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 \def\righthead{\sl\hfil SOLSTICE }
 \def\lefthead{\sl Winter, 1992 \hfil}
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 \centerline{\big SOLSTICE:}
 \centerline{\bf WINTER, 1992}
 \centerline{\bf Volume III, Number 2}
 \centerline{\bf Institute of Mathematical Geography}
 \centerline{\bf Ann Arbor, Michigan}
 \centerline{\bf SOLSTICE}
 \line{Founding Editor--in--Chief:  {\bf Sandra Lach Arlinghaus}. \hfil}
 \centerline{\bf EDITORIAL BOARD}
 \line{{\bf Geography} \hfil}
 \line{{\bf Michael Goodchild}, University of California, Santa Barbara. 
 \line{{\bf Daniel A. Griffith}, Syracuse University. \hfil}
 \line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment
  in School of Medicine.\hfil}
 \line{{\bf John D. Nystuen}, University of Michigan (College of
  Architecture and Urban Planning).}
 \line{{\bf Mathematics} \hfil}
 \line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil}
 \line{{\bf Neal Brand}, University of North Texas. \hfil}
 \line{{\bf Kenneth H. Rosen}, A. T. \& T. Bell Laboratories.
 \line{{\bf Engineering Applications} \hfil}
 \line{{\bf William D. Drake}, University of Michigan, \hfil}
 \line{{\bf Business} \hfil}
 \line{{\bf Robert F. Austin, Ph.D.} \hfil}
 \line{President, Austin Communications Education Services \hfil}
       The purpose of {\sl Solstice\/} is to promote  interaction
 between geography and mathematics.   Articles in which  elements
 of   one  discipline  are used to shed light on  the  other  are
 particularly sought.   Also welcome,  are original contributions
 that are purely geographical or purely mathematical.   These may
 be  prefaced  (by editor or author) with  commentary  suggesting
 directions  that  might  lead toward  the  desired  interaction.
 Individuals  wishing to submit articles,  either short or full--
 length,  as well as contributions for regular  features,  should
 send  them,  in triplicate,  directly to the  Editor--in--Chief.
 Contributed  articles  will  be refereed by  geographers  and/or
 mathematicians.   Invited articles will be screened by  suitable
 members of the editorial board.  IMaGe is open to having authors
 suggest, and furnish material for, new regular features.  

 The opinions expressed are those of the authors, alone, and the
 authors alone are responsible for the accuracy of the facts in
 the articles. 
 \noindent {\bf Send all correspondence to:
 Institute of Mathematical Geography, 2790 Briarcliff,
 Ann Arbor, MI 48105-1429, (313) 761-1231, IMaGe@UMICHUM,
 Suggested form for citation.  If standard referencing to the hardcopy
 in  the  IMaGe Monograph Series is not used (although we suggest that
 reference  to  that  hardcopy be included along with reference to the
 e-mailed copy from which  the hard copy is produced), then we suggest
 the following  format for  citation of the electronic copy.  Article,
 author, publisher (IMaGe) -- all the usual--plus a notation as to the
 time marked  electronically,  by the process of transmission,  at the
 top of the  recipients copy.   Note  when  it was sent from Ann Arbor
 (date and time to the second) and when you received it (date and time
 to the second)  and the field characters covered by the article  (for
 example FC=21345 to FC=37462).
       This  document is produced using the typesetting  program,
 {\TeX},  of Donald Knuth and the American Mathematical  Society.
 Notation  in  the electronic file is in accordance with that  of
 Knuth's   {\sl The {\TeX}book}.   The program is downloaded  for
 hard copy for on The University of Michigan's Xerox 9700 laser--
 printing  Xerox  machine,  using IMaGe's commercial account with 
 that University.
 Unless otherwise noted, all regular ``features"  are  written by
 the Editor--in--Chief.
       {\nn  Upon final acceptance,  authors will work with IMaGe
 to    get  manuscripts   into  a  format  well--suited  to   the
 requirements   of {\sl Solstice\/}.  Typically,  this would mean
 that  authors    would  submit    a  clean  ASCII  file  of  the
 manuscript,  as well as   hard copy,  figures,  and so forth (in
 camera--ready form).     Depending on the nature of the document
 and   on   the  changing    technology  used  to  produce   {\sl
 Solstice\/},   there  may  be  other    requirements  as   well.
 Currently,  the  text  is typeset using   {\TeX};  in that  way,
 mathematical formul{\ae} can be transmitted   as ASCII files and
 downloaded   faithfully   and   printed   out.    The     reader
 inexperienced  in the use of {\TeX} should note that  this    is
 not  a ``what--you--see--is--what--you--get"  display;  however,
 we  hope  that  such readers find {\TeX} easier to  learn  after
 exposure to {\sl Solstice\/}'s e-files written using {\TeX}!}
       {\nn  Copyright  will  be taken out in  the  name  of  the
 Institute of Mathematical Geography, and authors are required to
 transfer  copyright  to  IMaGe as a  condition  of  publication.
 There are no page charges; authors will be given  permission  to
 make reprints from the electronic file,  or to have IMaGe make a
 single master reprint for a nominal fee dependent on  manuscript
 length.   Hard  copy of {\sl Solstice\/} is  available at a cost
 of \$15.95 per year (plus  shipping  and  handling; hard copy is
 issued once yearly, in the Monograph series of the  Institute of
 Mathematical Geography.   Order directly from  IMaGe.  It is the
 desire of IMaGe to offer electronic copies to interested parties
 for free.  Whether  or  not  it  will  be  feasible  to continue
 distributing  complimentary electronic files remains to be seen.  
 Presently {\sl Solstice\/} is funded by IMaGe and by a  generous
 donation of computer time from a member  of the Editorial Board.
 Thank  you  for  participating  in  this  project  focusing   on 
 environmentally-sensitive publishing.}
 \copyright Copyright, December, 1992 by the
 Institute of Mathematical Geography.
 All rights reserved.
 {\bf ISBN: 1-877751-54-5}
 {\bf ISSN: 1059-5325} 
 \centerline{\bf SUMMARY OF CONTENT}
 \noindent{\bf  1.  A WORD OF WELCOME FROM A TO U}
 \noindent{\bf 2.  PRESS CLIPPINGS---SUMMARY}
 \noindent{\bf 3.  REPRINTS.}
 A.  {\bf What are mathematical models and what should they be ?}
 {\bf by Frank Harary}.
 \noindent Reprinted, with permission,
 from {\sl Biometrie-Praximetrie\/};
 full citation at the front of the article.
 1. What Are They? 2. Two Worlds:  Abstract and Empirical
 3. Two Worlds:  Two Levels   4.  Two Levels:  Derivation 
 and Selection.   5.  Research Schema.    6.  Sketches of
 Discovery.  7.  What Should They Be?
 B.  {\bf Where are we? 
 Comments on the concept of the ``center of population"}
 {\bf by Frank E. Barmore}
 Reprinted, with permission,
 from {\sl The Wisconsin Geographer\/};
 full citation at the front of the article.
 1.  Introduction  2.  Preliminary Remarks  3.  Census Bureau
 Center of Population Formul{\ae}    4.  Census Bureau Center
 of Population Description.  5. Agreement between Description
 and Formul{\ae}.  6.  Proposed  Definition of the  Center of
 Population.  7.  Summary.   8.  Appendix A.  9.  Appendix B. 
 10.  References.
 \noindent{\bf 4.  ARTICLE }
 {\bf The pelt of the Earth:  An essay on reactive diffusion}
 {\bf by Sandra L. Arlinghaus and John D. Nystuen}.  
 1.  Pattern formation:  Global views.  2.  Pattern formation:  
 Local views. 3.  References cited. 4.  Literature of apparent
 related interest.
 \noindent{\bf 5.  FEATURE}
 {\bf Meet New Solstice Board Member, William D. Drake}
 \noindent {\bf 6.  DOWNLOADING OF SOLSTICE}
 \noindent{\bf 7.  INDEX to Volumes I (1990),  II (1991),  and
 III (1992, Number 1) of {\sl Solstice}.}
 \noindent{\bf 8.  OTHER PUBLICATIONS OF IMaGe }
 \centerline{\bf 1.  A WORD OF WELCOME--FROM A TO U!}
 Welcome  to new subscribers from Alice Springs, Australia, to Ulm,
 Germany and to all those in between (alphabetically or otherwise)! 
 We  hope  you  enjoy  participating  in  this  means  of   journal
 distribution.  Instructions for downloading  the typesetting  have
 been repeated in this issue, at the end.  They are specific to the
 {\TeX} installation at The University of Michigan,  but apparently
 they  have  been  helpful  in  suggesting  to  others the sorts of
 commands  that  might  be  used  on their own particular mainframe
 installation of {\TeX}.  New subscribers might wish to  note  that
 the electronic files are typeset files---the mathematical notation
 will print out as typeset notation.  For example,
 when properly downloaded, will  print  out a typeset  summation as
 i goes from  one to  n symbol,  as a centered display on the page. 
 Complex notation is no barrier to this form of journal production. 

 Many  thanks  to  the  members  of  the  Editorial  Board  of {\sl
 Solstice\/};  with  the  publication  of this issue we welcome the
 addition to  that  Board  of  William D. Drake, Ph.D.  Engineering
 (Operations  Research),  and  Professor in various departments  of
 The  University of Michigan.  Bill  has a brief note later in this
 issue of {\sl Solstice\/} in which  he  introduces  himself,  some
 of his recent interests, and some of his students' interests.  
 \noindent{\bf 2.  PRESS CLIPPINGS---SUMMARY}

 Brief   write-ups   about {\sl Solstice\/}  have  appeared  in the
 following publications:

 \noindent 1.  {\bf Science}, ``Online Journals"  Briefings.  
 [by Joseph Palca]
 29 November 1991.  Vol. 254.
 \noindent 2. {\bf Science News}, ``Math for all seasons"
 by Ivars Peterson, January 25, 1992, Vol. 141, No. 4.
 \noindent 3. {\bf American Mathematical Monthly},
 ``Telegraphic Reviews" --- mentioned as
 ``one of the World's first electronic journals using {\TeX}," 
 September, 1992.
 \centerline{\bf Frank Harary}
 \centerline{Distinguished Professor of Computer Science}
 \centerline{New Mexico State University}
 \centerline{Professor Emeritus of Mathematics}
 \centerline{University of Michigan}
 \centerline{Reprinted with permission from }
 \centerline{\sl Biometrie-Praximetrie }
 \centerline{Vol. XII, 1-4, 1971, pp. 3-18}
 \centerline{Published by Soci\'et\'e Adolphe Quetelet}
 \centerline{Belgian Region of Biometric Society}
 \centerline{Av. de la Facult\'e, 22}
 \centerline{B-5030 Gembloux, Belgium}
 \centerline{At the time this paper was written,}
 \centerline{Frank Harary}
 \centerline{was also a member of the}
 \centerline{Research Center for Group Dynamics}
 \centerline{Institute for Social Research}
 \centerline{The University of Michigan}

 \noindent No matter what the area of scientific research, whether
 social or physical, mathematical thinking is involved, explicitly
 or implicitly. At the least, the precise formulation of a problem
 entails some aspect of set theory and logic.  Generally speaking,
 the  working  scientist  uses  the  term `mathematical model' for
 whatever branch of mathematics he  may be applying to his present
 problem.  On the other hand,  the  purist  mathematician-logician
 insists  strictly  on  the  use  of  `model'  to  mean  a certain
 interpretation of an abstract axiom system in the real world.

 We begin with a self-contained development of the concepts needed
 for  the  discussion  of  research  processes.  This leads to the
 distinction  between  the  real  and  abstract  world,   and  the
 interaction  between  them  by  interpretation  and  abstraction. 
 A similar, but conceptually different bifurcation is proposed for
 the two levels of research:   digging into the foundations versus
 extending  the  horizons  of knowledge.  These considerations are
 assembled  into  a  comprehensive Research Schema which enables a
 concise analysis of scientific discovery. Classical illustrations
 are provided, including true stories about Newton, Darwin, Freud,
 and  Einstein.   We  conclude with some subjective evaluations of
 acceptability of mathematical models.

 \noindent{\bf 1.  What Are They?}

 We have just noted  that  the word `model' has different meanings
 for  the  mathematician  and the scientist.  When a mathematician
 uses  the  word,  he  is  referring  to  the  physical  or social
 realization of his  theory.   On the other hand, when a scientist
 speaks of a mathematical model,  he means the area of mathematics
 which applies to his work.  Thus  one  (following Abraham Kaplan,
 oral communication)  could say as  a mnemonic aid that a model is
 always the  other fellow's system.   Contrariwise it also appears
 to be customary by usage to refer to ``research'' as what goes on
 in your own domain.

 In  order  to  define  a  model  rigorously,  it is convenient to
 develop  (as in Wilder [4] or in a more  elementary presentation,
 Richardson    [3])   several   notions   in  the  foundations  of 
 mathematics.   Recall  from  high  school  geometry that Euclid's
 axioms  are  about as follows (depending on which book you read). 
 The words ``point'' and ``line'' are undefined terms.

 $A_1$ (Axiom 1)  Every line is a collection of points.

 $A_2$  There exist at least two points.

 $A_3$  If $u$ and $v$ are points,  then there exists one and only
 one line containing $u$ and $v$.

 $A_4$  If $L$ is a line, then there exists a point not on $L$.

 $A_5$  If $L$  is a line,  and $v$  is  a  point not on $L$, then
 there  exists  one  and  only  one  line $L'$ containing $v$ which is 
 parallel to $L$, i.e., $L \cap L' = \emptyset $.

 Axiom 5 is the celebrated ``Parallel Postulate'' of Euclid.

 An  {\sl axiom system\/}  $\Sigma = (P, A)$ consists of two sets:
 a set $P$ of primitives and a set $A$ of axioms. {\sl Primitives}
 are the deliberately  undefined terms  upon which all definitions 
 in  the  system  are  based.  {\sl Axioms\/} are statements which
 are  assumed  to  be true, and from which other statements called
 {\sl theorems\/}, can be derived.  Primitives and axioms serve to
 avoid so-called circular definitions and circular reasoning. Each
 axiom in the system is an assertion about the primitives.

 Euclid's  axiom  system  consists  of two primitives, `point' and
 `line', and five axioms.  When Euclid developed geometry, he made
 a distinction between axioms and postulates. Both were statements
 whose truth was assumed,  but axioms were considered self-evident
 while  postulates  were  not!  This distinction eventually proved
 unnecessary and even undesirable,  and today axiom  and postulate
 are synonyms.

 We  shall  denote  by $T$ or $T(\Sigma )$ the set of all theorems 
 derivable   from   an  axiom  system  $\Sigma $.    Then  a  {\sl
 mathematical  system\/}  $(P, A, T)$  is an axiom system together
 with all theorems derivable from it.

 An {\sl independent axiom\/} $A$ of $\Sigma $ is one which cannot
 be derived from the remaining axioms.  An {\sl axiom system\/} is
 {\sl independent\/}  if every axiom is independent.  In is called 
 {\sl consistent\/} if  there are  no two contradictory statements
 in $T(\Sigma )$.
 One of the classical problems in 19th Century  mathematics was to
 determine whether or not Euclid's Parallel Postulate,  $A_5$, was
 independent.   The  consensus  of  opinion  was  that  $A_5$  was
 dependent,  that is,  it could be derived from   $A_1$ --- $A_4$. 
 Unsuccessful  attempts  to  derive  $A_5$  led  to  the discovery
 instead of non-euclidean geometry. The two types of non-euclidean
 geometry are now respectively called  {\sl hyperbolic geometry\/}
 (Bolyai-Lobachewski  independently)  in which  there can  be many
 parallels to a line through  a  point,  and   { \sl  elliptic
 geometry\/} (Riemann) in which there can be no such parallel.

 An  {\sl interpretation\/} of an axiom system is an assignment of
 meanings  to  its  primitives  which makes the axioms become true
 statements.   The  results  of  an interpretation of $\Sigma $ is
 called a {\sl model\/} for $\Sigma $.   This is the strict use of 
 `model' mentioned earlier.

 An  axiom system is called {\sl satisfiable\/} if it has at least
 one  model.   Two  models,  $M_1$ and $M_2$ of $\Sigma $ are {\sl
 isomorphic\/}  if  there  is  a  1-1  correspondence  between the
 elements  of  $M_1$  and  those  of  $M_2$  which preserves every
 $\Sigma$-statement.  In  a  {\sl categorical\/} axiom system, any
 two models are isomorphic.

 To illustrate, consider  an  axiom system with primitives $P=\{S,
 \circ \}$,  where  $S$  is  a  set  of integers, and $\circ $, is
 chosen  as  an  undefined  term for a binary operation denoted $a
 \circ b$, in order to avoid preconceived  notions that a familiar
 symbol like $a+b$ would  bring to mind.  The following statements
 $A_1$ --- $A_4$ are called {\sl group axioms\/},  and any set $S$
 on which they hold under the operation
 ${\circ }$ is called a {\sl group\/}.

 $A_1$  (Closure Law)  $S$ is closed under ${\circ }$, that is, if
 $a$ and $b$ are in $S$, $a {\circ } \, b$ is in $S$.

 $A_2$  (Associative Law)  Operation $\circ $ is associative, that
 is, $a \circ (b\circ c) = (a \circ b) \circ c $ for all $a$, $b$,
 and $c$ in $S$.

 $A_3$  (Identity Law)   There  is  a  unique  element $i$ in $S$,
 called the identity element, such that $a \circ i = i \circ a = a$
 for all $a$ in $S$.

 $A_4$  (Inverse Law)   For  every  $a$  in $S$, there is a unique
 element,  written $a^{-1}$ and called the {\sl inverse\/} of $a$,
 such that  $a \circ \, a^{-1}  =  a^{-1} \circ \, a  = i$.  Each of the
 four group axioms is independent,  and so this  is an independent
 axiom system.   To  verify that this axiom system is satisfiable,
 we now display a model.

