SOLSTICE:  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS.
Volume V, Number 1.  Summer, 1994.
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 \centerline{\big SOLSTICE:}
 \vskip.5cm
 
 \centerline{\bf AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS}
 \vskip5cm

 \centerline{\bf SUMMER, 1994}
 \vskip12cm

 \centerline{\bf Volume V, Number 1}
 \smallskip

 \centerline{\bf Institute of Mathematical Geography}
 \vskip.1cm

 \centerline{\bf Ann Arbor, Michigan}
 \vfill\eject
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 \hrule
 \smallskip
 \centerline{\bf SOLSTICE}
 \line{Founding Editor--in--Chief: 
      {\bf Sandra Lach Arlinghaus} \hfil}
 \line{Institute of Mathematical Geography and University of Michigan \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.\hfil}
 \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 Engineering Applications} \hfil}
 \line{{\bf William D. Drake},
        University of Michigan, \hfil}
 \smallskip
 \line{{\bf Education} \hfil}
 \line{{\bf Frederick L. Goodman},
        University of Michigan, \hfil}
 \smallskip
 \line{{\bf Business} \hfil}
 \line{{\bf Robert F. Austin, Ph.D.} \hfil}
 \line{President, Austin Communications Education Services \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.  

 The opinions expressed are those of the authors, alone, and the
 authors alone  are responsible for the accuracy of the facts in
 the articles. 
 \smallskip
 \noindent {\bf Send all correspondence to:}
 Sandra Arlinghaus, Institute of Mathematical Geography,
 2790 Briarcliff, Ann Arbor MI 48105.
 Solstice@um.cc.umich.edu; SArhaus@umich.edu
 \smallskip
 Suggested form for citation.   If  standard  referencing  to the
 hardcopy in the  IMaGe Monograph Series is not used (although we
 suggest that reference  to that  hardcopy be included along with
 reference  to  the  e-mailed  copy  from which  the hard copy is
 produced), then we suggest the following  format for citation of
 the electronic copy.  Article,  author, publisher (IMaGe) -- all
 the usual--plus a notation as to the time marked electronically, 
 by the process of transmission,  at the  top  of the  recipients
 copy.   Note  when  it was sent from Ann Arbor (date and time to
 the  second)  and  when  you  received  it (date and time to the
 second)  and  the  field characters covered by the article  (for
 example FC=21345 to FC=37462).
 
       This  document is produced using the typesetting  program,
 {\TeX},  of Donald Knuth and the American Mathematical  Society.
 Notation  in  the electronic file is in accordance with that  of
 Knuth's   {\sl The {\TeX}book}.   The program is downloaded  for
 hard copy for on The University of Michigan's Xerox 9700 laser--
 printing  Xerox  machine,  using IMaGe's commercial account with 
 that University.
 
 Unless otherwise noted, all regular ``features"  are  written by
 the Editor--in--Chief.
 \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 Copyright, June, 1994 by the
 Institute of Mathematical Geography.
 All rights reserved.
 \vskip1cm
 {\bf ISBN: 1-877751-56-1}
 {\bf ISSN: 1059-5325} 
 \vfill\eject
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 \centerline{\bf TABLE OF CONTENT}
 \smallskip
 \noindent{\bf  1.  WELCOME TO NEW READERS AND THANK YOU}
 \smallskip
 \noindent{\bf  2.  PRESS CLIPPINGS---SUMMARY}
 \smallskip
 \noindent{\bf 3.  REPRINTS}
 \smallskip
 \noindent{\bf Getting Infrastructure Built}
 \smallskip
 \noindent{\bf Virginia Ainslie and Jack Licate}
 \smallskip
 Transmitted as part 2 of 13.
 \smallskip
     Cleveland Infrastructure Team Shares Secrets of Success;
     What Difference Has the Partnership Approach Made?
     How Process Affects Products --- Moving Projects Faster
          Means Getting More Public Investment;
     How Can Local Communities Translate These Successes to
          Their Own Settings?
 \smallskip
 \noindent{\bf Center Here; Center There; Center, Center Everywhere} 
 \smallskip
 \noindent {\bf Frank E. Barmore}.
 \smallskip
 Transmitted as parts 3 and 4 of 13.
 \smallskip
 Reprinted from {\sl The Wisconsin Geographer\/}, Vol. 9, pp. 8-21,
 1993.
 Publication of the Wisconsin Geographical Society, reprinted here
 with permission of that Society.
 \smallskip
   Abstract;
   Introduction;
   Definition of Geographic Center;
   Geographic Center of a Curved Surface;
   Geographic Center of Wisconsin;
   Geographic Center of the Conterminous United States;
   Geographic Center of the United States;
   Summary and Recommendations;
   Appendix A:  Calculation of Wisconsin's Geographic Center;
   Appendix B:  Calculation of the Geographical Center of the
                Conterminous United States;
   References
 \smallskip
 \noindent{\bf  4.  ARTICLES}
 \smallskip
 \noindent{\bf Equal-Area Venn Diagrams of Two Circles:
               Their Use with Real-World Data}
 \smallskip
 \noindent{\bf Barton R. Burkhalter}
 \smallskip
 Transmitted as parts 5 and 6 of 13.
 \smallskip
     General Problem;
     Definition of the Two-Circle Problem;
     Analytic Strategy;
     Derivation of $B\%$ and $AB\%$ as a Function of 
           $r_B$ and $d_{AB}$.
 \smallskip
 \noindent{\bf  Los Angeles, 1994 --- A Spatial Scientific Study}
 \smallskip
 Transmitted as parts 7, 8, 9, 10, 11, 12 of 13 (with body of
 text in part 7 and supporting tables and computer program
 in five subsequent parts).
 \smallskip
 \noindent{\bf Sandra L. Arlinghaus, William C. Arlinghaus,
               Frank Harary, John D. Nystuen}
 \smallskip
     Los Angeles, 1994;
     Policy Implications;
     References.
     Tables and Complicated Figures.
 \smallskip

 \noindent{\bf 5.  DOWNLOADING OF SOLSTICE}
 \smallskip

 \noindent{\bf 6.  INDEX to Volumes I (1990),  II (1991),  
 III (1992), and IV (1993) of {\sl Solstice}.}

 \smallskip
 \noindent{\bf 7.  OTHER PUBLICATIONS OF IMaGe }

 \vfill\eject
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 \centerline{\bf 1.  WELCOME TO NEW READERS AND THANK YOU}

 Welcome to new subscribers!   We  hope  you  enjoy participating 
 in  this   means   of journal  distribution.   Instructions  for
 downloading  the typesetting have  been  repeated in this issue,
 near the end.  They are specific to the  {\TeX}  installation at
 The University of Michigan, but apparently they have been helpful 
 in suggesting to others the sorts of commands that might be used 
 on their own  particular  mainframe installation of {\TeX}.  New
 subscribers might wish to  note that  the  electronic  files are
 typeset files---the  mathematical notation  will  print  out  as 
 typeset notation.  For example,
 $$
 \Sigma_{i=1}^n
 $$
 when  properly downloaded, will print out a typeset summation as
 $i$  goes from one to $n$, as  a  centered  display on the page. 
 Complex  notation  is  no  barrier  to  this   form  of  journal
 production. 
 \vskip.5cm

      Thanks much to subscribers who have offered input.  Helpful
 suggestions  are  important  in trying to keep abreast, at least
 somewhat, of the constantly  changing  electronic  world.   Some
 suggestions from readers  have  already been implemented; others
 are being worked on.  Indeed,  it is particularly  helpful  when
 the reader  making  the suggestion  becomes actively involved in
 carrying it out.  We hope you continue to enjoy
 {\sl Solstice\/}.
 \smallskip
 %---------------------------------------------------------------
 %---------------------------------------------------------------
 \centerline{\bf 2.  PRESS CLIPPINGS---SUMMARY}

 \noindent
     Volume 72, Number 4, October 1993 issue of {\sl Papers in
 Regional Science:  The Journal of the Regional Science Association\/}
 carried an article by Gunther Maier and Andreas Wildberger
 entitled ``Wide Area Computer Networks and Scholarly Communication
 in Regional Science."  Maier and Wildberger noted that ``Only one 
 journal in this directory can be considered to be related to
 Regional Science, {\sl Solstice:  An Electronic Journal of Geography
 and Mathematics\/}."
 
      Beyond that,  brief  write-ups  about {\sl Solstice\/}  have 
 appeared in the following publications:

 \noindent 1.  {\bf Science}, ``Online Journals"  Briefings.  
 [by Joseph Palca]
 29 November 1991.  Vol. 254.
 \smallskip

 \noindent 2. {\bf Science News}, ``Math for all seasons"
 by Ivars Peterson, January 25, 1992, Vol. 141, No. 4.
 \smallskip

 \noindent 3.  {\bf Newsletter of the Association of American
 Geographers}, June, 1992.
 \smallskip

 \noindent 4. {\bf American Mathematical Monthly},
 ``Telegraphic Reviews" --- mentioned as
 ``one of the World's first electronic journals using {\TeX}," 
 September, 1992.
 \smallskip

 \noindent 5. {\bf Harvard Technology Window}, 1993.
 \smallskip

 \noindent 6.  {\bf Graduating Engineering Magazine}, 1993.

 \noindent 7.  {\bf Earth Surface Processes and Landforms},
 18(9), 1993, p. 874.

 \noindent 8.  {\bf On Internet}, 1994.

 If  you  have  read about {\sl Solstice\/} elsewhere, please let
 us know the correct citations (and add to those above).  Thanks.
 We are happy to share information  with all  and are delighted
 when others share with us, as well.  
 \vfill\eject

 Publications of the Institute of Mathematical Geography have,
 in addition, been reviewed or noted in 
 \smallskip
 
 1.  {\sl The Professional Geographer\/} published
 by the Association of American Geographers;
 \smallskip

 2.  The {\sl Urban Specialty Group Newsletter\/}
 of the Association of American Geographers;
 \smallskip

 3.  {\sl Mathematical Reviews\/} published by the
 American Mathematical Society;
 \smallskip

 4.  {\sl The American Mathematical Monthly\/} published
 by the Mathematical Association of America;
 \smallskip

 5.  {\sl Zentralblatt\/} fur Mathematik,  Springer-Verlag, Berlin
 \smallskip

 6.  {\sl Mathematics Magazine\/}, published by the Mathematical
 Association of America.
 \smallskip

 7.  {\sl Newsletter\/} of the Association of American Geographer.
 \smallskip

 8.  {\sl Journal of The Regional Science Association\/}.
 \smallskip

 9.  {\sl Journal of the American Statistical Association\/}.
 \smallskip

 \vfill\eject

 \centerline{\bf 3.  REPRINTS}
 \smallskip
 \centerline{ \bf Getting Infrastructure Built}
 \smallskip
 \smallskip
 \centerline{Virginia Ainslie}
 \centerline{Technical Liaison to Congress}
 \smallskip 
 \centerline{Jack Licate (Ph.D., Geography)}
 \centerline{Director, Build Up Greater Cleveland Program and}
 \centerline{Director of Federal Programs for the Greater 
 Cleveland Growth Association.}
 \smallskip
 \noindent Reprinted with permission from Land Development/Spring-Summer 
1994.
 \smallskip

     The profitability and success of a development project often
 hinge  on  the  timely  completion  of  improvements to adjacent
 highways,  bridges,  sewers,  and  transit  services.  In recent
 years, complex  environmental and construction requirements have
 increased  the  lead  time   and   costs   required   for   many
 infrastructure  improvements.   At  the  same  time,  the public
 funding needed to finance road widening,  interchange and bridge
 construction, sewer improvements,  and transit  development  has
 come  under  severe  budgetary  constraints  at  all  levels  of
 government.

     In  northeast  Ohio,  public and private sector leaders have
 entered  into  a  successful partnership to solve infrastructure
 development  problems.   The  lessons  learned  from Cleveland's
 partnership can be readily translated to other communities.

 \noindent{\bf Cleveland Infrastructure Team 
               Shares the Secrets of Success}

     Founded in 1983 and commonly referred to as ``BUGC", Build Up 
 Greater  Cleveland  is  a  unique  partnership  that  consists of
 elected and  appointed  officials  from local, state, and federal
 governments  as  well  as  dedicated  private  sector   executive
 volunteers from engineering,  banking, investment, manufacturing,
 utility, accounting, and law firms. The development community has
 actively  participated in all  aspects of BUGC's activities since
 the program was founded.  A team  effort  that was born of crisis
 but matured  over a decade of wrenching  economic upheaval,  BUGC
 has earned national recognition for its ability to attract public
 financing needed  for  the improvement, repair,  and construction
 of roads, bridges, and sewer, water, and transit facilities.

    The  first secret of BUGC's success is its systematic strategy
 for  gaining the commitment of public funding through coordinated
 and  simultaneous  advocacy  efforts  at  the  local,  state, and
 federal  levels.   The  strategy  calls  for  the  aggressive and
 persistent  pursuit  of  a fair and  equitable share of state and
 federal infrastructure investment. Central to the strategy is the
 involvement of elected officials.  Specifically,  these officials
 enacted legislation that maximizes the return of tax  dollars  to
 northeast Ohio.

     At the  federal level, the formula used to divide highway and
 bridge  funding  has  been  amended  in favor of Ohio and certain
 other  states  in  each  piece  of  major  surface transportation
 legislation  since  1987.   The  Ohio  Congressional delegation's
 leadership for this effort was based, in large part, on technical
 assistance and networking support from BUGC.

    For their part,  private sector  volunteers  help  develop the
 data  that  form  the  basis  for  BUGC's ``fair share"  advocacy
 efforts.   Corporate  expertise  has  been particularly useful in
 quantifying  the  public  benefits  of infrastructure investment. 
 Private sector  executives  also  play an  active role in various
 task forces charged with solving problems and  coordinating  road
 and bridge repair work with utilities.  In addition,  both public
 and private  sector  members  of BUGC are  involved in  long-term
 efforts to educate the public on the importance of infrastructure
 to the community and its economy.

