can pr~ condition the data of problems with smooth solutions in order not to introdi.:ce artificial discontinuities into the separation of variables solution. We solve P'=- tential problems with various boundary conditions and conclude with a calcu- lation of eigenfunctions for the two-dimensional Laplacian. Chapter 9 uses the eigenfunctions of the preceding chapter to find eigenfunc- tion expansion solutions of two- and three-dimensional heat, wave, and potenti.a:: equations. This text is written for advanced undergraduate and graduate students :.::. science and engineering with previous exposure to a course in engineering math- ematics, but not necessarily separation of variables. Basic prerequisites beyond calculus are familiarity with linear algebra, the concept of vector spaces of func- tions, norms and inner products, the ability to solve linear inhomogeneous first and second order ordinary differential equations, and some contact with practi- cal applications of partial differential equations. The book contains more material than can (and should) be taught in a cour~ on separation of variables. We have introduced the eigenfunction approach to our own students based on an early version of this text. We covered parts of Chapters 2-4 to lay the groundwork for an extensive discussion of Chapter 5. The remainder of the term was filled by working through selected examples involving the heat, wave, and potential equation. We believe that by term·s end the students had an appreciation that they could solve realistic problems. Since we view Chapters 2-5 as suitable for teaching separation of variables, we have included exercises to help deepen the reader's understanding of the eigenfunction approach. The examples of Chapters 6-8 and their exercise sets generally lend themselves for project assignments. This text will put a bigger burden on the instructor to choose topics and guide students than more elementary texts on separation of variables that start with product solutions. The instructor who subscribes to the view put forth in Chapter 5 should find this text workable. The more advanced applications, such as interface, inverse, and multidimensional problems, as well as the the more theoretical topics require more mathematical sophistication and may be skipped without breaking continuity. The book is also meant to serve as a reference text for the method of separa- tion of variables. We hope the many examples will guide the reader in deciding whether and how to apply the method to any given problem. The examples should help in interpreting computed solutions, and should give insight into those cases in which formal answers are useless because of lack of convergence or unacceptable oscillations. Chapters 1 and 9 are included to support the reference function. They do not include exercises. We hasten to add that this text is not a complete reference book. We do not attempt to characterize the equations and coordinate systems where a separation of variables is applicable. We do not even mention the various coordinate systems (beyond cartesian, polar, cylindrical, and spherical) in which the Laplacian is separable. We _have not scoured the literature for new and innovative applications of separation of variables. Moreover, the examples we do
  • 17. PREFACE xi include are often meant to show structure rather than represent reality because in general little attention is given to the proper scaling of the equations. There does not appear to exist any other source that could serve as a prac- tical reference book for the practicing engineer or scientist. We hope this book will alert the reader that separation of variables has more to offer than may be apparent from elementary texts. Finally, this text does not mention the implementation of our formulas and calculations on the computer, or do we provide numerical algorithms or pro- grams. Yet the text, and in particular our numerical examples, could not have been presented without access to symbolic and numerical packages such as !viaple, Mathematica, and Matlab. We consider our calculations and the graph- ical representation of their results routine and well within the competence of today's students and practitioners of science and engineering.
  • 19. Contents Acknowledgments v Preface vii 1 Potential, Heat, and Wave Equation 1 1.1 Overview . . . . . . . . . . . . . . 1.2 Classification of second order equations 2 1.3 Laplace's and Poisson's equation 3 1.4 The heat equation 11 1.5 The wave equation . 18 2 Basic Approximation Theory 25 2.1 Norms and inner products . . 26 2.2 Projection and best approximation 29 2.3 Important function spaces 34 3 Sturm-Liouville Problems 45 3.1 Sturm-Liouville problems for q/' = µ¢ 45 3.2 Sturm-Liouville problems for £¢ = µ¢ . . . 53 3.3 A Sturm-Liouville problem with an interface 59 4 Fourier Series 67 4.1 Introduction . 67 4.2 Convergence . 68 4.3 Convergence of Fourier series 74 4.4 Cosine and sine series . . . . 77 4.5 Operations on Fourier series . 80 4.6 Partial sums of the Fourier series and the Gibbs phenomenon 84 5 Eigenfunction Expansions for Equations in Two Independent Variables 95
  • 20. XiY 6 CONTENTS One-Dimensional Diffusion Equation 6.1 Applications of the eigenfunction expansion method Example 6.1 How many terms of the series solution are enough? Example 6.2 Determination of an unknown diffusivity from mea- sured data .. Example 6.3 Thermal waves Example 6.4 Matching a temperature history Example 6.5 Phase shift for a thermal wave . Example 6.6 Dynamic determination of a convective heat transfer coefficient from measured data . . . . . . . . . . . . Example 6.7 Radial heat flow in a sphere Example 6.8 A boundary layer problem . Example 6.9 The Black-Scholes equation Example 6.10 Radial heat flow in a disk . Example 6.11 Heat flow in a composite slab Example 6.12 Reaction-diffusion with blowup 6.2 Convergence of UN(x, t) to the analytic solution : 6.3 Influence _of the boundary conditions and Duhamel's solution 115 115 115 119 120. 125 129 131 134 137 139 142 146 148 151 155 7 One-Dimensional Wave Equation 161 7.1 Applicatfons of the eigenfunction expansion method - : 161 Example 7.1 A vibrating string with initial displacement . 161 Example 7.2 A vibrating string with initial velocity . . 166 Example 7.3 A forced wave and resonance . 168 Example 7.4 Wave propagation in a resistive medium. 171 Example 7.5 Oscillations of a hanging chain 175 Example 7.6 Symmetric pressure wave in a sphere 177 Example 7.7 Controlling the shape of a wave . . . 180 Example 7.8 The natural frequencies. of a uniform beam 182 Example 7.9 A system of wave equations . . . . . 185 7.2 Convergence of UN(x, t) to the analytic solution . 188 7.3 Eigenfunction expansions and Duhamel's principle . 190 8 Potential Problems in the Plane 195 8.1 Applications of the eigenfunction expansion method . 195 Example 8.1 The Dirichlet problem for the Laplacian on a rectanglel95 Example 8.2 Preconditioning for general boundary data . . 201 Example 8.3 Poisson's equation with Neumann boundary data 213 Example 8.4 A discontinuous potential 215 Example 8.5 Lubrication of a plane slider bearing . 218 Example 8.6 Lubrication of a step bearing . . 220 Example 8.7 The Dirichlet problem on an £-shaped domain 221 Example 8.8 Poisson's equation in polar coordinates . . . . 225 Example 8.9 Steady-state heat flow around an insulated pipe I 230 Example 8.10 Steady-state heat flow around an insulated pipe II 232
  • 21. CONTENTS xv Example 8.11 Poisson's equation on a triangle . 234 8.2 Eigenvalue problem for the two-dimensional Laplacian 237 Example 8.12 The eigenvalue problem for the Laplacian on a rectangle . . . . . . . . . . 237 Example 8.13 The Green's function for the Laplacian on a square 239 Example 8.14 The eigenvalue problem for the Laplacian on a disk 243 Example 8.15 The .eigenvalue problem for the Laplacian on the surface of a sphere . . . . . . . . . 244 8.3 Convergence of UN(x, y) to the analytic solution 247 9 Multidimensional Problems 255 9.1 Applications of the eigenfunction expansion method 255 Example 9.1 A diffusive pulse test . . . . . . . 255 Example 9.2 Standing waves on a circular membrane 258 Example 9.3 The potential inside a charged sphere 260 Example 9.4 Pressure in a porous slider bearing . . . 261 9.2 The eigenvalue problem for the Laplacian in ffi:.3 . . . 265 Example 9.5 An eigenvalue problem for quadrilaterals 265 Example 9.6 An eigenvalue problem for the Laplacian in a cylinder266 Example 9.7 Periodic heat flow in a cylinder 267 Example 9.8 An eigenvalue problem for the Laplacian in a sphere 269 Example 9.9 The eigenvalue problem for Schrodinger's equation with a spherically symmetric potential well . . . . 271 Bibliography 277 Index 279
  • 23. Chapter 1 Potential, Heat, and Wave Equation This chapter provides a quick look into the vast field of partial differential equations. The main goal is to extract some qualitative results on the three dominant equations of mathematical physics, the potential, heat, and wave equation on which our attention will be focused throughout this text. 1.1 Overview When processes that change smoothly with two or more independent variables are modeled mathematically, then partial differential equations arise. Most common are second order equations of the general form M M Lu= L aijUxix, + L biuxi +cu = F, xE DC IRM (1.1) i,j=l i=l where the coefficients and the source term may depend on the independent vari- ables {x1, ... ,xM}, on u, and on its derivatives. D is a given set in !RM (whose boundary will be denoted by [)D). The equation may reflect conservation and balance laws, empirical relationships, or may be purely phenomenological. Its solution is used to explain, predict, and control processes in a bewildering array of applications ranging from heat, mass, and fluid flow, migration of biological species, electrostatics, and molecular vibration to mortgage banking. In (1.1) C. is known as a partial differential operator that maps a smooth function u to the function C.u. Throughout this text a smooth function denotes a function with as many continuous derivatives as are necessary to carry out the operations to which it is subjected. lu = F is the equation to be solved. Given a partial differential equation and side constraints on its solution, typically initial and boundary conditions, it becomes a question of mathematical analysis to establish whether the problem has a solution, whether the solution