 One  model  for  this  system  is  the set $S_1 = \{1,-1\}$ under 
 multiplication $\times $.  Thus this is called a group of {\sl order\/}  
 i.e., having just two elements.  The identity element is 1,  each
 element has itself as an inverse, and $S$ is obviously closed and
 associative,  as  can  be  seen from the following multiplication

 \line{\hfil \phantom{$-1$}$\times$$\vert$\phantom{$-$}$1$\quad$-1$\hfil}
 \line{\hfil -----------------------\hfil}
 \line{\hfil \phantom{$\times$}$-1$$\vert$$-1$\quad\phantom{$-$}$1$\hfil}

 Another  model  for  this  axiom  system is the set $S_2=\{0,1\}$
 under  addition modulo 2.  We define the sum of $a$ and $b$ mod 2
 to be the  remainder  of  $a+b$  after division by 2.  Under this
 operation,  we  see  at  once  from  the next table that $S_2$ is
 closed  and  associative,  0 is the identity, and each element is
 again its own inverse.  Thus $S_2$ is also a group of order 2.

 \line{\hfil $+$ mod 2\phantom{1} $\vert$ $0$ \quad $1$ \hfil}
 \line{\hfil ----------------------- \hfil}
 \line{\hfil \phantom{$+$ mod 2}$0$$\vert$$0$ \quad $1$ \hfil}
 \line{\hfil \phantom{$+$ mod 2}$1$$\vert$$1$ \quad $0$ \hfil}
 More  generally,  one  can  take  $S$  to be the set  $\{0, 1, 2,
 \ldots ,  n-1\}$   and  $a \circ b$  to  mean $a+b\,\hbox{mod}\,n$. 
 Then for each positive  integer  $n$,  we get a distinct group of
 order  $n$.   Thus  the  above  axiom  system  for  groups is not
 categorical, since it has many non-isomorphic models.

 These  two  groups,  $S_1$ and $S_2$, are isomorphic since we can
 let  operation  $\times $  correspond with $+$ mod 2 and set
 $\{1, -1\}$ to correspond with $\{0, 1\}$.   All statements
 derivable from the axioms  still  hold.  That  the two models
 are isomorphic is also shown in the fact that their tables both
 have the following form:

 \line{\hfil $\circ$ \phantom{a} $\vert$ $a$ \quad $b$ \hfil}
 \line{\hfil ----------------------- \hfil}
 \line{\hfil \phantom{$\circ$}$a$ $\vert$ $a$ \quad $b$ \hfil}
 \line{\hfil \phantom{$\circ$}$b$ $\vert$ $b$ \quad $a$ \hfil}
 In fact, any pair of groups with two elements are isomorphic,  so
 it is customary to speak of ``the group of order two." 

 The  study of group theory was originally motivated by properties
 which  are  possessed  by  the  symmetries  of  a  configuration,
 whether it be  geometric,  algebraic, architectural, physical, or
 chemical. It is readily verified that symmetries satisfy the four
 group  axioms.   For  example,  the  inverse  of  a symmetry of a 
 configuration  is  the  corresponding  symmetry  mapping  done in

 \noindent{\bf 2.  Two Worlds:  Abstract and Empirical}

 The  realm  of  research  activity is naturally divided into two
 worlds:  the  abstract and the empirical.  The abstract world is
 generally regarded as the domain of the mathematician, logician,
 or purely  theoretical  physicist,  while the empirical world is
 inhabited  by  experimental  scientists   of   many   varieties: 
 physical,   social,   and  others.   [It  has  been  established
 empirically that the less  scientific a subject, the more likely
 it  is  that  its  practitioners call it a science.  Outstanding
 examples  include  (in  alphabetical  order):  divinity science,
 library  science,  military  science,   political  science,  and
 secretarial science.]  There is a growing tendency, however, for
 people to live in both worlds in these interdisciplinary times.

 Those  who  work  entirely  in the abstract world are engaged in
 deriving  new  theorems  either  from axioms or from an existing
 theory  or  coherent body of theorems.  Such results are usually
 expressed in  symbols rather than numbers, and rarely touch upon
 the real world.

 On the other hand, the inhabitants of the empirical world ``work
 for a living." Some live in laboratories and perform experiments
 in  order  to  collect  meaningful  data leading to a scientific

 The  two   worlds  are  shown in Figure 1.  The two loops, called
 theory building and experimentation, represent purely theoretical
 and purely experimental research.

 Figure 1 exhibits a symmetric pair of directed links between  the
 worlds,  the  first  of  which  can  be  called interpretation in
 accordance  with  the  use of this word in the preceding section. 
 In a confrontation  between these two worlds, the mathematician's
 theorems  become  predictions  about the real world, which can be
 tested  by  the  scientist.   If a  prediction  is verified by an
 appropriate  experiment,  the  scientist  feels  that the theorem
 really   works,   and   the   mathematician   has  found  a  {\sl

 \noindent{\bf Figure 1 --- Two worlds}.
 [Two rectangles, representing the two worlds, are linked by a left
 arrow  and  a  right  arrow.   The  left  rectangle  is  labelled
 ``Abstract";  the  right  rectangle  is  labelled   ``Empirical." 
 The right arrow, from  the  abstract world to the empirical world
 is labelled ``interpretation." The left arrow, from the empirical
 world to the  abstract  world is  labelled abstraction.  A  loop,
 labelled  ``theory building,"  is  attached to  the upper left of
 the abstract  world.  A  loop,   labelled  ``experimentation," is
 attached  to  the  upper  right  of  the empirical world.
 Inserted by Ed.] \endinsert

 If the predictions are entirely incorrect,  the  model  cannot be
 used.  However,  in cases where the predictions are not verified,
 yet  are  ``rather close"  to correct,  further abstraction is in
 order to construct a working model. This abstraction in the light
 of the  experiment  may suggest alternate hypotheses which should
 result  in  new  theorems.  These theorems hopefully will lead to
 better predictions than previously, and to a working model.

 \noindent{\bf 3.  Two Worlds:  Two Levels}

 Each  of  our  two worlds may be divided into two levels.   As we
 have  indicated, the upper level of the abstract world deals with
 the  development  of  mathematical  systems  by the derivation of
 theorems.  We  have  discussed interaction between worlds at this
 level by means of interpretation and abstraction. In this section
 we shall observe  that this same type of interaction can occur at
 the lower level.

 The  lower level of the abstract world deals with the foundations
 of mathematics, axioms, and logic.  The research activities might
 involve trying  to  prove consistency or independence of an axiom

 A  rather  esoteric  and  dramatic recent example of an important
 discovery  at  this level is given by the definitive work of Paul
 Cohen [1]. It is known (see Wilder [4], for example) that a $1-1$
 correspondence can be constructed  between  the  natural  numbers
 $1$, $2$, $\ldots$, and all  the  integers,  $\ldots$ $-3$, $-2$,
 $-1$, $0$, $1$, $2$, $\ldots$,  and  between the integers and the
 rational  numbers.   These  three sets of numbers are all said to
 have   the   same   (infinite)  {\sl   cardinality\/}   which  is
 conventionally denoted $\aleph_0$.

 It  is also known that there are more real numbers than integers. 
 The  real  line is sometimes called the {\sl continuum\/}, and so
 ${\bf c}$  is  written  for  the  number  of  reals.   The  {\sl
 continuum hypothesis\/} states that there is no infinite set with
 cardinality between $\aleph_0$ and ${\bf c}$. 

 Cohen  proved  that  the  continuum  hypothesis  (as  well as its
 negation) is consistent with the usual axioms of set theory. As a
 consequence,  it  is  independent  and  can neither be proved nor
 disproved in that axiom system.   Analogous to the development of
 non-euclidean geometry, two entirely different axiom systems have
 been created;  one  by  assuming  the  continuum  hypothesis, and
 the  other  by  taking  its  negation.   Cohen  also  proved  the
 independence of the ``axiom of choice."

 On  the  other  hand, the lower level of the empirical world also
 deals  with  foundations,  but  in  the form of the basic laws of
 science.   Kepler's  Laws  of  Planetary  Motion, Darwin's Law of
 Natural Selection,  Newton's Laws of Motion,  Kirchhoff's Laws of
 Electricity,  and  Einstein's Law  of Special Relativity  are all

 The  link  between  the  two  worlds at this lower level is quite
 analogous to that at the upper level.   Thus interpretation of an
 axiom  leads  to  a  basic  law  about  the  real world, while an
 abstraction,  a  coherent set of scientific laws becomes an axiom
 system.   The schematic representation of interaction between the
 two worlds is shown in Figure 2.
 \midinsert \vskip 3in
 \noindent{\bf Figure 2 --- Interaction between the two worlds}.
 [There are four rectangles in this figure, arranged at the upper
 left, upper right, lower left, and lower right.  The two uppers
 have  a  left arrow and right arrow linking them, as do the two
 lowers.   The  upper  left  rectangle  is labelled ``Theorems"; 
 the upper right,  ``Data";  the lower left, ``Axioms"; and, the
 lower right ``Laws."   The right arrow in each case is labelled
 ``interpretation."   The  left  arrow  in each case is labelled
 ``abstraction."   The  left hand side of the figure is labelled
 ``Abstract"; the right, ``Empirical."  There is a loop attached
 to each of the four rectangles. Ed.] \endinsert

 \noindent{\bf 4.  Two Levels:  Derivation and Selection}
 Having  discussed  interaction between the  two worlds,  we shall
 now  establish  links  between their upper and lower levels.  The
 process  of  climbing  from  the  lower level to the upper in the
 abstract  world  can  be  regarded as {\sl derivation\/}.  For we
 begin with an axiom system and then,  sometimes painfully, derive
 progressively  complicated  theorems  to  obtain  a  mathematical

 Now consider how one goes from the upper level to the lower. From
 an existing body of theorems, an axiom system is to be built.  To
 accomplish this, we select a body of particularly appropriate and
 fruitful  theorems  to  use  as  axioms.   This  process  of {\sl
 selection\/}  yields  a  small,  more  manageable  and often more
 powerful  system,  which  is  conducive  to the derivation of new

 Selection in the empirical world involves collecting and studying
 vast amounts of data, and observing a pattern which may suggest a
 general law.  Thus it is actually the {\sl induction\/} process.

 There  appears  to  be no direct link in the empirical world from
 the lower level to the upper.  Derivation does occur, and in fact
 uses the deduction  process,  but again and again we find that it 
 takes the ``long way around,"  as shown in Figure 3.   One begins
 with several scientific laws (lower right), and abstracts them to 
 formulas  (lower  left) from which theorems can be derived (upper
 left) which make  predictions about the real world (upper right). 
 It  is  convenient,  however,   to  draw  the  link  representing
 derivation directly as well, as we do later.

 \midinsert \vskip 2in
 \noindent{\bf Figure 3 --- Derivation in the empirical world}
 [The four rectangles of Figure 2, are linked with three arrows
 from Laws to Axioms to Theorems to Data. Ed.] \endinsert

 In general, innovative research is initiated in the upper  level,
 and particularly in the upper right quadrant.  This is due to the
 fact that the great majority of natural and fundamental questions
 arise from an attempt to observe or explain  empirical phenomena.  
 In fact, most research is done at the upper level, both right and
 left, while almost no one continuously remains at the lower level.

 For  example,  in  ancient  Egypt,  the  discovery  of  geometric 
 formulas  was  necessitated by the search for improved techniques
 in measuring and surveying. Problems in geometry were solved long
 before Euclid organized the subject in an axiomatic formulation.

 \noindent{\bf 5.  Research Schema}

 We  contend  that  the  above Research Schema represents all  the
 types  of  interaction  between the abstract and empirical worlds
 during the processes of research and discovery.  Its two diagonal
 links are  shortcuts  which  represent research processes that go
 directly to  ``opposite" quadrants.   There do not seem to be any
 directly ascending diagonal links.  

 It is rarely  but definitely possible  to predict scientific laws
 from a body of theorems without actually working with experimental
 data.  This  is  represented  by  the diagonal from upper left to
 lower right  in the  Research Schema.  We shall see that Einstein
 took  this  route  in  his  formulation  of the theory of special

 \midinsert \vskip 3in
 \noindent{\bf Figure 4.  Research schema}. [Draw Figure 2.  Label
 the loop on ``Theorems" as ``theory building"; that on ``Data" as 
 ``experimentation"; that on ``Axioms" as ``axiomatic archaeology";
 and, that on ``Laws" as ``empirical archaeology." Add up and down
 vertical arrows joining the rectangles; label the  downward arrow
 in each case as ``selection"; the upward as ``derivation."   Draw
 the  two  diagonals -- one  with  an  arrow to suggest going from
 ``Theorems" to ``Laws" and the other from ``Data" to ``Axioms."
 Ed.] \endinsert 

 The  shortcut  from  experimental  data  to  axioms,  skipping the
 formulation  of  laws,  occasionally occurs in the social sciences
 when  a  careful  analysis  of  data  patterns  produces  a set of
 formulas that can be taken as axioms.  These are then interpreted,
 and  hopefully  suggest  an  empirical  law, without the selection

 When considering  routes  between  the  two  worlds, one must also
 allow for traversing loops at any quadrant one or more times.  The
 upper  right  loop,  for  example,  when  traversed several times,
 indicates  repeated efforts in observation and collection of data,
 before attempting to select corresponding laws.  

 One  must  also note that the most direct route is not often taken
 in research.  This will become evident in the next section when we
 take a closer look at particular cases of discovery.

 \noindent{\bf 6.  Sketches of Discovery}

 We  shall  illustrate the Research Schema with the work of several
 men  who represent varied branches of science and mathematics.  We
 begin with Euclid, whose work in the axiomatization and derivation
 of what we now call euclidean geometry is represented schematically
 in Figure 5.   [It has been said that the ultimate recognition of a
 man's  contribution is conferred when his name is made an adjective
 and not capitalized.]

 \midinsert \vskip2in
 \noindent{\bf Figure 5 --- Euclid's Research Schema}.   [Draw three
 rectangles:    upper left,  upper right,  lower left.   Label them,
 respectively, ``Theorems of Geometry,"   ``Egyptian observations on
 measure," ``Axioms of Geometry."   Add a loop to the two rectangles
 on the left.  Join the upper left and  lower  left rectangles by an
 up arrow and a down arrow.   Draw an  arrow from the upper right to
 the upper left rectangle. Ed.] \endinsert

 {\sl Euclid\/}:    Although  Euclid  is  the acknowledged father of
 geometry, his main contribution was to its organization rather than
 to its derivation.   The early Egyptians already knew the rudiments
 of geometry,  including a form  of  the  pythagorean  theorem,  and
 formulas for the area and volume of many geometric figures. Thus we
 attribute  the  upper  right quadrant in Figure 5 to the Egyptians. 
 The emphasis on proof, however, was  introduced by the early Greeks
 and  Euclid's  contemporaries  developed  many  of  the theorems of
 geometry.   Euclid  selected  the  five  axioms above from existing
 results.   He then  proved  from these all the theorems of geometry
 then known and a few new ones, and presented a logical organization
 of  the  material  in  an  exhaustive  text.  By today's standards,
 Euclid's axiomatic work is not rigorous, but it was an  outstanding
 accomplishment for its time.

 \noindent{\bf Figure 6 --- Newton's Research Schema}.  
 [Draw four rectangles: upper left  --  ``Theorems  of  Calculus";
 upper right  --  left  half  labelled ``Verification" right  half
 labelled ``Collection of Data"; lower left -- Abstraction of Laws
 of Motion; lower right -- ``Laws of Motion."  Join the rectangles
 with arrows forming  a rectangular cycle  oriented in a clockwise
 direction.  Add a loop to  the  lower  left  rectangle; label the
 loop ``Formalization of Calculus by Cauchy."   Add  a loop to the
 upper  right  rectangle;  label  the  loop  ``Galileo."  Link the
 ``Galileo"  loop  to  the  down arrow as a dashed line separating
 ``Verification" from ``Collection of Data"  in  the  upper  right
 hand box. Ed.] \endinsert

 {\sl Newton\/}:  Unlike Euclid, Newton occupied every quadrant of
 the  Research Schema.   His  first work was on the upper level of
 the  empirical  world,  where  he  experimented  in chemistry and
 optics  while still a student.   Newton's most important results,
 however, were not derived from his own data, but from the work of
 those  before  him.   His  formulation  of the Laws of Motion was
 induced from Galileo's extensive experimentation. Hence we credit
 the  upper  right  loop  in  Newton's Research Schema to Galileo. 
 Newton's Laws of Motion have been stated as follows:

 \item{1.}Every body will continue in its state of rest or uniform 
 motion in  a straight  line unless it is compelled to change that
 state by impressed force.

 \item{2.}  The  rate of change of momentum is proportional to the
 impressed force and  takes  place  in the line in which the force

 \item{3.}   For  every  action,  there  is  an equal and opposite

 Newton  left  the  empirical  world  and  entered the abstract by
 expressing  his  laws  symbolically  as equations.  His work with
 these resulted in the discovery of both differential and integral
 calculus.  Others independently discovered these concepts, but it
 is  believed  that  only  Newton  and  Leibnitz  (who  discovered
 calculus   independently)   realized  that   differentiation  and
 integration were inverse processes.

 Calculus  did  not  become  mathematically precise until the next
 century  when  Cauchy  introduced the necessary concepts of limit
 and infinite sequence.  We draw a loop in the lower left quadrant
 of  Figure  6  to  represent Cauchy's work in the  foundations of

 This  new  branch  of  mathematics  readily  produced an abundant
 supply  of theorems.   The predictions which resulted were tested
 in the  laboratory,  and found  to be entirely correct within the
 range of current measuring instruments.

