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 \headline = {\ifnum\pageno=1 \hfil \else {\ifodd\pageno\righthead
   \else\lefthead\fi}\fi}
 \def\righthead{\sl\hfil SOLSTICE }
 \def\lefthead{\sl Winter, 1991 \hfil}
 \def\ref{\noindent\hang}
 \font\big = cmbx17
 \font\tn = cmr10
 \font\nn = cmr9 %The code has been kept simple to facilitate reading as 
e-mail
 \centerline{\big SOLSTICE:}
 \vskip.5cm
 \centerline{\bf  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS}
 \vskip5cm
 \centerline{\bf WINTER, 1991}
 \vskip12cm
 \centerline{\bf Volume II, Number 2}
 \smallskip
 \centerline{\bf Institute of Mathematical Geography}
 \vskip.1cm
 \centerline{\bf Ann Arbor, Michigan}
 \vfill\eject
 \hrule
 \smallskip
 \centerline{\bf SOLSTICE}
 \line{Founding Editor--in--Chief:  {\bf Sandra Lach Arlinghaus}. \hfil}
 \smallskip
 \centerline{\bf EDITORIAL BOARD}
 \smallskip
 \line{{\bf Geography} \hfil}
 \line{{\bf Michael Goodchild}, University of California, Santa Barbara. 
\hfil}
 \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).}
 \smallskip
 \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.
        \hfil}
 \smallskip
 \line{{\bf Business} \hfil}
 \line{{\bf Robert F. Austin, Ph.D.} \hfil}
 \line{President, Austin Communications Education Services \hfil}
 \line{Past-president, AM/FM International \hfil}

 \smallskip
 \hrule
 \smallskip
 
       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.  
 \vskip1in
 \noindent
 \vskip.2cm
 \noindent {\bf Send all correspondence to:}
 \vskip.1cm
 \centerline{\bf Institute of Mathematical Geography}
 \centerline{\bf 2790 Briarcliff}
 \centerline{\bf Ann Arbor, MI 48105-1429}
 \vskip.1cm
 \centerline{\bf (313) 761-1231}
 \centerline{\bf IMaGe@UMICHUM}
 \vfill\eject
 
       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.
 \smallskip
       {\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.}
 \vskip.5cm
 Copyright, December, 1991, Institute of Mathematical Geography.
 All rights reserved.
 \vskip1cm
 {\bf ISBN: 1-877751-53-7}
 {\bf ISSN: 1059-5325} 
 \vfill\eject
 \centerline{\bf SUMMARY OF CONTENT}
 \smallskip
 \noindent{\bf 1.   REPRINT.}
 \smallskip
 {\bf Saunders  Mac  Lane}.   {\bf  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.
 \smallskip
 \noindent
 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.
 \smallskip
 \noindent{\bf 2.  ARTICLE }
 \smallskip\noindent
 {\bf Robert F. Austin}.   {\bf  Digital  Maps  and  Data   Bases:
 Aesthetics versus Accuracy $^{\bf \star }$}
 \smallskip\noindent
 I.  Introduction.  II.  Basic Issues.  III.  Map Production.  IV.
 Digital Maps.  V.  Computerized Data Bases.  VI.  User Community.
 \smallskip
 \noindent{\bf 3.  FEATURES}
 \smallskip
 \item{i.}   {\bf Press Clipping} --- reprint of ``Briefing" from
             {\sl   Science\/}  mentioning  {\sl Solstice\/} --- 
             ``Online Journals," by Joseph Palca.
 \item{ii.}  {\bf  Educational  feature} --- Word  search  puzzle
             promoting mathematical and geographical awareness.
 \item{iii.} {\bf Software  Briefs} ---  Brief  descriptions   of
             software provided by the  creator.  Look for reviews
             of    the   software   in   subsequent   issues   of
             {\sl Solstice \/}. 

             a.  RangeMapper$^{\hbox{TM}}$  ---  version 1.4.
             Created  by Kenelm W. Philip,  Tundra Vole Software,
             Fairbanks, Alaska.  Program and Manual by  Kenelm W.
             Philip; commentary from the Manual.    
             ``A  utility  for  biological species range mapping,
             and similar mapping tasks in other fields."

             b.    ``XYNIMAP"   ---  created  by David H. Douglas, 
             University of Ottawa;  ``a comprehensive system  for 
             computer cartography and geo-spatial analysis."
             Preliminary Version. 
 \item{iv.}   Index to Volumes I (1990) and II (1991) of
             {\sl Solstice}.
 \item{v.}  Other  publications  of the Institute of Mathematical
             Geography.
 \item{vi.} Solution to Word Search Puzzle.
 \vfill\eject
 \centerline{\bf ACKNOWLEDGMENTS}
 \vskip.5cm
 
     The 1982 Ryerson Lecture, ``Proof, Truth, and Confusion," is
 republished  in  {\sl Solstice\/}  with  the  permission  of The
 University  of  Chicago  and  with  the  permission of Professor
 Saunders Mac Lane.

 The column ``Online Journals" is republished in {\sl Solstice \/}
 with   the  permission  of  the  American  Association  for  the 
 Advancement    of    Science.      It   appeared  originally  in
 {\sl Science \/}, ``Briefings,"   Vol. 254,  No. 5036  (Nov. 29,
 1991), p. 1291.  Copyright is held by the AAAS.
 \vfill\eject
 \noindent 1.  REPRINT
 \smallskip
 \centerline{\bf PROOF, TRUTH, AND CONFUSION}
 \vskip.2cm
 \centerline{Saunders Mac Lane}
 \centerline{Max Mason Distinguished Service Professor of Mathematics}
 \centerline{The University of Chicago}
 \smallskip
 \centerline{The 1982 Nora and Edward Ryerson Lecture}
 \centerline{at The University of Chicago.}
 \smallskip
 Copyright, 1982 by The University of Chicago,  republished  with
 permission of that University and of the author.
 \smallskip
 \centerline{\bf Saunders Mac Lane } %photograph of Mac Lane, c. 1982.
 \vfill\eject
 \centerline{\bf Introduction}
 \centerline{by}
 \centerline{Hanna H. Gray}
 \centerline{President of the University}
 
      Each year, one of the annual events at the University is the
 selection of the Nora and Edward Ryerson Lecturer.  The selection
 is made by a committee of faculty which receives nominations from
 their  faculty colleagues.   Each year,  this committee comes  up
 with an absolutely superb selection for the Ryerson Lecturer, and
 this year is triumphant confirmation of that generalization.
 
      The   selection   emanates  from  faculty   nomination   and
 discussion,  and  it is analogous to the process of the selection
 of faculty in this University,  representing a selection based on
 the   work  and  contribution,   on  the  high  esteem  for   the
 intellectual  imagination and breadth of a colleague.   The  peer
 review  process,  in this as in  faculty  appointments,  stresses
 scholarship  and research,  stresses the contribution of a member
 of  the faculty to the progress of  knowledge.  In  addition,  of
 course,  the  faculty appointment process looks also to  teaching
 and to institutional citizenship.
 
      If  I  had the nerve to fill out an A-21 form for  Professor
 Saunders  Mac  Lane,   I  would,  I  think,  be  creating  a  new
 mathematics  because I would award him 100 percent for  research,
 100  percent for teaching,  and 100 percent for  contribution  or
 citizenship.  Even I can add that up to 300 percent.
 
       Now,  of course,  in the evaluation of younger scholars for
 junior  appointments similar judgments are made.   They are based
 on  the same three categories,  and they are judgments about  the
 promise of continuing creativity,  continuing growth,  continuing
 intellectual contribution.   That judgment of the young  Saunders
 Mac Lane was made a very long time ago in Montclair,  New Jersey.
 Montclair,  New Jersey is the home of the Yale Club of Montclair.
 I  once  had the enormous privilege of being invited to the  Yale
 Club  of  Montclair where I was given something called  the  Yale
 Bowl, which had on it an inscription testifying that I had earned
 my ``Y" in the ``Big Game of Life."
 
      In 1929,  the young Saunders Mac Lane went to the Yale  Club
 of Montclair,  I was told---he was then finishing his senior year
 at  Yale---and there was a young dean of the Yale Law school  who
 was  leaving  in  order  to go to the University  of  Chicago  as
 president.   And the Yale Club of Montclair,  which usually  gave
 its awards to football players,  decided on this occasion to give
 recognition  to a young mathematician and to a young  law  school
 dean, and that was where Mr. Hutchins and Mr. Mac Lane met.
 
      As  Saunders  Mac Lane graduated  from  Yale,  Mr.  Hutchins
 encouraged him personally to come to Chicago.  And Saunders came.
 He had,  however,  neglected to take steps that are usually taken
 when one travels to enter another university, and the chairman of
 the  Department of Mathematics,  Mr.  Bliss,  had to say  to  him
 rather directly, ``Young man, you've got to apply first."
 
      He  did apply,  and fortunately he was accepted.   Within  a
 year,  he  had received his M.A.  from Chicago and had come  into
 contact   with  lots  of  extraordinary  people,   but  two  very
 extraordinary   people  in  particular.    One  was   the   great
 mathematician E.  H.  Moore, and the other was a graduate student
 in economics named Dorothy Jones,  who was in 1933 to become Mrs.
 Mac Lane.
 
      Now,  those  of us who  know Saunders think of him as a Hyde
 Parker,  and indeed as a Hyde Parker forever.  And, of course, he
 is a Hyde Parker,  and a Hyde Parker forever, but he had a period
 in  his  life,  I  have to tell you, after he had taken his M.A.,
 when  he  became  something  of an academic traveler.   We really
 ought to have been able  to  trace those travels  when  we  think
 about Saunders' choice of costume.
 
      Now,  that's  not easy to figure out today because  I  think
 that  necktie came out of a safe this morning.   But if you think
 about the flaming reds,  for example,  that Saunders affects, you
 are perhaps reminded of Cambridge,  Massachusetts.   If you think
 of  the Scottish plaids which he affects,  that's a  harder  one,
 because  I would say that that has to do with the great tradition
 which took him to New England and made him for a time a  resident
 of  Connecticut.   And then,  of course,  there is the Alpine hat
 which could only have come from Ithaca, New York.
      Saunders  received  his  doctorate from  the  University  of
 G\"ottingen  in 1934.   He had spent the years 1933-34  again  at
 Yale  as a Sterling Fellow.   He then spent two years at Harvard.
 He then spent a year at Cornell.   Then he came to the University
 of Chicago for a year.   And then again he moved,  called back to
 Harvard  as  an assistant professor,  and there he  rapidly  went
 through  the ranks.   Fortunately,  in 1947,  he returned to  the
 University  of  Chicago  and,  in  1963,  became  the  Max  Mason
 Distinguished  Service  Professor.   Between 1952  and  1958,  he
 succeeded  Marshall  Stone  as  chairman  of  the  Department  of
 Mathematics  for  two three-year terms,  and he  has  served  the
 University as he has his department with total dedication.
 
      Saunders  has  extended  his  role  beyond  our  University,
 serving  primarily  and  prominently  in  a  number  of  national
 scholarly   organizations  and  institutions  devoted  to   large
 questions  of  the  relationship  of learning to policy.   He was
 president of the Mathematical Association of America and received
 its  Distinguished  Service Award in 1975 in recognition  of  his
 sustained and active concern for the advancement of undergraduate
 mathematical teaching  and  undergraduate  mathematics.   He  was
 also  president of the American Mathematical Society in  1973-74.
 He  has  been  a  member  of  the   National  Science  Board  and 
 vice-president of the National Academy of Sciences.
 
      His  work  in  mathematics,   of  course,  has  been  widely
 recognized.  Alfred Putnam, who studied with Saunders at Harvard,
 had this to say of Saunders in a biographical sketch that he  has
 published.   He  wrote,  ``Beginning as a graduate student with a
 brief exposure to group extensions,  I've watched the development
 of Saunders Mac Lane's mathematics through homological algebra to
 category  theory.   Saunders  Mac Lane belongs in a  category  by
 himself."
 
      And  so  he  does.   So he does as  a  mathematican,  as  an
 academic citizen,  as a spokesman for the fundamental values  and
 principles  of  the University,  and,  of  course,  in  sartorial
 wonder.
 
      Now,  it  is to this category that we look for the Nora  and
 Edward  Ryerson  Lecturers.   When the Trustees  established  the
 lectureship  in  1973,   they  sought  a  way  to  celebrate  the
 relationship that the Ryersons and their family have had with our
 University---a  relationship of shared values and a commitment to
 learning at the most advanced level.
 
      Mr.  Ryerson  was  elected to the Board in 1923  and  became
 Chairman  of  the  Board in 1953.   Nora  Butler  Ryerson  was  a
 founding member,  if not {\sl the\/} founder, of the University's
 Women's  Board.   Both  embraced  a  civic trust  that  left  few
 institutions  in our city untouched,  and they passed  to  future
 generations   of  their  family  the  sense  of  engagement   and
 participation.
 
      Saunders  Mac  Lane,  through  his staunch  loyalty  to  our
 University,  his  broad interest in the community of scholars and
 their work, his distinguished scholarly career,  represents these
 values for us in a special way,  and,  of course,  he is entirely
 uncompromising also in his commitment to them.   It is a pleasure
 to introduce this year's Ryerson Lecturer, Saunders Mac Lane.
 \vfill\eject
 \centerline{\bf Proof, Truth, and Confusion}
 \smallskip
 \centerline{\bf Saunders Mac Lane}
 \smallskip
 \centerline{\bf I.  The Fit of Ideas}
 
      It  is an honor for a mathematician to stand here.   Let  me
 first  say  how  much I appreciate the initiative  taken  by  the
 trustees  on behalf of the Ryerson family in providing  for  this
 series of lectures,  which afford opportunity for a few fortunate
 faculty  members to present aspects of their scholarly work which
 might  be of interest to the whole university community.   In  my
 own case, though the detailed development of mathematics tends to
 be  highly  technical,   I find that there  are  some  underlying
 notions  from mathematics and its usage which can and will be  of
 general interest.   I will try to disentangle these and to relate
 them to the general interest.
 
      This  intent  accounts  for my  title.   Mathematicians  are
 concerned to find truth,  or,  more modestly,  to find a few  new
 truths.   In  reality,  the  best  that I and  my  colleagues  in
 mathematics can do is to find proofs which perhaps establish some
 truths.  We try to find the right proofs.  However, some of these
 proofs  and  the  techniques and numbers which embody  them  have
 turned  out to be so popular that they are applied where they  do
 not belong---with results which produce confusion.   For this,  I
 will try to cite examples and to draw conclusions.
 