    The second  secret of BUGC's success is its focus on improving
 the  process  by  which  projects  move  from  identified need to
 construction.   Between 1988 and 1993, the greater Cleveland area
 posted a 166 percent increase in the number of completed road and
 bridge projects.   This surge  resulted largely from the adoption
 of  new  procedures  that  reduced  project completion time by 44
 percent.  BUGC's  approach to  achieving performance to achieving
 performance has helped reshape  Cleveland's  skyline and,  at the
 same  time,  contributed  to  a  major  renaissance  in  regional
 economic development.

 \noindent{\bf What Difference Has the Partnership Approach Made?}

     Over  the last decade,  BUGC's advocacy  program has  yielded
 more  than  one  billion  dollars  in unprogrammed funds that the
 greater  Cleveland  area would not have otherwise received.  Much
 of the credit for the funds goes to northeast Ohio's congressional
 delegation.  The  delegation  orchestrated a multistate/multiyear
 coalition effort that has increased the return of Ohio's share of
 the federal  gas  tax  from 61  cents on the dollar in 1981 to 90
 cents on the dollar in 1993.  Changes  in  the  formulas  used to
 distribute federal highway and bridge dollars  have  also brought
 more than 1.5 billion dollars in new funds to Ohio  and more than
 two million dollars in new funds  to  greater Cleveland.   At the
 state level,  BUGC played a pivotal role in establishing the Ohio
 Public Works Commission and its  program of  needs-based  funding
 for  local  capital  assets.   This  effort  has  resulted  in an 
 estimated  annual  increase  of  14.5  million dollars for county
 roads, bridges,  and sewers.  BUGC worked locally for legislation
 that increased motor vehicle license tag fees, which now generate
 more  than  13 million dollars per year in road and bridge repair
 funds.

    BUGC  has lobbied successfully at the federal and state levels
 for  project-specific  funding  for  infrastructure  improvements
 essential to  Tower City Center, a new baseball stadium and arena
 project, the  Rock  and  Roll  Hall of Fame, and many other major
 development  projects.   BUGC  has  also been a key player in the
 cleanup of the Cuyahoga River, the overhaul and rehabilitation of
 Cleveland's transit  system, and the completion of the Interstate
 highway network in northeast Ohio.  BUGC's efforts have generated
 396  million  dollars  for  road  and  bridge improvements, 260.7
 million dollars for transit  development, 396 million dollars for
 water projects, and 341 million  dollars  for  sewer needs.  BUGC
 Chairman James M. Delaney of Deloitte and Touche points out, ``We
 now  are witnessing a shift in  investment from rehabilitation of
 existing  infrastructure  to  increased  expenditures   for   new
 facilities,  which  support  high  impact  economic   development
 projects."

     With  the  increase  in funding for necessary infrastructure
 repairs  and  improvements,  it  became clear to city and county
 engineers, the  Ohio  Department  of Transportation, the Greater
 Cleveland Regional  Transit  Authority,  and  the Northeast Ohio
 Regional Sewer  District  that  projects  were moving too slowly
 from design to construction.  The  various  agencies shared many
 problems, particularly the burden of  meeting  new  and  complex
 environmental  requirements while  processing  more  and  larger
 projects with a limited number of  professional  staff.  To meet
 this challenge, the several agencies worked  with private sector
 executives  to  develop,   test,   and   implement   performance
 improvement measures.  The payoff  has been a dramatic  increase
 in the agencies' ability to complete road and bridge projects at 
 a much faster pace.

    Whenever possible, environmental impact and engineering tasks 
 are  performed simultaneously rather than sequentially.  Scoping 
 meetings that include public and private sector participants are 
 conducted  early in  the project planning process.  The meetings
 sort  out  which  agency   or   entity   will   assume   primary
 responsibility  for  each   task   and   institute   cooperative
 mechanisms to ensure that projects  remain  on schedule and that
 problems  are  addressed  quickly.   The  application  of  value
 engineering techniques helps make certain that the right project
 is initiated for  the  right  reasons  within a time frame  that
 makes  sense  for all involved.  BUGC's recommendations for fast 
 tracking  highway  projects  have  worked  so  well the Governor
 George  Voinovich  has   encouraged   the  Ohio  Department   of
 Transportation and  the Ohio  Environmental Protection Agency to
 implement similar action on a statewide basis.

 \noindent {\bf How Process Affects Products ---
 Moving Projects Faster Means Getting More Public Investment}

   The fast-track approach has reduced costs, improved completion 
 times, and helped finance infrastructure projects.  Given that a
 great  deal  of federal and state financing for highway, bridge,
 sewer, and  transit project s is distributed  on a ``first come,
 first served" basis, it is  hustle --- and the  ability  to keep
 the bureaucratic pipelines full of ready-to-go projects --- that
 determines where public money is spent.

     Most  of  the  federal  funding  for highways and bridges is
 disbursed  with  time  constraints; that is, a jurisdiction must
 spend federal funds within a certain number of years or lose its
 funding allotment to other states. Therefore, a state department
 of transportation must, for  example, meet all federal and state
 planning  and  programming  requirements  while   spending   its
 allotment  of  federal  funds  within  the  stated  time  limit. 
 Accordingly,  cities  and  counties  with  ready-to-go  projects
 consistently  receive  funding.  On  the other hand, communities 
 that  fail  to  put  together ready-to-go projects are unable to
 attract anything close to their fair share of public investment.

 \noindent {\bf How Can Local Communities Translate
 These Successes to Their Own Settings?}

 \item{1.}  
 Develop a public/private infrastructure partnership team.  BUS-C 
 will  be pleased to provide you with advice and written material
 on creating a partnership and, in return, asks only for feedback
 on what  actions  you take and what works in your community.  In
 the  meantime, even if you have a partnership in place, consider
 the actions outlined below.

 \item{2.}
  Visit your metropolitan planning organization (MPO) to find out 
 about  the  availability  and requirements for state and federal
 funding.  If you do not live in an urban area, visit the nearest
 field  office  of  your  state  department  of   transportation. 
 Planners and engineers at this and other agencies can assist you
 in determining whether federal  participation is appropriate for
 a given project.  If participation is  appropriate,  staff  will
 tell you what steps are necessary to ensure federal financing.

 \item{\phantom{2.}}
     It  is  important  to recognize that projects can usually be
 completed more quickly in the absence of assistance. The federal
 government  conditions  the  receipt of funds on compliance with
 federal    standards   for   planning,   environmental,   public
 participation,  and  programming  actions.  For large, expensive
 public  works  improvements,  however,  federal   financing   is
 typically essential.
 \vfill\eject
 \item{\phantom{2.}}
   If your project can be successfully undertaken with only state
 and local financing, your MPO will advise you accordingly.  Your
 MPO  can  also greatly assist in guiding you through the federal
 and state  funding  processes  and  introducing  you  to the key
 players.

 \item{3.}  
     Identify the most  appropriate public  agency  or government
 sponsor  for  your  project and secure that party's agreement to
 fund  your  project.    Early  on,  talk  to  the  head  of  the
 appropriate agency about what you  need,  when  you need it, and
 the rationale for the project.   Ask  for and  follow the agency
 head's advice on how to ``feed the agency."  Identify  the staff
 members  who  will  be  assigned to your project and get to know 
 those individuals.  In  other words, find out who is responsible
 for your  project, what they need from you, and when you need to
 complete key  steps to  ensure adherence  to a mutually workable
 and realistic project schedule.

 \item{4.}   Be sure that elected officials are familiar with and
 support your project. Find out which public bodies must sign off
 on your project  and what specific actions are necessary.  Visit
 the appropriate elected  officials and describe how your project
 will  contribute  to  their  vision  of  and  priorities for the
 community.  Is a consent  ordinance  needed  from  city council? 
 Must the  MPO  board  of directors  include your project in  its
 short-  and  long-term  plans and project lists?  Does the state
 need to file project-related documents with the U. S. Department
 of  Transportation  or  U. S.  Environmental  Protection Agency? 
 Does the  state  legislature  assign  funds for projects such as
 yours?   Elected  officials  can  be  of  immense  assistance in
 spurring timely action by  public  agencies,  especially  if the
 officials  are  involved  in  the  process  early  and  consider
 themselves stakeholders in the project.

 \item{5.}
     Assume 100 percent proactive responsibility for keeping your 
 project moving.  If a delay occurs, identify the reason.  To the
 greatest  extent  possible, help the person who must resolve the
 problem get whatever he or she needs to move the project forward. 
 Keep all involved parties informed of project progress, and alert
 key public agency staff to changes as soon as possible.

 \item{6.}
 Say thank you often and keep your word; deliver on your promises.  
 Most  staff  at  public works agencies labor under hiring freezes
 and  have  not seen  a significant pay increase in years.  At the
 same time, they are responsible for large numbers of projects and
 must  comply  with  new  and  confusing  regulations.   They  are
 answerable to a diverse set of interests.  Let them know that you
 appreciate their efforts.

 \item{\phantom{6.}} 
     Encourage  public  agency  staff to let you know if a problem
 arises  or  if  you can do something to help keep your project on
 track.  If  staff  members  ask you for information, drawings, or
 legal or other  information,  tell  them when you will submit the
 requested materials to them.  Make sure that you deliver what you
 promised when you  promised.   Make  certain that any material is
 delivered in a form that  most  readily  serves  agency purposes. 
 Confirm  that  the  right  person  has  in  fact  received   your
 information.

 \item{\phantom{6.}}
     If the various steps look like a lot of work --- and for some
 projects  they  represent  a  full-time job --- remember that the
 Cleveland experience has proven to be well worth the effort.  The
 actions  described here have led to considerable success, but the
 process can be expanded and improved.  We look forward to hearing
 from  you  about  your  experiences  in  building  public/private
 partnerships.
 \vfill\eject
 To make comments or to request further information write or phone:

 Jack Licate, Director;
 Build Up Greater Cleveland;
 200 Tower City Center;
 50 Public Square;
 Cleveland, Ohio 44114; 216/621-3300
 \vfill\eject


 \centerline{ \bf Center Here; Center There;
                  Center, Center Everywhere!}
 \smallskip
 \smallskip
 \centerline{The Geographic Center of Wisconsin and the U.S.A.:}
 \centerline{Concepts, Comments, and Misconceptions}
 \smallskip 
 \centerline{Frank E. Barmore}
 \centerline{Department of Physics}
 \centerline{University of Wisconsin --- La Crosse}
 \smallskip
 \noindent Reprinted with permission from the Wisconsin Geographical
 Society:  {\sl The Wisconsin Geographer\/}, Volume 9, 1993, pp. 8-21.
 \smallskip
 \smallskip
 \noindent{\bf Abstract}

 Published  locations  of  geographic  centers  are  found  to  be
 inaccurate,  inconsistently  determined  and  in  serious need of
 revision.   The  definition  of  geographic  center is clarified.
 Methods of computation of two-dimensional distributions on curved
 surfaces  are  given.   An  accurate  location  for the center of
 Wisconsin is determined to be at
           latitude of $44^{\circ}38'04''$ N.,
           longitude of $89^{\circ}42'35''$ W.
 The uncertainty in the geographic center  of  the  United  States
 is discussed.  Recommendations for future further work are given.

 \noindent{\bf Introduction}

 For more than two thirds of a century the U.S. Geological  Survey
 has published information about the area and geographic center of
 the various states and the United States  (Douglas,  1923,  1930;
 Van Zandt,  1966,  1976;  and  pamphlets  of the  U.S. Geological
 Survey, 1967, 1991).  These publications are careful to point out
 the  uncertainty  and  limitation  of the data and  results.  For
 example, Douglas (1923, p. 221) states that ``That exact position 
 of the center of each State can not be  determined  from the data
 available, $\ldots $" and Van Zandt  (1966, p. 265)  states  that 
 ``There being  no  generally  accepted  definition of `geographic
 center' and no completely satisfactory method for determining it,
 a State or country may have as many  geographic  centers as there
 are definitions of the term." and,  ``Because  many factors, such
 as  the  curvature  of  the  earth,  large  bodies  of water, and
 irregular  surfaces,  affect  the  determination  of   geographic
 centers,  the  locality  of  the centers  should be considered as
 approximations only."   Since first  published,  the  information
 (with minor exceptions) has not been revised.

     Some  things  are  getting  better.  Adequate  data  are  now 
 available.  There  are  satisfactory  definitions.  Computers and
 powerful software are now widely available.  There are analytical
 means of taking into account the Earth's surface curvature. Large
 bodies of water are just as much a part  of  the whole as is  the
 land and should be included.  As a result, it is now  possible to
 determine ``geographic centers" to high accuracy. This paper will
 discuss these points and their impact on the determination of the
 geographic centers of Wisconsin and the United States.

 \noindent{\bf Definition of Geographic Center}

 The  lack  of  agreement  on  a  definition  of geographic center
 (center  of  area)  is  unfortunately  true.  Opinions range from
 despair  of  any  suitable  solution existing, expressed by Adams
 (1932),  to  enthusiasm  over  the  existence  of  an infinity of 
 centers, all equally valid (if not equally popular), outlined  by
 Neft (1966, p. 21).  I suggest that a reading  of the  literature
 will show that an intermediate  view  is widely held and a single
 definition  of  ``center"  is  agreed on:  {\bf The center of any
 distribution of things is the average  location of those things}.  
 It corresponds to the ``center of  gravity" or  ``balance  point"
 of the distribution.  In  Euclidean  spaces  the average location
 is most easily calculated  by  taking  the weighted vector sum of
 the location vectors  (vectors  whose magnitude and direction are
 the  distance  and  direction  of  the  various  things  in   the
 distribution)  and  dividing  by  the  total  weight   or   total
 population  of  the  things.   Such  a  center has the additional
 property that the sum of the squares of the distances between the
 center and the location of the various things in the distribution
 is minimum. This definition is equally suitable for distributions
 in one-, two- three-, or higher-dimensional Euclidean spaces.