 \midinsert \vskip2in
 \noindent{\bf Figure 7 --- Einstein's Research Schema}.
 [Draw four rectangles.  Label upper left: ``Theorems for Special
 Relativity";  upper right  --  ``Michelson-Morley  and  others";
 lower  left  --  ``Formulas";  and,  lower  right is split (by a
 dashed line) -- top half  ``Laws of Light  Motion,"  bottom half
 ``Special Relativity Theory."   Arrows  from upper left to upper
 right  --  ``prediction";  from  upper  right  to lower right --
 ``selection"; from lower right to lower left; from lower left to
 upper left -- ``derivation." Ed.] \endinsert

 {\sl Einstein\/}:   Eventually,  more accurate measuring devices 
 revealed  that  Newton's  Laws  of  Motion could not explain the
 behavior  of  light  on  either  the microscopic or astronomical
 level.   Furthermore,  the  Michelson-Morley  experiment  proved
 conclusively that ``ether" did not exist.  These discoveries led
 to  a  period  of  great activity in physics pioneered by Albert

 Like  Newton, Einstein's major work resulted from data collected
 by  scientists  before  him.   Einstein was a purely theoretical
 physicist,  and never worked  in the upper right quadrant of the
 Research Schema himself. But he certainly stimulated an enormous
 number of experiments there. He proposed the following empirical
 axiom system as laws of light motion:

 \item{1.} No physical object can travel faster than the speed of
 \item{2.}  The speed of light depends not at all on the relative
 positions  of  the  source  of  light and the observer, or their
 relative speeds.
 \item{3.}  The  mass  at a velocity $v$ of a particle equals its
 mass at  velocity 0  divided by $\sqrt{1-v^2/c^2}$, where $c$ is
 the speed of light.

 Einstein  abstracted  these  three laws to an axiom system, from
 which he  derived the body of theorems interpreted as the theory
 of special relativity. He found that in particular, his distance
 formulas  for  relativity  theory  were   related  to  those  of
 hyperbolic   non-euclidean   geometry;  thus  relativity  theory
 provides a physical model for hyperbolic geometry.  The Research
 Schema for this discovery is shown in Figure 7.  We begin in the
 upper right with the Michelson-Morley experiment, and then go to
 the  Laws  of  Motion  of  Light  in  the lower right, and their
 abstractions in the lower left. From there we go to the theorems
 of  special  relativity  in  the  upper left, and finally to the
 experimental   verification   in  the  upper  right  where  this
 cycle  started.  Einstein then went around this cycle again with
 his more refined theory of general relativity, which led to more
 precise predictions of physical measurements.

 \noindent{\bf Figure 8 --- Darwin's Research Schema}.
 [Draw  two  rectangles, one  above and one below.  The top one is
 labelled  ``Data";  the  bottom  one   is   labelled  ``Theory of
 Evolution." There are three loops attached to the top one.  There
 is a line linking the two rectangles. Ed.] \endinsert

 {\sl Darwin\/}:   Charles  Darwin  spent  most  of his life doing
 research in  only  one quadrant of the Research Schema, the upper
 right.   His  research  career  began when he became the official
 naturalist on the good ship Beagle, and embarked upon a five-year
 voyage.  He made observations  on all species of animals he could
 find,  and  took  voluminous  notes.  During the remainder of his
 life,  Darwin  analyzed  and classified these notes and all other
 available   information.    The   climax  of  his  work  was  the
 formulation  of  his  Law  of Natural Selection and his Theory of

 Darwin's  theory  asserts  that all animal species have descended
 from  a  common  origin.   The  variety  of  species results from
 ``natural  selection,"  in  which  those  animals  which are best
 adapted  to  their  environment   survive.    Due  to  occasional
 mutations,  certain  animals  in  a  species  are  better able to
 survive than others.  These mutations may be  passed  on to their
 offspring  who  in  turn  will  tend  to  survive  and reproduce,
 eventually  resulting  in  a new species which has been naturally

 \midinsert \vskip2in
 \noindent{\bf Figure 9  ---  Freud's  Research  Schema}. [Draw two
 rectangles,  one above and one below.  Label the top one ``Medical
 Practice." Label the bottom one ``Psychoanalytic Theory." Join the
 two rectangles with an up arrow and a down arrow.  There is a loop
 attached to the top rectangle. Ed.] \endinsert

 {\sl Freud\/}: Sigmund Freud, like Darwin, stayed in the empirical
 world.   In fact, their Research Schemata are quite alike, as seen
 in Figures 8 and 9. He began with a medical degree and turned from
 general  practice to specialization.  Freud (in collaboration with
 J. Breuer initially) did research in the treatment of ``hysterical"
 patients who  had  physical  symptoms  for which no physical cause
 could be found.  He inferred from the study of many cases that the
 symptoms could be traced  back to some repressed childhood trauma,
 and went on to develop the  concept  of  the subconscious together
 with  the  id,  ego, and superego.  First through hypnosis, and later
 through ``free association," Freud was  able to induce himself and
 his patients to recall these forgotten experiences,  and alleviate
 their symptoms.

 Much  of  the psychoanalytic theory which Freud developed is still
 highly controversial today,  although it has made a lasting impact
 on the development of many modern theories in psychology.

 There  has  been a highly publicized report of the proof of a deep
 and important theorem by a mathematician while  boarding  a bus in 
 Paris.  It  may  be  just  as  true as the anecdote about Newton's
 finding his law of gravitational attraction when an apple fell off
 its  tree  and  landed  on his head.  This sort of phenomenon does
 occur, but  fortunately is not an  intrinsic part of the discovery
 procedure.  In the words of Hans Zinsser,
 It  is  an  erroneous  impression, fostered by sensational popular
 biography, that scientific  discovery is often made by inspiration
 $\ldots$ .  This  is  rarely  the  case.  Even  Archimedes' sudden
 inspiration  in the bathtub; Descartes' geometrical discoveries in
 his  bed;  Darwin's  flash  of  lucidity  on  reading a passage in
 Malthus; Kekule's vision  of the closed carbon ring came to him on
 top of a London bus;  and  Einstein's  brilliant  solution  of the
 Michelson puzzle in the patent  office in Bern, were not messages
 out of the blue.  They were the  final co-ordinations, by minds of
 genius,  of  innumerable  accumulated  facts and impressions which
 lesser men could grasp only in their uncorrelated isolation, which
 --- by them --- were  seen in entirety and integrated into general
 principles. The scientist takes off from the manifold observations
 of  predecessors,  and  shows  his  intelligence,  if  any, by his
 ability to discriminate between the  important and the negligible,
 by selecting here and there  the  significant  steppingstones that
 will lead across the difficulties to new  understanding.   The one
 who  places  the  last  stone  and  steps across to the {\sl terra
 firma\/} of accomplished discovery gets all the  credit.  Only the
 initiated  know  and  honor  those  whose  patient  integrity  and
 devotion to exact observation  have  made  the last step possible. 

 When a researcher has become sufficiently steeped in his problem,
 he  has  amassed  enough  meaningful  data  (mathematicians  also
 accumulate  data  via  ``thought-experiments")  to  perceive  the
 proper  pattern  and  conceive the correct conjecture.  This is a
 necessary but not sufficient  step toward establishing a theorem. 
 A  proof,  which  is  valid,  must  be  supplied;  otherwise, the
 assertion remains a conjecture. The two talents of conjecture and
 proof appear to be quite separable.

 \noindent{\bf 7.  What Should They Be?}

 It  is  becoming more fashionable to use mathematical models as a
 powerful  analytic  device for advancing scientific research in a 
 remarkable variety  of  disciplines.  This usage is certainly not
 unwarranted,  since  models,  when used with care and discretion, 
 can and should be of great value in the clarification of existing
 problems and the formulation of important new ones. Unfortunately,
 it seems that models are misused all too often.  The word `model'
 is  sometimes  bandied  about by people with little conception of
 its real meaning simply  because  it is {\sl au courant\/}.  They
 don't  even  define  `model',  but use the word to suit their own

 A  model  need  not  be  impressively  confusing  in  order to be
 valuable.  In fact, one of the main contributions of a model lies
 in its ability to simplify a problem, and so it should be no more
 complicated than necessary.

 Neither should a model be symbol-rich but idea-poor. Models which
 hide miniscule content behind a mass of symbolic formulas tend to
 look impressive,  but add nothing.   ``Mystery is no criterion of
 knowledge." For example, a recent paper in a leading psychological
 journal had only one abstract idea: the number of elements in the
 union  of  two  sets is the sum of the number of elements in each
 minus the number they have in common. Alas, the author apparently
 did not recognize it as  the simplest special case of the Principle
 of Inclusion and Exclusion.

 Another  unfortunate  use  of  mathematical  models occurred in a
 published paper in sociology in which there were  ten  axioms and
 zero  theorems.   However, an interpretation was then given which
 resulted in ten ``empirical theorems," one for each axiom.  This 1-1
 correspondence  between  axioms  and  empirical  theorems  simply
 involves the preparation  of  axioms  which  will  yield  desired
 empirical assertions.

 Furthermore,  an  axiom  system should not be constructed for the 
 artificial purpose of deriving just one theorem which has already
 been verified statistically.   Clearly such a model only clutters
 the literature and does not involve genuine derivation.

 We do not wish to lay all the blame for the misuse of mathematical
 models  on  scholars  in  the  empirical  world; it occurs in the
 abstract  world  as  well.   The following passage by the eminent
 linguist  Gustave Herdan  [2] shows the dual roles the two worlds
 can play in the misuse of models.

 Without  going  into  details,  I  will  only  mention  a certain
 quantitative  relation  known to  linguists  as  the  `Zipf law'.
 Mathematicians believe in it as a law, because  they  think  that
 linguists have  established it as a relation of linguistic facts,
 and linguists  believe  in  it because they, on their part, think
 that mathematicians have established it to be a mathematical law. 
 As can be shown in five minutes,  it is  not a  law at all in the
 sense in which we speak of natural laws. \smallskip}

 Loosely  stated,  this  law  of  Zipf proposes a high correlation 
 between the frequency of use of words and their brevity.

 Another typical superficial use of  mathematical  models involves
 the  bland  assumption  that  the  most  elementary  parts  of an
 existing branch of  mathematics  apply  unchanged to a problem in
 social science.  Typical  examples  include  high school algebra,
 coordinate geometry,  matrix  manipulation, graph theory, and the
 probabilistic  theory  of  Markov  chains.   In  such models, the
 typical procedure is to assign empirical terms to the mathematical
 variables by way of interpretation  at the lower level.  Then the
 existing theorems and methods of  calculation  are  translated at
 the  upper  level  into  statements  which  are claimed to be new
 empirical findings.

 What,  then,  should  mathematical  models be?  We have suggested
 that  they  should  lead  to new theorems, but this is not always
 necessary.    The   precise  thinking  involved  in  the  careful
 formulation  of  an  axiom  system  will  lead  to  an   improved
 conceptualization of the empirical  phenomena at hand.   This  in
 turn can suggest the proper variables to measure, and  perhaps an
 approach to the measurement problem.
 Sometimes  an existing area of mathematics can be quite useful as
 a  mathematical model provided it is augmented by one or more new
 axioms  suggested by the real world.  The most productive models,
 however,  have  involved  derivation.   For  it is only after the
 derivation  of  new  theorems  that  unexpected  and far-reaching
 predictions can be made.  From a mathematician's viewpoint, it is
 best if derivation leads to  nontrivial  theorems, which actually
 qualify  for  publication  in  the  mathematical  literature.  To
 summarize, it is our personal and perhaps controversial contention
 that  mathematical  models  will  lead to significant and natural
 growth in both the abstract and empirical worlds.
 \noindent{\sl Acknowledgment.}

 Research  supported  in  part  by Grant MH22743 from the National
 Institute of Mental Health.
 \noindent{\bf References}

 \ref [1]    P. J. Cohen,   {\sl  Set  Theory  and  the  Continuum
 Hypothesis\/}, Benjamin, New York, 1966.

 \ref [2]  G. Herdan,  {\sl Sonderdruck aus Zeichen und System der
 Sprache\/}, Vol. 2, Akademie-Verlag, Berlin, 1962, p. 108.

 \ref [3]  M. Richardson, {\sl Fundamentals of Mathematics\/}, 3rd
 ed., Macmillan, New York, 1966.

 \ref [4]  R. L. Wilder,   {\sl Introduction to the Foundations of 
 Mathematics\/}, 2nd ed., Wiley, New York, 1965.
 \centerline{\bf 3B.  WHERE ARE WE?}
 \centerline{\bf Frank E. Barmore}
 \centerline{Associate Professor of Physics}
 \centerline{University of Wisconsin, La Crosse}
 \centerline{Reprinted, with permission, from}
 \centerline{\sl The Wisconsin Geographer}
 \centerline{Vol. 7, pp. 40-50, 1991}
 \centerline{A publication of The Wisconsin Geographical Society}
 \noindent {\bf 1.  Introduction}

 \noindent I was recently flabbergasted when I discovered how  the
 Bureau of the  Census  calculates  the location  of the center of
 population of the  United States following each decennial census. 
 I found that:

 The  center  of  population  is  the point at which an imaginary,
 flat, weightless, and rigid map of the United Sates would balance
 if weights  of  identical  value  were  placed on it so that each
 weight represented  the  location of one person on April 1, 1980. 
 Located at latitude  38 degrees, 8 minutes, 13 seconds north, and
 longitude  90  degrees,  34  minutes, 26 seconds west, $\ldots $. 
 The computation of the  center of population in 1980 was based on
 the 1980 population counts and the 1970 centers of population for
 counties.  County population centers have not been determined for
 1980.  The center is the point whose latitude (LAT) and longitude
 (LONG) satisfy the equations
 \hbox{LAT}={{\Sigma w_i \times \hbox{lat}_i}
            \over {\Sigma w_i}}, 
 \hbox{LONG}={{\Sigma w_i \times \hbox{long}_i \times 
             \hbox{Cos} (\hbox{lat}_i)} \over
              {\Sigma w_i \times \hbox{Cos}(\hbox{lat}_i)}},
 where  $\hbox{lat}_i$,  $\hbox{long}_i$, $w_i$ are  the latitude,
 longitude, and population, respectively, of the counties.  (U. S.
 Bureau of the Census, 1983, Appendix A, p. A-5.)
 The statements were surprising for a number of reasons.  First, as
 every good introductory physical geography text book points out, a
 flat map of the earth's curved surface is a {\sl distorted\/} map.
 Though  distortions  can  be  reduced  by  a  careful   choice  of
 projection, appropriately centered, some distortion always remains
 and cannot be eliminated.  Of course, one may define something any
 way one wishes,  but  at least it should be stated which method of
 projection  is  used  to  create  this  ``flat  map."  Second, the 
 formul{\ae}  given  suggest that  east-west distances are measured
 (reasonably,  though  arbitrarily)  along  parallels  of latitude,
 which are small circles.  But distances are usually measured along
 great circles and this would produce different results. Third, the
 formul{\ae}  yield  results  that  differ  from  results  of other
 accepted  definitions  of  the center of population.   Fourth, the
 formul{\ae} can produce some rather peculiar results.  These  will
 be discussed below.

 The  purpose  of this paper is to discuss some of the difficulties
 in  the  concept  of  the  ``center of population" when applied to
 populations  spread  over  enough  of the earth's surface that its
 curvature  is  noticeable.  A  more satisfactory definition of the
 center of population is proposed.
 \noindent{\bf 2.  Preliminary Remarks}

 When  considering the characteristics of a large group of anything
 distributed  over  a  region  it is often useful to concentrate on
 only the most  basic  characteristics,  the first three moments of
 the distribution:   1) the population, 2) the location, and 3) the
 spatial dispersion of the group.   This  paper concentrates on the
 second moment, the location. There are many ways location could be
 specified.   Almost  a  century  ago  Hayford  (1902) convincingly
 argued that the most appropriate measure of location of an area or
 population is a  statistic  called  the average (arithmetic mean). 
 Abler, {\it et al.\/}  (1971) agree:   ``When we ask {\sl where\/}
 questions about distributions  we almost  always desire an average
 location  which  represents  the  entire set."  When averaging the
 location, each location is ``weighted''  according to the specific
 characteristic of interest and the result  is the average location
 of  the weighting character.   The result is a  ``center of mass''
 if  the  weighting  character  is  mass,  a ``center of area'' (or
 geographic center)  if the weighting character is area, a ``center
 of  population''  if  the  weighting character is population, {\it
 etc\/}. Sviatlovsky and Eells (1937) have discussed in some detail
 the  use  and  significance  of  the  concept of the ``center'' in
 geographical regional analysis.

 Locations  can  be  described  as  vectors  whose  magnitudes  and
 directions are taken as the distances and directions of  the items
 whose center is to be calculated.  Then the  power and convenience
 of vector algebra can be used to calculate the center in one, two,
 three or even higher dimensional  spaces.  If the space is ``flat"
 (Euclidean)  then  the  process  is  quite straightforward, though
 tedious.  Also, by using vectors in a Euclidean space to represent
 the distances and directions of individuals in a population, there
 exist  several  interesting  and useful concepts.  First, when the
 center  of  the coordinate system used is {\sl at\/} the center of
 population then the vector  sum of the distances of all the people
 is  zero  and  the sum of  the squares of the distances of all the
 individuals is at a  minimum.   The  minimum  sum of the distances
 squared  when  measured  from  the average is not an accident, but
 rather,  the result of a fundamental mathematical relation between
 the two quantities. Sufficiently fundamental is this relation that
 Warntz and Neft (1960), for example, define the mean as the  place
 from which the sum of squares of the distances to each  member  of
 the  population  is  minimum.   Second,  when  the  center  of the
 coordinate system used is {\sl not at\/} the center of  population
 then  the  vector  sum  of  the  distances  of all the individuals
 provides  the distance  and direction  of the center of population
 from  the  center  of  the  coordinate  system.   I will use these
 characteristics later.