  • 24. 2 CHAPTER 1. POTENTIAL, HEAT, AND WAVE EQUATION is unique, and whether the solution changes continuously with the data of the problem. If that is the case, then the given problem for (1.1) is said to be well posed; if not then it is ill posed. We note here that the data of the problem are the coefficients of£, the source term F, any side conditions imposed on u, and the shape of D. However, dependence on the coefficients and on the shape of D will be ignored. Only continuous dependence with respect to the source term and the side conditions will define well posedness for our purposes. The technical aspects of in what sense a function u solves the problem and in what sense it changes with the data of the problem tend to be abstract and complex and constitute the mathematical theory of partial differential equations (e.g., [5]). Such theoretical studies are essential to establish that equation (1.1) and its side conditions are a consistent description of the processes under consid- eration and to characterize the behavior of its solution. Outside mathematics the validity of a mathematical model is often taken on faith and its solution is assumed to exist on "physical grounds." There the emphasis is entirely on solving the equation, analytically if possible, or approximately and numerically otherwise. Approximate solutions are the subject of this text. 1.2 Classification of second order equations The tools for the analysis and solution of (1.1) depend on the structure of the coefficient matrix A= {aij} in (1.1). By assuming that ux,x1 = ux1x, we can always write A in such a way that it is symmetric. For example, if the equation which arises in modeling a process is then it will be rewritten as so that A= G~) We can now introduce three broad classes of differential equations. Definition The operator £ given by M M Lu =L aijUxtx) + L biuXi +cu i,j=l i::::l is i) Elliptic at x= (x1 , ••. , XM) if all eigenvalues of the symmetric matrix A are nonzero and have the same algebraic sign,
  • 25. 1.3. LAPLACE'S AND POISSON'S EQUATION 3 ii) Hyperbolic at i if all eigenvalues of A are nonzero and one has a different algebraic sign from all others, ili) Parabolic at x if A has a zero eigenvalue. If A depends on u and its derivatives, then £ is elliptic, etc. at a given point relative to a specific function u. If the operator£ is elliptic at a point then (1.1) is an elliptic equation at that point. (As mnemonic we note that for M = 2 the level sets of AG~),(~~)) =constant, where (i, if') denotes the dot product of x and y, are elliptic, hyperbolic, and parabolic under the above conditions on the eigenvalues of A). The lower order ~erms in (1.1) do not affect the type of the equation, but in particular applica- tions they can dominate the behavior of the solution of (1.1). Each class of equations has its own admissible side conditions to make (Ll) well posed, and all solutions of the same class have, broadly speaking, common characteristics. We shall list some of them for the three dominant equations of mathematical physics: Laplace's equation, the heat equation, and the wave equation. 1.3 Laplace's and Poisson's equation The most extensively studied example of an elliptic equation is Laplace's equa- ~ion £u =1 ·Ju= 0 which arises in potential problems, steady-state heat conduction, irrotational flow, minimal surface problems, and myriad other applications. The operator £u is known as the Laplacian and is generally denoted by £u = 1 ·Ju= 72 u = 6.u. The last form is common in the mathematical literature and will be used con- sistently throughout this text. The Laplacian in cartesian coordinates assumes the forms i.i In polar coordinates (r, ()) M ~u. = LUxixi i=l ii) In cylindrical coordinates (r, (), z)
  • 26. Random documents with unrelated content Scribd suggests to you:
  • 27. evil282—save insofar as evil may be a beneficent penalty and discipline. At the same time, while advising the imprisonment or execution of heretics who did not believe in the Gods, Plato regarded with even greater detestation the man who taught that they could be persuaded or propitiated by individual prayer and sacrifice.283 Thus he would have struck alike at the freethinking few and at the multitude who held by the general religious beliefs of Greece, dealing damnation on all save his own clique, in a way that would have made Torquemada blench.284 In the face of such teaching as this, it may well be said that “Greek philosophy made incomparably greater advances in the earlier polemic period [of the Ionians] than after its friendly return to the poetry of Homer and Hesiod”285—that is, to their polytheistic basis. It is to be said for Plato, finally, that his embitterment at the downward course of things in Athens is a quite intelligible source for his own intellectual decadence: a very similar spectacle being seen in the case of our own great modern Utopist, Sir Thomas More. But Plato’s own writing bears witness that among the unbelievers against whom he declaimed there were wise and blameless citizens;286 while in the act of seeking to lay a religious basis for a good society he admitted the fundamental immorality of the religious basis of the whole of past Greek life. 3. Aristotle [384–322], like Sokrates, albeit in a very different way, rendered rather an indirect than a direct service to Freethought. Where Sokrates gave the critical or dialectic method or habit, “a process of eternal value and of universal application,”287 Aristotle supplied the great inspiration of system, partly correcting the Sokratic dogmatism on the possibilities of science by endless observation and speculation, though himself falling into scientific dogmatism only too often. That he was an unbeliever in the popular and Platonic religion is clear. Apart from the general rationalistic tenor of his works,288 there was a current understanding that the Peripatetic school denied the utility of prayer and sacrifice;289 and though the essentially partisan attempt of the anti- Macedonian party to impeach him for impiety may have turned largely on his hyperbolic hymn to his dead friend Hermeias (who was a eunuch, and as such held peculiarly unworthy of being addressed as on a level with semi-
  • 28. divine heroes),290 it could hardly have been undertaken at all unless he had given solider pretexts. The threatened prosecution he avoided by leaving the city, dying shortly afterwards. Siding as he did with the Macedonian faction, he had put himself out of touch with the democratic instincts of the Athenians, and so doubly failed to affect their thinking. But nonetheless the attack upon him by the democrats was a political stratagem. The prosecution for blasphemy had now become a recognized weapon in politics for all who had more piety than principle, and perhaps for some who had neither. And Aristotle, well aware of the temper of the population around him, had on the whole been so guarded in his utterance that a fantastic pretext had to be fastened on for his undoing. Prof. Bain (Practical Essays, p. 273), citing Grote’s remark on the “cautious prose compositions of Aristotle,” comments thus: “That is to say, the execution of Sokrates was always before his eyes; he had to pare his expressions so as not to give offence to Athenian orthodoxy. We can never know the full bearings of such a disturbing force. The editors of Aristotle complain of the corruption of his text: a far worse corruption lies behind. In Greece Sokrates alone had the courage of his opinions. While his views as to a future life, for example, are plain and frank, the real opinion of Aristotle on the question is an insoluble problem.” (See, however, the passage in the Metaphysics cited below.) The opinion of Grote and Bain as to Aristotle’s caution is fully coincided in by Lange, who writes (Gesch. des Mater. i, 63): “More conservative than Plato and Sokrates, Aristotle everywhere seeks to attach himself as closely as possible to tradition, to popular notions, to the ideas embodied in common speech, and his ethical postulates diverge as little as may be from the customary morals and laws of Greek States. He has therefore been at all times the favourite philosopher of conservative schools and movements.” It is clear, nevertheless, if we can be sure of his writings, that he was a monotheist, but a monotheist with no practical religion. “Excluding such a thing as divine interference with Nature, his theology, of course, excludes the possibility of revelation, inspiration, miracles, and grace.”291 In a passage in the Metaphysics, after elaborating his monistic conception of Nature, he dismisses in one or two terse sentences the whole current religion as a mass of myth framed to persuade the multitude, in the interest of law and order.292 His influence must thus have been to some extent, at least, favourable to rational science, though unhappily his own science is too often a blundering reaction against the surmises of earlier thinkers with
  • 29. a greater gift of intuition than he, who was rather a methodizer than a discoverer.293 What was worst in his thinking was its tendency to apriorism, which made it in a later age so adaptable to the purposes of the Roman Catholic Church. Thus his doctrines of the absolute levity of fire and of nature’s abhorrence of a vacuum set up a hypnotizing verbalism, and his dictum that the earth is the centre of the universe was fatally helpful to Christian obscurantism. For the rest, while guiltless of Plato’s fanaticism, he had no scheme of reform whatever, and was as far as any other Greek from the thought of raising the mass by instruction. His own science, indeed, was not progressive, save as regards his collation of facts in biology; and his political ideals were rather reactionary; his clear perception of the nature of the population problem leaving him in the earlier attitude of Malthus, and his lack of sympathetic energy making him a defender of slavery when other men had condemned it.294 He was in some aspects the greatest brain of the ancient world; and he left it, at the close of the great Grecian period, without much faith in man, while positing for the modern world its vaguest conception of Deity. Plato and Aristotle between them had reduced the ancient God-idea to a thin abstraction. Plato would not have it that God was the author of evil, thus leaving evil unaccounted for save by sorcery. Aristotle’s God does nothing at all, existing merely as a potentiality of thought. And yet upon those positions were to be founded the theisms of the later world. Plato had not striven, and Aristotle had failed, to create an adequate basis for thought in real science; and the world gravitated back to religion. [In previous editions I remarked that “the lack of fresh science, which was the proximate cause of the stagnation of Greek thought, has been explained like other things as a result of race qualities: ‘the Athenians,’ says Mr. Benn (The Greek Philosophers, i, 42), ‘had no genius for natural science: none of them were ever distinguished as savans.... It was, they thought, a miserable trifling [and] waste of time.... Pericles, indeed, thought differently....’ On the other hand, Lange decides (i, 6) “that with the freedom and boldness of the Hellenic spirit was combined ... the talent for scientific deduction. These contrary views,” I observed, “seem alike arbitrary. If Mr. Benn means that other Hellenes had what the Athenians lacked, the answer is that only special social conditions could have set up such a difference, and that it could not be innate, but must be a mere matter of usage.” Mr. Benn has explained to me that he does not dissent from this view, and that I had not rightly gathered his from the passage I quoted. In his later work, The Philosophy of Greece