 Almost a century ago, Hayford (1902) convincingly argued that the
 average location was the most appropriate center. D. I. Mendeleev
 (1907 and before) used formul{\ae} which may be  derived from the
 ``balance point" concept (derived by his son I. D. Mendeleev) for
 finding the geographic center and  population  center  of Russia. 
 Deetz (1918, p. 57) states that the  ``~`Geographic center of the
 United States'  is  here  considered  as a point analogous to the
 center of gravity of a spherical  surface  equally  weighted (per
 unit area) and of the outline of the country, and hence it may be
 found by means similar  to  those  employed to find the center of
 gravity."  All six  Geological Survey  publications, cited in the
 Introduction,  appeal to the  ``balance point" concept.  For more
 than a  century  the  U. S.  Bureau  of  the  Census has used the
 concept of a ``center of gravity" or ``balance point" as defining
 the U.S.A. population center (Barmore, 1991).

 In  spite  of  this  long  tradition, there are still dissenters.
 Kumler and Goodchild (1992,  p.~ 278) recommend that the point of
 minimum aggregate travel (M.A.T.) is the best  measure of  center
 of population.  I find it hard to accept some  of  their  reasons
 for this recommendation.  First, they say  that when  calculating
 the mean  or  average  location,  ``the  points,  or  people, are
 effectively weighted proportionally to their  distance  from  the
 center  ---  more  distant  people  have greater influence on the
 location of the mean center than people nearby."  But, it is {\bf
 location} that is being averaged  (weighted by  population), {\bf
 not  people}  being  averaged  (weighted   by   distance).   Each
 individual has exactly the same  weight in  finding  the  average
 location.  Second, they believe the M.A.T.  ``point does have one
 flaw --- it is insensitive to radial movement:  If a person moves
 1,000 kilometers directly toward  or  away from the mat [M.A.T.],
 the point will not move; if that  same person, however moves only
 a few kilometers in any other  direction  the [M.A.T.] point will
 move accordingly."  And this  shortcoming  ``is the least severe"
 shortcoming of the various  measures of center of population they
 discuss.   I  disagree.   Are  we  to  have  preferred  or  elite
 directions?  Shouldn't the center  of  a  distribution be equally
 sensitive to the motion of its component parts in any direction?

 I  suggest  that  the  term,  center,  should be reserved for the 
 average  (arithmetic  mean) location.  Other statistical concepts
 that  are  found to be useful should be labeled with names (other
 than center)  that  are  descriptive of what they represent.  For
 example, ``the point  of minimum aggregate travel," is just that;
 it should not be called the center.  To do otherwise is to invite 
 a  return  to  the confusion that existed earlier in this century
 when  the  point  of  minimum  aggregate  travel,  the center (or
 average) location and  the  median latitude (and/or longitude) of
 an area were often and incorrectly thought to be the same (Eells,
 1930).

 \noindent{\bf Geographic Center of a Curved Surface}

 As  mentioned  in  the Introduction, one difficulty that must  be
 dealt  with  is  the  curvature  of  the Earth's surface.  If the
 Earth's  surface  were  flat,  or  if  there  existed  a flat map
 projection which left area, distance,  and direction undistorted,
 the determination of geographic center of portions of the Earth's
 surface would be much simplified.  However, distributions  on the
 Earth's  curved  surface  are  spread  over   a   two-dimensional
 non-Euclidean space.  Traditionally there have been two different
 ways of responding to this problem.

 One response is to find a higher dimension space that is Euclidean
 in  which  to  embed  the non-Euclidean space.  Then the necessary
 calculations  can  be  carried  out  using  the familiar Euclidean
 geometry.  Thus, one can embed the two-dimensional Earth's surface
 in  a  three-dimensional  Earth's  surface  in a three-dimensional
 Euclidean  space  and  calculate  the  three-dimensional   average
 location,   balance   point,   or   ``center  of  gravity".   This
 three-dimensional  approach  is  equivalent  to  the  one sentence
 definition given by Deetz (1918)  and  results  in the formul{\ae}
 given and used by Mendeleev (1907) for  population  and geographic
 centers on a spherical Earth.   The  method can easily be extended
 for distributions  on  the  surface  of an ellipsoid of revolution
 representing  the  Earth, though the formul{\ae} are more complex. 
 The  resulting  centers  are  below  the  surface  and I find this
 distasteful.

 The  second  response is to adapt and restrict the calculations to
 the  two-dimensional  non-Euclidean  space.   As I have previously
 described  in  some  detail  (Barmore,  1991,  1992)  this  second 
 solution  is  preferable.   The  result is a method that restricts
 the  computations  of  average  location  and  the  outcome to the
 surface  of  a  sphere  or  an  ellipsoid of revolution which very
 closely approximates the Earth's surface.

 \noindent{\bf Geographic Center of Wisconsin}

 There  exists,  several hundred feet south of the geometric center
 of  the  City  of Pittsville, Wood Co., Wisconsin, a monument with
 the following text:
 \smallskip
 \smallskip
 \hrule
 \smallskip
 \centerline{\bf Center of the State of Wisconsin}
 \smallskip
 \smallskip
 \centerline{\sl In the early 1950's Governor Walter J. Kohler, Jr.}
 \centerline{\sl frequently visited the Pittsville area.}
 \centerline{\sl On one such trip he Proclaimed Pittsville to be}
 \centerline{\sl the exact center of the State by Official Proclamation}
 \centerline{\sl on the 27th of June, 1952.}
 \centerline{\sl Professional Land Surveyors established the corner}
 \centerline{\sl lying 250 feet North of where you are now standing.}
 \smallskip
 \smallskip
 \centerline{\nn This monument donated by the Central Chapter}
 \centerline{\nn of the Wisconsin Society of Land Surveyors}
 \centerline{\nn Erected July 1987}
 \smallskip
 \smallskip
 \centerline{\nn Wayside construction donated by Cedar
                 Corporation, Marshfield.}
 \centerline{\nn Dale Decker Surveying; Esser Trucking, Arpin;}
 \centerline{\nn Mid State Associates; People's State Bank, Pittsville.}
 \smallskip
 \hrule
 \smallskip
 \smallskip

 The text of the proclamation  (Kohler, 1952) gives no hint  of how
 or when it was determined  that  the  center  of  Wisconsin was at
 Pittsville.  The Geological Survey places the Wisconsin geographic
 center at ``9 miles southeast of Marshfield."  This point is 16 km
 from the Pittsville monument.

 The geographic centers of the various states were first  published
 by the U.S. Geological Survey (Douglas, 1923, p. 221-222).   Since
 then, and until as recently as 1991, the centers for  most of  the
 States  and  particularly  for Wisconsin  have remained unrevised.
 Thus, the most recently published center of Wisconsin reflects the
 boundaries  and  geographic  data  quality  as of 1923 or earlier. 
 Also, according to the very brief definition accompanying the list
 of centers and Adams' (1932) lament that no analytical process was
 available, the outcome is only approximate.  Thus, the results are
 of low accuracy.  Third, the Great Lakes and some islands were not
 included when determining the centers.  Thus, significant portions
 of Wisconsin were not included.  Clearly, these centers  are  ripe
 for revising.

 It is now possible to calculate the geographic center of Wisconsin
 to much higher accuracy.  I have determined the geographic  center
 of  Wisconsin  with  an  uncertainty  of  less  than  0.1 km.  The
 determination was done for the center of all land and  water areas
 including those portions of the Great Lakes within Wisconsin.  The
 center  is  in the east central portion of Sec. 19, R 7 E, T 25 N,
 in the  Town  of  Eau  Pleine, Portage Co.  A second determination
 was done for the center  of the land area and ``inland waters" for
 comparison  with  the  previous   determination   given   by   the 
 Geological Survey.  This ``center"  is near the center of Sec. 23,
 R 4 E, T 25 N, a little northeast of  the  northeast corner of the
 city  of  Auburndale  in Wood Co. and is about 8 km from the point
 published  by  the  Geological Survey.  Based on these results, it
 would be reasonable to assume that one could expect similar errors
 in the existing published locations of the other state centers and
 they are also in need of revision.  These and previous results for
 Wisconsin are given in Table 1 and displayed on a map in Figure 1. 
 (The computational details and assumptions are given in Appendix A).
 \vskip.5in
 \vskip.5in

 \topinsert
 \hrule
 \vskip.5cm
 \noindent{Table 1.  Wisconsin Geographic Center According to
                     Various Sources}
 \vskip.5cm
 \hrule
 \smallskip
 \settabs\+\quad
   &Center of land and ``inland waters"\quad
   &USGS 1923\quad
   &$44.5728^{\circ}$\quad
   &$89.7098^{\circ}$\quad
   &\cr %sample line
 \+&{\bf Description of Computation}&{\bf Source}
   &{\bf N. lat.}&{\bf W. long.}&  \cr
 \smallskip
 \+&Center of all of Wisconsin&this work
   &$44.6344^{\circ}$&$89.7098^{\circ}$ &\cr
 \+&Center of land and ``inland waters"&this work
   &$44.6351^{\circ}$&$89.9923^{\circ}$&  \cr
 \+&9 miles southeast of Marshfield, WI&USGS 1923
   &$44.5728^{\circ}$&$90.0441^{\circ}$&   \cr
 \+&On the Pittsville, WI monument&Gov. 1952
   &$44.4384^{\circ}$&$90.1301^{\circ}$& \cr
 \vskip.5cm
 \hrule
 \smallskip
 \hrule
 \smallskip\smallskip\smallskip
 \vskip.5cm
 \hrule
 \vskip.5cm
 \noindent{Table 2.  Geographic Center of the Conterminous United States}
 \vskip.5cm
 \hrule
 \smallskip
 \settabs\+\quad
   &On a Lambert Azimuthal Equal Area map\quad
   &Deetz 1918\quad
   &$39.7872^{\circ}$\quad
   &$98.9830^{\circ}$\quad
   &\cr %sample line
 \+&{\bf Description of Computation}&{\bf Source}
   &{\bf N. lat.}&{\bf W. long.}&  \cr
 \smallskip
 \+&On Clarke's (1866) ellipsoid surface&&&&\cr
 \+&\quad a) land \& inland waters only&this work  
   &$39.7872^{\circ}$&$98.9830^{\circ}$ &\cr
 \+&\quad b) all land \& water areas&this work
   &$39.9074^{\circ}$&$98.6843^{\circ}$&  \cr
 \+&In three dimensions&this work
   &$39.9020^{\circ}$&$98.6909^{\circ}$&  \cr
 \+&On a Lambert Azimuthal Equal Area map&this work
   &$39.8785^{\circ}$&$98.6593^{\circ}$& \cr
 \+&On Albers Equal Area Conic projection&this work
   &$39.8352^{\circ}$&$98.6896^{\circ}$& \cr
 \+&Analogue:  Balancing flat map (??)&Deetz 1918
   &$39.8333^{\circ}$&$98.5833^{\circ}$& \cr
 \vskip.5cm
 \hrule
 \smallskip
 \hrule
 \endinsert

 \topinsert \vskip6.3in
 \noindent{\bf Figure 1.}  Wisconsin Geographic Centers according to 
 various sources.  The point labeled ``CENTER OF WISCONSIN" is the
 center calculated for all the land and water area within the
 boundaries of Wisconsin.  The location uncertainty of the point
 is not noticeable on a map of this scale.  The point labeled
 ``CENTER, LAND ONLY" is the center calculated for all the land and
 ``inland waters" but excluding the portions of the Great Lakes
 lying within Wisconsin.  The location uncertainty of this point is
 not noticeable on a map of this scale.  The point labeled
 ``CENTER, USGS, 1923" is the center published by the U.S. Geological
 Survey since 1923.  The ``error bars" indicate the probable 
 uncertainty implied by the manner in which the various State center
 locations were stated.  The point labeled ``CENTER, KOHLER, 1952"
 inside the boundaries of Pittsville is the result of Governor
 Kohler's 1952 Official Proclamation.  The location uncertainty
 and method of determination of this point are unknown.
 \endinsert

 \noindent{\bf Geographic Center of the Conterminous United States}

 The geographic  center  of  the  ``Conterminous" United States (48
 States  and  the  District  of  Columbia)  is  widely published on 
 maps,  in  atlases  and  in  government  documents,  as being near
 Lebanon,  Smith  County, Kansas, at latitude of $39^{\circ} 50'$ N
 and at longitude of $98^{\circ} 35'$ W.

 All sources  for  this  and similar statements that can be traced,
 ultimately refer to a one sentence statement with a brief footnote
 published by Deetz (1918, p. 57) that reads:

 {\narrower\smallskip\noindent
 ``The Geographic center ($^*$) of the United States is
 approximately in latitude $39^{\circ} 50'$ and longitude
 $98^{\circ} 35'$.
 \smallskip
 \smallskip
 $(^*)$ `Geographic center of the United States' is here
 considered as a point analogous to the center of gravity
 of a spherical surface equally weighted (per unit area)
 and of the outline of the country, and hence it may be
 found by means similar to those employed to find the
 center of gravity"
 \smallskip}

 There  is  a hint as to how this might have been determined in the
 melancholy paper by Adams (1932) which states:

 {\narrower\smallskip\noindent
 ``A method that was used in the Coast and Geodetic
 Survey a number of years ago was the following:
 An equal-area map of the United States was constructed
 on thin cardboard and then the outline map was cut out
 along the various boundaries.  The center of gravity of
 this outline map was then determined." \smallskip}

 As  this was done in an analogue way (on what must have been a map
 of modest scale)  rather  than  calculated  in  a precise way, the
 result  is  probably  of  modest accuracy.  Note that: (a) It is a
 flat  (and  therefore  distorted)  map  not  a spherical map whose
 center was found.  (b) It is not stated which map  projection  was
 used  to  produce  the  map.  (c) It is not stated what boundaries
 were used.