 However, the surface of the earth is not ``flat,'' but ``curved,''
 and  though finite,  is without a boundary.  On such a surface one
 can get  into  difficulty  with  the  concept of average location. 
 Where, for  example, is the ``center of area'' (geographic center)
 of  the  earth's  entire  surface?  If one  chooses to preserve an
 earth surface provincialism it is not clear how one can modify the
 ``vector  representation''  of  distance  and  direction  for  the
 locations of individuals in a population. Some criteria are needed.

 I suggest that any reasonable definition of ``center of population''
 should  meet  at  least  the following standards:  (1) population 
 distributions which are symmetric about some central point should
 have  their  center  of  population at this central point and (2)
 distances should be  measured  as  true  distances, either on the
 surface along great circles or in three dimensions along straight
 lines.  It would also be desirable to have any definition satisfy
 additional restrictions:  a) it should correspond to one's common
 understanding of ``center of population,"  b) it should reduce to
 the  usual definition of ``center of population'' when there is no
 curvature  and  c)  it  should be easily extended to nonspherical 
 surfaces --- for example,  the  International  Ellipsoid  or  the

 In  the  examples  and  discussions  which follow, all angles and
 great circle arc lengths are given in degrees and decimal degrees.  
 Azimuths  of places from any point are measured from North toward
 East.  Latitudes and  longitudes are designated as North or South
 and East  or  West  respectively.  I have ignored the differences
 between  the  shape of the earth and a sphere, as does the Bureau
 of the  Census,  when  calculating  centers  of population (U. S.
 Bureau of the Census, 1973).  When calculating the 1980 center of
 population  of the  United States in the various examples, I have
 used the original published 1980 populations (U. S. Bureau of the
 Census, 1983)  and the unpublished 1980 centers of population for
 the fifty states  and  the District of Columbia (see Appendix A). 
 When calculating the  1910  and 1880 centers of population of the
 United States I have  used  the published populations and centers
 of population of the  various states and the District of Columbia
 (U.S. Bureau of the Census, 1913 and 1914).

 \noindent{\bf 3.  Census Bureau Center of Population Formul{\ae}}

 Imagine a circumpolar population uniformly distributed along, say,
 the 70th parallel of latitude north (see Fig. 1).  If longitude is
 measured from $180^{\circ}$ W through $0^{\circ}$ to $180^{\circ}$  
 E  then  the  Bureau  of  the Census formul{\ae} put the center of
 population  at  $70^{\circ}$ N  on  the  Greenwich meridian.  Yet,
 surely the center of  this  population  is at the North Pole.  The
 Bureau  of  the  Census  formul{\ae}  fail  to  meet the suggested

 \topinsert \vskip 4in
 \noindent{\bf Figure 1.}         Example  population  centers  and
 distributions used in Part 3 of the text. Figure available in hard
 copy only; content  should  be  clear  from text and from caption. 
 A  sphere  is  drawn  containing  parallels  and meridians.  Three
 figures  are  highlighted  on  this  sphere.   I:   a  circumpolar
 population distributed along the  70th parallel of latitude north. 
 II.   a symmetric  population  distribution  centered  at latitude
 $38^{\circ}$ N and longitude $90^{\circ}$ W.   This  population is
 spaced at eight locations around the perimeter of a circle.   III:
 a simple symmetric population distribution centered at $38^{\circ}$
 N and longitude $30^{\circ}$ W. This population is spaced at either
 end of  a line  segment  centered at III.  The precise locations of
 places in distributions II and III  are  listed  in Tables 1 and 2,

 This failure is not due to the choice of a circumpolar population.
 Even  at mid-latitudes the formul{\ae} fail to meet the suggested
 standard.   Consider  a  second  example,  a  collection of eight
 equally populated places located on a circle of 15 degrees of arc
 radius.   Center the circle at $38^{\circ}$ N and $90^{\circ}$ W.
 Choose the  position of the first place so  that its azimuth from
 the  center point is 15 degrees and each succeeding place has its
 azimuth 45 degrees greater than the preceding place (see Fig. 1). 
 The  latitude  and  longitude  of each of the eight places can be
 calculated  with  spherical  trigonometry.  The results are shown
 in Table 1.  When  the Bureau of the Census  formul{\ae} are used
 to calculate the center of population of the odd numbered places,
 then of the even numbered places and finally for all eight places
 the  results  differ.   Specifically, for the odd numbered places
 one finds:
 \hbox{LAT} = 37.2351\,\hbox{N}, \qquad \hbox{LONG} = 90.0146\,\hbox{W}.
 For the even numbered places one finds:
 \hbox{LAT} = 37.2155\,\hbox{N}, \qquad \hbox{LONG} = 89.9855\,\hbox{W}.
 While for all eight places one finds:
 \hbox{LAT} = 37.2253\,\hbox{N}, \qquad \hbox{LONG} = 90.0000\,\hbox{W}.
 Yet, surely the different symmetric distributions centered on the
 same  point  should  have  their center of population at the same
 place  and  surely  that  place  (in  this  example)   should  be
 $38^{\circ}$ N and $90^{\circ}$ W !  Once more, the Bureau of the
 Census formul{\ae} do not meet the suggested standard.
 \centerline{\bf Table 1.  Locations of places, Example II}
 \settabs\+\indent\qquad\qquad\qquad&Place\qquad&N. Latitude\qquad
                                    &W. Longitude\qquad&\cr
 \+&Place&N. Latitude&W. Longitude\cr

 For a third example, consider the simplest  possible  case:   two
 equally populated places equidistant and in  opposite  directions
 from  a  central  place.   Specifically,  place  the   center  at
 $38^{\circ}$ N  and $30^{\circ}$ W  and the two places 15 degrees
 of  arc  from  the  center  to the northeast (Az = 45) and to the
 southwest (Az = 225).  (See Fig. 1).   The latitude and longitude
 of  the  two  places  can be calculated and are shown in Table 2. 
 When the center of population is  calculated for this very simple
 population  distribution  of  two  places using the Bureau of the
 Census formul{\ae} the result is:
 \hbox{LAT} = 37.2057\,\hbox{N}, \qquad \hbox{LONG} = 29.9626\,\hbox{W}.
 Yet, surely the true center is at the central point: $38^{\circ}$
 N,  $30^{\circ}$ W !   And  yet  again,  the Bureau of the Census
 formul{\ae} do not meet the suggested standard.
 \centerline{\bf Table 2.  Locations of places, Example III}
 \settabs\+\indent\qquad\qquad\qquad&Place\qquad&N. Latitude\qquad
                                    &W. Longitude\qquad&\cr
 \+&Place&N. Latitude&W. Longitude\cr

 What the Bureau of the  Census  formul{\ae} calculate is not  the
 latitude  and longitude  of the center of population, but rather,
 two different  and separate statistics:   1) the average latitude
 of the population  and 2) the longitude of  the average east-west
 distance of the  population {\sl on a specific map projection\/}. 
 This  longitude  is   {\sl  neither\/}   the   average  longitude
 {\sl nor\/}  the  longitude  of  the  center  of population.  The
 formul{\ae} calculate the location of a  place that  differs from
 other common measures of the center, such as  the  median or mean
 location  (as  defined  by  Warntz  and  Neft,  1960, and used by
 Haggett, {\it et al.\/}, 1977).  The  result of the Bureau of the
 Census formul{\ae} is the latitude and longitude of a  place that
 cannot justifiably be named the  ``center of population''  as the
 examples above clearly demonstrate.

 For comparison with later examples and further discussion, I have
 calculated  the  1980  center  of population of the United States
 with the Bureau of the Census formul{\ae}.  The result is:
 \hbox{LAT} = 38.1376\,\hbox{N}, \qquad \hbox{LONG} = 90.5737\,\hbox{W}.
 This  differs  from  the location published  by the Bureau of the
 Census  (latitude   of   $38^{\circ}08'13''$  or   38.1369 N  and
 longitude of  $90^{\circ}34'26''$  or  90.5739 W) by one to three
 seconds of arc.  This very small difference results from my using
 the populations and  centers of the fifty states and the District
 of Columbia rather than  the much larger full list of populations
 and  centers  of  all  the  individual  counties  or  enumeration
 districts used by the Census  Bureau.   As the discussion and all
 the examples that follow are based on  the same set  of data this
 small difference is unimportant --- the  examples approximate and
 represent more extensive  computations  and  their  outcome  well
 enough.  The concerns in this paper are the  methods  used rather
 than the data on which they operate.

 \noindent{\bf 4.  Census Bureau Center of Population Description}

 The Bureau of the Census description of the  center of population
 does  not  give  the  map  projection  used.    If  the center of
 population (the  ``balance point'' mentioned in  the description)
 is  calculated  on  a  flat  map  constructed  using  various map
 projections the results vary. In order to demonstrate this I have
 calculated the 1980 centers of population  (the  balance point of
 the population distribution) for the U.S. using several different

 flat map projections.  The projections used  are  all well known,
 having been developed in the 18th century, the  16th  century and
 much earlier.  For the projections chosen,  descriptions given in
 numerous texts were sufficient for the derivation of the relevant
 formul{\ae} for laying out the projections.  Alternately, one may
 refer to detailed monographs, such as the one  by  Snyder (1987),
 for  the  appropriate  formul{\ae}.   The results for each of the
 selected projections are listed in Table 3 and displayed  in Fig.
 \centerline{\bf Table 3.  The 1980 center of population of the United 
 \centerline{\bf when using the Bureau of the Census prose definition}
 \centerline{\bf and various different map projections}
 \settabs\+&No.$\,\,$&Azimuthal Equal-Area (centered at 0N, 0W)$\,\,$
           &N. Latitude$\,\,$&W. Longitude\quad&\cr
 \+&&&\quad Center of Population&\cr
 \+&No.&Projection and Comments&N. Latitude&W. Longitude\cr
 \+&\phantom{1}1&Cylindrical Equal-Area&37.9818&90.4237\cr
 \+&\phantom{1}2&Equidistant Cylindrical (Plate 
 \+&\phantom{1}3&Sinusoidal (centered at 0 W)&38.1376&90.2532\cr 
 \+&\phantom{1}4&Sinusoidal (centered at 60 W)&38.1376&90.4655\cr 
 \+&\phantom{1}5&Sinusoidal (centered at 120 W)&38.1376&90.6778\cr 
 \+&\phantom{1}6&Sinusoidal (centered at 180 W)&38.1376&90.8901\cr 
 \+&\phantom{1}7&Equatorial Mercator&38.2945&90.4237\cr 
 \+&\phantom{1}8&Transverse Mercator (centered at 90 
 \+&\phantom{1}9&Azimuthal Equal-Area (centered at 
 \+&10&Stereographic (centered at N. Pole)&39.7137&90.4888\cr 
 \vskip 5in
 \noindent{\bf Figure 2.}  The ``Centers of population"  for  1980
 for the United States calculated according to various definitions.
 The center shown by an asterisk (*) and labeled COP was determined
 by the method proposed in this paper which takes the  curvature of
 the  earth's  surface  into account in an appropriate manner.  The
 place shown  as  an  open circle ($\circ$) and labelled BC is that
 published by the  Bureau  of  the  Census  as  the location of the
 center of population.   As  discussed  in  the text, this location
 should not be called  the  center  of population.  Places shown as
 solid circles and numbered  are those which result when the center
 of population is  calculated  on various map projections using the
 Bureau  of  the  Census prose definition of the center.  The prose
 definition  does not specify which projection should be used.  The
 numbers  refer  to  the list of projections given in Table 3.  The
 Illinois-Missouri  boundary,  shown  dashed,  was  taken  from The
 National Atlas (U. S.  Geological  Survey,  1970).  This figure is
 available in hard copy only;  its  content  should  be  clear from
 Table 3 when one also notes that  the  Census calculated ``Center"
 lies in Missouri as do centers 1, 2, 4, 5, 6, and 7 (from Table 3);
 the COP Center lies in  Illinois  as do centers 8, 9, and 10 (from
 Table 3); Center 3 appears to lie on the  border  between Illinois
 and Missouri.

 Note that the calculated centers depend not only on the projection
 chosen but also on the center and the orientation selected for the
 projection.   The  results  differ  as  little as they do from one
 another because  the projections chosen leave the United States in
 those portions of the resulting maps where the distortions are not
 extreme.   Indeed,  using  the  Bureau  of  the Census descriptive
 definition  of  the  center  of population, I believe, that, given
 sufficient time  and mischievousness, one could choose projections
 of various orientations  and center that would place the center of
 population {\sl any place one wished\/} !

 \noindent{\bf 5.  Agreement between Description and Formul{\ae}}

 As it happens, the description and the formul{\ae} currently given
 by  the  Bureau  of  the Census agree for one map projection --- a 
 normal  Sinusoidal  (Sanson-Flamsteed)   {\sl   with  the  central
 meridian  of  the  map  the  same as the meridian of the center of
 population\/}.   Indeed,  the Bureau of the Census formula for the
 longitude of the  center of population can be derived by answering
 the following question:   What  must  the  central  meridian for a
 normal  Sinusoidal  projection  be  in  order  for  the  center of
 population   (that  is,  the  balance  point  of  the   population
 distribution) to lie on the central meridian?

 But, why is the Sinusoidal projection (and associated formul{\ae})
 preferred?  If this projection holds special appeal, why isn't the
 latitude  of the center of population determined in a similar way?
 One  could  determine  the  longitude  {\sl  and\/}  latitude   by
 answering  the  following  question:   What  must the center for a
 Sinusoidal  projection  be  in  order for the center of population
 (the balance  point  of the population distribution) to lie at the
 center of the map projection?  The result, for 1980, is:
 \hbox{LAT} = 39.1825\,\hbox{N}, \qquad \hbox{LONG} = 90.4934\,\hbox{W}.

 The  Bureau  of  the  Census  first  calculated  the  ``center  of
 population''  of  the  United States  following the census of 1870
 (Walker, 1874).  Then,  as now, the concept of a balance point was
 stated as underlying the  computation of the center of population. 
 The description of the calculation method (the formul{\ae} are not
 displayed) indicates the method was very similar to that currently
 used. East-west locations were taken as the distance from the 67th
 meridian  west, measured along parallels of latitude.  North-south
 locations  were  taken  as  the  latitude  above the 24th parallel
 north.  Thus, the calculations of the center of population for the
 1879 census  are  equivalent to calculating the balance point on a
 Sinusoidal projection with its central meridian at $67^{\circ}$ W. 
 Since 1870, east-west distances were measured from other meridians,
 chosen to be near the estimated center of population.  In 1910 for
 example, the 86th meridian was chosen (U. S. Bureau of the Census,
 1913)  and  thus  the calculations of the center of population for
 the 1910 census are equivalent to calculating the balance point on
 a Sinusoidal projection with its central meridian at  $86^{\circ}$

 \noindent{\bf 6.  Proposed Definition of the Center of Population}

 One could avoid some of the problems discussed above by  avoiding
 the use of  the  ``statistic,"  the  average.   There  are  other
 measures of the ``center," such as the  median.   But as  Hayford
 (1902) pointed out long ago, there are  fundamental  difficulties
 with the concept of the median for two dimensional distributions. 
 There  is  no  unique  point  that  ``divides"  two   dimensional
 distributions in half.  Another possible measure is the  ``point
 of minimum aggregate travel" --- the  point  for which the sum of
 all the distances to the various individuals  would  be  minimum. 
 But, as Eells (1930) demonstrates, this  point  has some peculiar
 characteristics.  Also, as Court (1964) makes clear, the point of
 minimum aggregate travel is very  difficult  to  find.  There are
 now elegant,  very  powerful  and  very  general  techniques  for
 solving such problems (Kirkpatrick,  {\it et al.\/}, 1983).  But,
 this is beside the point.  The statistic that we want is one that
 reflects  where  people  {\sl   are\/},   not  where  they  might
 congregate with the least total travel. The appropriate statistic
 is the mean.

 Whether  calculating  the  mean  location or the point of minimum
 aggregate  travel,  an arbitrary  decision must be made:  are the
 calculations to be done  on the curved two dimensional surface of
 the globe (or appropriate flat map) or are they to be carried out
 in three dimensions?   If the  point of  minimum aggregate travel
 were calculated in three  dimensions the paths to be traveled and
 the  resulting  point  would lie below the earth's surface.  What
 value  would  there  be  in  finding  a point of least cumulative
 travel when the place to congregate and  the paths to be traveled
 are  inaccessible?   I conclude  that if one is interested in the
 point of minimum aggregate travel,  it should  be calculated {\sl
 on\/} the earth's surface.

 If  the  mean (average) location of a population is calculated in
 three  dimensions,  the  resulting  point  is  located  below the
 surface.   In the  case of the United States in 1980, I find this
 mean location at:
 \hbox{LAT} = 39.1823\,\hbox{N}, 
 \hbox{LONG}=90.3477\,W\,\, \hbox{and}\,\,\hbox{Depth}=0.0259 \times R
 Where  $R$ is  the  radius of  the sphere representing the earth.
 Taking  $R=6371$ km,  the depth is about 165 km.  But calculating
 the  average  location  of simple  surface distributions in three
 dimensions  can  yield  some  peculiar  results.   Consider three
 different equatorial population distributions, each consisting of
 four equally populated places with longitudes as follows:
 \centerline{Example IV: places at 50.00 W, 15.00 W, 15.00 E, 50.00 E.}
 \centerline{Example \phantom{I}V: places at 50.00 W, 15.00 W, 15.00 E,
 130.00 E.}
\centerline{Example VI: places at 62.23 W, 60.00 W, 60.00 E, 62.23 E.}
 In all three examples the center of population (average location)
 ends  up  at  the  same place --- on the equator at the Greenwich
 meridian --- and  differ  only  in their depth below the surface,
 if  at  all.  (Examples V and VI have centers at the same depth.) 
 But, we are largely confined to the surface of the earth and from
 this provincial point of view the center of population in example
 V should be far to the east of the centers in examples IV and VI. 
 I  find  it  unsatisfactory  for  populations  of  such different
 East-West distribution  to  have  ``centers" which differ only in
 depth or not at all.  I conclude that average locations should be
 calculated {\sl on\/} the earth's surface.