  • 30. considered in relation to the character and history of its people (1898), he has pointed out how, in the period of Hippias and Prodikos, “at Athens in particular young men threw themselves with ardour into the investigation of” problems of cosmography, astronomy, meteorology, and comparative anatomy (p. 138). The hindering forces were Athenian bigotry (pp. 113–14, 171) and the mischievous influence of Sokrates (pp. 165, 173). Speaking broadly, we may say that the Chaldeans were forward in astronomy because their climate favoured it to begin with, and religion and their superstitions did so later. Hippokrates of Kos became a great physician because, with natural capacity, he had the opportunity to compare many practices. The Athenians failed to carry on the sciences, not because the faculty or the taste was lacking among them, but because their political and artistic interests, for one thing, preoccupied them—e.g., Sokrates and Plato; and because, for another, their popular religion, popularly supported, menaced the students of physics. But the Ionians, who had savans, failed equally to progress after the Alexandrian period; the explanation being again not stoppage of faculty, but the advent of conditions unfavourable to the old intellectual life, which in any case, as we saw, had been first set up by Babylonian contacts. (Compare, on the ethnological theorem of Cousin, G. Bréton, Essai sur la poésie philos. en Grèce, p. 10.) On the other hand, Lange’s theory of gifts “innate” in the Hellenic mind in general is the old racial fallacy. Potentialities are “innate” in all populations, according to their culture stage, and it was their total environment that specialized the Greeks as a community.] § 9 The overthrow of the “free” political life of Athens was followed by a certain increase in intellectual activity, the result of throwing back the remaining store of energy on the life of the mind. By this time an almost open unbelief as to the current tales concerning the Gods would seem to have become general among educated people, the withdrawal of the old risk of impeachment by political factions being so far favourable to outspokenness. It is on record that the historian Ephoros (of Cumæ in Æolia: fl. 350 B.C.), who was a pupil of Isocrates, openly hinted in his work at his disbelief in the oracle of Apollo, and in fabulous traditions generally.295 In other directions there were similar signs of freethought. The new schools of philosophy founded by Zeno the Stoic (fl. 280: d. 263 or 259) and Epicurus (341–270), whatever their defects, compare not ill with
  • 31. those of Plato and Aristotle, exhibiting greater ethical sanity and sincerity if less metaphysical subtlety. Of metaphysics there had been enough for the age: what it needed was a rational philosophy of life. But the loss of political freedom, although thus for a time turned to account, was fatal to continuous progress. The first great thinkers had all been free men in a politically free environment: the atmosphere of cowed subjection, especially after the advent of the Romans, could not breed their like; and originative energy of the higher order soon disappeared. Sane as was the moral philosophy of Epicurus, and austere as was that of Zeno, they are alike static or quietist,296 the codes of a society seeking a regulating and sustaining principle rather than hopeful of new achievement or new truth. And the universal skepticism of Pyrrho has the same effect of suggesting that what is wanted is not progress, but balance. It is significant that he, who carried the Sokratic profession of Nescience to the typical extreme of doctrinal Nihilism, was made high-priest of his native town of Elis, and had statues erected in his honour.297 Considered as freethinkers, all three men tell at once of the critical and of the reactionary work done by the previous age. Pyrrho, the universal doubter, appears to have taken for granted, with the whole of his followers, such propositions as that some animals (not insects) are produced by parthenogenesis, that some live in the fire, and that the legend of the Phœnix is true.298 Such credences stood for the arrest of biological science in the Sokratic age, with Aristotle, so often mistakenly, at work; while, on the other hand, the Sokratic skepticism visibly motives the play of systematic doubt on the dogmas men had learned to question. Zeno, again, was substantially a monotheist; Epicurus, adopting but not greatly developing the science of Demokritos,299 turned the Gods into a far-off band of glorious spectres, untroubled by human needs, dwelling for ever in immortal calm, neither ruling nor caring to rule the world of men.300 In coming to this surprising compromise, Epicurus, indeed, probably did not carry with him the whole intelligence even of his own school. His friend, the second Metrodoros of Lampsakos, seems to have been the most stringent of all the censors of Homer, wholly ignoring his namesake’s attempts to clear the bard of impiety. “He even advised men not to be
  • 32. ashamed to confess their utter ignorance of Homer, to the extent of not knowing whether Hector was a Greek or a Trojan.”301 Such austerity towards myths can hardly have been compatible with the acceptance of the residuum of Epicurus. That, however, became the standing creed of the sect, and a fruitful theme of derision to its opponents. Doubtless the comfort of avoiding direct conflict with the popular beliefs had a good deal to do with the acceptance of the doctrine. This strange retention of the theorem of the existence of anthropomorphic Gods, with a flat denial that they did anything in the universe, might be termed the great peculiarity of average ancient rationalism, were it not that what makes it at all intelligible for us is just the similar practice of modern non-Christian theists. The Gods of antiquity were non-creative, but strivers and meddlers and answerers of prayer; and ancient rationalism relieved them of their striving and meddling, leaving them no active or governing function whatever, but for the most part cherishing their phantasms. The God of modern Christendom had been at once a creator and a governor, ruling, meddling, punishing, rewarding, and hearing prayer; and modern theism, unable to take the atheistic or agnostic plunge, relieves him of all interference in things human or cosmic, but retains him as a creative abstraction who somehow set up “law,” whether or not he made all things out of nothing. The psychological process in the two cases seems to be the same—an erection of æsthetic habit into a philosophic dogma, and an accommodation of phrase to popular prejudice. Whatever may have been the logical and psychological crudities of Epicureanism, however, it counted for much as a deliverance of men from superstitious fears; and nothing is more remarkable in the history of ancient philosophy than the affectionate reverence paid to the founder’s memory302 on this score through whole centuries. The powerful Lucretius sounds his highest note of praise in telling how this Greek had first of all men freed human life from the crashing load of religion, daring to pass the flaming ramparts of the world, and by his victory putting men on an equality with heaven.303 The laughter-loving Lucian two hundred years later grows gravely eloquent on the same theme.304 And for generations the effect of
  • 33. the Epicurean check on orthodoxy is seen in the whole intellectual life of the Greek world, already predisposed in that direction.305 The new schools of the Cynics and the Cyrenaics had alike shown the influence in their perfect freedom from all religious preoccupation, when they were not flatly dissenting from the popular beliefs. Antisthenes, the founder of the former school (fl. 400 B.C.), though a pupil of Sokrates, had been explicitly anti- polytheistic, and an opponent of anthropomorphism.306 Aristippos of Cyrene, also a pupil of Socrates, who a little later founded the Hedonic or Cyrenaic sect, seems to have put theology entirely aside. One of the later adherents of the school, Theodoros, was like Diagoras labelled “the Atheist”307 by reason of the directness of his opposition to religion; and in the Rome of Cicero he and Diagoras are the notorious atheists of history.308 To Theodoros, who had a large following, is attributed an influence over the thought of Epicurus,309 who, however, took the safer position of a verbal theism. The atheist is said to have been menaced by Athenian law in the time of Demetrius Phalereus, who protected him; and there is even a story that he was condemned to drink hemlock;310 but he was not of the type that meets martyrdom, though he might go far to provoke it.311 Roaming from court to court, he seems never to have stooped to flatter any of his entertainers. “You seem to me,” said the steward of Lysimachos of Thrace to him on one occasion, “to be the only man who ignores both Gods and kings.”312 In the same age the same freethinking temper is seen in Stilpo of Megara (fl. 307), of the school of Euclides, who is said to have been brought before the Areopagus for the offence of saying that the Pheidian statue of Athênê was “not a God,” and to have met the charge with the jest that she was in reality not a God but a Goddess; whereupon he was exiled.313 The stories told of him make it clear that he was an unbeliever, usually careful not to betray himself. Euclides, too, with his optimistic pantheism, was clearly a heretic; though his doctrine that evil is non-ens314 later became the creed of some Christians. Yet another professed atheist was the witty Bion of Borysthenes, pupil of Theodoros, of whom it is told, in a fashion familiar to our own time, that in sickness he grew pious through fear.315 Among his positions was a protest or rather satire against the doctrine that the Gods