 In  an attempt to reproduce Deetz's result, this geographic center
 was  recomputed  in  a  variety  of  ways.  If  only the areas and 
 centers  of  the  land  and ``inland waters" of the various states
 were  used  the  agreement was very poor.  However, if the list of
 areas and centers used was expanded to include the portions of the
 Great Lakes within  the  United  States and to include the various
 sounds, straits, bays and coastal waters that are not  part of the
 ``inland waters"  of  the  various  states, then  modest agreement
 could  be  achieved  (see  Appendix  B   for   details  of   these
 calculations). The results are summarized in Table 2 and displayed
 on a map in Figure 2.  Because of the low quality of the data used
 in  the  computation,  these  results  should  not  be  considered
 accurate.

 \topinsert \vskip6.3in
 \noindent{\bf Figure 2.}  Geographic Center of the Conterminous
 United States determined by various computational methods. 
 The point labeled ``CENTER, 48 STATES, ALL AREAS" is the
 average location of all the various areas that make up the
 conterminous United States calculated on the surface of
 Clarke's (1866) ellipsoid using the preferred method (Barmore,
 1991, 1992).  The point labeled ``CENTER, LAND \& INLAND WATERS ONLY"
 is the average location of the various areas (the Great Lakes
 and other ``non-inland waters" being excluded) that make up the
 conterminous United States calculated in the same manner.
 Because of the limited accuracy of the data used, neither of
 these locations nor the other center locations displayed in 
 this figure should be considered as accurate.  The point 
 labeled ``3-D" is the three-dimensional average location,
 projected onto the surface, of all the areas that make up
 the conterminous United States.  The point labeled ``DEETZ,
 USCGS, 1918" is the widely quoted result.  The points
 labeled ``L" and ``A" are the centers determined by an
 analytical computation that is equivalent to finding the 
 balance point of a Lambert Azimuthal Equal Area map and an
 Albers Conical Equal map, respectively, of all areas of the
 conterminous United States.
 \endinsert


 \noindent{\bf Geographic Center of the United States}

 Apparently, the  geographic  center  of  The  United  States  (50
 States and the District of Columbia)  was  determined by the U.S.
 Coast  and  Geodetic  Survey (ca. 1959) in a manner described, if
 nowhere  else,  in  several  news releases.  The accuracy of this
 result is questionable for reasons outlined below.

 The center of all 50 states was apparently determined, piecemeal, 
 as follows:   The  48  states were represented as being 3,022,400
 square miles in area at  the previously determined location given 
 by  Deetz  (1918)  at   latitude  $39^{\circ} 50'$ N.,  longitude
 $98^{\circ}35'$.   Alaska's  land  and ``inland waters" area were
 represented  as  being   586,400   square   miles   at   latitude
 $63^{\circ}50'$ N., longitude  $152^{\circ} 00'$ W.   The balance 
 point   of   these   two  areas  was  found  to  be  at  latitude 
 $44^{\circ} 59'$ N.,   longitude  $103^{\circ} 38'$ W.   on   the
 (presumed great circle) arc between them. Then when Hawaii joined
 the  Union,  the  process  was  repeated.   The  49  states  were
 represented  as  the  sum  of  the  previous two areas (3,608,800
 square miles) located at their  balance point.  Hawaii's land and
 ``inland  waters"  area  were represented as 6424 square miles at
 latitude  $20^{\circ} 15'$ N.,  longitude   $156^{\circ} 20'$  W. 
 The balance point of these two  areas was found to be at latitude 
 $44^{\circ} 58'$ N., longitude $103^{\circ} 46'$ W.

 If  the  Earth's  surface  were  flat, this procedure would be as
 accurate  as  the  data  used  would allow.  However, the surface
 is  not  flat,  but  curved.   When the distances are as large as
 those  between  the various states of the United States, ignoring
 the  curvature  can result in a substantial error (Barmore, 1991,
 1992).  If the center is to be determined with distances measured
 on  the  curved  surface of the Earth, it must be redone from the
 beginning with each addition.

 Another difficulty has to do with using data of mixed consistency. 
 The 3,022,400 square mile figure for the 48 States  is  the  land
 plus ``inland waters" only.  The location used  for  this area is
 apparently the center of a different area  --- the land, ``inland
 waters," {\bf  and  a substantial area of  ``non-inland waters}." 
 (These ``non-inland waters" have an  area  of about 74,364 square
 miles, 2.4\% of the 3,022,400  square mile figure [U.S. Bureau of
 the Census, 1940].)

 In  order  to  illustrate  the  differences  that can result, the
 geographic  center  for  the  entire United States was calculated
 various  ways.   The  results  are  summarized  in  Table  3  and
 displayed  on  a map in Figure 3.  the same methods and data were
 used that were used in the preceding example with  the  exception
 that the total of all land and all  water areas  for  Alaska  and
 Hawaii were those given most recently (U.S. Bureau of the Census,
 1992, table 340).  Because the centers  and  areas  used have not
 been  revised  (with  the  exception  of the Alaskan and Hawaiian
 areas) the results should not be taken as accurate.
 \vskip.5in
 \vskip.5in

 \topinsert
 \hrule
 \vskip.5cm
 \noindent{Table 3.  Geographic Center of the United States
                      (All 50 States)}
 \vskip.5cm
 \hrule
 \smallskip
 \settabs\+\quad
   &\quad ``inland waters" only.  For comparison. \quad
   &news release\quad
   &$45.4344^{\circ}$\quad
   &$104.3524^{\circ}$\quad
   &\cr %sample line
 \+&{\bf Description of Computation}&{\bf Source}
   &{\bf N. lat.}&{\bf W. long.}&  \cr
 \smallskip
 \+&On Clarke's (1866) ellipsoid surface&this work
   &$45.4344^{\circ}$&$104.3524^{\circ}$& \cr
 \+&In three dimensions&this work
   &$45.2517^{\circ}$&$104.1776^{\circ}$&  \cr
 \+&On ellipsoid surface.  Land and&
   && &  \cr
 \+&\quad ``inland waters" only.  For comparison.&this work
   &$44.9482^{\circ}$&$104.1189^{\circ}$& \cr
 \+&U.S. Coast and Geodetic Survey&news release
   &$44.9667^{\circ}$&$103.7667^{\circ}$& \cr
 \vskip.5cm
 \hrule
 \smallskip
 \hrule
 \smallskip\smallskip\smallskip
 \vskip.5cm
 \hrule
 \vskip.5cm
 \noindent{Table 4.  Wisconsin Geographic Center Calculated Two Ways}
 \vskip.5cm
 \hrule
 \smallskip
 \settabs\+\quad
   &On a two-dimensional non-Euclidean surface\quad
   &$44.6343739^{\circ}$\quad
   &$89.7097544^{\circ}$\quad
   &2.4 km\quad
   &\cr %sample line
 \+&{\bf Description of Computation}
   &{\bf N. lat.}&{\bf W. long.}&{\bf depth}&  \cr
 \smallskip
 \+&In a three-dimensional Euclidean volume
   &$44.6343739^{\circ}$&$89.7097544^{\circ}$&2.4 km &  \cr
 \+&On a two-dimensional non-Euclidean surface
   &$44.6343818^{\circ}$&$89.7097566^{\circ}$&0.0 km &  \cr
 \vskip.5cm
 \hrule
 \smallskip
 \hrule
 \endinsert

 \topinsert \vskip6.3in
 \noindent{\bf Figure 3.}  Geographic Center of the
 United States (all land and water areas of all 50 States
 and the District of Columbia) determined various ways.  The 
 point labeled ``CENTER, ALL AREAS OF USA" is the average location
 of all the various land and water areas that make up the
 United States calculated on the surface of Clarke's (1866)
 ellipsoid using the preferred method (Barmore, 1991, 1992).
 Because of the limited accuracy and limited internal
 consistency of the data used, neither this location nor
 the other center locations displayed in this figure should be
 considered as accurate.  The point labeled ``3-D" is the 
 three-dimensional average location, projected onto the surface,
 of the same areas, that make up the United States.  The point
 labeled ``USCGS, 1959" is the widely quoted result.  The point
 labeled ``CENTER, LAND \& INLAND WATERS ONLY" is the center
 of all land area combined with only the ``inland waters"
 area, the Great Lakes and ``non-inland waters" being excluded.
 This center, calculated on the Earth's curved surface, 
 corresponds most closely to the U.S. Coast and Geodetic 
 Survey procedure for determining the geographic center.  It is
 presented here for comparison.
 \endinsert

 \noindent{\bf Summary and Recommendations}

 The geographic centers and areas  of  the  various States and The
 United  States  are  in  serious  need  of  revision  for several
 reasons.  In the seventy years that have passed since the centers
 were determined, much has happened.  Mapping of the United States
 is  much  improved.   Computational  capability  is  now   widely
 available ---it should no longer be necessary to make compromises
 for  computational  reasons.   Data  on  land and water  area are
 much  improved.  It  should  now  be  possible  to   compute  the 
 location of the  various  centers  to  an accuracy of ca. $10$ m. 
 The following  recommendations are made for this revision and any 
 similar sort of statistical analysis.

 \item{I.}
 The  term  {\bf center}  of spatial distributions should be
 reserved for the average (arithmetic mean) location.  Other
 statistics of spatial  distributions  that  are found to be
 useful should be given other names to avoid confusion.

 \item{II.}
 If  the  distribution covers enough of a curved surface for
 the curvature of the surface to be noticeable, then special
 care  must  be  taken.   Unless appropriate compensation is
 made  for the Earth's surface curvature, these calculations
 may  not  be  properly  done using any flat map projection. 
 There  is  no  flat  map of the Earth's curved surface that
 leaves  area,  distance,  and  direction  undistorted.  For
 distributions on the surface of the earth, the computations
 of average  location  should be  carried out on the surface
 and the results  restricted to the  surface.  The method of
 doing this is outlined in some detail  elsewhere  (Barmore,
 1991, 1992).  Alternately, the computations  can be carried
 out in three dimensions using more familiar procedures, but
 the computation of two-dimensional distribution  statistics
 in two dimensions is preferable.

 \item{III.}  If geographic centers  of hierarchical sets of
 areas  are  presented,  they should be done in a consistent
 way so that comparisons are easy within a level and between
 levels.  It  should  be  possible  at any level to find the
 average of the larger group by averaging over its component
 parts.  In particular, if centers at one level for separate
 land  and  water  areas  are  given,  the  centers  for the
 subdivisions should  be  separated  in the same manner.  If
 ``non-inland waters"  are excluded at one level they should
 be excluded at all levels.

 \item{IV.} What is included (or excluded) should be clearly
 stated.  The  absence of any discussion of what is meant by
 the term ``North  America" makes meaningless the statements
 concerning the center of North America published by the U.S.
 Geological Survey (Douglas, 1930; and pamphlets by the U. S.
 Geological  Survey, 1967 and 1991).  Is Greenland included? 
 Are ``non-inland  waters"  included?  Are off-shore islands
 included?

 \vfill\eject


 \noindent{\bf Appendix A:  Calculation of Wisconsin's
                            Geographic Center}

 All areas and centers were determined assuming they  lay  on  the
 surface  of  Clarke's  (1866)  ellipsoid  ($a=6378.2064$  km  and
 $e=0.08227185$).

 The  State's  surface  and  adjacent  areas were divided into $30
 \times  60$  minute  quadrangles.   For  $30  \times  60$  minute
 quadrangles that lay completely inside the State  boundaries  (or
 had more than half their area within the  boundaries)  the  areas
 and centers were calculated using the ellipsoid geometry found in
 Bomford (1977).  These results  are  very accurate.  Wherever the
 boundary cut a quadrangle,  the areas and centers were determined
 from the 1:100000, $30 \times 60$ minute quadrangle maps published
 by the  Geological  Survey.   If  less than half the quadrangle's 
 area was within Wisconsin, only the portion within the  State was
 considered.  If more than half the quadrangle's  area  was within
 the State, the area and center of the portion to be excluded were
 determined and subtracted from the  previously  calculated values
 for the entire quadrangle.  This process minimizes the areas that
 had to be measured rather than calculated.

 The  areas  and  centers  that  had  to  be measured were done as
 follows:   a)   If  the  areas were composed of quadrilaterals or
 triangles, the areas and centers were calculated from measurements
 taken directly  from  the  map.  b)  If the areas were irregular,
 they  were  carefully  traced  onto  a  uniform sheet whose areal
 density  had  been  previously  determined  with  the  aid  of an
 electronic ``balance," cut out, reweighed to determine their area
 and suspended from several points to determine their centers.  c) 
 The latitude and longitude of the centers  were  then  determined
 directly from the geographic grid on the map.  d)  The areas were
 then  corrected  for  scale  changes.  The scale changes have two
 causes:  First, very small variations in scale resulting from the
 Universal Transverse Mercator projection (Snyder, 1987, p. 58-64). 
 Second, scale changes  due  to  expansion or shrinkage of the map
 paper caused by humidity  changes  (determined  from measurements
 of the 10000 m grid on the map).