 To  insist  that  calculation of the average location or point of 
 minimum aggregate  travel  must be done in three dimensions is no
 more (or less) reasonable than to insist that the only proper map
 projection  is  on  a  globe.   I  leave  to  others  the task of
 championing  the  computations  of  two  dimensional   population
 distribution  statistics  in  three  dimensions.  I believe it is
 legitimate   to   consider  the  population  distribution  a  two
 dimensional distribution and display its  characteristics  on the
 two dimensional surface of a globe or appropriate flat map.

 I  suggest  that  the  appropriate  definition  of  the center of
 population is  one similar to the descriptive definition given by
 the Bureau of the  Census  but  with one addition.  Specifically,
 the center of population is  the  point  at  which  an imaginary,
 flat,  weightless,  and  rigid  map  of  the  United  States {\sl
 constructed by a specific method of projection\/}  would  balance
 if weights of identical value were  placed  on it  so  that  each
 weight represented the location of one person $\ldots $.  It only
 remains for one to choose the specific type of projection.  It is
 distance  and  direction  which are central to any calculation of
 the  center of any population distribution.  Therefore, I suggest
 that  the only map projection (on which to find the balance point
 of  the  population  distribution)  is  one  where  distances and
 directions of  the individuals in the population are undistorted. 
 If one chooses  to  measure the distances and directions from the
 center  of  an  Azimuthal Equidistant map, or on the surface of a
 globe,  they  will  be  undistorted.  Then one can create vectors
 whose  magnitudes  and  directions  are  the  true  distances and
 directions of the various  populated places.  A vector sum can be
 done and the result is  {\sl an estimate\/}  of  the distance and
 direction of the center of  population.   It is  only an estimate
 because,  although  a  map  is  flat, the earth's surface is not. 
 However, it is a very good estimate,  {\sl and the closer the map
 projection's center is to the center of  population the better is
 the estimate\/}. Whether one chooses to carry out the computation
 on an Azimuthal Equidistant map or on the surface of a  sphere is
 immaterial, since the process is algebraically identical  and the
 results are numerically identical.

 Because  the calculating of the center only produces an estimate,
 the  procedure  must be  an iterative one, with the center of the
 projection in each iteration  being the estimate of the center of 
 population from the previous  iteration.   But the estimate is an
 excellent  estimate,  so  the  process  converges  rapidly.   The
 iteration continues until the center of population is as close to
 the center of the map  as one wishes.  When calculating the U. S.
 center of population in this manner I find:
 \centerline{Latitude of the 1980 U. S. center of population = 39.1980 N.}
 \centerline{Longitude of the 1980 U. S. center of population=90.4978 W.} 
 This point lies about 125 km from the center given by the Bureau of the 
 Census and is in Greene County, Illinois, about 14 km southwest by south of
 Carrollton, the county seat (see Figs. 2 and 3).

 The iterative calculation is not the computational nightmare that
 one  might  imagine  (see Appendix B).   Even  when  choosing the
 initial starting  point at latitude zero and longitude zero or at
 latitude $20^{\circ}$ S and longitude $20^{\circ}$ E, the process
 rapidly converges to within  0.000001  degrees  of  the answer in
 four or five iterations.  But one  knows  that the U.S. center of
 population  is  not  in  or  near Africa --- there is no point in
 beginning the computations there.  As the approximate  center  of
 population  can  be  guessed,  only  two  or three iterations are 
 necessary to calculate the center of population to ample accuracy.

 I  have  also  tested  this  procedure  on  the   three   example
 distributions  used  in  Part 3  above and find that in all three
 cases the process rapidly converges on the expected central point.
 Therefore,  I suggest that the proper definition of the center of
 population of the United States (or for any population distributed
 over a substantial portion of the earth's surface) is:

 The center of population is the point at which an imaginary, flat,
 weightless, and rigid map of the United States would balance if
 weights of identical value were placed on it so that each weight
 represented the location of one person on a specific date.  The map
 in question is an azimuthal equidistant map whose center is at the
 center of population which must be calculated by successive

 This suggested definition of the center  of  population  has  the
 following advantages:  (1)  The center of  populations  symmetric
 about some central point is at that central point.  (2)  The true
 distance of  each place  is used in the computation.  (Thus, this
 definition satisfies  the two suggested standards given in Part 2
 above.)  (3) The suggested definition of the center of population
 also satisfies two  of  the three additional restrictions desired
 and stated in Part 2  above:   (a) it corresponds to one's common
 understanding of  center  of  population in that it does find the
 balance  point  of  a  distribution --- though  one  must be very
 specific  about  how  the  distribution  is  displayed,  and  (b)
 mathematically there is a  correspondence to the usual definition
 of the center in the sense of the  average --- the  vector sum of
 the ``distances'' is zero when  measured  from the center and the
 sum  of  the  squares  of  the ``distances'' is  minimum when the
 distances  are  measured  from the center.  In addition, when the
 center of the map is not at the center of  population  the vector
 sum of the ``distances" points approximately  to  the  center  of
 population.  Finally, our definition would  reduce to  the  usual
 mathematical definition when there is no curvature.

 It is not clear that the definition suggested  can be extended to 
 non-spherical  curved  surfaces  and  thus satisfy the additional
 desired restriction (c) mentioned in Part II above.  I believe it
 would work for  the  center of population of the United States on
 an ellipsoid of  revolution  but  there could be difficulties for
 non-spherical  surfaces  in  general --- the  shortest   distance
 between two points may not be uniquely defined and one may end up
 with several centers of population, all equally legitimate.

 A  widely  known  use  of  the  decennial  centers  of population
 determined  by the Bureau of the Census is their display on a map
 of the United States depicting the historic westward shift of the
 population.  In  addition  to  this westward shift, these centers
 have slowly moved south  since the turn of the century.  By 1980,
 the center determined by the Bureau of the Census was more than a
 degree of latitude  ({\it ca\/}. 110 km)  south  of  where it was
 located in 1790. In contrast, the center of population calculated
 by the proposed method  has followed  a different path, diverging
 from the other path, and in  1980  was  located at about the same
 latitude  as  the  1790  center.   (The  smaller  the   east-west
 dispersion of the population, the smaller will be the  difference
 between  the  center  of  population  calculated  by the proposed
 method and the center the Bureau of the Census calculates.  Thus,
 one  would expect the locations calculated by either method to be
 about  the  same  in  1790,  before  extensive  westward national
 expansion occurred.)

 In  order  to  show  the increasing divergence of the two paths I
 have calculated the centers of population for 1910 and 1880 using
 the proposed method.  The results are shown in Fig. 3 and labeled
 COP.  Also  shown  (and labeled BC)  are the locations determined
 and published by the  Bureau of  the  Census  for the same years. 
 One can see that the average latitude of the population (which is
 what the Bureau of the Census calculates) has moved farther south
 than has the center of population.

 \midinsert \vskip 5in
 \noindent{\bf Figure 3.}  ``Centers of population" for 1880, 1910
 and 1980 for the United States calculated according to various
 definitions.  Centers shown by asterisks (*) and labeled COP
 were determined by the method proposed in this paper which takes
 the curvature of the earth's surface into account in an appropriate
 manner.  Places shown as open circles ($\circ $) and labeled BC
 are those published by the Bureau of the Census as the location of the
 center of population.  As discussed in the text, these locations should 
 not be called the centers of population.  State boundaries, shown
 dashed, were taken from {\sl The National Atlas\/} (U. S. Geological
 Survey, 1970).  This map is available in hard copy only; it does
 not transmit electronically.  Its content should be clear from the
 combination of the text and this caption.  

 \noindent{\bf 7.  Summary}

 For  more  than  a  century  the  Bureau of  the  Census has been
 calculating  and  displaying  on  maps  a place designated as the 
 ``center  of  population"  of the United States.  The method used
 in  this  computation  is  equivalent  to calculating the average
 location  of  the  population on a Sinusoidal map projection.  As
 indicated  in  the  previous  discussion,  such a method does not
 adequately take into account the curvature of the earth's surface. 
 As a result, what the  Bureau of the Census calculates should not
 be  called  the  ``center  of  population."   It is,  rather, the
 location of a point  that  has the  population's average latitude
 and the population's  average  distance (measured east-west along
 parallels of latitude) from an arbitrarily chosen meridian.

 A different method of calculating the center  of  population  has
 been proposed in this paper. Like the Bureau of the Census method
 of  calculation,  the  proposed method is based on the concept of
 the  balance  point  of  the  population  distribution  and  thus
 corresponds  to  one's  common  understanding  of the center.  In
 contrast to the  Bureau of the Census method, the proposed method
 takes into account the  curvature of  the earth's surface and map
 projection distortions in an  appropriate manner and  is based on
 measuring distances along great circles.

 When  calculated as proposed, the center of population's location
 differs  substantially  from that calculated by the Bureau of the
 Census. Not only is this true for 1980, but also for other census
 years, and the greater the east-west dispersion of the population,
 the greater will be the difference.
 \noindent{\bf 8.  Appendix A}

 The  unpublished  1980 population centers of the fifty states and
 the  District  of  Columbia  used  in  the  various examples were
 obtained from the Bureau of the Census.  As they are unpublished,
 a  complete list of the center latitudes and longitudes that were
 used in the computations discussed in this paper is supplied
 This  is  the data used as the representative example data set in
 ``Where  are  we?   Comments  on  the  concept  of the `center of
 population' "  by  Frank  E.  Barmore,  published  in  {\sl   The
 Wisconsin Geographer\/}, Vol. 7, pp. 40-50, (1991), a publication
 of the Wisconsin Geographical Society.

 The  table  below  shows the original state populations and also, 
 State Centers of Population supplied by the Bureau of the Census.
 The first coordinate for  the center of population is measured in
 degrees  of  longitude  west  of  the  prime meridian; the second
 coordinate  is  measured  in  degrees  of  latitude  north of the
 \settabs\+\indent\quad&North Carolina\qquad&Population\qquad
 \+&Place       &Population    &Center of       &Center of \cr
 \+&            &1980          &population      &population\cr
 \+&            &              &1980            &1980      \cr
 \+&Alabama      &\phantom{2}3,893,888 &\phantom{1}86.7750      &32.9923\cr
 \+&Alaska       &\phantom{23,}401,851       &148.4964          &61.3650\cr
 \+&Arizona      &\phantom{2}2,718,215      &111.7186           &33.3245\cr
 \+&Arkansas     &\phantom{2}2,286,435      &\phantom{1}92.4340 &34.9718\cr
 \+&California   &23,667,902     &119.4380         &35.4746\cr 
 \+&Colorado     &\phantom{2}2,889,964      &105.1809         &39.4868\cr
 \+&Connecticut  &\phantom{2}3,107,576      &\phantom{1}72.8760  &41.4906\cr
 \+&Delaware     &\phantom{23,}594,338       &\phantom{1}75.5636 &39.4450\cr
 \+&D. C.        &\phantom{23,}638,333       &\phantom{1}77.0088  &38.9074\cr
 \+&Florida      &\phantom{2}9,746,324      &\phantom{1}81.6735   &27.7948\cr
 \+&Georgia      &\phantom{2}5,463,105      &\phantom{1}83.8100  &33.1866\cr
 \+&Hawaii       &\phantom{23,}964,691       &157.6129         &21.2009\cr
 \+&Idaho        &\phantom{23,}943,935       &114.9358         &44.2072\cr
 \+&Illinois     &11,426,518     &\phantom{1}88.4070         &41.2073\cr
 \+&Indiana      &\phantom{2}5,490,224      &\phantom{1}86.2835 &40.1759\cr
 \+&Iowa         &\phantom{2}2,913,808      &\phantom{1}93.0582  &41.9858\cr
 \+&Kansas       &\phantom{2}2,363,679      &\phantom{1}96.6379  &38.4544\cr
 \+&Kentucky     &\phantom{2}3,660,777      &\phantom{1}85.2228  &37.7918\cr
 \+&Louisiana    &\phantom{2}4,205,900      &\phantom{1}91.4656  &30.7177\cr
 \+&Maine        &\phantom{2}1,124,660      &\phantom{1}69.6408  &44.4125\cr
 \+&Maryland     &\phantom{2}4,216,975      &\phantom{1}76.7904  &39.1598\cr
 \+&Massachusetts&\phantom{2}5,737,037      &\phantom{1}71.3844  &42.2792\cr
 \+&Michigan     &\phantom{2}9,262,078      &\phantom{1}84.1083  &42.8410\cr
 \+&Minnesota    &\phantom{2}4,075,970      &\phantom{1}93.6489  &45.2543\cr
 \+&Mississippi  &\phantom{2}2,520,638      &\phantom{1}89.6224  &32.5778\cr
 \+&Missouri     &\phantom{2}4,916,686      &\phantom{1}92.0799  &38.4815\cr
 \+&Montana      &\phantom{23,}786,690       &110.9159         &46.8610\cr
 \+&Nebraska     &\phantom{2}1,569,825      &\phantom{1}97.5697  &41.1991\cr
 \+&Nevada       &\phantom{23,}800,493       &116.7563         &37.5535\cr
 \+&New Hampshire&\phantom{23,}920,610       &\phantom{1}71.4735 &43.1783\cr
 \+&New Jersey   &\phantom{2}7,364,823      &\phantom{1}74.4172  &40.4640\cr
 \+&New Mexico   &\phantom{2}1,302,894      &106.2391         &34.6202\cr
 \+&New York     &17,558,072     &\phantom{1}74.7181         &41.5458\cr
 \+&North Carolina&\phantom{2}5,881,766     &\phantom{1}79.6756 &35.5676\cr
 \+&North Dakota &\phantom{23,}652,717       &\phantom{1}99.5101&47.4277\cr
 \+&Ohio         &10,797,630     &\phantom{1}82.7006         &40.5199\cr
 \+&Oklahoma     &\phantom{2}3,025,290      &\phantom{1}96.8876 &35.5880\cr
 \+&Oregon       &\phantom{2}2,633,105      &122.5648         &44.6942\cr
 \+&Pennsylvania &11,863,895     &\phantom{1}77.2024         &40.4699\cr
 \+&Rhode Island &\phantom{23,}947,154       &\phantom{1}71.4419  &41.7595\cr
 \+&South Carolina&\phantom{2}3,121,820     &\phantom{1}81.0355   &34.0472\cr
 \+&South Dakota  &\phantom{23,}690,768      &\phantom{1}99.0563  &44.1116\cr
 \+&Tennessee     &\phantom{2}4,591,120     &\phantom{1}86.4217   &35.7793\cr
 \+&Texas         &14,229,191    &\phantom{1}97.4571         &30.9925\cr
 \+&Utah          &\phantom{2}1,461,037     &111.8261         &40.5165\cr
 \+&Vermont       &\phantom{23,}511,456      &\phantom{1}72.8055  &44.0566\cr
 \+&Virginia      &\phantom{2}5,346,818     &\phantom{1}78.0021   &37.6381\cr
 \+&Washington    &\phantom{2}4,132,156     &121.5325         &47.3363\cr
 \+&West Virginia &\phantom{2}1,949,644     &\phantom{1}80.9407   &38.7202\cr
 \+&Wisconsin     &\phantom{2}4,705,767     &\phantom{1}88.9756    
 \+&Wyoming       &\phantom{23,}469,557      &106.9348         &42.6568\cr
 \noindent{\bf 9.  Appendix B}
 All the computations reported here were done  on  an  Apple  IIgs
 computer  using  the spread sheet in AppleWorks 3.0.  Computation
 time  for  the  problems  varied.  When calculating the center of
 population using the  proposed method, each iteration took  about
 85 seconds.  Computations using a more detailed list of populated
 places would take longer  in  direct  proportion to the number of
 places used.  Computers  and  software  with  enormously  greater
 speed   and   capability  are  widely  available.   There  is  no
 computational reason for not using the proposed method.
 \noindent{\bf 10.  References}

 \ref Abler, Ronald, J. S. Adams and P. Gould.  1971.  {\sl
 Spatial Organization:  The Geographer's View of the World\/}.
 Englewood Cliffs NJ, Prentice-Hall. p. 59.

 \ref Court, A. 1964.  The elusive point of minimum travel.  {\sl
 Annals of the Association of American Geographers\/} 54:  400-403.

 \ref Eells, W.C. 1930.  A mistaken conception of the center of
 population.  {\sl Journal of the American Statistical Association\/},
 New Series No. 169  25:33-40.

 \ref  Haggett, Peter, A. D. Cliff and A. Frey.  1977.  {\sl
 Locational Analysis in Human Geography:  Vol. II,
 Locational Methods, 2nd Ed.\/}  New York NY, Wiley, p. 312.

 \ref Hayford, John F.  1902.  What is the center of an area, or the
 center of a population?  {\sl Journal of the American Statistical
 Association\/}, New Series No. 58, 8:47-58.

 \ref Kirkpatrick, S., C. D. Gelatt, Jr. and M. P. Vecchi.  1983.
 Optimization by simulated annealing.  {\sl Science\/} 220:671-680.