      This involves a thesis as to the nature of  mathematics:   I
 contend  that this venerable subject is one which does reach  for
 truth,  but  by way of proof,  and does get proof,  by way of the
 concatenation of the right ideas.   The ideas which are  involved
 in  mathematics  are  those  ideas which are  formal  or  can  be
 formalized.  However, they are not purely formal; they arise from
 aspects of human activity or from problems arising in the advance
 of scientific knowledge.  The ideas of mathematics may not always
 lead  to truth;  for this reason it is important that good  ideas
 not  be  confused by needless compromise.   In brief,  the  ideas
 which matter are the ideas that fit.
 
      However,  the  fit may be problematical.   A friend of  mine
 with  a vacation home in Vermont wanted to suitably decorate  his
 barn,  and  so asked the local painter to put on the  door  ``the
 biggest  number which can be written on the broad side of a  barn
 door."  The painter complied, painting on the barn door a digit 9
 followed by as many further such digits as could be squeezed onto
 the  door  (Figure 1.a).   A competitor then claimed he could  do
 better by painting smaller 9's and so a bigger number.   A second
 competitor then rubbed out the first line and wrote instead:  The
 square of the number 9,  $\ldots$ (Figure 1.b).  Even that didn't
 last,  because  another  young fellow  proposed  the  paradoxical
 words,  ``One  plus the biggest number that can be written on the
 broad side of this barn door" (Figure 1.c).  At each moment, this
 produces a bigger number than anything before.   We may  conclude
 that  there is no such biggest number.   This may illustrate  the
 point  that  it is not easy to get the ideas that  fit---on  barn
 doors or otherwise.
 \midinsert \vskip 6in
 \noindent {\bf Figure 1}.   a.   The biggest number on  that barn 
 door.   b.   A  bigger  number  on  that  barn door.  c.  An even
 bigger number on that barn door.\endinsert
 \smallskip
 \centerline{\bf II.  Truth and Proof}
 \smallskip
 
      I  return to the ``truth" of my title.   When I was young  I
 believed  in  RMH--which  sometimes  stands  for  Robert  Maynard
 Hutchins,  who  to my great profit first encouraged me to come to
 Chicago---and which sometimes stands for the slogan, ``Reach Much
 Higher."   At  any rate,  when young I thought  that  mathematics
 could reach very much higher so as to achieve absolute truth.  At
 that time, {\sl Principia Mathematica\/} by Whitehead and Russell
 seemed  to  model  this  reach;  it claimed  to  provide  all  of
 mathematics firmly founded on the truths of logic.   The logic in
 {\sl Principia\/} was elaborate,  symbolic,  and hard to  follow.
 As  a  result,  it  took  me some years  to  discover  that  {\sl
 Principia    Mathematica\/}    was    not   a    {\sl    Practica
 Mathematica\/}---much  of  mathematics,  in  particular  most  of
 geometry,  simply  wasn't there in {\sl Principia\/}.   For  that
 matter,  what  {\sl  was\/} there didn't  come  exclusively  from
 logic.   Logic  could  provide  a framework and a  symbolism  for
 mathematics,  but it could not provide guidelines for a direction
 in which to develop.
 
      This limitation  was a shocking discovery.   Logic, even the
 best symbolic logic, did not provide all of absolute truth.  What
 did it provide instead?   It provided proof---the rigorous  proof
 of  one formal statement from another prior statement;  that  is,
 the deduction of theorems from axioms.  For such a deduction, one
 needed logic to provide the rules of inference.  In addition, one
 needed  the subject matter handled in the deductions:   the ideas
 used  in  the formulation of the axioms of  geometry  and  number
 theory, as well as the suggestions from outside mathematics as to
 what theorems might usefully be proved from these axioms.
 
       Deductive  logic  is important not because it  can  produce
 absolute  truth but because it can settle  controversy.   It  has
 settled many.  One notable example arose in topology, a branch of
 mathematics  which  studies qualitative properties  of  geometric
 objects such as spheres.   From this perspective, a smooth sphere
 and  a  crinkly  sphere would have the {\sl  same\/}  qualitative
 properties---and   we  would  consider  not  just  the   ordinary
 spheres---two-dimensional,    since    the   surface   has    two
 dimensions---but also the spheres of dimensions 3,  4, and higher
 (Figure 2).    For these spheres, topologists wished to calculate
 a  certain   number which measures the  connectivity---a  measure
 ``two   dimensions  up" from the dimension of  the  sphere.   The
 Soviet  topologist L. Pontrjagin in 1938 stated that this desired
 measure   was one.   Others thought instead that the measure  was
 two.   In a  related connection, the American reviewer of another
 paper  by   Pontrjagin wrote,  ``Both  theorems  (of  Pontrjagin)
 contradict a  previous statement of the reviewer.  It is not easy
 to  see  who is  wrong here."   Fortunately,  it was possible  to
 see.  With careful  analysis of the proof, Pontrjagin did see who
 was  wrong---and  in  1950 published a statement  correcting  his
 1938 error:   that the  measure of connectivity two dimensions up
 is {\sl not\/} one, but  two.
 \midinsert \vskip 4in
 \centerline{\bf Figure 2}
 \endinsert
 
      A few years ago,  the {\sl New York Times\/} carried an item
 about  a  similar  fundamental disagreement  between  a  Japanese
 topologist  and one of our own recent graduate students,  Raphael
 Zahler.  Analysis of the deductions showed that Zahler was right.
 There  lies the real role of logic:   it provides a formal  canon
 designed to disentangle such controversies.
 
      Truth  may  be  difficult  to  capture,  but  proof  can  be
 described  with complete accuracy.   Each mathematical  statement
 can be written as a word or sentence in a fixed  alphabet---using
 one letter for each primitive mathematical  notion and one letter
 for each logical connective.   A proof of a theorem is a sequence
 of  such  statements.   The initial statement must be one of  the
 axioms.   Each  subsequent  statement  not an  axiom  must  be  a
 consequence   of   prior  statements  in  the   sequence.    Here
 ``consequence"  means  ``consequence  according  to  one  of  the
 specified  rules of inference"---rules specified in  advance.   A
 typical  such  rule  is that  of  {\sl  modus  ponens\/}:   Given
 statments  ``S"  and ``S implies T," one may infer the  statement
 ``T."
 
      This  description gives a firm standard  of  proof.   Actual
 proofs may cut a few corners or leave out some obvious steps,  to
 be  filled  in  if and when needed.   Actual proofs may  even  be
 wrong.   However,  the formal description of a proof is  complete
 and  definitive.   It  provides a formal standard of  rigor,  not
 necessarily for absolute truth, but for absolute proof.
 
      There  is a surprising consequence:   no one  formal  system
 suffices  to  establish all of  mathematics.   Precisely  because
 there  is such a rigorous description of a ``proof" in a ``formal
 system," Kurt G\"odel was able to show that,  in each such system
 with  calculable rules of inference,  one could formulate in  the
 system  a sentence which was not decidable in  the  system---that
 is,  a  sentence  $G$ which can neither be proved  nor  disproved
 according  to  the specified rules of inference.   More  exactly,
 this  is the case for any system which contains the  numbers  and
 the rules of arithmetic,  and in which the rules of inference can
 be   explicitly   listed  or  numbered  in  the  fashion   called
 ``recursive."
 
      In  such  a  system,  all  statements  are  formal  and  are
 constructed  from  a fixed alphabet.   Hence we can  number  {\sl
 all\/} the possible proofs.   Moreover,  we can formulate  within
 the  system  a sentence which reads,  ``$n$ is the number of  the
 proof of the statement with the number $k$."  On this basis,  and
 adapting ideas illustrated by the paradox of the barn  door,  one
 then constructs another sentence $G=G(p)$ (with number $p$) which
 reads,  ``There  is no number which is the proof of the  sentence
 number  $p$."   This means in particular that this very  sentence
 $G$ cannot be proved in the system.   This is because $G$  itself
 states that ``there is no proof in the system for me"---hence $G$
 is true (Figure 3).  Hence, unless the system is inconsistent, it
 can  contain no refutation of $G$.   Thus in such a formal system
 we can write one statement (and hence many) which,  though  true,
 is simply undecidable, yes or  no, within the system.
 \midinsert \vskip 3in
 \centerline{\bf Figure 3}
 \endinsert
 
      This result is startling.  It may seem catastrophic---but it
 turns out to be not quite so disastrous.   It shows that there is
 an intrinsic limitation on what can be proved within {\sl any one
 formal\/}  system;  thus proof within one such system cannot give
 all of truth.  Very well then, as we shall see, there can be more
 than  one formal system and hence more than one way in  which  to
 reach by proof for the truth.
 \smallskip
 \centerline{\bf III.  Ideas and Theorems}
 \smallskip
 
      Some   observers  have  claimed  that  mathematics  is  just
 formalism.   They  are wrong.   A mathematical proof in  a  given
 formal  system  must be {\sl about\/} something,  but it  is  not
 about  the  outside world.   I say it is about ideas.   Thus  the
 formal system of Euclidean geometry is about certain ``pictorial"
 ideas:   point,  line,  triangle,  and congruence; in their turn,
 these ideas arose as means of formulating our spatial experiences
 of shape, size, and extent and our attempts to analyze motion and
 symmetry.
 
      Each branch of formal mathematics has a comparable origin in
 some human activities or in some branch of scientific  knowledge.
 In  each  such  case,  the  formal  mathematical  system  can  be
 understood as the realization of a few central ideas.
 
     Mathematics  is  built  upon a considerable variety  of  such
 ideas---in the calculus,  ideas about rate of change,  summation,
 and  limit;  in geometry,  ideas of  proximity,  smoothness,  and
 curvature.   To further illustrate what I mean here by ``idea," I
 choose  a  small  sample:    The  related  ideas  of  ``connect,"
 ``compose," and ``compare."
 
      To  ``Connect" means to join.   There are different ways  in
 which  mathematicians  have defined what it means for a piece  of
 space to be connected.  One definition says that a piece of space
 is  connected  if  it  does not fall apart  into  two  (or  more)
 suitably disjoint pieces.   Another definition says that a  piece
 of space is {\sl path-connected\/} if any two points in the piece
 can be joined {\sl within\/} the piece by a path---that is,  by a
 continuous  curve  lying wholly in the piece.   These two  formal
 explications  of the idea of ``connected" are  not  identical;  a
 piece of space which is path-connected is always connected in the
 first sense, but not necessarily vice-versa.  This simple case of
 divergence  illustrates the observation that the same  underlying
 pre-formal idea can have different formalizations.
 
      ``Compose" is the next idea.  To compose two numbers $x$ and
 $y$  by addition is to take their sum $x+y$;  to compose them  by
 multiplication  is  to take their product $xy$.   To compose  one
 motion  $L$ with a second motion $M$ is to follow $L$ by  $M$  to
 get the ``composite" motion which we write as $L \circ M$.   Thus
 to  rotate a wheel first by $25^{\circ}$ and then by $45^{\circ}$
 will  yield  after composition a rotation  by  $70^{\circ}$.   To
 compose  a path $L$ connecting a point $p$ to a point $q$ with  a
 path  $M$  connecting  $q$ to a third point $s$ is  to  form  the
 longer path $L \circ M$ which follows first $L$ and then $M$,  as
 in  the top of figure 4.   In all such cases of composition,  the
 result  of a composition $L \circ M \circ N$ of three  things  in
 succession  depends  on the factors composed and the sequence  or
 order in which they were taken---but {\sl not\/} on the  position
 of  the  parenthesis.   Thus  arises one of the  formal  laws  of
 composition, the associative law:
 $$
 L \circ (M \circ N) = (L \circ M) \circ N.
 $$
 However,  $L \circ M$ may very well differ from $M \circ L$!  The
 order matters.
 \midinsert \vskip3in
 \centerline{\bf Figure 4.}
 \endinsert
 
      The  third  sample idea is ``Compare."  One may compare  one
 triangle  with  another  as to size,  so as  to  study  congruent
 triangles.   One  may  compare one triangle with  another  as  to
 shape,  and  so study more generally similar triangles.   Another
 comparison is that by deformation:  Two paths in a piece of space
 may  be  compared  by  trying  to deform  the  first  path  in  a
 continuous  way into the second---as in Figure 4,  the  composite
 path  $L  \circ M$ from $p$ to $r$ can be deformed  smoothly  and
 continuously into the path $K$ also joining $p$ to $r$.
 