 This  process  created  a  collection   of   111   area   elements
 representing  the  State.   Over 87\% of  the area (represented by
 the 37 full $30 \times 60$ minute quadrangles) in the calculations
 of center have calculated areas and centers for which the accuracy
 is very high.  For the remaining  13\% of the area (represented by
 74 fractional areas averaging  325 sq.  km)  the  accuracy  of the
 areas is probably limited by how well the areas were corrected for
 scale changes caused by  humidity  changes.  As a check, the total
 area of land and  ``inland waters"  was found to be 145435.166 sq.
 km = 56152.8  sq.  miles.  This compares  favorably with the 56153
 sq. miles listed as the area of Wisconsin in the 1980 Census (U.S.
 Bureau of the Census, 1983).  Also, the total  area  of  Wisconsin
 (including the portion of the Great Lakes falling within Wisconsin)
 was found to be 169609.8 sq. km.  The  Bureau of the Census (1992)
 reports  the  total  area  of  Wisconsin to be 169653 sq. km.  The
 difference  of  43  sq.  km  may  be due to disagreement about the
 boundaries  of  the  State  in  Lake  Michigan.  I  have  used the
 boundaries  shown  on  the  1:100000  scale, $30 \times 60$ minute
 series maps published by the Geological Survey.  These boundaries,
 in turn, are in agreement with those given in Van Zandt (1976) and
 further clarified in the 1948 Compact between Michigan, Wisconsin,
 and Minnesota which finally settled the boundary (U.S. Statutes at
 Large, 1948).  Other sources show  a  different  boundary --- {\sl
 The National Atlas\/}  (U.S.  Geological  Survey, 1970, p. 17, 19,
 313) or  the  Geological  Survey  map, {\sl State of Wisconsin\/},
 1:500,000 scale, 1966 comp., 1968 ed., for example.  In the  worst
 possible case an error of this magnitude would shift the center of
 Wisconsin two or three seconds of arc or about 50 m on the surface.  

 The  State  center  was  then  calculated  by  finding the average
 location of the 111 area elements.  This calculation  was done two
 ways:   first  as  a  three-dimensional  volume  distribution  and
 second as a two-dimensional surface distribution  (Barmore,  1991,
 1992).   For  areas  the  size  of  Wisconsin,  there  is   little
 difference between the two results except for depth.  For example,
 see Table 4.  The difference is only a few hundredths of a  second
 of arc, and corresponds to a distance of one or two meters  on the
 surface.

 In order to provide a comparison for the Center of Wisconsin given
 by  Douglas  (1923),  that  included the land and ``inland waters"
 only, this center was also redetermined.  Therefore, the  process,
 outlined above, was repeated for a somewhat  different  collection
 of 111 area  elements  (30  full $30 \times 60$ minute quadrangles
 and 81 fractional areas averaging 197  sq.  km;  representing 82\%
 and  18\%,  respectively,  of the  areas used in the calculation). 
 These  area  elements  represent the area of the land and ``inland
 waters",  but  not  the  Great  Lakes,  within  the  boundaries of
 Wisconsin.

 \noindent{\bf Appendix B:  Calculation of the Geographical
 Center of the Conterminous United States}

 The  geographic  center  of  the  conterminous United  States was
 calculated  using  methods  previously  described.   The  centers
 calculated  on  the  curved  surface  in  two  dimensions or when
 treating  the  areas  as  a three-dimensional volume distribution
 assumed Clarke's (1866) ellipsoid (though the data quality hardly
 justifies such accuracy).  The centers calculated by distributing
 the areas on the  surface of various flat maps used equations for
 the projections given by Snyder (1987, p. 100-101, 185-187) for a
 spherical  earth.   The  Lambert  Azimuthal  Equal  Area  map was
 centered at $38^{\circ}$ N. latitude, $95^{\circ}$ W.  longitude, 
 following Deetz (1918, p. 57) and the Albers Equal Area map  used
 two standard parallels at $29^{\circ} 30'$ and $45^{\circ} 30'$ N.
 latitude as suggested by Deetz and Adams (1945, p. 94).

 The  data  used  consisted  of two parts.  The first part  was the
 areas of land and ``inland waters" and centers as given by Douglas
 (1923, p. 219, 222) for the 48 States and the District of Columbia. 
 If the example of Wisconsin is typical,  the accuracy of this data
 is not high.  More recent and  probably  better data were not used
 because the 1923 data for area  are nearly identical to that given
 by Gannet (1906, p. 7, 8) and thus more characteristic of the data
 available to Deetz than more  modern material.  The second part of
 the data was for the ``non-inland waters".  The areas included are
 those delineated  earlier (U.S. Bureau of the Census, 1942, Map I,
 and Table IV).  The  approximate  centers  for  these ``non-inland
 water"  areas  were  determined  from  maps  in  {\sl The National
 Atlas\/} (U.S. Geological Survey, 1970).

 Because  of  the  uncertainty in the areas and centers of the area
 elements whose locations were averaged to get these results,  they
 should not be considered accurate.

 \vfill\eject
 \noindent{\bf References}
 \smallskip
 \ref Adams, Oscar S.  1932. 
 ``Geographical Centers."
 {\sl The Military Engineer\/},
 Vol. XXIV, No. 138, pp. 586-7.

 \ref Barmore, Frank E.  1991.  
 ``Where Are We?  Comments on the Concept of the `Center of Population' "
 {\sl The Wisconsin Geographer\/},
 Vol. 7, 40-50. 
 (Reprinted (with the  example data set used and with several corrections)
 in
 {\sl Solstice:  An Electronic Journal of Geography and Mathematics\/},
 Vol. III, No. 2, pp. 22-38.  Winter 1992.
 (Inst. of Mathematical Geography, Ann Arbor, MI)).  

 \ref Barmore, Frank E.  1992. 
 ``The Earth Isn't Flat.  And It Isn't Round Either! 
 Some Significant and Little Known Effects
 of the Earth's Ellipsoidal Shape."
 {\sl The Wisconsin Geographer\/}, Vol. 8, 1-9. 
 (Reprinted in
 {\sl Solstice:  An Electronic Journal of Geography and Mathematics\/},
 Vol. IV, No. 1, pp. 26-38.  Summer 1993.
 (Inst. of Mathematical Geography, Ann Arbor, MI)).  

 \ref Bomford, G. 1977. 
 {\sl Geodesy\/}. 
 (Oxford UK, Clarendon Press)
 a reprinting (with corrections) of the 1971 3rd Ed.

 \ref Deetz, Charles H. 1918. 
 {\sl The Lambert Conformal Conic Projection
 with Two Standard Parallels, etc.\/} 
 (Special Publication No. 47). 
 Washington DC, U.S. Coast and Geodetic Survey.

 \ref Deetz, Charles H. and Oscar Adams, 1945. 
 {\sl Elements of Map Projection with Applications
 to Map and Chart Construction\/}
 (U.S. Coast and Geodetic Survey Special Publication 68, 
 5th Ed., 1944 revision). 
 Washington DC, U.S. Government Printing Office.

 \ref Douglas, Edward M. 1923. 
 {\sl Boundaries, Areas, Geographic Centers and Altitudes
 of The United States and the Several States, etc.\/} 
 (Bulletin 689). 
 Washington DC, U.S. Geological Survey.

 \ref  Douglas, Edward M. 1930. 
 {\sl Boundaries, Areas,  Geographic Centers and Altitudes
 of the United States and the Several States, etc.\/},
 2nd Edition (Bulletin 817). 
 Washington DC, U.S. Geological Survey.

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

 \ref  Gannett, Henry 1906. 
 {\sl The Areas of the United States,
 The States, and the Territories\/}
 (Geological Survey Bulletin 302). 
 Washington DC, U.S. Geological Survey.

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

 \ref Kohler, Walter J. Jr.  1952. 
 No trace of the Proclamation could be found in the papers of
 Gov. Kohler or in the State Archives housed in the State Historical
 Society of Wisconsin, Madison.  A copy of the Proclamation is on
 display in the City Council Chamber, Pittsville, Wisconsin.

 \ref Kumler, Mark P. and Michael F. Goodchild.  1992.
 ``The Population Center of Canada -- Just North of Toronto?!?"
 in Donald G. Janelle (Editor)
 {\sl Geographical Snapshots of North America; etc.\/}
 New York, NY, The Guilford Press. pp. 275-279.

 \ref Mendeleev, D.I.  1907. 
 {\sl K Poznaniyu Rossii\/},
 5th ed.  (St. Petersburg, A. S. Suvorina) p. 139.

 \ref Neft, David S.  1966. 
 {\sl Statistical Analysis for Areal Distributions\/}
 (Monograph Series Number Two),
 Philadelphia, PA, Regional Science Research Institute.

 \ref  Snyder, J.P.  1987. 
 {\sl Map Projections --- A Working Manual\/},
 (USGS Professional Paper, 1395).
 Washington DC, U.S. Government Printing Office.

 \ref U. S. Bureau of the Census.  1942.  
 {\sl Sixteenth Census of the United States:  1940. 
 Areas of the United States:  1940\/}. 
 Washington DC, U.S. Government Printing Office.

 \ref U. S. Bureau of the Census.  1983. 
 {\sl 1980 Census of Population, Vol. 1, Chapter A, Part 1\/}
 [PC80-1-A1].
 Washington DC, U.S. Department of Commerce,
 Bureau of the Census.  Table 11, pp. 1-47.

 \ref U. S. Bureau of the Census.  1992.
 {\sl Statistical Abstract of the United States:  1992\/}.
 (112th Ed.). 
 Washington DC, U.S. Government Printing Office.

 \ref U.S. Coast and Geodetic Survey.  ca. 1959. 
 News releases titled: 
 ``New Geographic Center of the United States," Aug.(?) 1958;
 ``Geographic Center of U.S. Moved Again with Admission of 50th State,"
 March 1959; and,
 ``Geographic Center of the United States," n.d.
 Copies supplied the Earth Sciences Information Center
 (ESIC), U.S. Geological Survey, Reston, VA.

 \ref U.S. Geological Survey.  1967. 
 {\sl Geographic Centers of the United States\/} (pamphlet).
 Washington DC, U.S. Geological Survey.

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

 \ref U.S. Geological Survey.  1991. 
 {\sl Elevations and Distances in the United States\/} (pamphlet). 
 Washington DC, U.S. Geological Survey.

 \ref U.S. Statutes at Large.  1948.  Vol. 62, p. 1152,
 Chapter 757.  S.J. Res. 206.  June 30, 1948.  Public Law 844.

 \ref Van Zandt, Franklin K.  1966. 
 {\sl Boundaries of the United States and the Several States\/}
 (Geological Survey Bulletin 1212). 
 Washington DC, U.S. Geological Survey.

 \ref Van Zandt, Franklin K. 1976. 
 {\sl Boundaries of the United States and the Several States\/}
 (Geological Survey  Professional Paper 909). 
 Washington DC, U.S. Geological Survey.

 \vfill\eject

 \centerline{\bf 4.  ARTICLES}
 \smallskip
 \centerline{\bf Equal-Area Venn Diagrams of Two Circles: 
               Their Use with Real-World Data.}
 \vskip.5cm
 \centerline{Barton R. Burkhalter}
 \centerline{Senior Program Officer}
 \centerline{Academy for Educational Development}
 \centerline{Washington, D.C.}
 \vskip.2cm
 \centerline{First draft 12/21/90; revised 2/18/91.}
 \centerline{Communicated by John D. Nystuen to Solstice, 5/3/93.}
 \vskip.5cm
 \noindent{\bf General Problem}

      We are concerned with  populations  whose  members  have two
 discrete  characteristics;  that  is,  characteristics  which are 
 either  present  or absent in each member.  In populations having
 two  characteristics,  the  populations  can  be described by the 
 proportion  of  the  population  that  has  both  characteristics
 present, one or the other (but not both) characteristics present,
 or  both  characteristics  absent.  One well-known way to present
 this  data  is  with  a  two  circle  Venn  Diagram in which each
 circle  represents  one  of the characteristics, the intersection
 of  the circles represents the members with both characteristics,
 and  the  region  outside both circles but within the universe of
 discourse  (depicted  as a bounded figure surrounding the circles
  ---  often  a  rectangle)  represents  the  members with neither
 characteristic.  Appropriate regions might then be labelled  with
 suitable percentages, whether or  not the  geometric intersection
 pattern  is  suggestive  of the  numeric partition of the sample.
 At this point, it may be useful to the reader to draw a two-circle
 Venn diagram.

     For example, consider a population of countries with national
 child   vaccination  programs.    Some  of  the countries in  the
 population use a campaign strategy, some a clinic-based strategy,
 some  use  both  strategies,  and  a  few  use  neither strategy. 
 Construct a Venn diagram to  represent  this  grouping  of  mixed
 strategies.   Draw  Circle  $\alpha $  on  the  left  and draw an
 intersecting Circle $\beta $ on the right.  Draw a rectangle that
 is large enough to easily contain all of the intersecting  circle
 configuration.   In  this  Venn  diagram,  Circle $\alpha $ could
 represent countries using a campaign strategy and Circle $\beta $
 could represent countries with a clinic-based strategy. Partition
 each  of  these circles according to their  intersection  pattern
 using the following notation.
 \vskip.5cm
 \hrule
 \vskip.5cm

 \noindent {\sl Notation}

 \noindent Let  the symbol $A$ denote the area of Circle $\alpha $
 that does NOT also lie within the Circle $\beta $.

 \noindent Let  the  symbol $B$ denote the area of Circle $\beta $
 that does NOT also lie within the Circle $\alpha $.

 \noindent Let  the  symbol  $AB$  denote  the  area  of the
 intersection of Circles $\alpha $ and $\beta $.
 \vskip.5cm
 \hrule
 \vskip.5cm

      In the vaccination program interpretation of the two  circle
 Venn  diagram,  Area  $A$  is proportional to the countries using
 only a  campaign strategy, Area $B$ to the countries using only a
 clinic-based  strategy, and Area $AB$ to the countries using both
 a campaign and a clinic-based  strategy.  Countries using neither
 strategy  are   not   represented;   indeed,   only   participant
 populations will be considered for the remainder of this analysis,
 although we do  note the existence of the logical category of the
 ``neither" class.