  • 34. punished children for the crimes of their fathers.316 In the other schools, Speusippos (fl. 343), the nephew of Plato, leant to monotheism;317 Strato of Lampsakos, the Peripatetic (fl. 290), called “the Naturalist,” taught sheer pantheism, anticipating Laplace in declaring that he had no need of the action of the Gods to account for the making of the world;318 Dikaiarchos (fl. 326–287), another disciple of Aristotle, denied the existence of separate souls, and the possibility of foretelling the future;319 and Aristo and Cleanthes, disciples of Zeno, varied likewise in the direction of pantheism; the latter’s monotheism, as expressed in his famous hymn, being one of several doctrines ascribed to him.320 Contemporary with Epicurus and Zeno and Pyrrho, too, was Evêmeros (Euhemerus), whose peculiar propaganda against Godism seems to imply theoretic atheism. As an atheist he was vilified in a manner familiar to modern ears, the Alexandrian poet Callimachus labelling him an “arrogant old man vomiting impious books.”321 His lost work, of which only a few extracts remain, undertook to prove that all the Gods had been simply famous men, deified after death; the proof, however, being by way of a fiction about old inscriptions found in an imaginary island.322 As above noted,323 the idea may have been borrowed from skeptical Phoenicians, the principle having already been monotheistically applied by the Bible-making Jews,324 though, on the other hand, it had been artistically and to all appearance uncritically acted on in the Homeric epopees. It may or may not then have been by way of deliberate or reasoning Evêmerism that certain early Greek and Roman deities were transformed, as we have seen, into heroes or hetairai.325 In any case, the principle seems to have had considerable vogue in the later Hellenistic world; but with the effect rather of paving the way for new cults than of setting up scientific rationalism in place of the old ones. Quite a number of writers like Palaiphatos, without going so far as Evêmeros, sought to reduce myths to natural possibilities and events, by way of mediating between the credulous and the incredulous.326 Their method is mostly the naïf one revived by the Abbé Banier in the eighteenth century of reducing marvels to verbal misconceptions. Thus for Palaiphatos the myth of Kerberos came from the facts that the city Trikarenos was commonly spoken of as a beautiful and
  • 35. great dog; and that Geryon, who lived there, had great dogs called Kerberoi; Actæon was “devoured by his dogs” in the sense that he neglected his affairs and wasted his time in hunting; the Amazons were shaved men, clad as were the women in Thrace, and so on.327 Palaiphatos and the Herakleitos who also wrote De Incredibilibus agree that Pasiphae’s bull was a man named Tauros; and the latter writer similarly explains that Scylla was a beautiful hetaira with avaricious hangers-on, and that the harpies were ladies of the same profession. If the method seems childish, it is to be remembered that as regards the explanation of supernatural events it was adhered to by German theologians of a century ago; and that its credulity in incredulity is still to be seen in the current view that every narrative in the sacred books is to be taken as necessarily standing for a fact of some kind. One of the inferrible effects of the Evêmerist method was to facilitate for the time the adoption of the Egyptian and eastern usage of deifying kings. It has been plausibly argued that this practice stands not so much for superstition as for skepticism, its opponents being precisely the orthodox believers, and its promoters those who had learned to doubt the actuality of the traditional Gods. Evêmerism would clinch such a tendency; and it is noteworthy that Evêmeros lived at the court of Kassander (319–296 B.C.) in a period in which every remaining member of the family of the deified Alexander had perished, mostly by violence; while the contemporary Ptolemy I of Egypt received the title of Sotêr, “Saviour,” from the people of Rhodes.328 It is to be observed, however, that while in the next generation Antiochus I of Syria received the same title, and his successor Antiochus II that of Theos, “God,” the usage passes away; Ptolemy III being named merely Evergetês, “the Benefactor” (of the priests), and even Antiochus III only “the Great.” Superstition was not to be ousted by a political exploitation of its machinery.329 In Athens the democracy, restored in a subordinate form by Kassander’s opponent, Demetrius Poliorkêtes (307 B.C.), actually tried to put down the philosophic schools, all of which, but the Aristotelian in particular, were anti-democratic, and doubtless also comparatively irreligious. Epicurus and some of his antagonists were exiled within a year of his opening his school
  • 36. (306 B.C.); but the law was repealed in the following year.330 Theophrastos, the head of the Aristotelian school, was indicted in the old fashion for impiety, which seems to have consisted in denouncing animal sacrifice.331 These repressive attempts, however, failed; and no others followed at Athens in that era; though in the next century the Epicureans seem to have been expelled from Lythos in Crete and from Messenê in the Peloponnesos, nominally for their atheism, in reality probably on political grounds.332 Thus Zeno was free to publish a treatise in which, besides far out-going Plato in schemes for dragooning the citizens into an ideal life, he proposed a State without temples or statues of the Gods or law courts or gymnasia.333 In the same age there is trace of “an interesting case of rationalism even in the Delphic oracle.”334 The people of the island of Astypalaia, plagued by hares or rabbits, solemnly consulted the oracle, which briefly advised them to keep dogs and take to hunting. About the same time we find Lachares, temporarily despot at Athens, plundering the shrine of Pallas of its gold.335 Even in the general public there must have been a strain of surviving rationalism; for among the fragments of Menander (fl. 300), who, in general, seems to have leant to a well-bred orthodoxy,336 there are some speeches savouring of skepticism and pantheism.337 It was in keeping with this general but mostly placid and non-polemic latitudinarianism that the New Academy, the second birth, or rather transformation, of the Platonic school, in the hands of Arkesilaos and the great Carneades (213–129), and later of the Carthaginian Clitomachos, should be marked by that species of skepticism thence called Academic—a skepticism which exposed the doubtfulness of current religious beliefs without going the Pyrrhonian length of denying that any beliefs could be proved, or even denying the existence of the Gods. For the arguments of Carneades against the Stoic doctrine of immortality see Cicero, De natura Deorum, iii, 12, 17; and for his argument against theism see Sextus Empiricus, Adv. Math. ix, 172, 183. Mr. Benn pronounces this criticism of theology “the most destructive that has ever appeared, the armoury whence religious skepticism ever since has been supplied” (The Philosophy of Greece, etc., p. 258). This seems an over-statement. But it is just to say, as does Mr. Whittaker (Priests, Philosophers, and Prophets, 1911, p. 60; cp. p.
  • 37. 86), that “there has never been a more drastic attack than that of Carneades, which furnished Cicero with the materials for his second book, On Divination”; and, as does Prof. Martha (Études Morales sur l’antiquité, 1889, p. 77), that no philosophic or religious school has been able to ignore the problems which Carneades raised. As against the essentially uncritical Stoics, the criticism of Carneades is sane and sound; and he has been termed by judicious moderns “the greatest skeptical mind of antiquity”338 and “the Bayle of Antiquity”;339 though he seems to have written nothing.340 There is such a concurrence of testimony as to the victorious power of his oratory and the invincible skill of his dialectic341 that he must be reckoned one of the great intellectual and rationalizing forces of his day, triumphing as he did in the two diverse arenas of Greece and Rome. His disciple and successor Clitomachos said of him, with Cicero’s assent, that he had achieved a labour of Hercules “in liberating our souls as it were of a fierce monster, credulity, conjecture, rash belief.”342 He was, in short, a mighty antagonist of thoughtless beliefs, clearing the ground for a rational life; and the fact that he was chosen with Diogenes the Peripatetic and Critolaos the Stoic to go to Rome to plead the cause of ruined Athens, mulcted in an enormous fine, proved that he was held in high honour at home. Athens, in short, was not at this stage “too superstitious.” Unreasoning faith was largely discredited by philosophy. On this basis, in a healthy environment, science and energy might have reared a constructive rationalism; and for a time astronomy, in the hands of Aristarchos of Samos (third century B.C.), Eratosthenes of Cyrene, the second keeper of the great Alexandrian library (2nd cent. B.C.), and above all of Hipparchos of Nikaia, who did most of his work in the island of Rhodes, was carried to a height of mastery which could not be maintained, and was re-attained only in modern times.343 Thus much could be accomplished by “endowment of research” as practised by the Ptolemies at Alexandria; and after science had declined with the decline of their polity, and still further under Roman rule, the new cosmopolitanism of the second century of the empire reverted to the principle of intelligent evocation, producing under the Antonines the “Second” School of Alexandria.