       Ideas   such   as  these  will  function   effectively   in
 mathematics   only after they have been formalized,  because then
 explicit   theorems about the ideas can be proved.   The idea  of
 composition   is  formalized by the concept  of  a  group,  which
 applies  to  those   compositions in which each thing  $L$  being
 composed  has an  ``inverse" thing or operation $L^{-1}$ so  that
 $L \circ L^{-1} =  1$.   One readily sets down axioms for a group
 of  ``things"  with   such composition.   The  axioms  are  quite
 simple,  but  the  concept   has  proven  to  be  extraordinarily
 fruitful.   There are very many  examples of groups:   Groups  of
 rotations,  groups of symmetry,   crystallographic groups, groups
 permuting  the roots of equations,   the gauge groups of physics,
 and  many others.   There is a sense  (analyzed by  Eilenberg-Mac
 Lane  in a series of papers) in which  any group can be built  up
 by successive extensions from certain  basic pieces,  called  the
 ``simple"  groups.   Specifically,  a  group  is said to be  {\sl
 simple\/}  when  it  cannot be collapsed into  a   smaller  group
 except in a trivial way.   A long-standing  conjecture  suggested
 that  the  number  of  elements in a  finite   simple  group  was
 necessarily  either  an even number or a  prime   number.   About
 twenty years ago, here at Chicago, Thompson and Feit succeeded in
 proving  this to be true (and I could take  pleasure in the  fact
 that  Thompson,   one  of  my  students,  had   achieved  such  a
 penetrating result).   The Thompson-Feit method  turned out to be
 so  suggestive and powerful that others have now  been able to go
 on to explicitly determine {\sl all\/} the finite  simple groups.
 For   example,   the  biggest  sporadic  one  has    $2^{46}\cdot
 3^{20}\cdot   5^9\cdot  7^6\cdot  11^2\cdot  13^3\cdot    17\cdot
 19\cdot  23\cdot  29\cdot 41\cdot 47\cdot 59\cdot  71$   elements
 (that  number is approximately 8 followed by  53  zeros).    This
 simple group is called the ``Monster" (Figure 5).  Another one of
 our   former  students has been able to make a  high  dimensional
 geometric  picture  which shows that this monster really  exists.
 He needed a space of dimension 196,884.
 \midinsert \vskip3in
 \centerline{\bf Figure 5}
 \endinsert
 
       Groups  also serve to measure the connectivity  of  spaces.
 In  particular, there are certain homology groups which count the
 presence of higher dimensional holes in space.   To start with, a
 piece $\chi $ of space is said to be {\sl simply connected\/}  if
 any closed path in the space $\chi$ can be deformed into a point.
 For example,  the surface of a sphere is simply connected, so its
 first homology group is zero;  however,  it has a non-zero second
 homology group---meaning the ``hole" represented by the inside of
 the  sphere.   These properties characterize the  two-dimensional
 sphere.   Long ago, the French mathematician Poincar\'e said that
 the  same  should hold for a three-dimenional sphere (Figure  6).
 This famous  conjecture has not yet been settled---but some years
 ago,  Smale   showed  that the characterization was  true  for  a
 sphere of  dimension 5 or higher.  Just during the last year, the
 Californian Michael Friedman,  in a long proof, showed that it is
 also  true for a sphere of dimension 4.   Except for  a  solution
 which was announced on April 1, nobody yet knows the answer for a
 three-dimensional sphere.  Proof advances, but slowly.
 \midinsert \vskip3in
 \centerline{\bf Figure 6}
 \endinsert
 \vfill\eject
 \centerline{\bf IV.  Sets and Functions}
 \smallskip
 
      As  already  indicated,   Whitehead  and  Russell,  by  {\sl
 Principia  Mathematica\/},  had suggested that  all  mathematical
 truth  could  be subsumed in one monster  formal  system.   Their
 system, corrupted as it was with ``types," was too complicated---
 but  others proposed a system based on the idea of a {\sl set\/}.
 A set is just a collection of things---nothing more.  Mathematics
 does  involve sets,  such as the set of all prime numbers or  the
 set  of  all  rational numbers between  1  and  2.   Mathematical
 objects can be defined in terms of sets.   For example,  a circle
 is the set  of all points  in the plane  at a fixed distance from
 the center, while a line can be described as the set of  all  its
 points.   Numbers  can be defined as sets---the number two is the
 set of all pairs;  an irrational number is the set of all smaller
 rational  numbers.   In this way numbers,  spatial  figures,  and
 everything  else  mathematical can be defined in  terms  of  sets
 (Figure  7).   All  that matters about a set $S$ is the  list  of
 those things $x$ which are members of $S$.   When this is so,  we
 write  $x\,  \epsilon \,  S$,  and  call  this  the  ``membership
 relation."
 \midinsert \vskip3in
 \centerline{\bf Figure 7}
 \endinsert
 
      There  are  axioms  (due  to  Zermelo  and  Fraenkel)  which
 adequately  formalize the properties of this membership relation.
 These  axioms claim to provide a formal foundation--I  call  this
 the grand set-theoretic doctrine---for all of mathematics.
 
      By 1940 or so this grand set-theoretic foundation had become
 so  prominent  in advanced mathematics that it  was  courageously
 taught  to  freshmen right here in the  Hutchins  college.   This
 teaching practice spread nationally to become the keystone of the
 ``New  Math."   As a result,  twenty years later sets came to  be
 taught in the kindergarten.   There is even that story about  the
 fond parents inquiring as to little Johnny's progress.   Yes said
 the teacher, he is doing well in math except that he can't manage
 to  write  the symbol $\epsilon$ when $x$ is a member of the  set
 $S$.
 
      Johnny was not the only one in trouble.  The grand  doctrine
 of  the  new math:   ``Everything is a set" came at the  cost  of
 making  artificial  and clumsy  definitions.   Moreover,  putting
 everything  in  one formal system of axioms for  set  theory  ran
 squarely into the difficulties presented by G\"odel's undecidable
 propositions.
 
      Fortunately,  just  about the time when sets reached down to
 the kindergarten,  an alternative approach to a system of  ``all"
 (better  ``most") of mathematics turned up.   This used again the
 idea  of  composition  for functions  $f:  S  \longrightarrow  T$
 sending the elements of a set $S$ to some of those of another set
 $T$.   Another  function  $g:  T \longrightarrow U$ can  then  be
 composed with $f$ to give a new function $g \circ f$ (Figure  8).
 It  sends an element of $S$ first by $f$ into $T$ and then by $g$
 into $U$.  The prevalence of many such compositions led Eilenberg
 and  Mac  Lane  in  1945 to define the  formal  axioms  for  such
 composition.   With no apologies to Aristotle, they called such a
 system a ``category"---because many types of mathematical objects
 did form such categories, and these properties were useful in the
 organization of mathematics.   Note especially that the intuitive
 idea  of ``composition"  has  several  different  formalizations:
 category and group.
 \midinsert \vskip4in
 \centerline{\bf Figure 8}
 \endinsert
 
      Then   in  1970  Lawvere  and  Tierney  made  a   surprising
 discovery:  that in treating a function $f:  S \longrightarrow T$
 one could forget all about the elements in $S$ and $T$, and write
 enough  axioms  on  composition alone  to  do  almost  everything
 otherwise  done  with sets and elements.   This formal system  is
 called   an   ``elementary  topos"---to  suggest  some   of   its
 connections  to  geometry and ``Top"-ology.    Their  success  in
 discovering  this  wholly new view of mathematics  emphasizes  my
 fundamental  observation:   That  the ideas  of  mathematics  are
 various and can be encapsulated in different formal systems.
 
      Waiting  to be developed,  there must be still other  formal
 systems for the foundation and organization of mathematics.
 \vfill\eject
 \centerline{\bf V.  Confusion via Surveys}
 \smallskip
 
      The  crux  of  any search for the right alternative  to  set
 theory  is the search for the right concatenation  of  ideas---in
 the  same  way  in which leading ideas in mathematics  have  been
 combined in the past to solve problems (the Poincar\'e conjecture
 on  spheres)  and to give new insights.   Thus it  was  with  our
 example,  where the related ideas of connection, composition, and
 comparison  came together in group theory,  in the application of
 groups to geometry, and in category theory.
      But  sometimes the wrong ideas are brought together,  or the
 right ideas are used in the wrong way.   Today the use of numbers
 and  of quantitative methods is so pervasive that many arrays  of
 numbers  and  of other mathematical techniques  are  deployed  in
 ways which do not fit.
 
      This  I  will illustrate by  some  examples.   Recently,  in
 connection  with my membership on the National Science  Board,  I
 came across  the work of one prominent social scientist  who  was
 promoting   (perhaps  with  reason)  the  use  of  computer-aided
 instruction  in  courses  for  college  students.   However,  the
 vehicle he chose for such instruction was the formal manipulation
 of  the elementary consequences  of the  Zermelo-Fraenkel  axioms
 for  set  theory---and the result was an emphasis on  superficial
 formalism  with no attention to ideas or  meaning.   It  was,  in
 short, computer-aided pedantry.
 
      ``Opinion  Surveys" provide another example of the confusion
 of ideas.  For some social and behavioral research, the necessary
 data  can  be obtained only by survey  methods,  and  responsible
 scientists  have developed careful techniques to  help  formulate
 the  survey  questions used to probe  for  facts.   Unfortunately
 these  techniques  are often used  carelessly---both  because  of
 commercial abuse, statistical malpractice, or poor formulation of
 survey questions.  First the malpractice:
 
      On  many surveys the percentage of response is uncomfortably
 low, with the result that the data acquired are incomplete.  This
 situation  has led the statisticains into very elaborate  studies
 of  means for approximately completing such  ``incomplete  data."
 One  recent  and  extensive  such publication  (by  the  National
 Research  Council)  seemed  to me technically  correct  but  very
 elaborate---perhaps  overdone,  and  in any event,  open  to  the
 misuse of too much massaging of data that are fatally incomplete.
 
      In   opinion surveys  touching  directly  on  the   academic
 profession some of the worst excesses are those exhibited by  the
 so-called  ``Survey  of the American Professoriate."   Successive
 versions  of  this  survey  are  replete  with  tendentious   and
 misleading  questions,  often  such likely to  ``create"  opinion
 rather than to measure actual existing opinions.   Despite heroic
 attempts  by others to suggest improvements,  the authors of this
 particular  survey have continued in their mistaken practices  in
 new   such  surveys---as  has  been  set  forth  with   righteous
 indignation  by Serge Lang in his publication {\sl The  File:   A
 Case Study in Correction\/}.
 
      That otherwise useful publication {\sl Science Indicators\/}
 from  the National Science Board makes excessive use  of  opinion
 surveys.   The  most  recent report of the series  ({\sl  Science
 Indicators  1980\/}) coupled results from a new,  more  carefully
 constructed  opinion survey with a simple continuation of  poorly
 formulated   questions  taken  from  previous  and  less  careful
 surveys.
 
      The main new opinion survey commissioned for this NSB report
 used  an elaborate design---but this design still  involved  some
 basic  misconceptions  about  science and  some  questions  about
 science so formulated as to distort the opinions which were to be
 surveyed.   For  example,  its  Question 71 first  observes  that
 ``Science  and technology can be directed toward solving problems
 in  many  different  areas"---while I would  claim  that  science
 cannot  be  ``directed"  in the fashion  intended  by  government
 bureaucrats.   The  question  then lists fourteen areas  and asks
 ``Which  three areas on the list would you {\sl most\/}  like  to
 receive  science and technology funding from your tax money?"  Of
 the  fourteen  areas,  some  had little to  do  with  science  or
 technology  and  much  to  do with  the  political  and  economic
 structure   of  society  (for  example,   controlling  pollution,
 reducing crime, and conserving energy).  Only one of the fourteen
 dealt with basic knowledge.  With an unbalanced list of questions
 like this, the report goes on to claim that the answers ``suggest
 that  the  public interest tends to focus on  the  practical  and
 immediate  rather  than  on results that are  remote  from  daily
 life."  This may be so,  but it cannot be demonstrated by answers
 to  a  survey questionnaire which itself is so constructed as  to
 focus on the ``practical and immediate."
 
      To get comparisons of opinions across time,  new surveys try
 to  continue questions which have been used before---and so often
 use older questions of a clearly misleading character.   In  {\sl
 Science  Indicators\/},  a typical such previous question is  the
 hopelessly general one, ``Do you feel that science and technology
 have  changed  life  for the better or the worse?"   The  current
 version  of this question does still more to lead the  respondent
 to a negative answer.   It reads, ``Is future scientific research
 more  likely  to  cause problems than to find  solutions  to  our
 problems?"  It is no wonder that this latter slanted question, in
 the  1979  survey,  had only 60\% answers favorable  to  science,
 while  the  earlier one had 75\% favorable in 1974  and  71\%  in
 1976.
 
      Surveys also may pose questions which the respondents are in
 no position to answer.  For instance, one question in this survey
 probed   the   respondents'   expectations  of   scientific   and
 technological  achievements:   ``During the next 25 years or  so,
 would you say it is very likely,  possible but not too likely, or
 not likely at all that researchers will discover a way to predict
 when  and  where earthquakes will occur?"  How  can  the  general
 public have a useful or informed opinion on this highly technical
 and  speculative question?   The question brought answers of 57\%
 ``very  likely," 34\% ``possible," and 7\% ``not likely."   After
 giving  these figures,  the text obscures the careful  tripartite
 posture  of  the  question  as stated by lumping  the  first  two
 categories together in the following summary: ``About 9 out of 10
 consider it possible or very likely $\ldots$"
 
      The   other  five  questions  asking  for  similar   25-year
 predictions (for example,  a cure for the common forms of cancer)
 are not much better.
 
      In  sum,  the public opinion surveys currently used in  {\sl
 Science  Indicators\/}  are  poorly  constructed  and  carelessly
 reported.  By emphasizing remote and speculative uses of science,
 the  thrust  of the questions misrepresents the  very  nature  of
 scientific  method.   (There  are worse  misrepresentations,  for
 example,  in  a  report  for  GAO  (General  Accounting  Office),
 mistitled  {\sl  Science  Indicators:    Improvements  Needed  in
 Design, Construction, and Interpretation\/}).
 
      To  summarize:   Opinion  surveys may attempt to  reduce  to
 numbers both nebulous opinions and other qualities not easily  so
 reducible.   It  would  be wiser if their use were restricted  to
 those things which are properly numerical.
 
      My  own chief experience with other unhappy attempts to  use
 mathematical ideas where they do not fit comes from studying many
 of  the reports of the National Research Council (in  brief,  the
 NRC).   I  recently  served for eight years as  chairman  of  the
 Report Review Committee for this Council.   This Council operates
 under the auspices of the National Academy of Sciences,  which by
 its  charter  from  the government is  required  to  provide,  on
 request,  advice on questions of science or art.   There are many
 such requests.  Each year, to this end, the NRC publishes several
 hundred  reports,  aimed to apply scientific knowledge to various
 questions of public policy.   Some of these policy questions  are
 hard or even impossible of solution,  so it may not be surprising
 that the desire to get a solution and to make it precise may lead
 to  the use of quantitative methods which do not fit.   This lack
 of fit can be better understood at the hand of some examples.
 \smallskip
 \centerline{\bf VI.  Cost-Benefit and Regression}
 \smallskip
 
      Before  making  a difficult decision,  it may be helpful  to
 list  off the advantages and the disadvantages of  each  possible
 course  of  action,  trying to weigh the one against  the  other.
 Since a purely qualitative weighing of plus against minus may not
 be  objective (or at any rate can't be done on a computer), there
 has grown up a quantitative cost-benefit analysis,  in which both
 the costs and the benefits of the action are reduced to a  common
 unit---to  dollars  or  to  some other  such  ``numeraire."   The
 comparison  of  different  actions and thus  perhaps  a  decision
 between them can then be made in terms of a number,  such as  the
 ratio of cost to benefit.
 
      In  simple cases or for isolated actions this may work well;
 I am told that it did so function in some of its initial uses  in
 decisions about plans for water resources.  However, the types of
 decisions   considered  in   NRC  reports  were  usually  not  so
 straight-forward.   I studied many such reports which did attempt
 to  use cost-benefit  analysis.   In  every  such case which came
 to  my attention  in eight years,  these attempts at quantitative
 cost-benefit analysis were failures.
 