      Data such as this  is  sometimes illustrated with bar or pie 
 charts.  Such illustrations  are  inadequate  because they do not
 allow easy portrayal on a single diagram of both the intersecting
 areas  and  the  total  percentage  of  each characteristic.  The
 advantage of using a diagram of  intersecting  circles is that it
 portrays all the data clearly in a single diagram.

      The  objective  here  is to draw the intersecting circles so
 that  the  different  areas are exactly proportional to the data,
 in much the way that an equal area map is drawn so that different
 areas are exactly proportional to the size of the landmass. Equal
 area  graphic  displays,  be  they maps or diagrams, are critical
 in making accurate visual comparisons of mapped or plotted data.

      The  remainder  of  this  paper is devoted to displaying the
 detail  of  the  calculations  required  to  construct equal-area
 Venn  diagrams  from  real-world   data.    Fundamentally,  these
 calculations  rest  on  the  problem  of finding the radii of the
 circles  and  the  location  of  their centers, given the various
 common (intersecting) and non-common areas.
 \vskip.5cm
 \noindent{\bf Definition of the two-circle problem}

      Given  two intersecting circles, $\alpha $ and $\beta $, and
 given their common and  non-common  areas,  $AB$,  $A$  and  $B$,
 respectively.   Find  the radii of both circles, $r_A$ and $r_B$,
 and the distance between  the centers of the two circles $d_{AB}$
 such that the centers can  be  located and the circles drawn.  It
 is clear that if the center of Circle $\alpha $ is at the origin,
 the Cartesian coordinates of  the center of  Circle $\beta $  are
 $(d_{AB}, 0)$.

      We  can  transform  the  three areas into percentages of the
 total  area  covered  by  the two intersecting circles  by noting
 that the total area covered is $A+B+AB$.  
 $$ 
 A\% = 100 \times (A/(A+B+AB)) \leqno (1)
 $$
 gives the $A$-only area percent;
 $$ 
 B\% = 100 \times (B/(A+B+AB)) \leqno (2)
 $$
 gives the $B$-only area percent;
 $$ 
 AB\% = 100 \times (AB/(A+B+AB)) \leqno (3)
 $$
 gives the $AB$ area percent;
 
 \noindent No generality is lost by requiring Circle $\alpha $  to
 be  the  larger  circle  and  by standardizing the size of Circle
 $\alpha $ by setting its radius equal to 1:  $r_A=1$. (Naturally,
 this assumes that $\alpha >\beta$ and that the bigger  real-world
 characteristic is assigned to Circle $\alpha $.) As a result, the 
 area  $A+AB$  of  Circle  $\alpha $ is $\pi $.   The  problem can
 now be restated, more simply, as follows.

 Given:  $A\%$, $B\%$, and $AB\%$, where $A\%+B\%+AB\% = 100$.

 Find:  $r_B$, the radius of Circle $\beta $, and $d_{AB}$, the
        distance between the centers.
 \vskip.5cm
 \noindent{\bf Analytic Strategy}

      In order to solve the two-circle problem,  define  the  {\sl
 chord} of the intersection to be the straight  line  joining  the
 two  points  where  the  perimeters  of the two circles intersect
 --assuming here, and throughout  the  remainder of the text, that
 one of the two circles is not fully  contained  within the other. 
 There are two situations that arise:  one in which the chord lies
 between the two centers  and  a second in which the chord lies to
 one side of both centers.  To  visualize  this relationship, draw
 one pair of circles with a relatively small area of intersection;
 in this case the chord lies between the centers. In what follows,
 this  configuration  will  be  referred to as one of type Case I.  
 Alternatively,  draw two circles with  a  relatively  large  area
 of  overlap;  in  this  case  the  chord lies on one side of both
 centers.  In what follows, this configuration will be referred to
 as one of type Case II.

     Starting with $A\%$, $B\%$, and $AB\%$, it is straightforward
 to  derive  $r_B$,  but  not  to  derive $d_{AB}$.  Therefore, we 
 reverse  the  situation and seek  the function that yields $A\%$,
 $B\%$,  and  $AB\%$  given  $r_A$,  $r_B$, and $d_{AB}$.  Several
 derivations  are  possible.  The simplest one (not using integral
 calculus) is presented below to maximize accessibility of content.  

      Functions for $A\%$, $B\%$, and  $AB\%$  in  terms of $r_A$,
 $r_B$,  and  $d_{AB}$  were  obtained for Case I and for Case II. 
 These  functions  are   sufficiently  complex  to  obstruct   the
 derivation of an inverse  function  that would yield  $d_{AB}$ in 
 terms  of  $A\%$,  $B\%$,  and $AB\%$.  Consequently, a numerical 
 approach  was used in which $d_{AB}$ was calculated for a grid of
 values  of  $B\%$  and  $AB\%$.   The  results  are  presented in
 Tables 1 and 2.  The  value  $r_A$  is  assumed  equal to one and
 $r_B$ is readily calculated from $A\%$, $B\%$, and $AB\%$. 

      With  these  results,  several  options  are  available   to 
 estimate $d_{AB}$  from  $A\%$, $B\%$, and $AB\%$.  The preferred
 option  depends  on  the  accuracy  desired.   Option  1  entails
 interpolating  from  the  data  in  Tables  1  and  2.  Option  2
 entails  using  a  polynomial  in  $B\%$ and $AB\%$ (obtained via
 regression) to estimate  $d_{AB}$.  Options 1 and 2 are the least
 accurate,  both  giving  answers  within  one  percent   accuracy
 relative  to the radius of the largest circle (Circle $\alpha $).
 Option  3,  which  will  yield  $d_{AB}$ to any desired accuracy,
 entails  searching  by  trial  and  error  using  the   functions
 $B\%  =  f_1(d_{AB}, r_B)$  and  $AB\%  = f_2(d_{AB}, r_B)$.  The 
 trial  and  error  search  is greatly simplified by the fact that
 $r_B$ can be calculated directly from $B\%$ and $AB\%$. 
 \vskip.5cm
 \noindent{\bf Derivation of $B\%$ and $AB\%$ as a function of 
               $r_B$ and $d_{AB}$}
 \vskip.2cm
 \noindent{\sl General formul{\ae}}

 Heron's  formula  for  the  area  of  a triangle is based  on the
 lengths,  $a$,  $b$,  and  $c$, of its sides.  Let $S=(1/2)\times
 (a+b+c)$.  Then the area of the triangle is:
 $$
 (S \times (S-a) \times (S-b) \times (S-c))^{1/2} \leqno (4)
 $$
 A  sector of a circle is the pie-shaped wedge cut from the center
 of  the  circle  out  to  the edge.  The region of overlap of two
 intersecting  circles  is  called  a  ``lune."   A  sector can be
 decomposed into a triangle  and  a lune split longitudinally.  We
 refer to the triangular portion  as  the  ``triangle" of a sector
 and to the lunar portion as  the ``segment"  of  a  sector.   The
 formula for the area of a sector of a circle  with  central angle
 $Q$ (measured in radians) and radius $r$ is:
 $$
 (1/2) \times (Q/\pi )\times(\pi \times r^2)
 =(1/2)\times (Q\times r^2).  \leqno (5)
 $$
 The  formula  for  the  area  of  the corresponding triangle of a
 sector is:
 $$
 (1/2)\times r^2\times \hbox{sin}\,Q. \leqno(6)
 $$
 The  formula  for  the  area  of  the  corresponding segment of a
 sector is:
 $$
 (1/2)\times (Q\times r^2) - (1/2) \times r^2\times \hbox{sin}\,Q
  = (1/2)\times r^2 \times (Q - \hbox{sin}\,Q).  \leqno(7)
 $$
 \vskip.5cm
 \noindent{\sl Case I:  Chord lies between the two centers}

      In Case I, the chord of the lune  separates  the  centers of
 circles  $\alpha $  and  $\beta $.  The  distance $d_{AB}$ is the
 distance  between  the  two  centers,  measured along the line of 
 centers.  Form a triangle using  the line  of centers as one side
 of  length  $d_{AB}$.  The  second  side is formed by joining the
 center  of  circle $\alpha $ to the top intersection point of the
 lune; the acute angle enclosed by  the  line  of centers and this
 side  has  measure  $Q_A$  which  is  $1/2$  of the central angle
 subtending  the  chord  of  the  lune  from  the center of circle
 $\alpha $.  In a similar  fashion,  join  the  center  of  circle
 $\beta $ to the same third vertex to complete the  triangle.  The
 acute angle enclosed between the line of centers  and  this  side
 has measure $Q_B$ which is $1/2$ of the central angle  subtending
 the chord of the lune from the  center  of  circle $\beta $.  Let
 $h$ denote the altitude of  this  triangle from the vertex of the 
 lune  to  the  line  of  centers.   Let $X$ denote the horizontal
 distance from the center of Circle  $\alpha $ to the intersection
 with the chord.  Let $Z$ denote  the  area  of  the triangle with
 sides  of  lengths  $r_A=1$,  $r_B$, and $d_{AB}$.  Let $K_A$ and
 $K_B$  denote  areas  of  the  sectors  in  circles $\alpha $ and
 $\beta $ subtended by the chord.  Let $L_A$  and $L_B$ denote the
 areas of the triangles of these two sectors.   Finally, let $M_A$
 and $M_B$ denote the areas of  the  segments  of the two sectors. 
 Then given $r_A$, $r_B$, and $d_{AB}$,  find  $h$, $B\%$, $AB\%$,
 and $A\%$ as follows.

 From equation (4),
 $$ 
 S=(1+r_B+d_{AB})/2. \leqno(8)
 $$
 From equations (4) and (8), we get the area  of  the  triangle as
 $$
 Z=(S\times (S-1)\times (S-r_B)\times (S-d_{AB}))^{1/2}.
 \leqno (9)
 $$
 From equation (9),
 $$
 h=2\times Z/d_{AB}, \leqno(10)
 $$
 because $Z=(1/2)\times h \times d_{AB}$.  

 From equation (10),
 $$
 Q_A=\hbox{Arcsin}\,(h/r_A), \leqno(11)
 $$
 because $\hbox{Sin}\,Q_A = h/r_A$; also from equation (10),
 $$
 Q_B=\hbox{Arcsin}\,(h/r_B), \leqno(12)
 $$
 because $\hbox{Sin}\,Q_B = h/r_B$.

 From equations (5) and (11), the sector $A$ area is
 $$
 K_A = {r_A}^2\times (2\times Q_A)/2 = Q_A, \leqno(13)
 $$
 and from equations (5) and (12), the sector $B$ area is 
 $$
 K_B = {r_B}^2\times (2\times Q_B)/2 = Q_B. \leqno(14)
 $$

 From equation (9), the sum of the areas of the two sectors is
 $$
 L_A + L_B = 2\times Z.  \leqno(15)
 $$

 To find the area $AB$ of the  intersecting  area  (lune), view it
 as the sum of the two segments of the two sectors.  From equation
 (7):
 $
 AB = M_A + M_B
    =(K_A - L_A) + (K_B - L_B)
    =(K_A + K_B) - (L_A + L_B)
 $
 so that from equations (9), (13), (14), (15),
 $
 AB = Q_A + Q_B \times {r_B}^2 - 2 \times Z.
 $
 Using equations (11) and (12), it follows that
 $
 AB = \hbox{Arcsin}(h/r_A) + \hbox{Arcsin}(h/rB) \times {r_B}^2
      -2\times Z
 $
 and finally, noting that $r_A$=1, that
 $$
 AB = \hbox{Arcsin}(h) +\hbox{Arcsin}(h/r_B) \times {r_B}^2
      - 2\times Z.  \leqno(16)
 $$
 The $B$-only area is found by subtracting  the  area  of the lune 
 from the area of the whole circle as
 $$
 B=(\pi \times {r_B}^2)-(AB).  \leqno(17)
 $$
 Subtracting  out  the  extra intersection, the total area covered
 by the circles, denoted TOTAL, is (from equation (16))
 $$
 \hbox{TOTAL} = (\pi) +(\pi \times {r_B}^2) - AB.  \leqno(18)
 $$
 [Some  may  recognize  the  formula  in  (18)  as one form of the 
 Principle of Inclusion and Exclusion--ed.]
 
 From equations (16) and (18) it follows that
 $$
 AB\% = 100 \times AB/\hbox{TOTAL};  \leqno(19)
 $$
 from equations (17) and (18) it follows that
 $$
 B\% = 100 \times B/ \hbox{TOTAL};  \leqno(20)
 $$
 and from equations (19) and (20) it follows that
 $$
 A\% = 100 - AB\% - B\%.  \leqno(21)
 $$
 These results hold for $d_{AB}$ greater  than  or  equal to  $X$,
 the distance from the origin to the chord, but not  greater  than
 $r_A +r_B$, that is:
 $$
 X \leq d_{AB} \leq r_A+r_B,  \leqno(22)
 $$
 where, from equation (11), $X = \hbox{Cos}\,(Q_A/2)$.
 \vfill\eject

 \noindent{\sl Case II:  Chord to one side of both centers}
  
 The  same  definitions  apply  as in the previous section, except
 in relation to the  following  situation.   Draw two intersecting
 circles and  associated  lines,  labelling them as follows.  Draw
 the larger of the two circles on the left.  Insert the center  of
 the large circle as a distinguished dot.   Draw a smaller  circle
 intersecting the larger one in such a way that the center  of the
 large circle is contained within the smaller circle.  Much of the
 small circle is therefore necessarily contained within  the large
 circle.  Note the center of the small circle as a dot.  Draw  the
 chord joining the two intersection points of the small  and large
 circles;  half  of  it  has  length  $h$.   Draw the line segment
 joining  the  two  circle  centers, of length $d_{AB}$ and extend
 the  segment  to  intersect  the  chord.   The  small  circle now
 contains a right triangle which in turn contains a  triangle with
 an obtuse angle.  Label the radius of the larger circle as $r_A$;
 label  the  radius  of  the  smaller  circle as $r_B$.  Label the
 constructed central angle in the larger circle as $Q_A$  and  the
 constructed central angle in the  smaller  circle  as $Q_B$.  The
 area  of  the  obtuse  triangle  is  $Z_1$  and  the  area of the
 difference between the right triangle and the obtuse triangle  is
 $Z_2$.  
  