 \ref Snyder, J. P. 1987.  {\sl Map Projections --- A Working Manual.  
 (USGS Professional Paper 1395)\/}.  Washington DC, U. S. 
 Government Printing Office.  pp. 41, 58, 77, 91, 157, 185, 195, 247.

 \ref Sviatlovsky, E. E. and W. C. Eells.  1937.  The centrographical
 method and regional analysis.  {\sl The Geographical Review\/},

 \ref U. S. Bureau of the Census.  1983.  {\sl 1980 Census of Population, 
 Vol. I\/}, Chapter A, Part 1 [PC80-1-A1].  Washington DC, U. S. 
 Dept. of Commerce, Bureau of the Census.  Appendix A, p. A-5 and
 Table 8, pp. 1-43.

 \ref U. S. Bureau of the Census.  1973.  {\sl 1970 Census of Population and
 Housing:  Procedural History\/} [PHC(R)-1].  Washington DC, U. S. Dept. of
 Commerce, Bureau of the Census.  Appendix B (Computation of the
 1970 U. S. center of population), pp. 3-50.

 \ref U. S. Bureau of the Census.  1914.  {\sl Statistical Atlas of the
 United States\/}.  Washington DC, Government Printing Office.  pp. 29-32.

 \ref U. S. Bureau of the Census.  1913.  {\sl Thirteenth Census of the
 United States taken in the year 1910.  Vol. I:  Population, 1910\/}.
 Washington DC, Government Printing Office.  pp. 30, 46.

 \ref U. S. Geological Survey.  1970.  {\sl The National Atlas of the United
 States of America\/}.  Washington DC, U. S. Dept. of the Interior, 
 Geological Survey.

 \ref Walker, F. A. 1874.  The progress of the nation.  In {\sl
 Statistical Atlas of the United States Based on the Results
 of the Ninth Census, 1870, with ... \/}.  Francis A. Walker,
 compiler.  New York NY, J. Bien, Lithographer.  List of 
 Memoirs and Discussions, Part II, pp. 5, 6.

 \ref Warntz, W. and D. Neft.  1960.  Contributions to a Statistical
 Methodology for Areal Distributions.  {\sl Journal of Regional
 Science\/}, 2:47-66.
 NOTE: The original article contains several printing errors which
 might  cause  misunderstanding.   These  were  corrected  in  the
 reprints of the article.  The {\sl Solstice\/}  copy was prepared
 from  the  corrected  reprint.   The  errors  in  question in the
 original include:  i.  the longitudes of places in Examples IV, V
 and VI  were incorrect and ii. near the end of part 6 of the text
 a date of 1790 was incorrectly given as 1970.  Also, the location,
 about  14 km  southwest  by  south of Carrollton, was incorrectly
 given as about 7 km southeast of Carrollton.
 \centerline{\bf 4.  THE PELT OF THE EARTH:}
 \centerline{\bf Sandra L. Arlinghaus, John D. Nystuen}
 \centerline{Founding Director, Institute of Mathematical Geography}
 \centerline{Ann Arbor, MI}
 \centerline{Professor of Geography and Urban Planning}
 \centerline{The University of Michigan}

 \noindent Reactive diffusion (see the many references to  Murray)
 is  an  idea  that  draws  on the concept that boundary shape can
 influence  the  spatial  pattern  of  the  developing  forms  and
 processes interior to that boundary.  A second idea is  involved. 
 Once  a  natural  diffusive  process  has  been at work, there is
 reaction to it, altering the shape of the  underlying  diffusion. 
 Reactive  diffusion  is  thus  a dynamic process that is, to some
 extent, self-adjusting to change.

 This  sort  of  idea is one that has met with many expressions in
 the past --- in  the  biological  as  well as in the geographical
 landscape (Arlinghaus, Nystuen, and  Woldenberg, 1992).  Boundary
 shape can determine how matter and energy travel within a  closed
 system.  Standing  waves  can  be created in this manner, be they
 standing  waves  of  translation  of  pigment  on  animal  coats,
 producing  striped  animals;  or standing waves of oscillation of
 water,  producing  seiches  as  water-stripes in reaction to lake
 depth  and  coastline  shape of the containing vessel (e.g., Lake
 Michigan or Lake Geneva).  One might even be tempted to speculate
 on a possible role for seiche-like stripes in  the  ``parting  of
 the Red Sea."

 Xu, Vest, and Murray  (1983)  created  mock  animal  outlines  on
 laminar  plates  shaped like two-dimensional pelts --- as ``maps"
 of  three-dimensional  animals;  when  small  adjustments  in the
 outlines were made, vibrational patterns formed in a surface dust
 placed within these outlines created various spotted and  striped
 patterns as a reaction to the boundary shape.  Indeed, a circular
 drum head boundary offers one way for the roll of the  drum  wave
 of noise to interact with the boundary; using  a fractal boundary
 for  the  drum  head  can  produce  a vastly different pattern of
 resonating pockets of drum roll (Science, 1991).  The  continuing
 work of  Batty  and Longley  (1985 and later)  in  using  fractal
 concepts to  track  the pattern of the urban fringe might also be
 (but has not yet been,  to  our  knowledge) cast in the framework
 of reactive diffusion.   Three-dimensional solids,  covered  with 
 a coat of spots, might also have their spot  patterns  determined
 by some underlying vibrational process that  causes the substance
 of the spots on the surface  to  react with the three-dimensional
 volume  over which  the  surface  is stretched.  Thus, the calico
 cat and the  earth  might  have  a great deal in common when land
 masses, driven by tectonic rather than by biological rhythms, are
 seen as the calico spots on the pelt of the earth.

 For example, the burn pattern created by random lightning strikes
 in  a  forest, and the reaction of firefighters to these strikes,
 displays a clear case of reactive diffusion and pattern formation
 on the  earth.   For, in  the absence of firefighters, the random
 strikes start fires  which  coalesce  to form  an advancing front 
 that  may  ultimately burn the  entire region.  When firefighters
 enter  the  scene,  they  work to confine the random strikes; the
 fire may  leap  the  barriers  they create, and when it does, the
 firefighters talk about it and react by moving to control the new
 hotspot.   Ultimately,   the   spread   of   communication  among
 firefighters, in response to the leapfrogging character of forest
 fires, produces  a  forest  spotted with burnt dark patches.  The
 reaction of  firefighters  to  the diffusion of information about
 the location  of  fires produces characteristic, and predictable,
 patterns on the earth.
 \noindent{\bf 1.  Pattern Formation:  Global Views}

 Nystuen noted (1966) that ``spatial  processes  depend  upon  the
 shape of  the  partitions created by their boundary patterns.  If
 the boundary shape  is  changed  the  process  itself is changed. 
 In fact,  the  very  existence  of  the process may depend on the
 boundary  shape."   The  biologist  Joseph  Birdsell  noted  that
 coastline shape has affected genetic diversity in  the Australian
 aborigine population (1950); migration patterns  forced away from
 the concave-up portions of the northern coastline were dispersed,
 while  those  forced  away  from  the  concave-down portions were 
 focused.  With  dispersal  of  hunting and gathering came genetic
 complexity; with  focusing  came  genetic inbreeding.  Arlinghaus
 (1977; 1986) drew  on  ideas  from  Birdsell and Nystuen in using 
 boundary shape of a limited access  arterial to suggest where new
 pockets of population concentration  and  dispersion  will appear
 relative to the concavity of the arterial.   In  all of these, as
 with  reactive  diffusion,  there  is  an  adjustment  of process
 (geographical or biological) to boundary,  with  implications for
 the spatial organization of associated human activity coming as a
 reaction to that adjustment.

 Indeed, even in medi{\ae}val guilds, retail services clustered in 
 pockets  across  the  geographical  landscape,  as  ``stripes" or
 ``spots" of commercial activity, in reaction to the  diffusion of
 information  as  to  type  of  service  available  (Vance, 1980). 
 Similar  urban  patterns  are  evident   in   modern   developing
 countries;  and  this  context  thus suggests, very generally, an
 interesting human  dimension  in  exploring  global  urban change
 (Drake, 1993; Meadows {\it et al.\/} 1992).

 In  a  classical  urban  context,  one  might  imagine Harris and
 Ullman's  ``multiple nuclei"  model  recast within the replicable
 theoretical framework of reactive diffusion.  As diffusion causes
 change  surrounding  and  within the nuclei, there is a reaction,
 and  the  nuclei  shift,  or  new  nuclei spring up.  The spatial
 evidence  of  reactive  diffusion  might  be  substituted for the
 historical evidence on which Harris and Ullman (1945) based their
 observational model,  pulling  the  multiple nuclei model more in
 line with the  earlier  spatial  models of Burgess and Hoyt.  The
 multiple nuclei  pattern  appears  as  a reaction to incompatible
 land  uses;  it   arises   from   an  alternative  resistance  to 
 residential and commercial land  uses in which further employment
 centers leapfrog over existing  urban  neighborhoods,  leading to
 extensive additional urban growth.

 Within  the Detroit metropolitan region, for example, the complex 
 changing  nature  of  the  local political scene coupled with the 
 increasing crime rates associated  with downtown  Detroit,  often
 encapsulated quite simply in the minds of many Detroiters  by the
 closing  of  the  downtown  Detroit  Hudson's  store,  led to the
 consequent reaction of many businesses to  move  to the  suburbs. 
 Thus, suburban Southfield became an early hub of  urban  reaction
 in  the  Detroit  metropolitan  region --- here,  a  new  nucleus
 emerged. Efforts to restore the prominence of the downtown on the
 Detroit  River  are  typified by the Renaissance Center --- here, 
 the old nucleus shifts toward the River banks.

 Indeed, the characterization of the collapse of the central city
 in  terms  of  the  failure  of  the  downtown  headquarters of
 Hudson's  department  store  may not be a strictly simple-minded
 view.   Like  the stars, the life of the city may take different
 paths --- at one  time  a  center  may  be a viable unit, and at
 another time the relative size and density of the urban area may
 cause inner city collapse.  In a central place context, in which
 the ``threshold" of a firm refers to the minimum number of sales
 which allows the firm to succeed  and give an adequate return to
 its owners, the situation with Hudson's was simply a matter that
 the buying population at the center was too  small  to  meet the
 threshold number.  Related central  place  terminology  involves
 the notions of the maximum range of a good and the minimum range
 of a good.  The  maximum  range  is  the  absolute  limit on the
 demand  of  a  good --- beyond  this limit, transportation costs
 reduce demand for the good to zero.  The  minimum  range  is the
 distance over which the firm must ship its  goods to include the
 threshold populations.  A logical consequence,  all  else  being
 equal, is that the minimum range of a good is  less in a densely
 settled region than it is in a sparsely settled one.

 Thus,  the  common  sense  notion  of ``how can a big store like
 Hudson's fail in downtown Detroit?" can be translated as follows. 
 Migration of the  affluent population to the suburbs reduced the
 number of potential  customers in the center.  The minimum range
 therefore needed  to be  extended outward from downtown in order
 to include the  threshold  number  of  customers.  But, suburban
 Hudson stores were already  in  place and also worked to attract
 those customers that the downtown branch now required to succeed. 
 The three large suburban stores competed with the downtown store
 for  these  customers,  won  them  over, and the  downtown store
 failed.  Stability in competition (Hotelling, 1929) was restored
 when  the  ``empty"  center  was  divided  among  the peripheral
 competitors,  in  a  sort  of  central  place  (two-dimensional)
 Hotelling  model.   This  sort of geometric view is a minimalist
 approach --- a best-case  scenario;  when additional social (and
 other)  issues  are  superimposed,  acceleration  along the path
 to collapse is more likely.  When  one  next considers that this 
 pattern  will  repeat  on the periphery of these suburban stores
 and within the maximum  ranges  of  the various goods, a sort of
 leapfrogging  of   circular/hexagonal  trade  areas  occurs  and
 suggests, once again, a conceptual context of reactive diffusion
 as an alternative, and addition to traditional spatial analysis.

 Unlike earlier models of urban ecologists (Burgess, 1925;  Hoyt,
 1939),  this  sort  of  urban  view  of  the  world  is  not   a
 generalization  of  a  particular  example --- that is why it is
 important to see reactive diffusion cast in the  geographical as
 well as  the  biological  (or other)  realms.   The  pattern  of
 clusters  of  urban  activity  on a regional part of the earth's
 surface is one that is produced in reaction to the  diffusion of
 urban process.
 \noindent{\bf 2.  Pattern Formation:  Local Views}

 Some  current  urban  research  strives  to develop indices that
 offer an  easy means for replication of experiments and that are
 sensitive to the role of boundary. Thus, Morrill (1991) proposed
 an index of segregation, modified by boundary considerations, to
 quantify  urban  spatial  segregation.   Wong  (1992)   modifies
 Morrill's  indices  by  arguing  that the length of the boundary
 separating adjacent urban  areal  units, as well as the shape of
 these adjacent units, is significant in determining segregation. 
 Indices  such  as  these,  that  already  are  sensitive to some
 boundary considerations,  may offer one  means  to  tighten  the
 focus of application of the concept  of  reactive  diffusion  in
 various specific urban situations.

 Often reactions to incompatible urban land uses are circumscribed 
 by the boundary of the system of local jurisprudence.  When these 
 reactions  fit  reasonably  well  within   the   laws,  competing
 commercial  and  residential  land  uses are in relative harmony. 
 Laws, such as the apocryphal ``it is  illegal to tie an alligator
 to a parking meter"  suggest  a reaction to an unusual situation. 
 When  that  reaction  is  passed  as  law,  it  diffuses  to  the
 population of the surrounding  area and may disturb the sensitive
 balance between incompatible land uses.

 Perhaps  the  most  difficult  situation  of  this  sort  is  in
 establishing rules  (legal,  ethical,  or otherwise) to position
 locally unwanted land uses  (``lulus").   Human  laws  permit or
 forbid  institutional  boundaries that can influence how process
 works.  Typically, a lulu, such as an adult bookstore or a toxic
 waste  site,  causes  a  strong  local  reaction   around   this
 ``hotspot."   This  reaction  is  confined   and   suppressed by
 municipal  authorities  using  the  local  legal system as their
 ``hose"  or  ``barrier"  to confine  the effects of the unwanted
 activity.   As with the forest fire example, the lulu leapfrogs,
 and yet another hotspot of locally unwanted activity occurs.

 Reactive  diffusion  offers  an attractive conceptual context in
 which  to  examine  pattern  formation on the pelt of the earth: 
 from  local  scenarios  that  mimic  the  forest fire example to
 global  scenarios  that  examine  entire  closed   and   bounded
 surfaces.   Beyond  this  essay,  the  next  step is to use this
 context in specific urban or physical settings.
 \noindent{\bf 3.  References Cited}

 \ref  Arlinghaus, S. L. 1986. ``Concavity and human settlement patterns,"
 {\sl Essays on Mathematical Geography\/}, Monograph \#3.  Ann Arbor,
 MI:  Institute of Mathematical Geography; 1977.

 \ref Arlinghaus, S. L., Nystuen, J. D., and Woldenberg, M. J. 1992.
 ``An application of graphical analysis to semidesert soils."  
 {\sl Geographical Review\/}, American Geographical Society.
 July, pp. 244-252.

 \ref Batty, M. and Longley, P.  1985.  ``The fractal simulation
 of urban sructure."  {\sl Papers in Planning Research\/} 92, Univ. of 
 Wales Institute of Science and Technology, Colum Drive,
 Cardiff, CF1 3EU.

 \ref Birdsell, J. B. 1950. ``Some implications of the genetical concept
 of race in terms of spatial analysis,"  {\sl Symposia on
 Quantitative Biology\/}, Vol. 15, Origin and Evolution of Man.
 Long Island, New York:  The Biological Laboratory, Cold Springs

 \ref Drake,  W. D.  forthcoming, 1993.  Towards building a theory of
 population - environment dynamics:  a family of transitions.  In
 {\sl Population - Environment Dynamics\/}, Ann Arbor:  University of 
 Michigan Press.

 \ref Hotelling, H.  1929.  ``Stability in competition,"
 {\sl Economic Journal\/}, 39:  41-57.

 \ref Hoyt, H. W. 1939.  According to Hoyt (1966), Washington D. C.:
 Homer Hoyt Associates.

 \ref Harris, C. D. and Ullman, E. L. 1945.  ``The nature of cities,"
 {\sl Annals of the American Academy of Political and Social
 Science\/}, CCXLII, Nov. 1945, pp. 7-17.

 \ref Meadows, D. H., Meadows, D. L., and Randers, J. 1992.  
 {\sl Beyond the Limits\/}.  Post Mills, VT:  Chelsea Green
 Publishing Company.

 \ref Morrill, R. L. 1991.  ``On the measure of geographic segregation."
 {\sl Geography Research Forum\/} 11, 25-36.

 \ref Nystuen, J. D. 1966. ``Effects of boundary shape and the concept of 
 local convexity."  Discussion Paper 10.  Michigan Inter-University
 Community of Mathematical Geographers (John D. Nystuen, ed.), Ann
 Arbor, MI.  (Reprinted by the Institute of Mathematical 
 Geography, 1986).

 \ref Park, R. E. and Burgess, E. W. 1925.  {\sl The City\/}.
 Chicago:  University of Chicago Press.

 \ref Vance, J. E. 1977. {\sl This Scene of Man:  The Role and
 Structure of the City in the Geography of Western Civilization \/}.
 New York:  Harper's College Press.

 \ref Wong, D. W. S. 1991. ``Spatial indices of segregation."
 Preliminary version, National Meetings, Regional Science Association,
 1991.  Forthcoming in {\sl Urban Geography\/}, 1992.

 \ref Xu, Youren; Vest, Charles M.; Murray, James D. 1983. ``Holographic
 interferometry used to demonstrate a theory of pattern formation
 in animal coats."  {\sl Applied Optics\/} 15 Nov., Vol. 22,
 No. 22, pp. 3479-3483.