      In  most cases,  these failures could have been anticipated.
 Sometimes  the intended cost-benefit analysis was not  an  actual
 numerical  analysis  but just a pious hope.   For  instance,  one
 study  tried to describe ways to keep clean air  somewhere  ``way
 out west."  In this case, there weren't enough dependable data to
 arrive  at  any numbers for either the costs or the  benefits  of
 that  clean  air.   Hence  the report initially included  a  long
 chapter describing how these costs and benefits {\sl might\/}  be
 calculated---although  it  really seemed more likely  that  there
 never  would  be data good enough to get dependable  numbers  for
 such a calculation.
 
      There  are also cost-benefit calculations which must  factor
 in  the  value of the human lives which might be saved by  making
 (or not making) this or that decision.   In such cases, the value
 ascribed  to one human life can vary by a factor of  10,  ranging
 from  one hundred thousand to one million dollars.   Much of  the
 variation  depends on whether one gets the value of that life  in
 terms  of  discounted  future earnings  or  by  something  called
 implicit  self-valuation  of  future  satisfaction.   However,  I
 strongly  suspect  that  whatever the method,  there  isn't  {\sl
 any\/}  one  number which can adequately represent the  value  of
 human  life for such cost-benefit purposes.   Our lives  and  our
 leisures  are too various and their value (to us or to others) is
 not  monetary.   The  consequence is that  decisions  which  deal
 substantially  with  actions looking to the potential  saving  of
 lives  cannot  be based in any satisfactory way  on  cost-benefit
 analysis.
 
      Another aspect of cost-benefit methods came to my attention
 just yesterday, in the course of a thesis defense.  Cost-benefit
 methods attend only to gross measures, in a strictly utilitarian
 way, and give no real weight to the distribution of benefits (or
 of costs) between individuals.
 
      Another striking example of the problems attending the  use
 of cost-benefit analysis in policy studies is provided by a 1974
 NRC study,  ``Air  Quality  and  Automobile  Emission  Control,"
 prepared for the Committee on Public Works, U.S.  Senate.   That
 committee was considering the imposition of  various  levels  of
 emission controls on automobiles;  it requested  advice  on  the
 merits of such controls, and in particular wanted  a  comparison
 of the costs and the benefits of such control.
 
      Some benefits of the control of automobile emissions are to
 be found in cleaner air and some in better health (less exposure
 to irritating smog).  A number of studies of such health effects
 had  been  done; the   NRC  committee  examined   them  all  and
 considered all but one  of them inadequate.   The  one  adequate
 study was for students in a nursing school  in  the  Los Angeles 
 area.   Each  student  carefully  recorded daily discomforts and
 illnesses; these records were then correlated with the  observed
 level of smog in Los Angeles. The results of this one study were
 then extrapolated  by  the  NRC  committee  to  the whole of the
 United States  in  order  to  estimate  the  health  benefits of
 decreasing smog!  It was never clear to me  why  Los Angeles  is 
 typical  or  how  such  a  wide extrapolation can be dependable.  
 Just as in the case of saving lives, the benefits of good health
 can hardly be reduced to numbers.
 
    Some other difficulties with this particular study concern the
 use of regression,  a  mathematical  topic  with  a  considerable
 history.  Mathematics deals repeatedly with the way in which  one
 quantity  $y$  may depend upon  one or more other quantities $x$.
 When such  a $y$  is an  explicitly  given  function  of $x$, the
 differential calculus has made extraordinarily effective  use  of
 the concept of a derivative $dy/dx = y'$; in the first  instance,
 the  use  of  the  derivative  amounts  to approximating $y$ by a
 linear function, such as $y=ax + c$, choosing $a$  to be  a value
 of the  derivative $y'$.  The number $a$ then is units of $y$ per
 unit of $x$  and  measures the number of units change  in $y$ due 
 (at $x$)  to  a  one-unit  change  in $x$.  For certain  purposes  
 these  linear  approximations work very well, but in other cases,  
 the  calculus  goes on to use higher stages of  approximation ---
 quadratic,  cubic,  and  even  an  infinite  series of successive
 powers of $x$.
 
     But a variable quantity $y$ involved  in a policy question is
 likely to depend not just on one $x$, but on a  whole  string  of
 other quantities $x$, $z$, and so on.  Moreover,  the  fashion of
 this dependence can be quite complex.   One  approximation  is to
 again try to express  $y$ as a constant  $a$  times  $x$  plus  a
 constant $b$ times $z$ and so on---in brief to  express  $y$ as a
 linear function
 $$
 y=ax + bz + \cdots
 $$
 with coefficients $a$, $b$, $\ldots $  which  are  not yet known.
 Given enough data, the famous  method  of  ``least  squares" will
 provide the ``best" values of the constants  $a$,  $b$, $\ldots $
 to make the formula  fit  the  given  data.   In  particular, the
 coefficient $a$ estimates the number of units change in  $y$  per
 unit change in $x$---holding the other  quantities  constant  (if
 one can).
 \midinsert \vskip 3in
 \centerline{\bf Figure 9}
 \endinsert
 
      This process is called a multiple  ``regression" of  $y$  on
 $x$, $z$, $\ldots $.    This  curious  choice  of  a  word has an
 explanation.  It was first used  by  Galton  in  his  studies  of
 inheritance.  He noted that tall fathers had  sons  not  quite so
 tall---thus height had ``regressed on the mean."
 
      This  technique  of  regression  has been amply developed by
 statisticians and others; it is now popular in some  cost-benefit
 analyses.  For example, with the control of  auto  emission,  how
 does one determine the benefit of the resulting clean air?
 
      Clean air cannot be purchased on the market, so the benefits
 of cleaner air might be measured by ``shadow"  prices  found from
 property values, on the grounds that homes in a region where  the
 air is clear should command higher prices than  comparable  homes
 where the air is thick.  In the NRC study,  the prices of  houses
 in various subregions of  greater  Boston  were  noted  and  then
 expressed  as a (linear)  function  of  some  thirteen  different
 measured variables thought to influence these prices:  Clean air,
 proximity  to schools,  good  transportation,  proximity  to  the
 Charles River, and so on. The constants in this linear expression
 of house prices  were  then  determined  by  regression.  In this
 equation,  the  coefficient  $a$  for the  variable  representing
 ``clean air" (of units of dollars per measure of  smog-free  air)
 was then held  to  give  the  ``shadow price" for clean air.  The
 resulting shadow price from this and one  other  such  regression
 was then extrapolated to the whole U.S.A. to give  a  measure  of
 the benefit of cleaner air to be provided  by  the  proposed auto
 emission control.
 \vskip.1cm
 \smallskip
 \hrule
 \smallskip \quad
 In Boston: House \$ = a (Smog) + b (Charles River)
                                + $\cdots$ one dozen more
 \smallskip
 \hrule
 \smallskip
 \vskip.1cm
 
      This is surely a brash attempt to get a number, cost what it
 may.  In my considered judgment, the result is nonsense.   It  is
 not clear that buyers of houses monitor the clean air before they
 sign the mortgage.  A nebulous (or even an airy) quantity said to
 depend on thirteen other  variables  is  not  likely  to  be well
 grasped  by  any  linear  function  of  those  variables.    Some
 variables may have quadratic effects, and there  could  be  cross
 effects  between  different  variables.   That  list  of thirteen
 variables  may  have  duplicates  or  may  very  well  miss  some
 variables which  should  be  there.   Moreover, the  coefficients  
 in that function are likely to be still more uncertain  than  the
 known costs the equation estimates.   These coefficients  are not
 shadow prices;  they are shadowy numbers, not worthy  of  serious
 regard.  They employ a mathematics which does not fit.
 
      The  difficulties  which have been noted in interpreting the
 coefficients  in  some  regressions  are  by  no  means new.  For
 example, you can find them discussed with vigor and clarity in  a
 text   by   Mosteller   and   Tukey,   {\sl   Data  Analysis  and
 Regression\/},  kept  here  on  permanent  reserve in the Eckhart 
 Library.  I trust that such reserve has not kept it from the eyes
 of  economists or other  users  of  regression.  What with canned
 formulas from  other  sources  and fast computers, any big set of
 data  can  be analyzed by regression---but that doesn't guarantee
 that the  results will fit!
 
      I  have  not  studied  the  extensive  academic literature on
 cost-benefit analysis, but these and other  flagrant  examples  of
 the misuse of these analyses in  NCR  reports leave me disquieted.
 Current political dogma may create pressure for more  cost-benefit
 analysis. In Congress, the House is now considering a ``Regulatory
 Reform  Bill"  which  requires  that   independent  and  executive
 agencies of the government  make a  cost-benefit  analysis  before
 issuing  any new regulation (except for those  health  and  safety
 regulations required by law).  It is high time  that  academicians  
 and politicians give  more  serious thought to the limitations  of 
 such methods of analysis.
 
      The future is inscrutable.  However, people  are  curious, so
 fashion usually provides some  method  for  its  scrutiny.   These
 methods may range from consultation with the Oracle  at  Delphi to
 opinion polls to the examination of the entrails of a  sacrificial
 animal.  Now,  thanks  to  the  existence  of fast computers, some
 economists  can  scrutinize  the  future  without  entrailing such
 sacrifice.  The  short-term  predictions by econometric models can
 be sold at high prices, though I am told that some of these models
 deliver more dependable short-term predictions  when  the original
 modeler is at hand to suitably massage the output figures.
 
      At the NRC, my chief contact with  projection  was  on a very
 much longer time scale---econometric  projections  of  the  energy
 future of the United States going forward for fifty years or more.
 This was done in connection with a massive NRC study called CONAES
 (for the Committee on Nuclear and Alternative Energy Systems). For
 this  study,  there  was  not  just  one econometric projection of
 energy needs,  but  a half  dozen  such  models,  with  a  variety 
 of time horizons.  Now projections for a span of  forty  or  fifty 
 years cannot possibly take account of  unexpected  events  such as
 wars, oil cartels,  depressions,  or even the discovery of new oil 
 fields. Since the present differs drastically from the past, there
 is  little  or no hope of checking a fifty-year projection against
 fifty  years  of actual  past  development.    Consequently,  this
 particular NRC study  did not check theory against fact, but  just 
 theory against theory---by asking  just how  much  agreement there
 was between  the half-dozen models.  It hardly  seemed  reasonable
 to me to  conclude that agreement---even a  perfect  agreement ---
 in  the  results   of  several  fictive  models   can  be  of  any 
 predictive  value.    In  the case of the CONAES report, there was
 even a  proposal  to  use the thirty-five-year projection of those
 models to assess the future economic value of the breeder reactor. 
 Such  assessment breeds total futility.  All told, despite the use
 of  fast  computers  and  multiple models,  the ambiguities of the 
 models being computed still leave the future dark and inscrutable.
 
      Projections over time into an unknown future are not the only
 examples of policy-promoted projection of the unknown.  Many other
 types  of   extrapolation   can   be   stimulated---for   example,
 extrapolation designed to estimate risks.   Since  it  is  claimed
 society has become more risk-averse, there is great demand to make
 studies of future risks, as in the reports of the NRC Committee on
 the Biological Effects of Ionizing  Radiation  (BEIR  for  short).
 The third report of this committee,  a report  commonly  known  as
 ``BEIR III," dealt with extrapolation, another kind of projection.
  Data available from Hiroshima and  Nagasaki  give  the numbers of
 cancers caused by high dosages of radiation.  For present puropses
 one wants rather the effect of  low  doses,  on  which  there  are
 little or no data.  To estimate this effect, one may  assume  that
 the effect $E$ is proportional to dosage $D$---so that $E=kD$  for
 some constant $k$.  Alternatively, on may assume that  the  effect
 is quadratic so that $E$ depends both on $D$ and  on  $D^2$.  Then
 the curve giving $E$ as a  function of  $D$ is  parabolic  (Figure
 10).  The constants involved---such as  the proportionality factor
 $k$---are then  chosen  to get  the  best  fit  of the line or the
 parabola to the high dosage data.  The resulting  formula  is then
 used to calculate the effect at low dosage.  Quite naturally,  the
 linear  formula and the quadratic one give substantially different
 results by this extrapolation; this is the cause  of  considerable
 controversy.  Is the  linear formula right?   Does  the  choice of 
 formula  depend  on  the  type  of cancer considered?  There is no
 secure and scientific answer to these  pressing  policy questions. 
 In particular, the mathematical methods themselves cannot possibly
 produce  an  answer.  Mathematical  models  such as these  may  be 
 internally consistent, but  that  doesn't imply that they must fit
 the facts.  Here, as in the case  of  regression,  the  assumption 
 that the variables of interest are connected by a  linear equation
 is gratuitous and misleading.
 \midinsert \vskip4in
 \centerline{\bf Figure 10}
 \endinsert
 
      That BEIR III report deals with  just  one  of many different
 kinds of risks that plague mankind.  There  are  many  others that
 might be estimated, by extrapolation or otherwise.  From all these
 cases there has arisen some hope that  there  might  be  effective
 general principles underlying such cases--and  so  constituting  a
 general subject of ``risk analysis."  The  hope  to  get  at  such
 generality   may   resemble   the  process  of  generalization  so
 successful  in  mathematics,  where  properties  of  numbers  have 
 been  widely  extended  to form the subject of number  theory  and 
 properties  of  specific  groups have led to general group theory. 
 However,  I  am doubtful  that there can yet be a generalized such 
 ``risk analysis" --- and this I judge  from  another  current  NRC
 report.
 
      This report arose as follows:  The  various  public  concerns
 about risks were reflected in Congress,  so  a  committee  of  the
 Congress  instructed  the  National  Science  Foundation  (NSF) to
 establish  a  program  supporting  research on risk analysis.  The
 NSF,  in its turn,  did not know how to go about choosing projects
 in  such  a  speculative field---so it asked the National Research
 Council for advice on how to do this.  The NRC, again in turn, set 
 up a committee of experts on risk analysis. This committee, in its
 turn,  prepared  a  descriptive  report  on   risk  analysis  ``in
 general."  The report also commented  on  specific  cases  of risk 
 analysis.  For example,  there were extensive comments on the BEIR
 III report---but  these  comments  did  not  illuminate  the  BEIR 
 III  problem  of  extrapolation  and   made   no   other  specific
 suggestions.  The  report had little of positive value to help the
 NSF decide which projects in risk analysis to fund. Such a general
 study of  risk  analysis  is clearly interdisciplinary, but I must
 conclude that it is not yet disciplined.
 