     The following formul{\ae} can then be readily deduced.
 From equation (4),
 $$
 S_1 = (r_A + r_B +d_{AB})/2 = (1 + r_B +d_{AB})/2;  \leqno(23)
 $$
 from equations (5) and (23),
 $$
 Z_1 =
 (S_1 \times (S_1-1)\times (S_1-r_B)\times (S_1-d_{AB}))^{1/2};
 \leqno(24)
 $$
 from equation (24),
 $$
 h = 2\times Z_1/d_{AB}  \leqno(25)
 $$
 because $Z_1=(1/2)\times d_{AB} \times h$; from equation (25)
 $$
 Q_A = \hbox{Arcsin}\,(h), \leqno(26)
 $$
 because $\hbox{Sin}\,(Q_A) =h/r_A = h$; from equation (26)
 $$
 Q_B = \hbox{Arcsin}\,(h/r_B),  \leqno(27)
 $$
 because $\hbox{Sin}\,(Q_B) = h/r_B$; from equations (25) and (27)
 $$
 Z_2 = (1/2)\times h\times r_B\times \hbox{Cos}\,(Q_B)); \leqno(28)
 $$
 from equations (5) and (26)
 $$
 K_A = \hbox{Sector $A$ area} 
     = (1/2)\times (2\times Q_A) \times {r_A}^2 
     = Q_A;  \leqno(29)
 $$
 from equations (24) and (28)
 $$
 L_A=\hbox{Triangle Area of Sector $A$}
    =2\times (Z_1 + Z_2);  \leqno(30)
 $$
 from equations (7), (29), (30)
 $$
 M_A=\hbox{Segment Area of Sector $A$}
    =K_A-L_A; \leqno(31)
 $$
 from equations (5) and (27),
 $$
 K_B=\hbox{Sector $B$ Area}
    =(1/2)\times (2\times Q_B)\times {r_B}^2
    =Q_B\times {r_B}^2; \leqno(32)
 $$
 from equation (28)
 $$
 L_B=\hbox{Triangle Area of Sector $B$}
    =2\times Z_2;  \leqno(33)
 $$
 from equations (32) and (33)
 $$
 M_B=\hbox{Segment Area of Sector $B$}
    =K_B-L_B;  \leqno(34)
 $$
 from equations (31) and (34)
 $$
 \hbox{Area}\,W = M_B-M_A; \leqno(35)
 $$
 from equation (35)
 $$
 AB = \hbox{Area of Circle $B$}\, - \,\hbox{Area $W$}
    =\pi \times {r_B}^2-W; \leqno(36)
 $$
 thus,
 $$
 B=W; \leqno(37)
 $$
 from equation (35)
 $$
 \hbox{TOTAL, the total area covered by the circles}\,
 $$
 $$
 = \hbox{Area of Circle $A$} + \hbox{Area $W$}
 = \pi + W; \leqno(38)
 $$
 from equations (36) and (38)
 $$
 AB\%=100\times AB/\hbox{TOTAL};  \leqno(39)
 $$
 from equations (37) and (38)
 $$
 B\%=100\times B/ \hbox{TOTAL};  \leqno(40)
 $$
 and from equations (39) and (40)
 $$
 A\%=100-AB\%-B\%.  \leqno(41)
 $$
 These results hold for $d_{AB}$ greater  than  or  equal to zero,
 but not greater than  $X$,  the  distance  from the origin to the
 chord, that is:
 $$
 0 \leq d_{AB} \leq X, \leqno(42)
 $$
 where from equation (26) $X=\hbox{cos}\,(Q_A)$.
 \vfill\eject

 \noindent{\bf Methods for Computing $r_B$ and $d_{AB}$.}

 The  computation  of  $r_B$  given  $A\%$,  $B\%$, and $AB\%$  is
 straightforward.  However, this is not the case for  $d_{AB}$  in
 light of the fact that we did not obtain a function for  $d_{AB}$
 in terms of $A\%$, $B\%$, and $AB\%$.  We present three numerical
 methods for estimating $d_{AB}$, each with  a  different level of
 accuracy.   First,  however,  we  derive  $AB\%$  and  $B\%$ as a
 function of $A$, $B$, and $AB$, and $r_B$ as a function of $AB\%$
 and $B\%$.  These derivations  of $AB\%$, $B\%$ and $r_B$ are the
 same for all three methods of estimating $d_{AB}$.

 \noindent{\sl $r_B$ as a Function of $B\%$ and $AB\%$}

 Let $A$, $B$, and $AB$ be the $B$-circle only area, the $A$-circle
 only  area,  and the area of intersection, respectively,  and  let
 $TA=A+B+AB$; then,
 $$
 A\%=100\times A/TA,\quad A=A\%\times TA/100; \leqno(43)
 $$
 $$
 B\%=100\times B/TA,\quad B=B\%\times TA/100; \leqno(44)
 $$
 $$
 AB\%=100\times AB/TA,\quad AB=AB\%\times TA/100; \leqno(45)
 $$
 $$
 \hbox{Circle $A$ area} = A+AB=\pi;  \leqno(46)
 $$
 $$
 \hbox{Circle $B$ area} = B+AB=\pi \times {r_B}^2.  \leqno(47)
 $$
 Substituting equation (46) in equation (47):
 $B+AB = \pi \times {r_B}^2 = (A + AB)\times {r_B}^2$,
 $$
 {r_B}^2 = (B+AB)/(A+AB).  \leqno(48)
 $$
 Substituting equations (43), (44), and (45) in equation (48):
 $$
 {r_B}^2={{(B\%\times TA)/100+(AB\%\times TA)/100} \over
          {(A\%\times TA)/100+(AB\%\times TA)/100}}
        ={{B\% + AB\%} \over {A\% + AB\%}}.  \leqno(49)
 $$
 $$
 A\% =100-B\%-AB\%, \leqno(50)
 $$
 because $A\% +B\% +AB\% =100$.

 \noindent  Substituting equation (50) into equation (49):
 $$
 {r_B}^2={{B\%+AB\%} \over {(100-B\%-AB\%)+AB\%}}
        ={{B\%+AB\%} \over {100 -B\%}}. 
 $$
 Thus,
 $$
 r_B = ((B\%+AB\%)/(100-B\%))^{1/2}.  \leqno(51)
 $$
 \vfill\eject


 \noindent{\sl Look-up Table Method for Estimating $d_{AB}$}

 Table 1 (at end of article)  gives  the  value  of  $d_{AB}$ to 6
 decimal  places  for  all  values of $AB\%$ from 0 to 100 and for
 $B\%$ from 0 to 50 in 5  percentage  point  increments for values
 of $d_{AB}$ from 0 to $r_A + r_B$.  Note from  Table 1  that some
 of  the  values are calculated using the procedure for Case I and
 some using the Case II procedure.

 The  procedure used to obtain the values in Table 1 is summarized
 here.   For  each  combination  of  $B\%$  and $AB\%$ in Table 1,
 calculate  $d_{AB}$  as  follows.   First  calculate  $r_B$ using
 equation  (51).   Then  guess  a  value  for  $d_{AB}$  that   is
 approximately  correct,  and  guess  whether  Case  I  or Case II 
 applies.  (In  most  areas  of  the table this is obvious.)  Then
 calculate the values of $B\%$ and $AB\%$ using the guessed  value
 of $d_{AB}$, the calculated  value of $r_B$ and either  equations
 (8) through (20) in Case I, or equations  (23)  through  (40)  in
 Case II.  Then adjust  the  guessed value of $d_{AB}$ up  or down
 and  recalculate  until the resulting values of $B\%$ and  $AB\%$
 approximate the desired values as closely as desired (six decimal
 points) in Table 1.  Check the final value of $d_{AB}$ to be sure
 the correct calculation procedure was  used  (Case  I or II) with
 the inequalities (22) or (42).

 Table 2 contains values of $d_{AB}$ for values of $AB\%$ in the 0
 to 10 range.  Between 0 and 5, $AB\%$ is in increments of 1. This
 table was produced because of the large and non-linear increments
 in $d_{AB}$ in this range of $AB\%$.

 If  the  given  values  of  $B\%$  and  $AB\%$  are  one  of  the 
 combinations  found  in  Table  1  or  Table 2, then the value of
 $d_{AB}$ can be obtained directly from the tables to six  decimal
 point accuracy.  If the exact values of $B\%$ and  $AB\%$ are not
 in either table, then an interpolation procedure can be used.  In
 Table 1, the procedure would be as follows.

 (a)  Assume  the  given  values of $B\%$ and $AB\%$ are not along
 the lower diagonal of the  table,  so  that  they  are bounded by
 table  values  of  $B\%$  and  $AB\%$  at four corners forming  a
 rectangle within the table.  Let $d(i,j)$ be the value of $d_{AB}$
 for any values of $AB\%$ (or $i$) and $B\%$  (or $j$), within the
 defined range.  Then $d(AB\%,B\%)$ is  the  value  of $d_{AB}$ at
 the given values of $B\%$ and $AB\%$.   If  $e$  is  the value of
 $AB\%$ just less than the given $AB\%$, $f$ is the value of $AB\%$
 just greater than the given $AB\%$,  $g$  is  the  value of $B\%$
 just less than the given $B\%$, and  $h$  is  the  value of $B\%$
 just greater than the given $B\%$, then,
 $$
 d_K = \hbox{estimated value of $d$ at point $K$}
 $$
 $$
     = d(e,g)+(d(f,g)-d(e,g)) \times (AB\% -e)/(f-e), \leqno(52)
 $$
 where  $K$  is  the  intersection  point of a horizontal  through 
 $d(i,j)$ with the vertical line through $g$.
 $$
 d_L = \hbox{estimated value of $d$ at point $L$}
 $$
 $$
     = d(e,h)+(d(f,h)-d(e,h)) \times (AB\% -e)/(f-e), \leqno(53)
 $$
 where  $L$  is  the  intersection  point of a horizontal  through 
 $d(i,j)$ with the vertical line through $h$.
 $$
 \hbox{Estimate of}\,\,\,
 d(AB\%,B\%)=d_K+(d_L-d_K)\times (B\%-g)/(h-g).  \leqno(54)
 $$

 (b) Assume  the  given  values  of  $AB\%$ and $B\%$ are near the
 lower diagonal  of  the  defined  range such that the location is
 bounded  by  only  three  table values (rather than by four table
 values,  as in  case  (a)  above).   Use the same labelling as in
 case (a)  above  for  the  bounding table entries; note, however,
 that d(f,h) is not defined  in this case (because of the nearness
 of  the  table  entry  to the lower diagonal).  At the boundaries
 where $B\%=0$  or  $AB\%=0$,  equation (54) holds.  Otherwise, we
 have:
 $$
 d_K = \hbox{estimated value of $d$ at point $K$}
 $$
 $$
     = d(e,g)+(d(f,g)-d(e,g)) \times (AB\% -e)/(f-e), \leqno(55)
 $$
 where  $K$  is  the  intersection  point of a horizontal  through 
 $d(i,j)$ with the vertical line through $g$.
 $$
 d_L = \hbox{estimated value of $d$ at point $L$}
 $$
 $$
     = d(e,h)+(d(e,h)-d(e,g)) \times (B\% -g)/(h-g), \leqno(56)
 $$
 where  $L$  is  the  intersection  point  of  a vertical  through 
 $d(i,j)$ with the horizontal line through $e$.
 $$
 d(AB\%,B\%)=d(e,g)+(d_K-d(e,g))+(k_L-d(e,g))
            =d_K+d_L-d(e,g)
 $$
 $$
            = d(e,g)+(d(f,g)-d(e,g))\times (AB\%-e)/(f-e)
                    +(d(e,h)-d(e,g))\times (B\%-g)/(h-g).  \leqno(57)
 $$ 

 The  use  of  formulas  (54) and (57) in conjunction with Table 1
 will  generally  produce  answers for $d_{AB}$ within 0.01 of the 
 correct figures,  with  the  exception  of  the  range for $AB\%$
 from 0 to 5.  In  some  areas  of  this  range,  particularly for
 $B\%$ greater than 45, the error can be over 0.05.  For  example,
 this  method  produces  an  estimated  value for  $d(2.5, 47.5) =
 1.7394$, compared to the correct value of  1.7927,  an  error  of
 0.053.  (This error is 5.3\% of the radius  of  circle $A$, which
 is  1,  and  is $100*0.053/1.7927=3\%$ of the  correct  value  of
 $d_{AB}$.)

 If  Table  2  is  used  for values of $AB\%$ between 0 and 5, the
 error can be reduced to  less  than 0.02 in the worst cases.  For
 example,  the  use  of  Table  2  produces an estimated value for
 $d(0.5,49.5)=1.93016$, compared to the correct value of  1.91299,
 an error of 0.01717.  (This is 1.7\% of the radius  of circle $A$
 and 0.9\% of the correct value of $d_{AB}$.)