  • 38. But the social conditions remained fundamentally bad; and the earlier greatness was never recovered. “History records not one astronomer of note in the three centuries between Hipparchos and Ptolemy”; and Ptolemy (fl. 140 C.E.) not only retrograded into astronomical error, but elaborated on oriental lines a baseless fabric of astrology.344 Other science mostly decayed likewise. The Greek world, already led to lower intellectual levels by the sudden ease and wealth opened up to it through the conquests of Alexander and the rule of his successors, was cast still lower by the Roman conquest. Pliny, extolling Hipparchos with little comprehension of his work, must needs pronounce him to have “dared a thing displeasing to God” in numbering the stars for posterity.345 In the air of imperialism, stirred by no other, original thought could not arise; and the mass of the Greek-speaking populations, rich and poor, gravitated to the level of the intellectual346 and emotional life of more or less well-fed slaves. In this society there rapidly multiplied private religious associations—thiasoi, eranoi, orgeones—in which men and women, denied political life, found new bonds of union and grounds of division in cultivating worships, mostly oriental, which stimulated the religious sense and sentiment.347 Such was the soil in which Christianity took root and flourished; while philosophy, after the freethinking epoch following on the fall of Athenian power, gradually reverted to one or other form of mystical theism or theosophy, of which the most successful was the Neo-Platonism of Alexandria.348 When the theosophic Julian rejoiced that Epicureanism had disappeared,349 he was exulting in a symptom of the intellectual decline that made possible the triumph of the faith he most opposed. Christianity furthered a decadence thus begun under the auspices of pagan imperialism; and “the fifth century of the Christian era witnessed an almost total extinction of the sciences in Alexandria”350—an admission which disposes of the dispute as to the guilt of the Arabs in destroying the great library. Here and there, through the centuries, the old intellectual flame burns whitely enough: the noble figure of Epictetus in the first century of the new era, and that of the brilliant Lucian in the second, in their widely different ways remind us that the evolved faculty was still there if the
  • 39. circumstances had been such as to evoke it. Menippos in the first century B.C. had played a similar part to that of Lucian, in whose freethinking dialogues he so often figures; but with less of subtlety and intellectuality. Lucian’s was indeed a mind of the rarest lucidity; and the argumentation of his dialogue Zeus Tragædos covers every one of the main aspects of the theistic problem. There is no dubiety as to his atheistic conclusion, which is smilingly implicit in the reminder he puts in the mouth of Hermes, that, though a few men may adopt the atheistic view, “there will always be plenty of others who think the contrary—the majority of the Greeks, the ignorant many, the populace, and all the barbarians.” But the moral doctrine of Epictetus is one of endurance and resignation; and the almost unvarying raillery of Lucian, making mere perpetual sport of the now moribund Olympian Gods, was hardly better fitted than the all-round skepticism of the school of Sextus Empiricus to inspire positive and progressive thinking. This latter school, described by Cicero as dispersed and extinct in his day,351 appears to have been revived in the first century by Ænesidemos, who taught at Alexandria.352 It seems to have been through him in particular that the Pyrrhonic system took the clear-cut form in which it is presented at the close of the second century by the accomplished Sextus “Empiricus”—that is, the empirical (i.e., experiential) physician,353 who lived at Alexandria and Athens (fl. 175–205 C.E.). As a whole, the school continued to discredit dogmatism without promoting knowledge. Sextus, it is true, strikes acutely and systematically at ill-founded beliefs, and so makes for reason;354 but, like the whole Pyrrhonian school, he has no idea of a method which shall reach sounder conclusions. As the Stoics had inculcated the control of the passions as such, so the skeptics undertook to make men rise above the prejudices and presuppositions which swayed them no less blindly than ever did their passions. But Sextus follows a purely skeptical method, never rising from the destruction of false beliefs to the establishment of true. His aim is ataraxia, a philosophic calm of non- belief in any dogmatic affirmation beyond the positing of phenomena as such; and while such an attitude is beneficently exclusive of all fanaticism, it unfortunately never makes any impression on the more intolerant fanatic,
  • 40. who is shaken only by giving him a measure of critical truth in place of his error. And as Sextus addressed himself to the students of philosophy, not to the simple believers in the Gods, he had no wide influence.355 Avowedly accepting the normal view of moral obligations while rejecting dogmatic theories of their basis, the doctrine of the strict skeptics had the effect, from Pyrrho onwards, of giving the same acceptance to the common religion, merely rejecting the philosophic pretence of justifying it. Taken by themselves, the arguments against current theism in the third book of the Hypotyposes356 are unanswerable; but, when bracketed with other arguments against the ordinary belief in causation, they had the effect of leaving theism on a par with that belief. Against religious beliefs in particular, therefore, they had no wide destructive effect. Lucian, again, thought soundly and sincerely on life; his praise of the men whose memories he respected, as Epicurus and Demonax (if the Life of Demonax attributed to him be really his), is grave and heartfelt; and his ridicule of the discredited Gods was perfectly right so far as it went. It is certain that the unbelievers and the skeptics alike held their own with the believers in the matter of right living.357 In the period of declining pagan belief, the maxim that superstition was a good thing for the people must have wrought a quantity and a kind of corruption that no amount of ridicule of religion could ever approach. Polybius (fl. 150 B.C.) agrees with his complacent Roman masters that their greatness is largely due to the carefully cultivated superstition of their populace, and charges with rashness and folly those who would uproot the growth;358 and Strabo, writing under Tiberius—unless it be a later interpolator of his work— confidently lays down the same principle of governmental deceit,359 though in an apparently quite genuine passage he vehemently protests the incredibility of the traditional tales about Apollo.360 So far had the doctrine evolved since Plato preached it. But to countervail it there needed more than a ridicule which after all reached only the class who had already cast off the beliefs derided, leaving the multitude unenlightened. The lack of the needed machinery of enlightenment was, of course, part of the general failure of the Græco-Roman civilization; and no one man’s efforts could have availed, even if any man of the age could have grasped the whole
  • 41. situation. Rather the principle of esoteric enlightenment, the ideal of secret knowledge, took stronger hold as the mass grew more and more comprehensively superstitious. Even at the beginning of the Christian era the view that Homer’s deities were allegorical beings was freshly propounded in the writings of Herakleides and Cornutus (Phornutus); but it served only as a kind of mystical Gnosis, on all fours with Christian Gnosticism, and was finally taken up by Neo-Platonists, who were no nearer rationalism for adopting it.361 So with the rationalism to which we have so many uneasy or hostile allusions in Plutarch. We find him resenting the scoffs of Epicureans at the doctrine of Providence, and recoiling from the “abyss of impiety”362 opened up by those who say that “Aphrodite is simply desire, and Hermes eloquence, and the Muses the arts and sciences, and Athênê wisdom, and Dionysos merely wine, Hephaistos fire, and Dêmêtêr corn”;363 and in his essay On Superstition he regretfully recognizes the existence of many rational atheists, confessing that their state of mind is better than that of the superstitious who abound around him, with their “impure purifications and unclean cleansings,” their barbaric rites, and their evil Gods. But the unbelievers, with their keen contempt for popular folly, availed as little against it as Plutarch himself, with his doctrine of a just mean. The one effectual cure would have been widened knowledge; and of such an evolution the social conditions did not permit. To return to a state of admiration for the total outcome of Greek thought, then, it is necessary to pass from the standpoint of simple analysis to that of comparison. It is in contrast with the relatively slight achievement of the other ancient civilizations that the Greek, at its height, still stands out for posterity as a wonderful growth. That which, tried by the test of ideals, is as a whole only one more tragic chapter in the record of human frustration, yet contains within it light and leading as well as warning; and for long ages it was as a lost Paradise to a darkened world. It has been not untruly said that “the Greek spirit is immortal, because it was free”:364 free not as science can now conceive freedom, but in contrast with the spiritual bondage of Jewry and Egypt, the half-barbaric tradition of imperial Babylon, and the
  • 42. short flight of mental life in Rome. Above all, it was ever in virtue of the freedom that the high things were accomplished; and it was ever the falling away from freedom, the tyranny either of common ignorance or of mindless power, that wrought decadence. There is a danger, too, of injustice in comparing Athens with later States. When a high authority pronounces that “the religious views of the Demos were of the narrowest kind,”365 he is not to be gainsaid; but the further verdict that “hardly any people has sinned more heavily against the liberty of science” is unduly lenient to Christian civilization. The heaviest sins of that against science, indeed, lie at the door of the Catholic Church; but to make that an exoneration of the modern “peoples” as against the ancient would be to load the scales. And even apart from the Catholic Church, which practically suppressed all science for a thousand years, the attitude of Protestant leaders and Protestant peoples, from Luther down to the second half of the nineteenth century, has been one of hatred and persecution towards all science that clashed with the sacred books.366 In the Greek world there was more scientific discussion in the three hundred years down to Epicurus than took place in the whole of Christian Europe in thirteen hundred; and the amount of actual violence used towards innovators in the pagan period, though lamentable enough, was trifling in comparison with that recorded in Christian history, to say nothing of the frightful annals of witch-burning, to which there is no parallel in civilized heathen history. The critic, too, goes on to admit that, while “Sokrates, Anaxagoras, and Aristotle fell victims in different degrees to the bigotry of the populace,” “of course their offence was political rather than religious. They were condemned not as heretics, but as innovators in the state religion.” And, as we have seen, all three of the men named taught in freedom for many years till political faction turned popular bigotry against them. The true measure of Athenian narrowness is not to be reached, therefore, without keeping in view the long series of modern outrages and maledictions against the makers and introducers of new machinery, and the multitude of such episodes as the treatment of Priestley in Christian Birmingham, little more than a century ago. On a full comparison the Greeks come out not ill.