      These and other examples of unsatisfactory reports may  serve
 to illustrate the confusion resulting  from  questionable  uses of
 quantitative methods or of mathematical models.  But why are there
 so many cases of such confusion?  Perhaps the troubled history  of
 that report on risk analysis  is  typical.   A  practical  problem
 appears; many people are concerned, and so is the Congress or  the
 Administration. Since the problem is intractable, but does involve
 some science, it is passed on to the scientists, perhaps to  those
 at the NRC.  Some of these problems  can be---and are---adequately
 treated.  For others there is not yet any adequate technique---and
 so those techniques which happen to be available (opinion surveys,
 cost-benefit  analysis,  regression,  projection,   extrapolation,
 decision analysis, and others) get applied to contexts where  they
 do not fit.  Confusion arises when the wrong idea is used, whether
 for political reasons or otherwise.
 
      There  are also  political  reasons for  such confusion.  Our
 representatives, meeting  in  that  exclusively  political city of
 Washington, represent a variety of sharply different interests and
 constituencies.  To get  something  done,  a  compromise  must  be
 struck.  This happens in many ways.  One which I have seen, to  my
 sorrow,  is  the  adjustment  of  the  onerous  and   bureaucratic
 regulations of the OMB (Office of  Management  and  Budget)  about
 cost  principles  for  universities.   Their  Circular  A-21   now
 requires faculty members to report the percentage distribution  of
 their various university activities, with results  to  add  up  to
 100\%, on a ``Personnel Activity Report Form" (PAR!). Such numbers
 are meaningless; they are fictions fostered by accountants. Use of
 such numbers makes for extra  paperwork---but  it  also  tends  to
 relocate some control of scientific research from universities  to
 the government bureaucrats.  For  A-21,  there was recently a vast
 attempt at improvement,  combining  all  parties:  the  government
 bureaucrats, their accountants, university financial officers, and
 a  few  faculty.   What  resuted?   A  compromise,  and not a very
 brilliant one.
 
      Thus government policy, when it requires scientific advice on
 matters that are intrinsically uncertain, is likely to  fall  into
 the government mold: compromise.  And that, I believe, is a source
 of confusion.
 \smallskip
 \centerline{\bf VIII.  Fuzzy Sets and Fuzzy Thoughts}
 \smallskip
 
     The misuse of numbers and equations to project  the future  or
 to extrapolate  risks  is  by  no  means  limited  to the National
 Research Council.  Within the academic community itself there  can
 be similar fads and fancies.  Recently I have been reminded of one
 curious such case:  The doctrine of ``fuzzy" sets.
 
      How can a set be fuzzy?  Recall that a set $S$ is  completely
 determined  by  knowing  what  things  $x$  belong to $S$ (thus $x
 \,  \epsilon  \,  S$)  and  what  things  do  not  so belong.  But
 sometimes,  it  is  said,  one  may  not know whether or not $x \, 
 \epsilon \, S$.   So  for  a  fuzzy  set  $F$  one  knows only the
 likelihood (call it  $\lambda (x)$)  that  the  thing  $x$  is  in 
 the fuzzy set $F$.  This  measure  of  likelihood may range from 0
 ($x$ is certainly not in $F$)  all  the way to 1 ($x$ is certainly
 in $F$).  Now  I  might  have  said  that  $\lambda  (x)$  is  the
 probability that $x$ is in $F$, to make  this definition a part of
 the  well-established  mathematical  theory  of  probability.  The
 proponents  do  not  so  formulate  it, because their intention is
 different  and  much more ambitious:   Replace  sets everywhere by
 fuzzy sets!
 
    By the grand set-theoretic doctrine, every mathematical concept
 can be defined in terms of sets, hence this  replacement  is  very
 extensive.  It even turns out that many mathematical concepts  can
 be fuzzed up in several ways, say, by varying the fuzzy meaning to
 be    attached    to   the   standard   set-theoretic   operations
 (intersection, union, etc.)  of the usual Boolean algebra of sets.  
 And  so  this  replacement  doctrine  has   already   produced   a
 considerable literature:  on  fuzzy  logic,  fuzzy  graphs,  fuzzy 
 pattern  recognition, fuzzy systems theory, and the like.  Much of 
 this  work  carries  large  claims for  applications of this fuzzy
 theory.   In  those  cases  which  I  have  studied,  none  of the
 applications  seem  to  be  real; they  do not answer any standing
 problems  or  provide  any  new  techniques for specific practical
 situations.   For  example, one  recent  book  is   entitled  {\sl
 Applications  of Fuzzy Sets to  Systems  Analysis\/}.  The  actual 
 content  of  the  book  is a sequence of formal fuzzy restatements
 of standard mathematical formulations of materials on programming,
 automata,  algorithms,  and  (even!)  categories,  but there is no
 example of  specific  use of such fuzzy restatement.  One reviewer
 (in {\sl  Mathematical Reviews\/}) noted a ``minimal use  or  lack 
 of   instructive  examples---the  title   of  the   book  purports
 applications."  Another more recent book on fuzzy decision  theory
 states as one of  its six conclusions, ``It  is  a great pity that
 there  exist  only  very  few  practical  applications   of  fuzzy
 decision theories, and even practical  examples  to illustrate the
 theories are scarce."  This leads me to suspect that the initially
 ingenious  idea  of  a  fuzzy  set  has  been  overdeveloped  in a
 confusing  outpouring  of  words  coupled  with spurious claims to 
 importance.
 
      There are other examples---cybernetics, catastrophe theory---
 where  an  originally  ingenious  new  idea  has   been   expanded
 uncritically to lead to meaningless confusion.
 \smallskip
 \centerline{\bf IX.  Compromise Is Confusing}
 \smallskip
 
      But enough  of  such troubling examples of confusion.  Let me
 summarize where we have come.  As with any branch of learning, the
 real substance of mathematics resides in the ideas.  The ideas  of
 mathematics are those which can be formalized and which have  been
 developed  to  fit issues arising in science or in human activity.
 Truth in mathematics is approached by way of proof  in  formalized
 systems.   However,  because  of  the  paradoxical  kinds of self-
 reference exhibited by the barn door and Kurt G\"odel,  there  can
 be no  single formal system which subsumes all mathematical proof.
 To  boot,  the  older  dogmas  that  ``everything  is  logic"   or
 ``everything is a set"  now  have  competition--``everything  is a
 function."  However, such questions of foundation are  but  a very
 small part of mathematical activity,  which  continues  to  try to
 combine the right ideas to attack substantive problems.   Of these
 I have touched on only a few examples:  Finding all simple groups,
 putting groups together by extension, and  characterizing  spheres
 by their connectivity.  In such cases,  subtle  ideas,  fitted  by
 hand to the problem, can lead to triumph.
 
      Numerical and mathematical methods can be used for  practical
 problems.  However, because of political pressures, the desire for
 compromise, or the simple  desire  for  more  publication,  formal
 ideas may be applied in practical cases where the ideas simply  do
 not  fit.   Then  confusion  arises --- whether  from   misleading
 formulation  of  questions  in  opinion  surveys,   from  nebulous 
 calculations  of  airy  benefits, by regression, by extrapolation,
 or otherwise.  As the case of fuzzy sets indicates, such confusion
 is  not  fundamentally  a  trouble  caused  by  the  organizations
 issuing reports, but is occasioned by academicians making careless
 use of good ideas where they do not fit.
 
      As Francis Bacon once said, ``Truth ariseth more readily from
 error  than  from  confusion."   There  remains  to  us, then, the
 pursuit  of  truth,  by  way  of proof, the concatenation of those
 ideas which fit, and the beauty which results when they do fit.
 \vskip.2cm
 
 \midinsert
 
       If only Longfellow were here to do justice to the situation:
 \smallskip
 \centerline{\bf Tell Me Not in Fuzzy Numbers}
 \smallskip
 \centerline{In the time of Ronald Reagan}
 \centerline{Calculations reigned supreme}
 \centerline{With a quantitative measure}
 \centerline{Of each qualitative dream}
 \smallskip
 \centerline{With opinion polls, regressions}
 \centerline{No nuances can be lost}
 \centerline{As we calculate those numbers}
 \centerline{For each benefit and cost}
 \smallskip
 \centerline{Though his budget will not balance}
 \centerline{You must keep percents of time}
 \centerline{If they won't sum to one hundred}
 \centerline{He will disallow each dime.}
 \endinsert
 \vfill\eject
 \centerline{\bf References}
 \smallskip
 \ref ``Air Quality and Automobile Emission Control."  A report
 prepared by the Commission on Natural Resources, National Academy
 of Sciences for the Committee on Public Works, U. S. Senate.
 September 1974.  Volume 4.  {\sl The Costs and Benefits of
 Automobile Emission Control\/}.
 
 \ref Eilenberg, S. and Mac Lane, S.  ``Group Extensions and Homology,"
 {\sl Annals of Math.\/} 43 (1941), 758-831.
 
 \ref Eilenberg, S. and Mac Lane, S.  ``General Theory of Natural
 Equivalences" (category theory), {\sl Trans. Am. Math. Soc.\/}
 vol. 28 (1945), 231-294.
 
 \ref Energy Modeling for an Uncertain Future:  Study of Nuclear and
 Alternative Energy Systems.  A series:  Supporting Paper \#1,
 225pp. National Academy Press, 1978.
 
 \ref Feit, Walter and Thompson, John G. ``Solvability of Groups
 of Odd Order," {\sl Pacific J. Math.\/} 13 (1968), 775-1029.
 
 \ref Kickert, Walter J. M.  Fuzzy Theories in Decision-making:
 A Critical Review.  Martinus Nijhoff, Social Sciences Division,
 Leyden, 1978.  182 pp.  Reviewed in {\sl Mathematical Reviews\/}
 vol. 81f (June 1981), \#90006.
 
 \ref Ladd, Everett C. and Lipset, Seymour M.  The 1977 Survey of the
 American Professoriate.
 
 \ref Lang, Serge.  The File:  A Case Study in Correction.  712 pp.
 Springer Verlag, New York, 1981.
 
 \ref Lawvere, R. W. ``Toposes, Algebraic Geometry, and Logic,"
 {\sl Springer Lecture Notes in Math.\/} No. 274 (1972).
 
 \ref Mosteller, F. and Tukey, John W. Data Analysis and Regression.
 586 pp. Addison-Wesley, Reading, Mass. (1977).
 
 \ref Negoita, C. V. and Ralescu, D. A.  Applications of Fuzzy Sets
 to Systems Analysis, 1975.  John Wiley and Sons, New York.  191pp.
 Reviewed in {\sl Mathematical Reviews\/} vol. 58 (1979),
 \#9442a.
 
 \ref Science Indicators:  Improvements Needed in Design, Construction,
 and Interpretation.  Report by the Comptroller General of the
 United States.  PAD 79-35, September 25, 1979.
 
 \ref Whitehead, A. N. and Russell, Bertrand.  {\sl Principia
 Mathematica\/}, vol.1, 2nd edition.  Cambridge University Press,
 1925.  674 pp.
 \vfill\eject
 The Ryerson Lecture was given April 20, 1982 in the Glen A. Lloyd
 Auditorium of the Laird Bell Law Quadrangle.
 \smallskip
 \smallskip
 {\nn  The Nora and Edward Ryerson Lectures were established by the
 trustees of the University in December 1972.  They are intended to
 give a  member of the faculty the opportunity each year to lecture
 to an  audience from the entire University on a significant aspect
 of his or her research and study.  The president of the University
 appoints the lecturer on the recommendation of a faculty committee
 which  solicits  individual  nominations  from  each member of the
 faculty  during the winter quarter preceding the academic year for
 which the appointment is made.}
 \smallskip
 {\sl The Ryerson Lecturers have been:}
 \smallskip
 \line{1973-74: 
       John Hope Franklin, ``The Historian and Public Policy" \hfil}
 \line{1974-75: 
       S. Chandrasekhar, ``Shakespeare, Newton, and Beethoven: \hfil}
 \line{\qquad \qquad Patterns of Creativity" \hfil}
 \line{1975-76:  
       Philip B. Kurland, ``The Private I: Some Reflections on \hfil}
 \line{\qquad \qquad Privacy and the Constitution" \hfil}
 \line{1976-77:  
       Robert E. Streeter, ``WASPs and Other Endangered Species" \hfil}
 \line{1977-78: 
       Dr. Albert Dorfman, ``Answers Without Questions and \hfil}
 \line{\qquad \qquad Questions Without Answers" \hfil}
 \line{1978-79: 
       Stephen Toulmin, ``The Inwardness of Mental Life" \hfil}
 \line{1979-80: 
       Erica Reiner,  ``Thirty Pieces of Silver" \hfil}
 \line{1980-81: 
       James M. Gustafson, ``Say Something Theological!" \hfil}
 \vfill\eject
 \noindent{\bf ARTICLE}
 \vskip.5cm
 \centerline{\bf DIGITAL MAPS AND DATA BASES:}
 \centerline{\bf AESTHETICS VERSUS ACCURACY $^{\bf \star }$}
 \vskip.5cm
 \centerline{Robert F. Austin, Ph.D.}
 \centerline{President}
 \centerline{Austin Communications Education Services}
 \centerline{28 Booth Boulevard}
 \centerline{Safety Harbor, FL  34695-5242}
 \centerline{Past-president, AM-FM International}
 \smallskip
 \centerline{\bf I. Introduction}
 
 Of the many courses lectured by Immanuel Kant  at  the  University  of
 K\"onigsberg, legend has it that one of the  most  frequently  offered
 was a course on natural  philosophy  (that  is,  physical  geography).   
 It was argued  that individuals  could  acquire understanding  through 
 three  distinct  perspectives:    the  perspective of formal logic and
 mathematics, the perspective of time (history),  and  the  perspective  
 of  space (geography).  The last of these --- the perspective of space
 --- acknowledges the importance of distance, site characteristics, and
 relative  location  in  describing  the  relationships  among  several
 objects or facilities.
 
 Maps are the primary means of representing such relationships.    Maps
 are  analytical  tools  which depict spatial relationships and portray
 objects from the perspective of space.   The power of  maps  rests  on
 their synoptic representation of complex phenomena.  To paraphrase the
 Confucian wisdom, a map is worth a thousand words.
 