 \noindent{\bf 4.  Literature of Apparent Related Interest}

 \ref Bard, Jonathan B. L. 1977. ``A unity underlying the different zebra
 striping patterns."  {\sl Journal of Zoology\/}, Vol. 183, part 4,
 pp. 527-539.

 \ref Boal, F. W. 1972.  ``Close together and far apart:
 Religious and class divisions in Belfast."  {\sl Community Forum\/},
 Vol. 3, No. 2, pp. 3-11.

 \ref Boal, F. W. and Livingstone 1986.  ``Protestants in Belfast:
 A view from the inside."  {\sl Contemporary Review\/}, 248: 169-75.

 \ref Boyce, R. R., and W. A. V. Clark.  1964.  ``The concept of 
 shape in geography."  {\sl Geographical Review\/} 54, 561-572.

 \ref Dewdney, A. K. ``A home computer laboratory in which balls
 become gases, liquids and critical masses."  Computer Recreations,
 {\sl Scientific American\/} pp. 114-117.

 \ref Freedman, David H. 1991. ``A chaotic cat takes a swipe at quantum
 mechanics."  {\sl Science\/}, Vol. 253, p. 626.

 \ref Gierer, A. 1981. ``Some physical, mathematical and evolutionary aspects
 of biological pattern formation."  {\sl Philosophical Transactions
 Royal Society, London\/}, Series B, 295, pp. 429-440.

 \ref Hagerstrand, T. 1967.  {\sl Innovation diffusion as a spatial 
  Chicago:  University of Chicago Press.

 \ref Kennedy, S., and W. Tobler.  1983.  ``Geographic interpolation."
 {\sl Geographical Analysis\/} 15, 151-156.

 \ref Murray, J. D. 1981. ``A pre-pattern formation mechanism for animal
 coat markings."  {\sl Journal of Theoretical Biology\/}.  Vol. 88, 
 No. 1, pp. 161-199.

 \ref Murray, J. D.  1988.  How the leopard gets its spots.
 {\sl Scientific American\/} 258:80-87.

 \ref Murray, J. D. 1981.  ``Introductory remarks" (to an entire volume
 devoted to pattern formation) {\sl Philosophical Transactions Royal
 Society, London\/}, Series B 295, pp. 427-428.

 \ref Murray, J. D. 1989.  Mathematical Biology.  Springer-Verlag, 

 \ref  Murray, J. D. 1981. ``On pattern formation mechanisms for 
 lepidopteran wing patterns and mammalian coat patterns."
 {\sl Philosophical Transactions of the Royal Society, London\/}
 Series B, Vol. 295, No. 1078, pp. 473-496; Oct. 7.

 \ref Murray, J. D. and P. K. Maini. 1986.  ``A new approach to the 
 of pattern and form in embryology."  {\sl Science Progress\/}, Vol. 70,
 No. 280, part 4, 539-553.

 \ref Nordbeck, S. 1965. ``The Law of Allometric Growth."  Discussion Paper
 \#7.  Michigan Inter-University Community of Mathematical Geographers
 (John D. Nystuen, ed.), Ann Arbor, MI.  (Reprinted by Institute
 of Mathematical Geography, 1986).

 \ref Pool, R. 1991.  Did Turing discover how the leopard got its spots?
 {\sl Science\/} 251:627.

 \ref Shaw, L. J. and Murray, J. D. 1990. ``Analysis of a model for 
 skin patterns."  SIAM Journal of Applied Mathematics, pp. 

 \ref Tobler, W. R. 1969. ``The spectrum of U. S. 40,"  {\sl Papers of
 the Regional Science Association\/}, Vol. XXIII, pp. 45-52.

 \ref Turing, A. M.  1952.  ``The chemical basis of morphogenesis."
 {\sl Philosophical Transactions of the Royal Society, London\/}, 
 Series B, 237, pp. 37-72.

 \ref White, M. J. 1983.  ``The measurement of spatial segregation."
 {\sl American Journal of Sociology\/} 88, 1008-1018.

 \ref Wolpert, L. 1981. ``Positional information and pattern formation."
 {\sl Philosophical Transactions Royal Society of London\/},
 Series B 295, pp. 441-450.  1981.

 \centerline{\bf 5.  FEATURE}
 \centerline{\bf Meet New Solstice Board Member}
 \centerline{\bf William D. Drake}
 \centerline{\bf The University of Michigan}
 \centerline{Professor of Resource Policy and Planning}
 \centerline{School of Natural Resources and the Environment}
 \centerline{Professor of Population Planning and International Health}
 \centerline{School of Public Health, The University of Michigan}
 \centerline{Professor of Urban, Technological and Environmental Planning}
 \centerline{College of Architecture and Urban Planning}

 Bill  Drake  teaches  courses  on  the Global Environment and  on
 Population-Environment Dynamics.   Much of his research portfolio
 is drawn from  ongoing projects in the developing world.  Many of
 these projects have  been  underway  for  several  years and have
 focused   on   the   problems   of  rural  community  development
 particularly relating to reducing child malnutrition.

 Recently,  he has  authored articles and was co-editor for a book
 on  population - en\-vi\-ron\-ment dynamics.  During  the fall of
 1992,  Drake  and  S. Arlinghaus  offered a  course  on  the same
 subject  which  has  resulted in  a monograph.  The focus of this
 course  is  captured  in  its  name  {\sl  Population-Environment
 Dynamics:  Toward Building  a  Theory\/}.   The effort draws upon
 recent work carried out as part  of  the University of Michigan's
 Population - Environment   Dynamics   Project.    Ten    graduate 
 students and  two  faculty  participated  formally,  and  several
 other students and faculty  sat  in  from  time-to-time, with one
 visitor attending every session.  Seminar  participants came from
 many    disciplinary    backgrounds   ranging   from   population  
 planning,  economics,  engineering,  biology,   remote   sensing,  
 geography,   natural   resources,     sociology,    international   
 health,  business administration  to  mathematics.   In  addition
 to U.S. students,  the  course  was  enriched  by colleagues from
 Mexico, Nepal, Taiwan, and Nigeria.

 The  monograph  serves  as  a  kind of a ``time capsule"--what do
 students  in  1992  think  will  be  issues of great significance
 in  the  current,  recently  identified,  need  to study ``global
 change"?  Here are the sectors of that ``capsule":

 \line{\bf Dawn M. Anderson \hfil}
 \line{The Historical Transition of Forest Stock Depletion in Costa Rica
 \line{\bf Katharine A. Duderstadt \hfil}
 \line{The Energy Sector of Population-Environment Dynamics in China \hfil}
 \line{\bf Eugene A. Fosnight \hfil}
 \line{Population Transition and Changing Land Cover and Land Use in Senegal
 \line{\bf Katharine Hornbarger \hfil} 
 \line{The Energy Crisis in India:  Options for a Sound Environment \hfil}
 \line{\bf Deepak Khatry \hfil}
 \line{An Analysis of the Major Sectoral Transitions in Nepal's Middle Hills
 \line{and their Relationship with Forest Degradation \hfil}
 \line{\bf Catherine MacFarlane \hfil}
 \line{The Interrelationship Between the Forestry Sector \hfil}
 \line{and Population-Environment Dynamics in Haiti \hfil}
 \line{\bf Gary Stahl \hfil}
 \line{Transition to Peace: \hfil}
 \line{Environmental Impacts of Downsizing the U.S. Nuclear Weapons Complex
      \hfil }
 \line{\bf Stephen Uche \hfil}
 \line{Population and Forestry Dynamics:  At the Crossroads in Nigeria \hfil}
 \line{\bf Hurng-jyuhn Wang \hfil}
 \line{The Cultivated Land-Rural Industrialization-Urbanization-Population
 Dynamics in Taiwan \hfil}
 \noindent This section shows the exact set of commands that  work  to 
 download {\sl Solstice\/} on The University of Michigan's  Xerox  9700.  
 Because different universities will have different installations 
 of {\TeX}, this is only a rough guideline which {\sl might\/} be 
 of use to the reader. 
 First step is to concatenate the files you received via
 bitnet/internet.  Simply piece them together in your computer,
 one after another, in the order in which they are numbered,
 starting with the number, ``1."

 The files you have received are ASCII files;  the concatenated file
 is used to form the .tex file from which the .dvi file (device
 independent) file is formed.  The words ``percent-sign" and 
 ``backslash" are written out in the example below; the
 user should type them symbolically.
  \# create -t.tex
 \# percent-sign t from pc c:backslash words backslash
    solstice.tex to mts -t.tex char notab
     [this command sends my file, solstice.tex, which I did as
      a WordStar (subdirectory, ``words") ASCII file to the
 \# run *tex par=-t.tex
     [there may be some underfull boxes that generally cause no
      problem; there should be no other ``error" messages in the
      typesetting--the files you receive were already tested.] 

 \# run *dvixer par=-t.dvi
 \# control *print* onesided
 \# run *pagepr scards=-t.xer, par=paper=plain
  \noindent{\bf 7.  SUMMARY OF CONTENT}

 \noindent IMaGe is working to establish a Bulletin Board on which back
 issues of {\sl Solstice\/} can be posted.  Subscribers will be notified
 when this service is available.
 \noindent {\bf Volume III, Number 1, Summer, 1992}

 \noindent{\bf 1.  ARTICLES.}
 {\bf Harry L. Stern}. 
 {\bf Computing Areas of Regions With Discretely Defined Boundaries}.
 1. Introduction 2. General Formulation 3. The Plane 4.  The Sphere
 5.  Numerical Example and Remarks.  Appendix--Fortran Program.
 \noindent{\bf 2.  NOTE }
 {\bf Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg}.  
 {\bf  The Quadratic World of Kinematic Waves}
 \noindent{\bf 3.  SOFTWARE REVIEW}
 RangeMapper$^{\hbox{TM}}$  ---  version 1.4.
 Created  by {\bf Kenelm W. Philip},  Tundra Vole Software,
 Fairbanks, Alaska.  Program and Manual by  {\bf Kenelm W. Philip}.
 Reviewed by {\bf Yung-Jaan Lee}, University of Michigan.
 \noindent{\bf 4.  PRESS CLIPPINGS}
 \noindent{\bf 5.  INDEX to Volumes I (1990) and II (1991) of
             {\sl Solstice}.}
 \noindent {\bf Volume II, Number 1, Summer, 1991}
 \noindent 1.  ARTICLE

 Sandra L. Arlinghaus, David Barr, John D. Nystuen.
 {\sl The Spatial Shadow:  Light and Dark --- Whole and Part\/}

      This account of some of the projects of sculptor David Barr
 attempts to place them in a formal, systematic, spatial  setting
 based  on  the  postulates  of  the  science of space of William
 Kingdon Clifford (reprinted in {\sl Solstice\/}, Vol. I, No. 1.).
 \noindent 2.  FEATURES

 \item{i}  Construction Zone --- The logistic curve.
 \item{ii.} Educational feature --- Lectures on ``Spatial Theory"
 \noindent {\bf Volume II, Number 2, Winter, 1991}
 \noindent 1.  REPRINT

 Saunders Mac Lane, ``Proof, Truth, and Confusion."  Given as the
 Nora and Edward Ryerson Lecture at The University of Chicago in
 1982.  Republished with permission of The University of Chicago
 and of the author.

 I.  The Fit of Ideas.  II.  Truth and Proof.  III.  Ideas and Theorems.
 IV.  Sets and Functions.  V.  Confusion via Surveys.
 VI.  Cost-benefit and Regression.  VII.  Projection, Extrapolation,
 and Risk.  VIII.  Fuzzy Sets and Fuzzy Thoughts.  IX.  Compromise
 is Confusing.

 \noindent 2.  ARTICLE

 Robert F. Austin.  ``Digital Maps and Data Bases:  
 Aesthetics versus Accuracy."

 I.  Introduction.  II. Basic Issues.  III. Map Production.
 IV.  Digital Maps.  V.  Computerized Data Bases.  VI.  User

 \noindent 3.  FEATURES

 Press clipping; Word Search Puzzle; Software Briefs.
 \noindent{\bf INDEX to Volume I (1990) of {\sl Solstice}.}
 \noindent{\bf Volume I, Number 1, Summer, 1990}

 \noindent 1.  REPRINT

 William Kingdon Clifford, {\sl Postulates of the Science of Space\/}

      This reprint of a portion of  Clifford's  lectures  to  the
 Royal  Institution in the 1870's suggests many geographic topics
 of concern in the last half of the twentieth century.   Look for
 connections  to  boundary  issues,  to  scale problems, to self-
 similarity and fractals, and to non-Euclidean  geometries  (from
 those based on denial of Euclid's parallel  postulate  to  those
 based on a sort of mechanical ``polishing").  What else did,  or
 might, this classic essay foreshadow?

 \noindent 2.  ARTICLES.

 Sandra L. Arlinghaus, {\sl Beyond the Fractal.}  

     An original article.  The fractal notion of  self-similarity
 is  useful  for  characterizing  change  in  scale;  the  reason
 fractals are effective in the geometry of central  place  theory 
 is  because  that  geometry  is hierarchical in nature.  Thus, a
 natural place to look for other connections of this  sort  is to
 other geographical concepts that are also hierarchical.   Within
 this fractal context, this article examines the case of  spatial
     When the idea of diffusion is extended to see ``adopters" of
 an innovation as ``attractors" of new adopters,  a  Julia set is 
 introduced as a possible axis against which to measure one class
 of geographic phenomena.   Beyond the fractal  context,  fractal
 concepts,  such  as  ``compression"  and  ``space-filling"   are
 considered in a broader graph-theoretic setting.
 William C. Arlinghaus, {\sl Groups, Graphs, and God}

      An original article based on a talk given  before  a MIdwest
 GrapH TheorY (MIGHTY) meeting.  The author,  an  algebraic  graph
 theorist, ties his research interests to a broader  philosophical
 realm,  suggesting  the  breadth  of  range  to  which  algebraic
 structure might be applied.

     The  fact  that  almost  all  graphs  are rigid (have trivial
 automorphism groups) is exploited to argue probabilistically  for
 the  existence  of  God.  This  is  presented  with the idea that 
 applications  of  mathematics  need  not be limited to scientific
 \noindent 3.  FEATURES
 \item{i.}  Theorem Museum --- Desargues's  Two  Triangle  Theorem 
            from projective geometry.
 \item{ii.} Construction Zone --- a centrally symmetric hexagon is
            derived from an arbitrary convex hexagon.
 \item{iii.} Reference Corner --- Point set theory and topology.
 \item{iv.}  Educational Feature --- Crossword puzzle on spices.
 \item{v.}   Solution to crossword puzzle.
 \noindent{\bf Volume I, Number 2, Winter, 1990}
 \noindent 1.  REPRINT

 John D. Nystuen (1974), {\sl A City of Strangers:  Spatial Aspects
 of Alienation in the Detroit Metropolitan Region\/}.  

     This paper examines the urban shift from ``people space" to 
 ``machine space" (see R. Horvath,  {\sl Geographical Review\/},
 April, 1974) in the Detroit metropolitan  region  of 1974.   As
 with Clifford's {\sl Postulates\/}, reprinted in the last issue
 of {\sl Solstice\/}, note  the  timely  quality  of many of the 

 \noindent 2.  ARTICLES

 Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical

      Linkage  between  scale  and  dimension  is made using the 
 Fallacy of Division and the Fallacy of Composition in a fractal
 Sandra Lach Arlinghaus, {\sl Parallels between Parallels\/}.

      The earth's sun introduces a symmetry in the perception of 
 its trajectory in the sky that naturally partitions the earth's
 surface  into  zones  of  affine  and hyperbolic geometry.  The
 affine zones, with  single  geometric  parallels,  are  located 
 north and south of the  geographic  parallels.   The hyperbolic
 zone, with multiple geometric parallels, is located between the
 geographic  tropical  parallels.   Evidence  of  this geometric
 partition is suggested in the geographic environment --- in the
 design of houses and of gameboards.
 Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen.
 {\sl The Hedetniemi Matrix Sum:  A Real-world Application\/}.

     In a recent paper, we presented an algorithm for finding the
 shortest distance between any two nodes in a network of $n$ nodes
 when  given  only  distances between adjacent nodes [Arlinghaus, 
 Arlinghaus, Nystuen,  {\sl Geographical  Analysis\/}, 1990].  In
 that  previous   research,  we  applied  the  algorithm  to  the
 generalized  road  network  graph surrounding San Francisco Bay.  
 Here,  we  examine consequent changes in matrix entires when the
 underlying  adjacency pattern of the road network was altered by 
 the  1989  earthquake  that closed the San Francisco --- Oakland
 Bay Bridge.
 Sandra Lach Arlinghaus, {\sl Fractal Geometry  of Infinite Pixel
 Sequences:  ``Su\-per\--def\-in\-i\-tion" Resolution\/}?