  • 43. It was, in fact, impossible that the Greeks should either stifle or persecute science or freethought as it was either stifled or persecuted by ancient Jews (who had almost no science by reason of their theology) or by modern Christians, simply because the Greeks had no anti-scientific hieratic literature. It remains profoundly significant for science that the ancient civilization which on the smallest area evolved the most admirable life, which most completely transcended all the sources from which it originally drew, and left a record by which men are still charmed and taught, was a civilization as nearly as might be without Sacred Books, without an organized priesthood, and with the largest measure of democratic freedom that the ancient world ever saw.
  • 44. 1 2 3 4 5 6 7 8 9 10 11 12 Cp. Tiele, Outlines, pp. 205, 207, 212. ↑ Cp. E. Meyer, Geschichte des Alterthums, ii, 533. ↑ Cp. K. O. Müller, Literature of Ancient Greece, ed. 1847, p. 77. ↑ Duncker, Gesch. des Alterth. 2 Aufl. iii, 209–10, 252–54, 319 sq.; E. Meyer, Gesch. des Alterth. ii, 181, 365, 369, 377, 380, 535 (see also ii, 100, 102, 105, 106, 115 note, etc.); W. Christ, Gesch. der griech. Lit. 3te Aufl. p. 12; Gruppe, Die griech. Culte und Mythen, 1887, p. 165 sq. ↑ E. Curtius, Griech. Gesch. i, 28, 29, 35, 40, 41, 101, 203, etc.; Meyer, ii, 369. ↑ See the able and learned essay of S. Reinach, Le Mirage Orientate, reprinted from L’Anthropologie, 1893. I do not find that its arguments affect any of the positions here taken up. See pp. 40–41. ↑ Meyer, ii. 369; Benn, The Philosophy of Greece, 1898, p. 42. ↑ Cp. Bury, History of Greece, ed. 1906, pp. vi, 10, 27, 32–34, 40, etc.; Burrows, The Discoveries in Crete, 1907, ch. ix; Maisch, Manual of Greek Antiquities, Eng. tr. §§ 8, 9, 10, 60; H. R. Hall, The Oldest Civilization of Greece, 1901, pp. 31, 32. ↑ Cp. K. O. Müller, Hist. of the Doric Race, Eng. tr. 1830, i, 8–10; Busolt, Griech. Gesch. 1885, i, 33; Grote, Hist. of Greece, 10-vol. ed. 1888, iii, 3–5, 35–44; Duncker, iii, 136, n.; E. Meyer, Gesch. des Alterthums, i, 299–310 (§§ 250–58); E. Curtius, i, 29; Schömann, Griech. Alterthümer, as cited, i, 2–3, 89; Burrows, ch. ix. ↑ Cp. Meyer, ii, 97; and his art. “Baal” in Roscher’s Ausführl. Lex. Mythol. i, 2867. ↑ The fallacy of this tradition, as commonly put, was well shown by Renouvier long ago —Manuel de philosophie ancienne, 1844, i, 3–13. Cp. Ritter, as cited below. ↑ Cp. on one side, Ritter, Hist. of Anc. Philos. Eng. tr. i, 151; Renan, Études d’hist. religieuse, pp. 47–48; Zeller, Hist. of Greek Philos. Eng. tr. 1881, i, 43–49; and on the other, Ueberweg, Hist. of Philos. Eng. tr. i, 31, and the weighty criticism of Lange, Gesch. des Materialismus, i, 126–27 (Eng. tr. i, 9, note 5). ↑
  • 45. 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Cp. Curtius, i, 125; Bury, introd. and ch. i. ↑ Cp. Bury, as cited. ↑ As to the primary mixture of “Pelasgians” and Hellenes, cp. Busolt, i, 27–32; Curtius, i, 27; Schömann, i, 3–4; Thirlwall, Hist. of Greece, ed. 1839, i, 51–52, 116. K. O. Müller (Doric Race, Eng. tr. i, 10) and Thirlwall, who follows him (i, 45–47), decide that the Thracians cannot have been very different from the Hellenes in dialect, else they could not have influenced the latter as they did. This position is clearly untenable, whatever may have been the ethnological facts. It would entirely negate the possibility of reaction between Greeks, Kelts, Egyptians, Semites, Romans, Persians, and Hindus. ↑ Murray, Four Stages of Greek Religion, 1912, p. 59. ↑ Cp. Meyer, Gesch. des Alt. ii, 583. ↑ The question is discussed at some length in the author’s Evolution of States, 1912. ↑ Lit. of Anc. Greece, pp. 41–47. The discussion of the Homeric problem is, of course, alien to the present inquiry. ↑ Introd. to Scientif. Mythol. Eng. tr. pp. 180, 181, 291. Cp. Curtius, i, 126. ↑ Cp. Curtius, i, 107, as to the absence in Homer of any distinction between Greeks and barbarians; and Grote, 10-vol. ed. 1888, iii, 37–38, as to the same feature in Archilochos. ↑ Duncker, Gesch. des Alt., as cited, iii. 209–10; pp. 257, 319 sq. Cp. K. O. Müller, as last cited, pp. 181, 193; Curtius, i, 43–49, 53, 54, 107, 365, 373, 377, etc.; Grote, iii, 39– 41; and Meyer, ii, 104. ↑ Duncker, iii, 214; Curtius, i, 155, 121; Grote, iii, 279–80. ↑ Busolt, Griech. Gesch. 1885, i, 171–72. Cp. pp. 32–34; and Curtius, i, 42. ↑ On the general question cp. Gruppe, Die griechischen Culte und Mythen, pp. 151 ff., 157, 158 ff., 656 ff., 672 ff. ↑ Preller, Griech. Mythol. 2 Aufl. i, 260; Tiele, Outlines, p. 211; R. Brown, Jr., Semit. Influ. in Hellenic Mythol. 1898, p. 130; Murray, Hist. of Anc. Greek Lit. p. 35; H. R. Hall, Oldest Civilization of Greece, 1901, p. 290. ↑ See Tiele, Outlines, pp. 210, 212. Cp., again, Curtius, Griech. Gesch. i, 95, as to the probability that the “twelve Gods” were adjusted to the confederations of twelve cities; and again p. 126. ↑
  • 46. 28 29 30 31 32 33 34 35 36 37 38 39 40 “Even the title ‘king’ (Αναξ) seems to have been borrowed by the Greek from Phrygian.... It is expressly recorded that τύραννος is a Lydian word. Βασιλεύς (‘king’) resists all attempts to explain it as a purely Greek formation, and the termination assimilates it to certain Phrygian words.” (Prof. Ramsay, in Encyc. Brit. art. Phrygia). In this connection note the number of names containing Anax (Anaximenes, Anaximandros, Anaxagoras, etc.) among the Ionian Greeks. ↑ iv, 561 sq. ↑ It is now agreed that this is merely a guess. The document, further, has been redacted and interpolated. ↑ Prehist. Antiq. of the Aryan Peoples, Eng. tr. p. 423. Wilamowitz holds that the verses Od. xi, 566–631, are interpolations made later than 600 B.C. ↑ Tiele, Outlines, p. 209; Preller, p. 263. ↑ Meyer says on the contrary (Gesch. des Alt. ii, 103, Anm.) that “Kronos is certainly a Greek figure”; but he cannot be supposed to dispute that the Greek Kronos cult is grafted on a Semitic one. ↑ Sayce, Hibbert Lectures, pp. 54, 181. Cp. Cox, Mythol. of the Aryan Nations, p. 260, note. It has not, however, been noted in the discussions on Semelê that Semlje is the Slavic name for the Earth as Goddess. Ranke, History of Servia, Eng. tr. p. 43. ↑ Iliad, xiv, 201, 302. ↑ Sayce, Hibbert Lectures, p. 367 sq.; Ancient Empires, p. 158. Note p. 387 in the Lectures as to the Assyrian influence, and p. 391 as to the Homeric notion in particular. Cp. W. Christ, Gesch. der griech. Literatur, § 68. ↑ It is unnecessary to examine here the view of Herodotos that many of the Greek cults were borrowed from Egypt. Herodotos reasoned from analogies, with no exact historical knowledge. But cp. Renouvier, Manuel, i, 67, as to probable Egyptian influence. ↑ Cp. Meyer, ii, §§ 453–60, as to the eastern initiative of Orphic theology. ↑ It is noteworthy that the traditional doctrine associated with the name of Orpheus included a similar materialistic theory of the beginning of things. Athenagoras, Apol. c. 19. Cp. Renouvier, Manuel de philos. anc. i, 69–72; and Meyer, ii, 743. ↑ Cp. Meyer, ii, 726. As to the oriental elements in Hesiod see further Gruppe, Die griechischen Culte und Mythen, 1887, pp. 577, 587, 589, 593. ↑