 {\narrower\smallskip\noindent
       ``Recently  it  has become common to convert spatial phenomena
        to digital form and store the data on tapes or discs.   These
        data  can then be manipulated by a computer to supply answers
        to  questions that formerly required  a  drawn  map $\ldots $
        This  stored  geographic  information  is  referred  to  as a
        [data base]." [1] \smallskip}
 
 The  maps  produced  from  such  data  bases  are termed digital maps.
 Computerized data bases, which may  be  queried and  used  by  several
 people  simultaneously,  and  digital  maps  are  of immense value  to 
 engineers, comptrollers, planners,  and managers.   The combination of
 a digital map and data base is worth a thousand ``mega-words."
 
 The  advantages  of  digital  maps over manually drafted maps are most
 apparent in situations of  frequent  growth  or  change.   Among these
 advantages  are  the  ease  and speed of revision and  the  fact  that
 special  purpose  maps  can  be produced in small volume at reasonable
 cost. Moreover, digital maps offer greater precision in representation
 and  analysis.   ``As  more  governmental bodies  [and other agencies]
 expend the necessary one-time capital investment,  and begin  to  reap
 the vast rewards of computer-assisted record and  map keeping,  others
 are likely to follow quickly."[2]
 \smallskip
 \centerline{\bf II. Basic Issues}
 
 After  an  agency or firm has decided to convert its manual records to
 digital map and data base form, several issues must be addressed.
 
 The  first,  and  most  important  question  the agency must answer is
 related  to  the  source  documents to be used by the mapping firm. In
 general  terms  the  choice  is  between   cartographic   sources  and
 mechanically drafted sources.
 
 Cartography is defined as:
 
 {\narrower\smallskip\noindent
       ``The  art,  science  and technology of making maps,  together
        with their study as scientific documents and  works  of  art.
        In  this  context maps may be regarded as including all types
        of  maps,  plans,  charts,  and  sections,  three-dimensional
        models  and  globes  representing  the Earth or any celestial
        body at any scale."[3] \smallskip}
 
 In  particular,   cartography  is  concerned  with  the  accurate  and
 consistent  depiction  on  a flat surface of activities occurring on a
 sphere.
 
 It  is  not possible to duplicate, without distortion, the features on
 the  surface of a sphere on any object other than a sphere.  A surface
 of constant positive curvature may be represented on a surface of zero
 curvature only if distortion is introduced in the representation.   As 
 a  simple  illustration  of  this  fact,   consider   the  problem  of
 ``flattening" an orange peel: it will tear.  If the orange was made of
 rubber,  it would be possible to flatten it without tearing,  but  not
 without distortion of another kind --- a topological transformation.
 
 The  methods by which cartographers represent the surface of the earth
 on  a  flat  piece  of  paper  are known as map projections.   For any
 particular  purpose,   the  selection   of  a   particular  projection
 (transformation) is based on the  properties  of  a  sphere  that  the
 projection  loses or retains.   Every method of mapping large areas is
 affected,  whether it is continuous mapping  or  facet  mapping.    No
 coherent,   distortion-free  transformation  exists,  nor,  given  the
 theorems  of  mathematics,  can  it  ever  exist.  [4,  5]    However,
 cartographers can identify projections that suit a client's particular
 purpose.
 
 Quite often appropriate  cartographic  source  documents  already  are
 available  to  a public utility and mapping firm team.   Indeed,  such
 sources may have served as the base for the construction  of  existing
 records.   In other cases, it may be necessary for the mapping firm to
 perform  an  aerial  photographic  survey  and  to   translate   these
 photographs   into  cartographic  documents  ---  a process  known  as
 photogrammetry.
 
 It is also possible to produce maps from non-cartographic sources such
 as tax assessor sheets.   Certainly the most  common  non-cartographic
 sources  are  mechanically  drafted  cadastral  maps  and  engineering
 drawings  or  plans.   These  documents  have  been  defined   by  the
 International Association of Assessing Officers:
 
 {\narrower\smallskip\noindent
             map,  cadastral  ---  A  map  showing  the boundaries of
             subdivisions  of land,  usually  with  the  bearings and
             lengths  thereof  and  the  areas  of individual tracts, 
             for purposes of describing and recording ownership.
             \smallskip}
 \smallskip
 
 {\narrower\smallskip\noindent
             map, engineering --- A  map  showing information that is
             essential  for   planning  an  engineering   project  or
             development and for estimating its cost.  An engineering
             map  is  usually  a  large-scale  map of a comparatively
             small area or of a route. [6]
             \smallskip}
 
 Although  such  drawings  have  some  value  for  small  area  design,
 engineering,  and planning purposes,  there are  a number  of problems
 associated with their use as source documents for large area  mapping.
 The most critical of these is related to accuracy; tax assessor sheets
 in the United States,  for example, are designed to be used as indices
 only and are subordinate to  actual  legal  descriptions.    They  are
 highly  stylized  and,  despite  their  appearance  and  name,  highly
 inaccurate in terms of geographic placement.
 
 Small plans ``look" correct primarily because they correspond  to  the
 limited  range  of  vision of human beings at ground level.   However,
 these  documents  also  suffer  from   the   transformation   problem.
 Non-uniform, interpretive, subjective  corrections by a draftsman make
 this  problem  appear  to  vanish  on  individual  sheets.   But  such
 corrections preclude accurately merging sheets for a large area.
 
 In the language of the philosophy of science,  the distinction is  one
 between  an  iconic model and a symbolic model.   An iconic model (the
 mechanically drafted plan) is designed to look,  in some  metaphorical
 fashion, like the object of study. Often, the closer the similarity in
 appearance,  the  less  valuable the model for analytical purposes.  A
 symbolic  model  (the  map)  is  designed  to  facilitate quantitative
 measurements of characteristics of interest to analysts, managers, and
 engineers.
 
 A second issue that must be considered by public utilities is the  use
 to  which the digital maps will be put.   This will determine the type
 of output the mapping firm will generate.   This also  will  determine
 the accuracy levels needed. [7]  In general terms, the types of output
 products  correspond  to  the types of input products: maps and plans.
 [8]    In  our  experience,  public  utility  clients  generally  have
 expressed  a  preference  for  digital   maps   related   to   spatial
 relationship  data  bases  because  they  facilitate the more accurate
 analysis  of  physical  plant  attributes  and  distributions  over  a
 large area in a geographic information system.
 
 The primary purpose of most digital map and data base conversion  work
 is  to  provide a means to manage corporate assets.   Often the actual
 maps produced are used for  index  only,  not  for  scaled  or  direct
 measurement.  This is in part a function of the distortion inherent to
 any mechanical production or reproduction process,  in part a function
 of the  demonstrated  superiority  of  a  fully  digital,  displayable
 linked-attribute  data base management system (see Section 5),  and in
 part a function of the distortion inherent to all map projections (the
 transformation problem previously discussed.)
 
 In some cases,  an agency may wish to construct a geographic data base
 that  will  support  a  computerized  plan  generation  and facilities
 management system.   As an example,  consider the case where  facility
 data  will  be superimposed on a merged cadastral and land base.   The
 data base must guarantee the  geographic  locations  of  features  and
 their  connectivity,  relationships,  and other characteristics.   The
 final digital plan and data base may include  information  on  street,
 road,  and highway names,  centerlines, and rights-of-ways; political,
 legal,  and natural boundaries;  township,  range,  and section lines;
 river,  stream,  and  creek  centerlines  and names; and legal lot and
 parcel lines and numbers, among other data. [9]
 \smallskip
 \centerline{\bf III. Map Production}
 
 Regardless  of the type of source document or output product,  several
 stages in cartographic production remain relatively constant.    These
 are  considered  first  in  a general manner and then as they apply to
 digital map production per se.
 
 First,  we must define the purpose and accuracy standards of the  map.
 For example, will the map be used for scaled measurement?  Or will the
 map  be used as an index?   Second,  we must identify the features and
 activities to be mapped.   The nature of these features will influence
 the  amount  of  detail  appropriate for the base map and the finished
 map.   The strength of a map may be diminished by displaying too  much
 detail.
 
 Third,  we  must  prepare or obtain a land base or base map.   In this
 regard, it is important to consider the variety of map projections and
 coordinate  systems  available  for  particular tasks.  Using a widely
 accepted  system  such  as  the  UTM grid or  latitude  and  longitude
 coordinates  has  a  number  of  advantages,  including  ease  of data
 exchange and reduced production time and cost.
 
 The next step is to collect and compile the data to be  mapped.    The
 basic  rule  is  to  compile  data  at  the  most  detailed  level  of
 measurement  possible  and  to  aggregate   the  data  only  at  later 
 analytical  stages.   Finally,  we  must design and construct the map. 
 This  is  a  two  step  process  that  involves:  (a)  the  design  of
 symbols,  patterns,  legends, and other cartographic devices,  and (b)
 the location and actual placement of the features and activities.
 \smallskip
 \centerline{\bf IV. Digital Maps }
 
 As in traditional cartography the first step in constructing a digital
 map  is to establish accuracy levels and to determine which attributes
 should be displayed and which should simply be stored.
 
 A  displayable  linked-attribute  data  base  system (discussed below)
 allows for the  construction of a fully digital geographic information
 system for data management,  as well as for the construction of  index
 and general route maps. For such maps, placing items of plant ``on the
 right side of the street" generally is adequate: in the real world,  a
 utility pole 60 feet tall is clearly visible at an intersection.   The
 critical  attributes  of each item of plant appear as numbers or words
 on the map and also as  manipulable  information in the data base.  On
 the other hand,  if the map  is  to  be  used  for scaled measurement,
 accurate placement of items of plant is paramount.  It should be noted
 that this second approach implies  considerable  supplementary  manual
 adjustment and therefore substantially higher production costs.
 The next question is the method of land base construction. Land bases,  
 or  base  maps,  may  be  constructed in a variety of ways.  It may be 
 possible  to  develop an accurate land  base  by  digitizing  existing 
 source documents.  The information may be captured by board digitizing
 vectors  (line strings and endpoints) or by raster scanning.   If  the
 quality  of the source documents is high,  these methods are extremely
 cost effective.
 
 If the quality  of  the  available  source  documents  is  unknown  or
 suspect,  it is common to conduct an aerial photographic survey and to
 compile the photographs into a ``model" of the region (the air  photos
 are  rectified  to  generate  a  plane view of the photographed region
 similar to a map projection).  These models are then  stereo digitized
 and  used   as  highly   accurate   source  documents  for  land  base  
 construction  using  the  digitizing  or scanning methods noted above.  
 (+ 10 feet accuracy is standard,  but + 1 foot accuracy is  possible.)   
 Although  more  expensive than working from existing source documents, 
 such  an  approach  guarantees  extreme  accuracy.   Moreover, the end
 product  often  has  resale  value  which will offset the initial cost
 incurred by the end user.
 
 Data  sources  also  may  be combined to form a hybrid land base.  For
 example, individual assessor sheets at a wide variety of scales can be
 overlaid  on  a  stereo  digitized  base.  The stereo digitized street
 centerline network can be modified to agree with cadastral maps so the
 assessor data will scale properly and satisfy aesthetic criteria.
 
 Because such a hybrid is,  by definition, unique, it is appropriate to
 discuss in detail at the outset of the project the problems likely  to
 be  encountered  in  production.   Disadvantages  of  such an approach
 include the substantial cost to make the map ``look"  like the  source
 documents, many difficult production problems (e.g., warp of cadastral
 data and fitting cadastral information to an accurate land base),  and
 the volume of source documents required.
 
 Nevertheless,  some utilities wish to use a hybrid approach because it
 gives  them  a product that is internally acceptable from an aesthetic
 viewpoint.   We have encountered many situations where end  users  are
 uncomfortable  with  the computer-plotted version of a geographic data
 base because it does not ``look" like the product they have  used  for
 many years.  The issue of user acceptance is of critical importance to
 the  ultimate success of a conversion project.   The cost and problems
 associated  with  a  hybrid  approach may be justified in the long run
 because the end user is comfortable with the ``look" of  the  product.
 However,  care  must  be  taken to avoid simply computerizing existing
 problems.
 
 The process of generating a hybrid combines land base construction and
 data compilation.   When a data base  management  system  approach  is
 used,  the  distinction between these two production processes is much
 clearer.  After a land base is assembled, a decision is made to define
 some  selection  of  ``attributes"  of facilities or items of plant as
 displayable. Displayable attributes are those attributes that actually
 will be plotted on a map.   Other attributes may be stored in the data
 base,  but  not,  as  a  matter  of  course,  be  displayed  on  maps.
 (Information  of  interest  to  comptrollers  may  be of little use to
 engineers.   Conversely,   attributes   that   facilitate  engineering
 procedures may be unimportant to comptrollers.)
 
 Once agreement on the attributes to be displayed is reached,  the data
 are coded and laid out on the source document. Some of the information
 will  be  recorded interactively at a board digitizing station.  Other
 information  will  be  keypunched  and bulk loaded at the construction
 phase.   After   construction,    updating   normally   is   performed
 interactively.   One  significant  advantage of a data base management
 system approach is  its ability  to  grow  or  change  with  technical
 advances.
 
 {\narrower\smallskip\noindent
       ``Maps  today  are  strongly  functional  in  that  they   are
        designed,  like  a  bridge or a house,  for a purpose.  Their
        primary purpose is to convey information or to `get across' a
        geographical  concept or relationship $\ldots $  The mapmaker
        is  essentially  a  faithful  recorder  of  given  facts, and
        geographical integrity cannot be  compromised  to  any  great 
        degree.  Nevertheless,  the   range   of  creativity  through 
        scale,  generalization,  and  graphic manipulation  available 
        to the cartographer is comparatively great."[10]
        \smallskip}
 
 Given its pragmatic character,  it may be surprising to learn that the
 physical appearance of a map is a common point of disagreement.   Most
 frequently  such  disagreement  arises   because   different  sets  of
 aesthetic principles have been applied by  the  client and the mapping
 firm.
 