    Comparison of space-filling qualities of square and hexagonal
 \noindent 3.  FEATURES
 \item{i.}       Construction  Zone ---  Feigenbaum's  number;  a
 triangular coordinatization of the Euclidean plane.
 \item{ii.}  A three-axis coordinatization of the plane.
 \noindent{\bf 8.  OTHER PUBLICATIONS OF IMaGe }
 \centerline{\bf MONOGRAPH SERIES}
 \centerline{\sl Scholarly Monographs--Original Material, refereed}
 \centerline{Prices on request, exclusive of shipping and handling;}
 \centerline{payable in U.S. funds on a U.S. bank, only.}
 Monographs are printed by {\bf Digicopy} on 100\% recycled paper
 of archival quality; both hard and soft cover is available.
 \vskip 0.2cm
 1.  Sandra L. Arlinghaus and John D. Nystuen.  {\it Mathematical
 Geography and Global Art:  the Mathematics of David Barr's ``Four
 Corners Project\/},'' 1986. 
 \vskip 0.1cm
 This monograph contains Nystuen's calculations, actually used
 by Barr to position his abstract tetrahedral sculpture
 within the earth.  Placement of the sculpture vertices in Easter
 Island, South Africa, Greenland, and Indonesia was chronicled in
 film by The Archives of American Art for The Smithsonian
 Institution.  In addition to the archival material, this 
 monograph also contains Arlinghaus's solutions to broader theoretical
 questions--was Barr's choice of a tetrahedron unique within his
 initial constraints, and, within the set of Platonic solids?
 \vskip 0.2cm
 2.  Sandra L. Arlinghaus.  {\it Down the Mail Tubes:  the Pressured
 Postal Era, 1853-1984\/}, 1986. 
 \vskip 0.1cm

 The history of the pneumatic post, in Europe and in the
 United States, is examined for the lessons it might offer to the
 technological scenes of the late twentieth century.  As Sylvia L.
 Thrupp, Alice Freeman Palmer Professor Emeritus of History, The
 University of Michigan, commented in her review of this work
 ``Such brief comment does far less than justice to the 
 intelligence and the stimulating quality of the author's writing,
 or to the breadth of her reading.  The detail of her accounts of
 the interest of American private enterprise, in New York and
 other large cities on this continent, in pushing for construction
 of large tubes in systems to be leased to the government, brings
 out contrast between American and European views of how the new
 technology should be managed.  This and many other sections of
 the monograph will set readers on new tracks of thought.'' 
 \vskip 0.2cm
 3.  Sandra L. Arlinghaus.  {\it Essays on Mathematical Geography\/},
 \vskip 0.1cm

 A collection of essays intended to show the range of power
 in applying pure mathematics to human systems.  There are two types of 
essay:  those which employ traditional mathematical
 proof, and those which do not.  As mathematical proof may itself
 be regarded as art, the former style of essay might represent
 ``traditional'' art, and the latter, ``surrealist'' art.  Essay
 titles are:  ``The well-tempered map projection,'' ``Antipodal
 graphs,'' ``Analogue clocks,'' ``Steiner transformations,'' ``Concavity
 and urban settlement patterns,'' ``Measuring the vertical city,''
 ``Fad and permanence in human systems,'' ``Topological exploration
 in geography,'' ``A space for thought,'' and ``Chaos in human
 systems--the Heine-Borel Theorem.''
 \vskip 0.2cm
 4.  Robert F. Austin, {\it A Historical Gazetteer of Southeast Asia\/},
 \vskip 0.1cm

 Dr. Austin's Gazetteer draws geographic coordinates of Southeast
 Asian place-names together with references to these
 place-names as they have appeared in historical and literary
 documents.  This book is of obvious use to historians and to
 historical geographers specializing in Southeast Asia.  At a
 deeper level, it might serve as a valuable source in establishing
 place-name linkages which have remained previously unnoticed, in 
 documents describing trade or other communications connections,
 because of variation in place-name nomenclature.
 \vskip 0.2cm
 5.  Sandra L. Arlinghaus, {\it Essays on Mathematical Geography--II\/},
 \vskip 0.1cm

 Written in the same format as IMaGe Monograph \#3, that seeks to use
 ``pure'' mathematics in real-world settings, this volume
 contains the following material:  ``Frontispiece--the Atlantic
 Drainage Tree,'' ``Getting a Handel on Water-Graphs,'' ``Terror in Transit:
 A Graph Theoretic Approach to the Passive Defense of Urban Networks,''
 ``Terrae Antipodum,'' ``Urban Inversion,'' 
``Fractals:  Constructions, Speculations,
 and Concepts,'' ``Solar Woks,'' ``A Pneumatic Postal Plan:  The 
 Chambered Interchange and ZIPPR Code,'' ``Endpiece.''
 \vskip 0.2cm
 6.  Pierre Hanjoul, Hubert Beguin, and Jean-Claude Thill, {\it Theoretical
 Market Areas Under Euclidean Distance\/}, 1988. 
 (English language text; Abstracts written in French and in English.) 
 \vskip 0.1cm
 Though already initiated by Rau in 1841, the economic theory of the 
shape of 
 two-dimensional market areas has long remained concerned with a
 representation of transportation costs as linear in distance. 
 In the general gravity model, to which
 the theory also applies, this corresponds to a decreasing exponential
 function of distance deterrence.  Other transportation cost and
 distance deterrence functions also appear in the literature, however.
 They have not always been considered from the viewpoint of the shape
 of the market areas they generate, and their disparity asks the
 question whether other types of functions would not be worth
 being investigated.  There is thus a need for a general theory
 of market areas:  the present work aims at filling this gap, 
 in the case of a duopoly competing inside the Euclidean plane
 endowed with Euclidean distance. \vskip 0.1cm

 (Bien qu'\'ebauch\'ee par Rau d\`es 1841, la th\'eorie \'economique
 de la forme des
 aires de march\'e planaires s'est longtemps content\'ee de l'hypoth\`ese
 de co\^uts de transport proportionnels \`a la distance.  Dans le mod\`ele
 gravitaire g\'en\'eralis\'e, auquel on peut \'etendre cette th\'eorie, ceci
 correspond au choix d'une exponentielle d\'ecroissante comme fonction de
 dissuasion de la distance.  D'autres fonctions de co\^ut de transport
 ou de dissuasion de la distance apparaissent cependant dans la 
 La forme des aires de march\'e qu'elles engendrent n'a pas toujours \'et\'e
 \'etudi\'ee ; par ailleurs, leur vari\'et\'e am\`ene \`a se demander
 si d'autres fonctions encore ne m\'eriteraient pas d'\^etre examin\'ees. 
 Il para\^it donc utile
 de disposer d'une th\'eorie g\'en\'erale des aires de march\'e : ce \`a
 quoi s'attache ce travail en cas de duopole, dans le cadre
 du plan euclidien muni d'une distance euclidienne.)
 \vskip 0.2cm
 7.  Keith J. Tinkler, Editor, {\it Nystuen---Dacey Nodal Analysis\/}, 1988.

 Professor Tinkler's volume displays the use of this graph theoretical
 tool in geography, from the original Nystuen---Dacey article, to a 
 bibliography of uses, to original uses by Tinkler.  Some reprinted 
 material is included, but by far the larger part is of previously 
 material.  (Unless otherwise noted, all items listed below are previously
 unpublished.)  Contents:  `` `Foreward' " by Nystuen, 1988; 
 ``Preface" by Tinkler,
 1988; ``Statistics for Nystuen---Dacey Nodal Analysis," by Tinkler, 1979;
 Review of Nodal Analysis literature by Tinkler (pre--1979, reprinted with
 permission; post---1979,
 new as of 1988); FORTRAN program listing for Nodal Analysis by Tinkler;
 ``A graph theory
 interpretation of nodal regions'' by John D. Nystuen and Michael F. Dacey,
 reprinted with permission, 1961; Nystuen---Dacey data concerning
 telephone flows in Washington and Missouri, 
 1958, 1959 with comment by Nystuen, 1988; 
 ``The expected distribution of nodality in random (p, q) 
 graphs and multigraphs,'' by Tinkler, 1976.
 8.  James W. Fonseca, {\it The Urban Rank--size Hierarchy:  A Mathematical
 Interpretation\/}, 1989.

 The urban rank--size hierarchy can be characterized as an equiangular spiral
 of the form $r=ae^{\theta \, \hbox{cot}\alpha}$.  An equiangular spiral
 can also be constructed from a Fibonacci sequence.  The urban rank--size
 hierarchy is thus shown to mirror the properties derived from Fibonacci
 characteristics such as rank--additive properties.  A new method of
 structuring the urban rank--size hierarchy is explored which essentially
 parallels that of the traditional rank--size hierarchy below rank 11.
 Above rank 11 this method may help explain the frequently noted
 concavity of the rank--size distribution at the upper levels.  The
 research suggests that the simple rank--size rule with the exponent equal
 to 1 is not merely a special case, but rather a theoretically justified norm
 against which deviant cases may be measured.  The spiral distribution model
 allows conceptualization of a new view of the urban rank--size hierarchy in
 which the three largest cities share functions in a Fibonacci hierarchy.
 9.  Sandra L. Arlinghaus, {\it An Atlas of Steiner Networks \/}, 1989. 

 A Steiner network is a tree of minimum total length joining a prescribed,
 finite, number of locations; often new locations are introduced into the 
 prescribed set to determine the minimum tree.  This Atlas explains the
 mathematical detail behind the Steiner construction for prescribed sets
 of n locations and displays the steps, visually, in a series of 
Figures.  The
 proof of the Steiner construction is by mathematical induction, and enough
 steps in the early part of the induction are displayed completely that the
 reader who is well--trained in Euclidean geometry, and familiar with 
 concepts from graph theory and elementary number theory, should be able to
 replicate the constructions for full as well as for degenerate Steiner 
 10.  Daniel A. Griffith, {\it Simulating $K=3$ Christaller Central Place
 Structures:  An Algorithm Using A Constant Elasticity of Substitution
 Consumption Function\/}, 1989.

 An algorithm is presented that uses BASICA or GWBASIC on IBM compatible
 machines.  This algorithm simulates Christaller $K=3$ central place
 structures, for a four--level hierarchy.  It is based upon earlier published
 work by the author.  A description of the spatial theory, mathematics, and
 sample output runs appears in the monograph.  A digital version is available
 from the author, free of charge, upon request; this request must be
 accompanied by a 5.5--inch formatted diskette.  This algorithm has been 
 developed for use in Social Science classroom laboratory situations,
 and is designed to (a) cultivate a deeper understanding of central place
 theory, (b) allow parameters of a central place system to be altered and
 then graphic and tabular results attributable to these changes viewed,
 without experiencing the tedium of massive calculations, and (c) help
 promote a better comprehension of the complex role distance
 plays in the space--economy.  The algorithm also should facilitate
 intensive numerical research on central place structures; it is expected
 that even the sample simulation results will reveal interesting insights 
 abstract central place theory.

 The background spatial theory concerns demand and competition in the
 space--economy; both linear and non--linear spatial demand functions are
 discussed.  The mathematics is concerned with (a) integration of non--linear
 spatial demand cones on a continuous demand surface, using a constant
 elasticity of substitution consumption function, (b) solving for roots of
 polynomials, (c) numerical approximations to integration and root
 extraction, and (d) multinomial discriminant function classification of 
 commodities into central place hierarchy levels.  Sample
 output is presented for contrived data sets, constructed from
 artificial and empirical information, with the wide range of all possible
 central place structures being generated.  These examples should facilitate
 implementation testing.  Students are able to vary single or multiple
 parameters of the problem, permitting a study of how certain changes 
 themselves within the context of a theoretical central place structure. 
 Hierarchical classification criteria may be changed, demand elasticities may
 or may not vary and can take on a wide range of non--negative values, the
 uniform transport cost may be set at any positive level, assorted fixed
 costs and variable costs may be introduced, again within a rich range of
 non--negative possibilities, and the number of commodities can be 
 altered.  Directions for algorithm execution are summarized.  An ASCII 
 version of the algorithm, written directly from GWBASIC, is included in 
an appendix; hence, it is free of typing errors.
 11.  Sandra L. Arlinghaus and John D. Nystuen,
      {\it Environmental Effects on Bus Durability\/}, 1990.  

  This monograph draws on the authors'
 previous publications on ``Climatic" and ``Terrain" effects on bus 
 Material on these two topics is selected, and reprinted, from
 three published papers that appeared in the {\sl Transportation
 Research Record\/} and in the {\sl Geographical Review\/}.  New
 material concerning ``congestion" effects is examined at the
 national level, to determine ``dense," ``intermediate," and
 ``sparse" classes of congestion, and at the local level of
 congestion in Ann Arbor (as suggestive of how one  might use local data).
 This material is drawn together in a single volume, along with a summary
 of the consequences of all three effects simultaneously, in order to suggest
 direction for more highly automated studies that should follow naturally
 with the release of the 1990 U. S. Census data.
 12.  Daniel A. Griffith, Editor.
 {\sl Spatial Statistics:  Past, Present, and Future\/},  1990. 

     Proceedings of a Symposium of the same name held at Syracuse
 University in Summer, 1989.  Content includes a Preface by Griffith and 
 following papers:  

 \line{{\sl Brian Ripley}, ``Gibbsian interaction models"; \hfil}
 \line{{\sl J. Keith Ord}, ``Statistical methods for point pattern data";
 \line{{\sl Luc Anselin}, ``What is special about spatial data"; \hfil}
 \line{{\sl Robert P. Haining}, ``Models in human geography: \hfil}
 \line{\qquad problems in specifying,
 estimating, and validating models for spatial data"; \hfil}
 \line{{\sl R. J. Martin},
 ``The role of spatial statistics in geographic modelling"; \hfil}
 \line{{\sl Daniel Wartenberg}, \hfil }
 \line{``Exploratory spatial analyses:  outliers,
 leverage points, and influence functions"; \hfil}
 \line{{\sl J. H. P. Paelinck},
 ``Some new estimators in spatial econometrics"; \hfil}
 \line{{\sl Daniel A. Griffith}, \hfil }
 \line{``A numerical simplification for estimating parameters of 
 spatial autoregressive models"; \hfil}
 \line{{\sl Kanti V. Mardia}
 ``Maximum likelihood estimation for spatial models"; \hfil}
 \line{{\sl Ashish Sen}, ``Distribution of spatial correlation statistics";
 \line{{\sl Sylvia Richardson},  
 ``Some remarks on the testing of association between spatial 
 \line{{\sl Graham J. G. Upton}, ``Information from regional data";\hfil}
 \line{{\sl Patrick Doreian},
 ``Network autocorrelation models:  problems and prospects." \hfil}

 Each chapter is preceded by an ``Editor's Preface" and followed by a 
Discussion and, in some cases, by an author's Rejoinder to the Discussion.
 13.  Sandra L. Arlinghaus, Editor.
 {\sl Solstice---I\/},  1990. 
 14.  Sandra L. Arlinghaus, {\sl Essays on Mathematical Geography--III\/},
 A continuation of the series.  Essays in this volume are:  Table for
 central place fractals; Tiling according to the ``Administrative"
 Principle; Moir\'e maps; Triangle partitioning; An enumeration of
 candidate Steiner networks; A topological generation gap; 
 Synthetic centers of gravity:  A conjecture.
 15.  Sandra L. Arlinghaus, Editor, {\sl Solstice---II\/}, 1991.
 \centerline{\it Editor, Daniel A. Griffith}
 \centerline{\it Professor of Geography}
 \centerline{\it Syracuse University}
 \centerline{Founder as an IMaGe series:  Sandra L. Arlinghaus}
 \noindent 1.  {\sl Spatial Regression Analysis on the PC:
 Spatial Statistics Using Minitab}.  1989.  
 \centerline{\it Editor of MICMG Series, John D. Nystuen}
 \centerline{\it Professor of Geography and Urban Planning}
 \centerline{\it The University of Michigan}
 \noindent 1.  {\sl Reprint of the Papers of the Michigan InterUniversity
 Community of Mathematical Geographers.}  Editor, John D. Nystuen.
 Contents--original editor:  John D. Nystuen.
 \noindent 1.  Arthur Getis, ``Temporal land use pattern analysis with the
 use of nearest neighbor and quadrat methods."  July, 1963
 \noindent 2.  Marc Anderson, ``A working bibliography of mathematical
 geography."  September, 1963.
 \noindent 3.  William Bunge, ``Patterns of location."  February, 1964.
 \noindent 4.  Michael F. Dacey, ``Imperfections in the uniform plane."
 June, 1964.
 \noindent 5.  Robert S. Yuill, A simulation study of barrier effects
 in spatial diffusion problems."  April, 1965.
 \noindent 6.  William Warntz, ``A note on surfaces and paths and
 applications to geographical problems."  May, 1965.
 \noindent 7.  Stig Nordbeck, ``The law of allometric growth."
 June, 1965.
 \noindent 8.  Waldo R. Tobler, ``Numerical map generalization;"
 and Waldo R. Tobler, ``Notes on the analysis of geographical
 distributions."  January, 1966.
 \noindent 9.  Peter R. Gould, ``On mental maps."  September, 1966.
 \noindent 10.  John D. Nystuen, ``Effects of boundary shape and the
 concept of local convexity;"  Julian Perkal, ``On the length of
 empirical curves;" and Julian Perkal, ``An attempt at
 objective generalization."  December, 1966.
 \noindent 11. E. Casetti and R. K. Semple, ``A method for the
 stepwise separation of spatial trends."  April, 1968.
 \noindent 12.  W. Bunge, R. Guyot, A. Karlin, R. Martin, W. Pattison,
 W. Tobler, S. Toulmin, and W. Warntz, ``The philosophy of maps."
 June, 1968.
 \centerline{\bf Reprints of out-of-print textbooks.}
 \centerline{\bf Printer and obtainer of copyright permission:  Digicopy}
 \centerline{Inquire for cost of reproduction---include class size}
 1.  Allen K. Philbrick.  {\sl This Human World}.
 Publications of the Institute of Mathematical Geography have
 been reviewed in 
 \item{1.} {\sl The Professional Geographer\/} published
 by the Association of American Geographers;
 \item{2.}  {\sl The Urban Specialty Group Newsletter\/}
 of the Association of American Geographers;
 \item{3.}  {\sl Mathematical Reviews\/} published by the
 American Mathematical Society;
 \item{4.}  {\sl The American Mathematical Monthly\/} published
 by the Mathematical Association of America;
 \item{5.}  {\sl Zentralblatt\/}  Springer-Verlag, Berlin
 \item{6.}  {\sl Mathematics Magazine \/}, published by the Mathematical
 Association of America.