  • 47. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Cp. however, Bury (Hist. of Greece, pp. 6, 65), who assumes that the Greeks brought the hexameter with them to Hellas. Contrast Murray, Four Stages, p. 61. ↑ Mahaffy, History of Classical Greek Literature, 1880, i, 15. ↑ Id. p. 16. Cp. W. Christ, as cited, p. 79. ↑ Mahaffy, pp. 16–17. ↑ Od. xviii, 352. ↑ Od. vi, 240; Il. v, 185. ↑ Od. xxii, 39. ↑ In Od. xiv, 18, αντίθεοι means not “opposed to the Gods,” but “God-like,” in the ordinary Homeric sense of noble-looking or richly attired, as men in the presence of the Gods. Cp. vi, 241. Yet a Scholiast on a former passage took it in the sense of God- opposing. Clarke’s ed. in loc. Liddell and Scott give no use of ἄθεος, in the sense of denying the Gods, before Plato (Apol. 26 C. etc.), or in the sense of ungodly before Pindar (P. iv, 288) and Æschylus (Eumen. 151). For Sophocles it has the force of “God- forsaken”—Oedip. Tyr. 254 (245), 661 (640), 1360 (1326). Cp. Electra, 1181 (1162). But already before Plato we find the terms ἄπιστος and ἄθεος, “faithless” or “infidel” and “atheist,” used as terms of moral aspersion, quite in the Christian manner (Euripides, Helena, 1147), where there is no question of incredulity. ↑ Cp. Lang, Myth, Ritual, and Religion, 2nd ed. i, 14–15. and cit. there from Professor Jebb. ↑ Cp. Meyer, Gesch. des Alterthums, ii, 724–27; Grote, as cited, i, 279–81. ↑ Meyer, ii, 724, 727. ↑ The tradition is confused. Stesichoros is said first to have aspersed Helen, whereupon she, as Goddess, struck him with blindness: thereafter he published a retractation, in which he declared that she had never been at Troy, an eidolon or phantasm taking her name; and on this his sight was restored. We can but divine through the legend the probable reality, the documents being lost. See Grote, as cited, for the details. For the eulogies of Stesichoros by ancient writers, see Girard, Sentiment religieux en Grèce, 1869, pp. 175– 79. ↑ Cp. Meyer (1901), iii. § 244. ↑ Ol. i, 42–57, 80–85. ↑
  • 48. 55 56 57 58 59 60 61 62 63 64 65 66 67 68 Ol. ix, 54–61. ↑ He dedicated statues to Zeus, Apollo, and Hermes. Pausanias, ix, 16, 17. ↑ Herodot. ii. 53. ↑ A ruler of Libyan stock, and so led by old Libyan connections to make friends with Greeks. He reigned over fifty years, and the Greek connection grew very close. Curtius, i, 344–45. Cp. Grote, i, 144–55. ↑ Grote, 10-vol. ed. 1888, i, 307, 326, 329, 413. Cp. i, 27–30; ii, 52; iii, 39–41, etc. ↑ K. O. Müller, Introd. to Mythology, p. 192. ↑ “Then one [of the Persians] who before had in nowise believed in [or, recognized the existence of] the Gods, offered prayer and supplication, doing obeisance to Earth and Heaven” (Persae, 497–99). ↑ Agamemnon, 370–372. This is commonly supposed to be a reference to Diagoras the Melian (below, p. 159). ↑ Agam. 170–72 (160–62). ↑ So Whittaker, Priests, Philosophers, and Prophets, 1911, pp. 42–43. ↑ So Buckley, in Bohn trans. of Æschylus, p. 100. He characterizes as a “skeptical formula” the phrase “Zeus, whoever he may be”; but goes on to show that such formulas were grounded on the Semitic notion that the true name of God was concealed from man. ↑ Grote, ed. 1888, vii, 8–21. See the whole exposition of the exceptionally interesting 67th chapter. ↑ Cp. Meyer, ii, 431; K. O. Müller, Introd. to Mythol. pp. 189–92; Duncker, p. 340; Curtius, i, 384; Thirlwall, i, 200–203; Burckhardt, Griech. Culturgesch. 1898, ii. 19. As to the ancient beginnings of a priestly organization, see Curtius, i, 92–94, 97. As to the effects of its absence, see Heeren, Polit. Hist. of Anc. Greece, Eng. tr. 1829, pp. 59–63; Burckhardt, as cited, ii, 31–32; Meyer, as last cited; Zeller, Philos. der Griechen, 3te Aufl. i, 44 sq. Lange’s criticism of Zeller’s statement (Gesch. des Materialismus, 3te Aufl. i, 124–26, note 2) practically concedes the proposition. The influence of a few powerful priestly families is not denied. The point is that they remained isolated. ↑ Cp. K. O. MÜller, Introd. to Mythol. p. 195; Curtius, i, 387, 389, 392; Duncker, iii, 519–21, 563; Thirlwall, i, 204; Barthélemy St. Hilaire, préf. to tr. of Metaphys. of Aristotle, p. 14. Professor Gilbert Murray, noting that Homer and Hesiod treated the Gods as
  • 49. 69 70 71 72 73 74 75 76 77 78 79 elements of romance, or as facts to be catalogued, asks: “Where is the literature of religion: the literature which treated the Gods as Gods? It must,” he adds, “have existed”; and he holds that we “can see that the religious writings were both early and multitudinous” (Hist. of Anc. Greek Lit. p. 62; cp. Meyer and Mahaffy as cited above, pp. 125–26. “Writings” is not here to be taken literally; the early hymns were unwritten). The priestly hymns and oracles and mystery-rituals in question were never collected; but perhaps we may form some idea of their nature from the “Homeridian” and Orphic hymns to the Gods, and those of the Alexandrian antiquary Callimachus. It is further to be inferred that they enter into the Hesiodic Theogony. (Decharme, p. 3, citing Bergk.) ↑ Meyer, ii, 426; Curtius, i, 390–91, 417; Thirlwall, i, 204; Grote, i, 48–49. ↑ Meyer, ii, 410–14. ↑ Cp. Curtius, i, 392–400, 416; Duncker, iii, 529. ↑ Curtius, i, 112; Meyer, ii, 366. ↑ Curtius, i, 201, 204, 205, 381; Grote, iii, 5; Lange, Gesch. des Materialismus, 3te Aufl. i, 23 (Eng. tr. i, 23). ↑ Herodotos, i, 170; Diogenes Laërtius, Thales, ch. i. ↑ On the essentially anti-religious rationalism of the whole Ionian movement, cp. Meyer, ii, 753–57. ↑ The First Philosophers of Greece, by A. Fairbanks, 1898, pp. 2, 3, 6. This compilation usefully supplies a revised text of the ancient philosophic fragments, with a translation of these and of the passages on the early thinkers by the later, and by the epitomists. A good conspectus of the remains of the early Greek thinkers is supplied also in Grote’s Plato and the other Companions of Sokrates, ch. i; and a valuable critical analysis of the sources in Prof. J. Burnet’s Early Greek Philosophy. ↑ Cp. Lange, Gesch. des Mat. i, 126 (Eng. tr. i, 8, n.). Mr. Benn (The Greek Philosophers, i, 8) and Prof. Decharme (p. 39) seem to read this as a profession of belief in deities in the ordinary sense. But cp. R. W. Mackay, The Progress of the Intellect, 1850, i, 338. Burnet (ch. i, § 11) doubts the authenticity of this saying, but thinks it “extremely probable that Thales did say that the magnet and amber had souls.” ↑ Mackay, as cited, p. 331. ↑ Fairbanks, p. 4. ↑