 The general question of aesthetics is not at all simple;  as  the  art
 historian  Ivins  has  argued,   the  notion   of   simple   geometric
 relationships is not invariant in aesthetic assessments.  [11] Indeed,
 aesthetic  issues  are  involved  in  both   the   creation   and  the
 appreciation or perception of a work of art.  Within cartography,  the 
 term ``aesthetics" is reserved for consideration of  the  placement of
 elements such as compass rose,  legends,  and scales;  the  balance of
 these elements vis-a-vis the map object;  the selection  of type faces
 from the range  of  standard  or customary fonts; and similar elements  
 of visual display.  To equate ``correct"  with cartographic aesthetics
 is inappropriate, except in the limited sense that some  features  are 
 required  by cartographic convention  (e.g.,  italic  type  fonts  for
 bodies  of  water).   Styles  as to what is ``correct" from a creative
 standpoint  also  vary  from  one academic discipline to another:  the
 geographer prefers a fine-line drawing  while  the  urban planner uses
 heavy  lines  to  focus  attention and the landscape architect employs
 pictorial  symbols.   Differences  in  styles  of map creation help to
 condition the manner in which maps are appreciated  by consumers; that
 is, aesthetics is more  than  simply  giving  the consumer  what he is
 used to.  For example, many public utilities have developed a sense of
 aesthetics  conditioned   by   experience   with   manually   drafted,
 subjective,  and  highly  symbolic  plans.   Unless  discussed  at the
 beginning  of  a  conversion  project  this  point  can  become   most
 difficult,  because a mapping  firm  must  assume  that  map  accuracy 
 takes  precedence  over  map  symbology,  visual  appearance,  and the
 superficial aesthetics of the perception of appearance. Often in cases
 of this  sort,  the  aesthetics  of  appreciation of the consumer give
 way to concerns of accuracy on the part of the mapping  firm  which in
 turn may rest on the aesthetics of cartographic creation.  
 
 \centerline{\bf V. Computerized Data Bases}
 
 {\narrower\smallskip\noindent
        ``Computer  systems  are  increasingly  used  to aid in the
         management of  information,  and  as a result,  new  kinds
         of data-oriented  software  and  hardware  are  needed  to 
         enhance  the  ability  of  the  computer to carry out this
         task.   [Data base  systems are]  computer systems devoted
         to  the  management  of relatively  persistent data.   The
         computer software employed in a data base system is called
         a data base management system (DBMS)."  [12]
         \smallskip}
 
 Of  the  several  methods  of  classifying data bases used by software
 engineers, one  most  important  dichotomy  is  that  between  network
 (hierarchical) and relational data bases.   [13]  Certainly  the  most
 useful  approach  for many users is the relational data base,  because
 this approach permits a larger variety of queries.   Regardless of the
 approach,  the method of manipulating the data base remains a critical
 issue.
 
 As  noted  in  the quotation of Blasgen [12], the software used with a 
 data  base  is  known  generically  as  a  data base management system
 (DBMS).  Such  a  system  generally  will  have  provisions  for  data 
 structure  definition as well as for data base creation,  maintenance,
 query, and  verification.  Blasgen  observed that in 1981 an estimated
 50 companies were marketing 54 different DBMS packages.  [14]
 
 In our experience,  as noted  earlier,  most  public  utility  clients
 prefer working with a ``displayable linked-attribute" DBMS.  This term
 describes a system in which selected attributes or characteristics  of
 the  company's physical plant are stored in the data base,  where they
 may be manipulated by users and also displayed  on  the  digital  map.
 For example,  the age,  length,  size, and identification number for a
 piece of cable may be stored and displayed.   Because  the  length  of
 cable  in  a  given  span is known from installation and stored in the
 data base, scaled measurement is unnecessary.
 
 A second approach to building and maintaining a data  base  is  termed
 the ``hybrid" approach.  Consider the following example.
 
 Analytics  technicians  select  National Geodetic Survey monuments and
 photo identifiable points which provide a network for accurate control
 of the area.   On-site field  survey  crews  accurately  survey  these
 points  and  target  them  for aerial photography.   After the flight,
 analytics technicians assemble these points into an  accurate  control
 network  and  place  them in a digital file.   Using the network file,
 features defined in contract specifications are stereo digitized  from
 the photography in a digital format.  The major features are portrayed
 in  the  form of a detailed centerline network of interstate highways,
 public roads and private roads.   The file then is divided into facets
 (corresponding to individual maps) and plotted at a scale of 1:100'.
 
 Tax  assessor  sheets and the 1:100' stereo digitized centerline plots
 are joined at the next production phase.   The 1:100' plot is overlaid
 on individual assessor sheets, intersections are held for control, and
 the  stereo digitized road centerline network is adjusted to match the
 cadastral maps.   The revised centerlines and additional features such
 as  rights-of-ways,   lot  lines,   boundaries,  and  text  are  board
 digitized.
 
 The product is the best of the digital and mechanical worlds.   It  is
 accomplished  in  digital  format  so future modifications can be made
 easily,  and  it  is  aesthetically  pleasing  ---  it  looks  like an
 engineering  drawing  or cadastral map.   The information is accurate, 
 the data can be scaled, and bearings can be extracted.
 
 The success of endeavors of this type depends on an  excellent  vendor
 and client relationship.  For such a project, it is recommended that a
 test or pilot study in a pre-selected area be completed prior to final
 contract  negotiations.    It  is  the responsibility of the client to
 define  his  needs  as  accurately  as  possible  and   convey   these 
 requirements  in  understandable  language   to  the  vendor.    Vague
 terminology  and  ambiguous  specifications  can  compound  production
 problems.   The vendor,  based  on his background and experience, must
 make meaningful suggestions  to  the  client as early in production as
 possible as  alternative options may be considered.
 \smallskip
 \centerline{\bf VI. User Community }
 
 In  summary,  a  comprehensive  digital  map  and data system has many
 advantages.  In  order  to  fully realize these advantages, users must
 consider and resolve several  questions  related  to actual needs  and
 aesthetic conditioning.  Users should understand that digital maps and
 data bases  constructed  using  highly  accurate  aerial  photographic
 source  documents  will  not,  as  a rule,  look like familiar graphic
 products.  They must consider the  distinctions  between  mechanically
 drafted products and cartographic products and decide at the outset of
 the  project  whether they are comfortable with the graphic
 specifications.
 
 Close communication between the utility company and the  mapping  firm
 is  critical.    The  more  carefully  specified the project is at the
 beginning,  the fewer the changes that will be required.   Changes  in
 specifications  made  during  production,  no  matter how trivial they
 appear, generally affect production schedules and costs adversely.

      Thus, the creation of a digital map involves not only the mastery 
 of current technology, in order to produce an ``accurate" map,  but it
 also  involves  an  awareness  of  aesthetics,  as well.  Attention to
 aesthetics,  as appreciation  of the map by the consumer will ensure a
 satisfied  client;  to this end,  considerable education of the client
 with  attention  to  close communication is appropriate.  At the other
 end, the mapping firm needs to consider the aesthetics of map creation.
 When the aesthetics of creation help to guide the choice of technology,
 an accurate and satisfying digital map is generally the end product.
 \vfill\eject
 \centerline{\bf References}
 
 \ref 1. Robinson,  A., R. Sale and J. Morrison (1978),
 {\sl Elements of Cartography\/} (4th edition),
 New York: John Wiley, p. 4.
 
 \ref 2.  Robinson, A., et al., p. 272.
 
 \ref 3.  International Cartographic Association
 (Commission II, E. Meynen, Chairman) (1973),
 {\sl Multilingual Dictionary of Technical Terms in Cartography\/},
 Wiesbaden: Franz Steiner Verlag GMBH.
 
 \ref 4.  P. W.  McDonnell, Jr. (1979),
 {\sl Introduction to Map Projections\/},
 New York and Basel: Marcel Dekker, Inc.
 
 \ref 5.  H. S. M. Coxeter,  (1974),
 {\sl Projective Geometry\/}, (second edition),
 Toronto; University of Toronto Press.
 
 \ref 6.    American Congress on Surveying and
 Mapping and American Society of Civil Engineers, 
 Joint Committee (1978),
 {\sl Definitions of Surveying and Associates Terms\/} (rev.),
 p. 101.
 
 \ref 7.  ``Symposium on the National Map Accuracy Standard,"
 {\sl Surveying and Mapping\/}, 1960, v.20,  n.4,  pp.  427-457,
 and M. M.  Thompson and G. H. Rosenfield,  ``On  Map Accuracy
 Specifications," {\sl Surveying and Mapping\/},
 1971, v.31, n.1, pp. 57-64.
 
 \ref 8.  Cuff, D.J. and M.T.  Mattson (1982),
 {\sl Thematic Maps: Their Design and Production\/},
 New York and London: Methuen.
 
 \ref 9.    For  additional discussion of this approach,
 see Easton,  C.H. (1975),
 ``The Land Records Information System in Forsyth County, 
 North Carolina," pp.  261-267  in 
 {\sl International Property Assessment Administration\/},
 Chicago: International  Association  of  Assessing Officers.
 
 \ref 10.  Robinson, A., et al., p.7.
 See also: Amheim, R. (1976), ``The Perception of Maps,"
 {\sl The American Cartographer\/}, 1976, v.3, n.1, pp. 5-10.
 
 \ref 11.  Irvins, W. Jr. (1964),
 {\sl Art and Geometry: A Study in Space Intuitions\/},
 New  York:  Dover 
 (reprint  of  1949 Harvard University Press edition).
 \ref 12.  Blasgen, M. W.,
 ``Data Base Systems,"
 {\sl Science\/}, 1982,  v.215,  12 February.
 
 \ref 13.   Codd,  E.,
 ``Relational Data Base: A Practical Foundation for Productivity,"
 {\sl Communications\/} of the ACM, 1982, v.25, n.2.
 
 \ref 14.  Blasgen [12].
 \vfill\eject
 \centerline{\bf $^{\bf \star }$ Acknowledgments}
 
 An earlier version of this essay appeared as a Chicago  Aerial  Survey
 Production Report.   Comments  on  the previous version by Mr.  Harold
 Flynn are gratefully acknowledged,  as  is  the  support  provided  by
 Geonex Corporation during its preparation.  The author wishes to thank
 an  anonymous  referee  for  comments  useful in preparing the current
 version.
 \vfill\eject
 \noindent{\bf FEATURES}
 \vskip.5cm
 \noindent{\bf i.  Press Clipping}
     
 {\sl Science\/}, November 29, 1991, Vol.  254,  No. 5036,  copyright, 
 the   American   Association  for  the  Advancement of Science.  Many
 thanks to Joseph Palca at {\sl Science\/} for his continuing interest 
 in  online  journals.   The  citation  appeared  in  ``Briefings" and
 is entitled ``{\bf  Online Journals}," by Joseph Palca.

 \centerline{\bf Online Journals}

          ``When the AAAS and OCLC Online Computer Library Center
          announced  the  scheduled  debut next year of their new
          journal---{\sl The On\-line Journal of Current Clinical
          Trials\/} --- they said it would be the  world's  first
          peer-reviewed, on\-line science journal ({\sl Science\/},
          27 September, p. 1480).   Since  then,  two other  such 
          journals   have   made    their   presence   known   to
          {\sl Science\/}.  They are {\sl Solstice: An Electronic
          Journal of Geography and Mathematics\/},  published  by
          Sandra Lach Arlinghaus of the Institute of Mathematical
          Geography in Ann Arbor, Michigan, and {\sl Flora Online
          \/}, published by Richard H. Zander,  curator of botany
          at the Buffalo Museum of Science. Both have been around
          for about 2 years and are available  free  over several
          popular research computer networks."

 NOTE:   Readers  wishing  to contact Richard Zander can do so at
 bitnet address: 

 VISBMS@UBVMS
 \vfill\eject
 \noindent{\bf ii. Word Search Puzzle}

 The  point of this puzzle is to develop familiarity  (dispelling fear) 
 with  a  selection of  words possibly not familiar to student readers. 
 The words are embedded in the jumble of letters below; not all letters
 in  this  array  are  part  of  a word in the list and other words may
 appear in the puzzle.  Words from the list may be written from left to
 right, from right to left, from top to bottom,  from bottom to top, or
 diagonally (in any direction).  Solution is on the last  page  of this
 issue.

 \centerline{\bf WORDS IN THE PUZZLE}

 $$\vbox{\settabs 3 \columns
 \+ Algorithm             &Gnomonic          &Parallel       \cr
 \+ Asymptote             &Graticule         &Polyconic      \cr
 \+ Azimuthal             &Integral          &Projection     \cr
 \+ Circumpolar           &Inverse           &Rotation       \cr
 \+ Circumscribe          &Jacobian          &Solstice       \cr
 \+ Conic                 &Lambert           &Stereographic  \cr
 \+ Converge              &Latitude          &Tangent        \cr
 \+ Curvature             &Logarithm         &Translation    \cr   
 \+ Cylindrical           &Longitude         &Vector         \cr
 \+ Divergent             &Matrix            &               
\cr                
 \+ Equatorial            &Mercator          &               \cr
 \+ Equidistant           &Meridian          &               \cr
 \+ Equinox               &Norm              &               \cr
 \+ Exponent              &Oblique           &               \cr
 \+ Fractal               &Orthogonal        &               \cr}$$
 $$\vbox{\settabs 26 \columns                     
 \+      P&O&R&T&H&P&O&L&Y&C&O&N&I&C&I&H&P&A&R&G&O&E&R&E&T&S \cr
 \+      E&L&O&N&G&I&T&U&D&E&E&Q&U&I&D&I&S&T&A&N&T&N&E&B&C&C \cr
 \+      Q&A&L&G&O&R&I&T&H&M&L&O&D&I&V&E&R&G&E&N&T&G&G&I&U&V \cr
 \+      U&H&S&L&G&H&T&M&I&H&P&R&C&Y&M&S&A&N&E&G&R&R&P&R&E&R \cr
 \+      A&T&E&Y&R&M&E&T&O&T&P&M&Y&S&A&P&L&N&U&O&A&R&V&C&E&O \cr
 \+      T&U&L&A&M&B&E&R&T&I&A&C&I&N&O&M&O&N&G&T&T&A&T&S&A&T \cr
 \+      O&M&R&O&N&P&T&R&B&R&E&N&N&R&O&P&P&X&I&C&T&O&I&M&L&A \cr
 \+      A&I&X&M&R&T&T&P&M&A&S&C&G&C&X&C&M&C&A&U&R&R&T&U&N&T \cr
 \+      L&Z&R&S&A&T&Z&O&L&G&C&M&D&E&N&U&U&C&R&A&U&N&E&C&O&I \cr
 \+      R&A&T&F&E&T&R&S&T&O&M&U&Q&Q&N&L&C&E&R&F&E&G&L&R&A&O \cr
 \+      I&R&C&V&R&E&R&G&N&L&E&U&V&U&E&T&R&A&N&S&L&A&T&I&O&N \cr
 \+      P&O&O&I&T&A&R&I&S&T&I&L&C&A&T&O&I&N&M&U&T&H&A&C&Z&I \cr
 \+      M&T&N&O&R&E&C&X&X&N&W&F&A&T&B&N&C&A&J&I&O&C&A&R&N&N \cr
 \+      E&A&V&E&R&D&A&T&O&F&J&A&C&O&B&I&A&N&T&G&E&R&G&T&A&V \cr
 \+      T&C&E&T&S&A&N&X&A&R&S&X&A&R&R&T&G&U&O&A&N&R&E&V&B&E \cr
 \+      G&R&R&A&M&R&S&I&A&L&X&S&E&I&P&R&D&N&D&G&E&G&A&R&V&R \cr
 \+      R&E&G&E&C&I&T&S&L&O&S&T&E&A&Z&E&A&M&U&T&R&H&A&L&S&S \cr
 \+      A&M&E&R&I&D&I&A&N&Y&L&A&A&L&E&L&L&A&R&A&P&R&A&L&T&E \cr
 \+      I&S&G&A&L&R&N&O&I&T&C&E&J&O&R&P&O&B&L&I&Q&U&E&C&A&L \cr}$$
 \vfill\eject
 \noindent{\bf iii. Software  Briefs} ---  Brief descriptions  of
             software provided by the  creator.  Look for reviews
             of    the   software   in   subsequent   issues   of
             {\sl Solstice \/}.   The  Institute  of Mathematical
             Geography (IMaGe) makes  no claim as to the accuracy
             of  the  statements  made  by  the   creators;   the
             appearance of their comments in {\sl Solstice\/}  is 
             {\bf not} an endorsement, either direct or indirect, 
             of the product by IMaGe or by anyone associated with
             either  IMaGe or {\sl Solstice\/}.   These ``Briefs"
             are simply  presented as a way for software creators
             to share infomation,  in an  e-journal,  with  other
             possibly interested parties.

 a.  RangeMapper$^{\hbox{TM}}$  ---  version 1.4.
             Created  by Kenelm W. Philip,  Tundra Vole Software,
             Fairbanks, Alaska.  Program and Manual by  Kenelm W.
             Philip; commentary below from the Manual.    
             ``A  utility  for  biological species range mapping,
             and similar mapping tasks in other fields."
             FNKWP@ALASKA
             
      ``RangeMapper$^{\hbox{TM}}$  is a Macintosh mapping utility
 program  designed specifically for the field or museum biologist
 who wants  to be  able to produce,  rapidly and easily,  species
 range maps for various organisms.  The program may also  be used
 for  mapping  other  kinds  of  data,  in medical, sociological,
 geological, geophysical, biological, etc. applications.

 The program is aimed at people whose mapping needs cover sizable
 areas.  The most accurate data files in the map base are derived
 from  the  CIA  mapping files, which are suitable for displaying
 regions  down  to  20-30  miles  or  so in linear extent without
 showing a `polygon' effect from data  point spacing.    The data
 files  also include the  Micro World Data Bank files,  which are
 suitable for  mapping on a whole-world scale and down to regions
 perhaps 500 miles in linear extent.

 On a Macintosh II or other machines in the Macintosh II family,
 most  maps  will  be  plotted  in  well under one minute.  Once
 plotted,  they  can  be  saved to disk and re-loaded as needed.
 In either case, data from properly formatted latitude/longitude
 textfiles can be read by the  program  and plotted to the maps. 
 If  your  species  data  (including  lat/long  coordinates  for
 collecting  sites) are stored in a computer database, it should
 be easy to arrange to export, for any given species, a textfile
 of lat/long coordinates that RangeMapper can read directly ---
 thus obtaining your range maps in one step from your database.

 In  conjunction  with the word processor NISUS, RangeMapper may
 be used as a map-based visual interface to a text database.

 The  current  version  (1.4) of RangeMapper is set up for world
 mapping   in   six   projections  ---  north  polar  azimuthal,
 cylindrical,   Mercator,   orthographic,   stereographic,   and
 Lambert azimuthal equal-area.   The north  polar azimuthal maps
 are  quite  usable  down  to  the  southern limits of the lower
 48  states  and  equivalent  latitudes  in  Eurasia,   and  are
 excellent   for   higher-resolution  mapping  of  Alaska.   The
 cylindrical and Mercator projections can show the entire  world
 (barring the extreme polar regions  in  Mercator),  centered at
 any  longitude.   The  last  three  projections  show up to one
 hemisphere, which may be centered at any point  on  the earth's
 surface.

 The map data files from the  Micro  World  Data  Bank cover the
 entire world.  The  only  CIA  file presently incorporated into
 RangeMapper is the  Alaska  file  (approximately 150,000 points
 for  coastlines,  islands,  rivers,  and  lakes).   The  entire
 continent of North America will be added from CIA files  later,
 permitting mapping of the U.S. and Canada to the same precision
 as can be obtained using the current file for Alaska.

 Other continents, and higher-precision files covering the U.S.,
 may be added later."
 \vfill\eject
 b.         ``XYNIMAP"   ---  created  by David H. Douglas, 
             University of Ottawa;  ``a comprehensive system  for 
             computer cartography and geo-spatial analysis."
             Preliminary Version. 
             DHDAD@UOTTAWA
 
 ``XYNIMAP is a comprehensive system for computer cartography
 and geo-spatial analysis, that does a lot of things, but not
 everything, that other packages do. If you give  it a chance
 you will find it does a number of things better  than  other
 packages.  The diskette contains all  manuals and  operating
 instructions. It is meant for PC computers.  PC-XT 286, 386,
 486.
 
 \centerline{COMPONENTS}
 
 XYNITIZE:    An  interactive  map  digitizing system (with a
 different way of interacting with the user). 
 
 BNDRYNET:  A program to convert a mass of intersecting lines
 into a  topology to represent the polygons that are visually
 evident. In other words BNDRYNET is a cartographic spaghetti
 to polygon converter.
 
 CONSURF: A contour to grid digital elevation model program.
 
 POLYGRID: A polygon to grid converter.
 
 XYBINASC \& XYASCBIN:   Programs  to  convert  a XYNIMAP {\sl
 stream  feature\/} file back and forth from compressed binary
 to readable (therefore editable) ASCII files.
 
 GDEMIDRI:   A  program  to  convert  a  XYNIMAP  grid digital
 elevation model to an IDRISI .img and .doc files.
 
 XYTALLY:  A program to read a XYNIMAP stream feature file and
 produce a printout of various measures: (lengths of lines and
 areas of regions).
 
 {\bf The following are tested workable and distributable programs }
 but I am just not ready to put them out just yet. 
 
 XYNIDISP:  A comprehensive display system for the PC computer
 with EGA or VGA graphics adapter cards.
 
 XYNIDRAW:  A  comprehensive  display  system for line drawing
 plotters
 
 VUBLOK:   A  particularly robust perspective view map program
 for   grid   digital  elevation  models.    It  produces  the
 traditional fishnet display and shaded relief.
 
 PILLAR:   A program to display a geographical distribution by
 an image of  standing  vertical  pillars  on the surface of a
 perspective  view  of  a  base  map.  The  program curves the
 surface to a realistic projection.
 
 PROCIR: A proportional circle display program."
 \vfill\eject
 \noindent{\bf iv.  Index to Volumes I (1990) and II (1991) of
             {\sl Solstice}.}
 \vskip.5cm
 \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
 diffusion.
     
     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.
 \smallskip
 \smallskip
 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
 ones.
 \smallskip
 \noindent 3.  FEATURES
 \smallskip
 \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 --- Crossward puzzle on spices.
 \item{v.}   Solution to crossword puzzle.
 \smallskip
 \noindent {\bf 4.  SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE}
 \smallskip
 \noindent{\bf Volume I, Number 2, Winter, 1990}
 \smallskip
 \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 
 observations.

 \noindent 2.  ARTICLES

 Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical
 Harmony\/}.

      Linkage  between  scale  and  dimension  is made using the 
 Fallacy of Division and the Fallacy of Composition in a fractal
 setting.
 \smallskip
 \smallskip
 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.
 \smallskip
 \smallskip
 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.
 \smallskip
 \smallskip
 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
 pixels.
 \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.
 \smallskip
 \noindent {\bf Volume II, Number 1, Summer, 1991}
 \smallskip 
 \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.).
 \smallskip
 \smallskip
 \noindent 2.  FEATURES

 \item{i}  Construction Zone --- The logistic curve.
 \item{ii.} Educational feature --- Lectures on ``Spatial Theory"
 \vfill\eject
 \noindent{\bf v. Other publications of }
 \noindent{\bf the Institute of Mathematical Geography. }
 \noindent{\bf 1991}
 \smallskip
 \centerline{\it INSTITUTE OF MATHEMATICAL GEOGRAPHY (IMaGe)}
 \centerline{\it 2790 BRIARCLIFF}
 \centerline{\it ANN ARBOR, MI 48105-1429; U.S.A.}
 \centerline{(313) 761-1231; IMaGe@UMICHUM}
 \vskip 0.2cm
 \centerline{\it ``Imagination is more important than knowledge"}
 \centerline{\it A. Einstein}
 \vskip.2cm
 \centerline{\bf MONOGRAPH SERIES}
 \centerline{\sl Scholarly Monographs--Original Material}
 \centerline{Prices on request, exclusive of shipping and handling;}
 \centerline{payable in U.S. funds on a U.S. bank, only.}
 \smallskip
 Monographs are printed by {\bf Digicopy} on 100\% recycled paper
 of achival 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\/},
 1986.
 \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\/},
 1986.  
 \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\/},
 1987.  
 \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 
litt\'erature.
 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.
 \vskip.1cm

 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 
unpublished
 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.
 \vskip.2cm
 8.  James W. Fonseca, {\it The Urban Rank--size Hierarchy:  A Mathematical
 Interpretation\/}, 1989.
 \vskip.1cm

 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.
 \vskip.2cm
 9.  Sandra L. Arlinghaus, {\it An Atlas of Steiner Networks\/}, 1989. 
\vskip.1cm 

 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 
trees.
 \vskip.2cm
 10.  Daniel A. Griffith, {\it Simulating $K=3$ Christaller Central Place
 Structures:  An Algorithm Using A Constant Elasticity of Substitution
 Consumption Function\/}, 1989.
 \vskip.1cm

 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 
into
 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 
manifest
 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.
 \vskip.2cm
 11.  Sandra L. Arlinghaus and John D. Nystuen,
      {\it Environmental Effects on Bus Durability\/}, 1990.  
 \vskip.1cm

  This monograph draws on the authors'
 previous publications on ``Climatic" and ``Terrain" effects on bus 
durability.
 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.
 \vskip.2cm
 12.  Daniel A. Griffith, Editor.
 {\sl Spatial Statistics:  Past, Present, and Future\/},  1990. 
 \vskip.1cm

     Proceedings of a Symposium of the same name held at Syracuse
 University in Summer, 1989.  Content includes a Preface by Griffith and 
the 
 following papers:  
 \vskip.1cm

 \line{{\sl Brian Ripley}, ``Gibbsian interaction models"; \hfil}
 \line{{\sl J. Keith Ord}, ``Statistical methods for point pattern data"; 
\hfil}
 \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";\hfil}
 \line{{\sl Sylvia Richardson},  
 ``Some remarks on the testing of association between spatial 
processes";\hfil}
 \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.
 \vskip.2cm
 13.  Sandra L. Arlinghaus, Editor.
 {\sl Solstice---I\/},  1990. 
 \vfill\eject
 \centerline{\it DISCUSSION PAPERS--ORIGINAL}
 \centerline{\it Editor, Daniel A. Griffith}
 \centerline{\it Professor of Geography}
 \centerline{\it Syracuse University}
 \centerline{Founder as an IMaGe series:  Sandra L. Arlinghaus}
 \smallskip
 \noindent 1.  {\sl Spatial Regression Analysis on the PC:
 Spatial Statistics Using Minitab}.  1989.  
 \vfill\eject
 \centerline{\it DISCUSSION PAPERS--REPRINTS}
 \centerline{\it Editor of MICMG Series, John D. Nystuen}
 \centerline{\it Professor of Geography and Urban Planning}
 \centerline{\it The University of Michigan}
 \smallskip
 \noindent 1.  {\sl Reprint of the Papers of the Michigan InterUniversity
 Community of Mathematical Geographers.}  Editor, John D. Nystuen.
 \smallskip
 Contents--original editor:  John D. Nystuen.
 \smallskip
 \noindent 1.  Arthur Getis, ``Temporal land use pattern analysis with the
 use of nearest neighbor and quadrat methods."  July, 1963
 \smallskip
 \noindent 2.  Marc Anderson, ``A working bibliography of mathematical
 geography."  September, 1963.
 \smallskip
 \noindent 3.  William Bunge, ``Patterns of location."  February, 1964.
 \smallskip
 \noindent 4.  Michael F. Dacey, ``Imperfections in the uniform plane."
 June, 1964.
 \smallskip 
 \noindent 5.  Robert S. Yuill, A simulation study of barrier effects
 in spatial diffusion problems."  April, 1965.
 \smallskip
 \noindent 6.  William Warntz, ``A note on surfaces and paths and
 applications to geographical problems."  May, 1965.
 \smallskip
 \noindent 7.  Stig Nordbeck, ``The law of allometric growth."
 June, 1965.
 \smallskip
 \noindent 8.  Waldo R. Tobler, ``Numerical map generalization;"
 and Waldo R. Tobler, ``Notes on the analysis of geographical
 distributions."  January, 1966.
 \smallskip
 \noindent 9.  Peter R. Gould, ``On mental maps."  September, 1966.
 \smallskip
 \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.
 \smallskip
 \noindent 11. E. Casetti and R. K. Semple, ``A method for the
 stepwise separation of spatial terends."  April, 1968.
 \smallskip
 \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.
 \vfill\eject
 \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}
 \smallskip
 1.  Allen K. Philbrick.  {\sl This Human World}.
 \smallskip 
 \vfill\eject
 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.
 \vfill\eject
 \noindent{\bf vi. Solution to Word Search Puzzle.}
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