  • 50. 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 Diogenes Laërtius, Thales, ch. 9. ↑ Fairbanks, pp. 3, 7. ↑ Herodotos, i, 74. ↑ Cp. Burnet, Early Greek Philos. 2nd. ed. introd. § 3. To Thales is ascribed by the Greeks the “discovery” of the constellation Ursus Major. Diog. ch. 2. As it was called “Phoenike” by the Greeks, his knowledge would be of Phoenician derivation. Cp. Humboldt, Kosmos, Bohn tr. iii, 160. ↑ Diog. Laërt. ch. 3. On this cp. Burnet, introd. § 6. ↑ Herod. i, 170. Cp. Diog. Laërt. ch. 3. ↑ Diog. Laërt. ch. 9. ↑ Cp. Burnet, p. 57. ↑ Fairbanks, pp. 9–10. Mr. Benn (Greek Philosophers, i, 9) decides that the early philosophers, while realizing that ex nihilo nihil fit, had not grasped the complementary truth that nothing can be annihilated. But even if the teaching ascribed to Anaximandros be set aside as contradictory (since he spoke of generation and destruction within the infinite), we have the statement of Diogenes Laërtius (bk. ix, ch. 9, § 57) that Diogenes of Apollonia, pupil of Anaximenes, gave the full Lucretian formula. ↑ Diogenes Laërtius, however (ii, 2), makes him agree with Thales. ↑ Fairbanks, pp. 9–16. Diogenes makes him the inventor of the gnomon and of the first map and globe, as well as a maker of clocks. Cp. Grote, i, 330, note. ↑ See below, p. 158, as to Demokritos’ statement concerning the Eastern currency of scientific views which, when put by Anaxagoras, scandalized the Greeks. ↑ Fairbanks, pp. 17–22. ↑ See Windelband, Hist. of Anc. Philos. Eng. tr. 1900, p. 25, citing Diels and Wilamowitz-Möllendorf. Cp. Burnet, introd. § 14. ↑ It will be observed that Mr. Cornford’s book, though somewhat loosely speculative is very freshly suggestive. It is well worth study, alongside of the work of Prof. Burnet, by those interested in the scientific presentation of the evolution of thought. ↑ Diog. Laërt. ix, 19; Fairbanks, p. 76. ↑ Herodotos, i, 163–67; Grote, iii, 421; Meyer, ii, § 438. ↑
  • 51. 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 Cp. Guillaume Bréton, Essai sur la poésie philosophique en Grèce, 1882, pp. 23–25. The life period of Xenophanes is still uncertain. Meyer (ii, § 466) and Windelband (Hist. of Anc. Philos. Eng. tr. p. 47) still adhere to the chronology which puts him in the century 570–470, making him a young man at the foundation of Elea. ↑ Cousin, developed by G. Bréton, work cited, p. 31 sq., traces Xenophanes’s doctrine of the unity of things to the school of Pythagoras. It clearly had antecedents. But Xenophanes is recorded to have argued against Pythagoras as well as Thales and Epimenides (Diog. Laërt. ix, 2, §§ 18, 20). ↑ Metaphysics, i, 5; cp. Fairbanks, pp. 79–80. ↑ One of several so entitled in that age. Cp. Burnet, introd. § 7. ↑ Metaph., as cited; Plato, Soph. 242 D. ↑ Long fragment in Athenæus, xi, 7; Burnet, p. 130. ↑ Burnet, p. 141. ↑ Cp. Burnet, p. 131. ↑ Fairbanks, p. 67, Fr. 5, 6; Clem. Alex. Stromata, bk. v, Wilson’s tr. ii, 285–86. Cp. bk. vii, c. 4. ↑ Fairbanks, Fr. 7. ↑ Cicero, De divinatione, i, 3, 5; Aetius, De placitis reliquiæ, in Fairbanks, p. 85. ↑ Aristotle, Rhetoric, ii, 23, § 27. A similar saying is attributed to Herakleitos, on slight authority (Fairbanks, p. 54). ↑ Cicero, Academica, ii, 39; Lactantius, Div. Inst. iii, 23. Anaxagoras and Demokritos held the same view. Diog. Laërt, bk. ii, ch. iii, iv (§ 8); Pseudo-Plutarch, De placitis philosoph. ii, 25. ↑ Cp. Mackay, Progress of the Intellect, i, 340. ↑ Diog. Laërt. in life of Pyrrho, bk. ix, ch. xi, 8 (§ 72). The passage, however, is uncertain. See Fairbanks, p. 70. ↑ Fairbanks. Fr. 1. Fairbanks translates with Zeller: “The whole [of God].” Grote: “The whole Kosmos, or the whole God.” It should be noted that the original in Sextus Empiricus (Adv. Math. ix, 144) is given without the name of Xenophanes, and the ascription is modern. ↑
  • 52. 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 Grote, as last cited, p. 18. ↑ Fairbanks, Fr. 19. In Athenæus, x, 413. ↑ Polybius, iv, 40; Sextus Empiricus, Adversus Mathematicos, viii, 126; Fairbanks, pp. 25, 27; Frag. 4, 14. Cp. 92, 111, 113. ↑ Diog. Laërt. ix, i, 2. ↑ Fairbanks, Fr. 134. ↑ Id. Frag. 36, 67. ↑ Id. Frag. 43, 44, 46, 62. ↑ Diog. Laërt. last cited. This saying is by some ascribed to the later Herakleides (see Fairbanks, Fr. 119 and note); but it does not seem to be in his vein, which is wholly pro- Homeric. ↑ Clem. Alex. Protrept. ch. 2, Wilson’s tr. p. 41. The passage is obscure, but Mr. Fairbanks’s translation (Fr. 127) is excessively so. ↑ Clemens, as cited, p.32; Fairbanks, Fr. 124, 125, 130. Cp. Burnet, p. 139. ↑ Fairbanks, Fr. 21. ↑ Cp. Burnet, pp. 175–90. ↑ Theaetetus, 180 D. See good estimates of Parmenides in Benn’s Greek Philosophers, i, 17–19, and Philosophy of Greece in Relation to the Character of its People, pp. 83–95; in J. A. Symonds’s Studies of the Greek Poets, 3rd ed. 1893, vol. i, ch. 6; and in Zeller, i, 580 sq. ↑ Plutarch, Perikles, ch. 26. ↑ Mr. Benn finally gives very high praise to Melissos (Philos. of Greece, pp. 91–92); as does Prof. Burnet (Early Gr. Philos. p. 378). He held strongly by the Ionian conception of the eternity of matter. Fairbanks, p. 125. ↑ Diog. Laërt. bk. ix, ch. iv, 3 (§ 24). ↑ Diog. Laërt. ix, 3 (§ 21). ↑ As to this see Windelband, Hist. Anc. Philos. pp. 91–92. ↑ Cp. Mackay, Progress of the Intellect, i. 340. ↑
  • 53. 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 “The difference between the Ionians and Eleatæ was this: the former endeavoured to trace an idea among phenomena by aid of observation; the latter evaded the difficulty by dogmatically asserting the objective existence of an idea” (Mackay, as last cited). ↑ Cp. Mackay, i, 352–53, as to the survival of veneration of the heavenly bodies in the various schools. ↑ Grote, i, 350. ↑ Meyer, ii, 9, 759 (§§ 5, 465). ↑ Id. §§ 6, 466. ↑ Jevons, Hist. of Greek Lit. 1886, p. 210. ↑ Compare Meyer, ii, § 502, as to the close resemblances between Pythagoreanism and Orphicism. ↑ Meyer, i, 186; ii, 635. ↑ Fairbanks, pp. 145, 151, 155, etc. ↑ Id. p. 143. ↑ Id. p. 154. ↑ Prof. Burnet insists (introd. p. 30) that “the” Greeks must be reckoned good observers because their later sculptors were so. As well say that artists make the best men of science. ↑ Metaph. i, 5; Fairbanks, p. 136. “It is quite safe to attribute the substance of the First Book of Euclid to Pythagoras.” Burnet, Early Greek Philos. 2nd ed. p. 117. ↑ Diog. Laërt. Philolaos (bk. viii, ch. 7). ↑ L. U. K. Hist. of Astron. p. 20; A. Berry’s Short Hist. of Astron. 1898, p. 25; Narrien’s Histor. Acc. of the Orig. and Prog. of Astron. 1850, p. 163. ↑ See Benn, Greek Philosophers, i, 11. ↑ Diog. Laërt. in life of Philolaos; Cicero, Academica, ii, 39. Cicero, following Theophrastus, is explicit as to the teaching of Hiketas. ↑ Hippolytos, Ref. of all Heresies, i, 13. Cp. Renouvier, Manuel de la philos. anc. i, 201, 205, 238–39. ↑ Pseudo-Plutarch, De Placitis Philosoph. iii, 13, 14. ↑
  • 54. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookultra.com