 \noindent{\sl Polynomial Estimation of $d_{AB}$}

 The  regression  formul{\ae}  were  used to obtain polynomials in
 $AB\%$ and $B\%$ that estimated $d_{AB}$, in effect interpolating 
 for values of $AB\%$ and $B\%$ between the grid points in Table 1.
 The range of $AB\%$ and $B\%$ was  separated into three subranges
 and  a  polynomial  was  obtained  for  each subrange.  The three
 subranges  are  specified  below  and  also  denoted graphically,
 using a variety of typefaces, in Table 3.
 $$
 \hbox{Subrange 1:}\, AB\% > 5 \,\hbox{and}\, B\%\geq 5. \leqno(58)
 $$
 $$
 \hbox{Subrange 2:}\, 0 \leq AB\% \leq 5 \,
 \hbox{and}\, 0 \leq B\% \leq 50. \leqno(59)
 $$
 $$
 \hbox{Subrange 3:}\, 0 \leq AB\% \leq 100 \,
 \hbox{and}\, 0 \leq B\% < 5. \leqno(60)
 $$
 The polynomials obtained for each subrange are given below.
 
 \noindent Polynomial 1 for subrange 1:
 $$
 \hbox{est}\, {d_{AB}}^1=c_0+c_1X_1+c_2X_2+c_3X_3+c_4X_4
                            +c_5X_5+c_6X_6+c_7X_7+c_8X_8,  \leqno(61)
 $$
 where
 
 \qquad $c_0=\phantom{-}0.994388189$

 \qquad $c_1=\phantom{-}0.003790799$, $X_1=AB\%$

 \qquad $c_2=         -0.001818030$, $X_2=B\%$
 
 \qquad $c_3=         -0.129148003$, $X_3=(AB\%)^{1/2}$

 \qquad $c_4=\phantom{-}0.130891455$, $X_4=(B\%)^{1/2}$

 \qquad $c_5=         -0.000147200$, $X_5=(AB\%)\times (B\%)$

 \qquad $c_6=         -0.000017449$, $X_6=(AB\%)^2$

 \qquad $c_7=\phantom{-}0.000081024$, $X_7=(B\%)^2$

 \qquad $c_8=         -0.004913375$, $X_8=(AB\%\times B\%)^{1/2}$.

 \noindent Polynomial 2 for subrange 2:
 $$
 \hbox{est}\, {d_{AB}}^2=c_0+c_1X_1+c_2X_2+c_3X_3+c_4X_4
                            +c_5X_5+c_6X_6+c_7X_7, \leqno(62)
 $$
 where
 
 \qquad $c_0=\phantom{-}1.003584849$

 \qquad $c_1=         -0.009650203$, $X_1=AB\%$

 \qquad $c_2=\phantom{-}0.002712922$, $X_2=B\%$
 
 \qquad $c_3=         -0.089520075$, $X_3=(AB\%)^{1/2}$

 \qquad $c_4=\phantom{-}0.093223275$, $X_4=(B\%)^{1/2}$

 \qquad $c_5=         -0.000366121$, $X_5=(AB\%)\times (B\%)$

 \qquad $c_6=         -0.000521608$, $X_6=(AB\%)^2$

 \qquad $c_7=\phantom{-}0.000075950$, $X_7=(B\%)^2$.
 
 \noindent Polynomial 3 for subrange 3:
 $$
 \hbox{est}\, {d_{AB}}^3=c_0+c_1X_1+c_2X_2+c_3X_3+c_4X_4
                            +c_5X_5+c_6X_6+c_7X_7+c_8X_8,  \leqno(63)
 $$
 where
 
 \qquad $c_0=\phantom{-}1.009984781$

 \qquad $c_1=\phantom{-}0.001043059$, $X_1=AB\%$

 \qquad $c_2=\phantom{-}0.009094475$, $X_2=B\%$
 
 \qquad $c_3=         -0.106641306$, $X_3=(AB\%)^{1/2}$

 \qquad $c_4=\phantom{-}0.088497298$, $X_4=(B\%)^{1/2}$

 \qquad $c_5=         -0.000104701$, $X_5=(AB\%)\times (B\%)$

 \qquad $c_6=         -0.000052845$, $X_6=(AB\%)^2$

 \qquad $c_7=         -0.000084727$, $X_7=(B\%)^2$

 \qquad $c_8=         -0.007209240$, $X_8=(AB\%\times B\%)^{1/2}$.

 \noindent{\sl Numerical Search on the Inverse Function}

 The value of $d_{AB}$ can be obtained to any desired accuracy for
 any combination of $AB\%$ and $B\%$ in  the  defined range  using
 the same procedure as was used to derive  Table 1.  We  summarize
 the procedure below.

 \item{(1)}
 Given $A$, $B$, and $AB$.

 \item{(2)}
 Calculate $B\%$ using equation (44).

 \item{(3)}
 Calculate $AB\%$ using equation (45).

 \item{(4)}
 Calculate $r_B$ using equation (51).

 \item{(5)}
 Estimate an approximate value for $d_{AB}$ using Table 1.

 \item{(6)}
 Estimate whether  Case I  or  Case II applies for the  calculated
 values of $B\%$ and $AB\%$ using Tables 1 and 2.

 \item{(7)}
 Calculate  estimated  values  of  $AB\%$  and  $B\%$  using   the
 calculated  value  of  $r_B$  from step 4, the estimated value of
 $d_{AB}$ from step 5 and equations (19) and  (20)  for  Case I or
 equations (39) and (40) for Case II.

 \item{(8)}
 If  the estimated value of $B\%$ obtained in step 7 is too small,
 increase  the  estimated value of $d_{AB}$ and recalculate $AB\%$
 and  $B\%$  by  recycling  through  step  7; if $B\%$ is too big,
 reduce the estimated value of $d_{AB}$ and recycle  through  step
 7.  The size of the adjustment depends on the  approximate  slope
 in the region of concern.  For example, if we were in a region of
 Table 1 where $d_{AB}$ increased 0.025 while $B\%$ was increasing
 by 5 (as is approximately the case for $B\%$  between  10  and 15
 and  $AB\%$=60),  then  adjust  $d_{AB}$  by 0.005 for each error
 increment of 1 in $B\%$.  In this way,  the error in $B\%$ can be
 made  as  small  as desired by continued recycling.  The value of
 $AB\%$ converges to the desired value along with $B\%$.

 \item{(9)}
 Check  to  be  sure  that  the  proper case (I or II) was used by 
 applying the inequalities (22) or (42).
 \vfill\eject
 \smallskip
 {\bf Editor's Note:}

 In  the  original  submission  the  author  also  considered Venn
 diagrams  of   three   circles, noting that the three-circle Venn
 diagram  contains  insufficient  degrees  of freedom to provide a
 general solution to a three characteristic situation.  The reader  
 interested   in generalizations of the two-circle case might wish
 to  examine  the  literature  of  Boolean  algebra,  particularly
 Karnaugh maps used in the minimization of switching circuits.

 More  detail is presented in this presentation than  would  be  in
 traditional publications, suggesting yet another avenue to explore
 in the dissemination of information across disciplinary boundaries
 and one way to offer detail that might  be  required by  engineers
 in the field  to  implement  abstract ideas presented in journals. 
 The  increase  in  cost,  to  present extra detail that may not be
 necessary to all, is minuscule in an electronic format.
 \smallskip
 \smallskip

 {\bf Author's Note:}

 The author wishes to thank  anonymous referees for suggesting  the
 viewpoint of ``equal area" Venn diagrams, and for substantial help
 in making the context of the problem reflect this viewpoint.
 \vfill\eject
 \vskip.5cm
 \centerline{TABLE 1}
 \centerline{TWO INTERSECTING CIRCLES PROBLEM}
 \centerline{Distance between centers (d),
             given $AB\%$ and $B$-ONLY $\%$}
 \smallskip
 \smallskip
 \hrule
 \vskip.5cm
 {\ee \settabs\+
   &{\ee $AB\%$}
   &{\be 0.776393}\quad&{\be 0.868866}\quad&{\be 0.657898}\quad&{\be 
0.590844}
   &{\ee 0.395288}    &{\ee 1.307136}    &{\ee 1.374962}    &{\ee 1.289705}
   &{\ee 1.350925}    &{\ee 1.413929}    &{\ee 50}&\cr %sample line
 \+&&&&&&{\ee $B$}-Only {\ee $\%$}&&
     &&&&&&&& \cr
 \+&$AB\%$&00&05&10&15&20&25&30&35&40&45&50 \cr
 \+&00 &1             &1.229399          &1.33333       &1.42008    
       &1.5           &1.57735           &1.65464       &1.7337      
       &1.8163        &1.90453           &2 \cr
 \+&05 &{\be 0.776393}&0.982242      &1.082793      &1.164457   
       &1.23779           &1.307136          &1.374962      &1.443     
       &1.5127            &1.58547 & \cr
 \+&10 &{\be 0.683773}&{\be 0.868866}&0.961981      &1.037539
       &1.10493       &1.168068      &1.229156      &1.289705
       &1.350925      &1.413929&  \cr
 \+&15 &{\be 0.612702}&{\be 0.782479}&0.86877       &0.938611
       &1.0005        &1.05796       &1.112964      &1.166795
       &1.220433&& \cr
 \+&20 &{\be 0.552787}&{\be 0.710053}&{\be 0.79011} &0.8546
       &0.9113        &0.96341       &1.012665      &1.060145 
       &1.106585 && \cr
 \+&25 &{\be 0.5}     &{\be 0.646459}&{\be 0.72074} &0.78016
       &0.831894      &0.878849      &0.922537      &0.963822
       &&& \cr
 \+&30 &{\be 0.452278}&{\be 0.589054}&{\be 0.657898}&0.712452
       &0.759365      &0.801269      &0.839453      &0.874543
       &&& \cr
 \+&35 &{\be 0.408393}&{\be 0.536264}&{\be 0.599914}&0.649738
       &0.691905      &0.728788      &0.761409
       &&&& \cr
 \+&40 &{\be 0.367545}&{\be 0.487058}&{\be 0.545676}
       &{\be 0.590844}
       &0.628273      &0.660049      &0.686956
       &&&& \cr
 \+&45 &{\be 0.32918} &{\be 0.440711}&{\be 0.494394}
       &{\be 0.534916}
       &0.567539      &0.594046
       &&&&& \cr
 \+&50 &{\be 0.292894}&{\be 0.396685}&{\be 0.445464}
       &{\be 0.48128}
       &0.508937      &0.529864
       &&&&& \cr
 \+&55 &{\be 0.258381}&{\be 0.354559}&{\be 0.398393}&0.429363
       &0.451767
       &&&&&& \cr
 \+&60 &{\be 0.225404}&{\be 0.313984}&{\be 0.352752}&0.378617
       &0.395288
       &&&&&& \cr
 \+&65 &{\be 0.193775}&{\be 0.27465} &{\be 0.308123}&0.328451
       &&&&&&& \cr
 \+&70 &{\be 0.16334} &{\be 0.236231}&{\be 0.264046}&0.278098
       &&&&&&& \cr
 \+&75 &{\be 0.133975}&{\be 0.198489}&{\be 0.219922}
       &&&&&&&& \cr
 \+&80 &{\be 0.105573}&{\be 0.160916}&0.174756
       &&&&&&&& \cr
 \+&85 &{\be 0.078049}&{\be 0.122853}
       &&&&&&&&& \cr
 \+&90 &{\be 0.051317}&0.082698
       &&&&&&&&& \cr
 \+&95 &{\be 0.025321}
       &&&&&&&&&& \cr
 \+&100&{\be 0}
       &&&&&&&&&& \cr }
 \vskip.5cm
 \hrule
 \vskip.5cm
 \centerline{Case I:  Chord joining intersection points lies between
                      the two centers}
 \centerline{\bf Case II:  Chord lies to one side of both centers.}
 \vfill\eject

 \vskip.5cm
 \centerline{TABLE 2}
 \centerline{DATA TABLE FOR TWO-CIRCLE  PROBLEM}
 \centerline{Distance between centers (d),
             given $AB\%$ and $B$-ONLY $\%$}
 \centerline{for $AB\% = 00 - 05$}
 \smallskip
 \smallskip
 \hrule
 \vskip.5cm
 \settabs\+\quad
   &$AB\%$\quad 
   &{\bf 0.776397}$\,$&{\bf 0.856395}$\,$&{\bf 0.897278}$\,$&{\bf 
0.929644}$\,$
   &0.957428$\,$      &0.982242$\,$   &\cr %sample line
 \+&&&$B$-Only $\%$&&&& & \cr
 \+&$AB\%$&00&01&02&03&04&05& \cr
 \+&00 &1             &1.100503      &1.142857      &1.175863    
       &1.204124      &1.229399      & \cr
 \+&01 &{\bf 0.9}     &0.996625      &1.041666      &1.076413   
       &1.105877      &1.132029      & \cr
 \+&02 &{\bf 0.858578}&{\bf 0.949095}&0.993166      &1.027487
       &1.056694      &1.082652      & \cr
 \+&03 &{\bf 0.826794}&{\bf 0.91298} &{\bf 0.95593} &0.989618
       &1.018366      &1.044         & \cr
 \+&04 &{\bf 0.8}     &{\bf 0.882799}&{\bf 0.924676}&{\bf 0.957696}
       &0.985978      &1.011199      & \cr
 \+&05 &{\bf 0.776397}&{\bf 0.856395}&{\bf 0.897278}&{\bf 0.929644}
       &0.957428      &0.982242      & \cr
 \vskip.5cm
 \hrule
 \vskip.5cm

 \smallskip
 \smallskip
 \hrule
 \vskip.5cm
 \settabs\+\quad
   &$AB\%$\quad 
   &1.082793$\,$      &1.164457$\,$
   &1.2270744$\,$     &1.307136$\,$      &1.374962$\,$      &1.480464$\,$
   &1.552075$\,$         
   &\cr %sample line
 \+&&&&$B$-Only $\%$&&&&& \cr
 \+&$AB\%$&10&15&20&25&30&35&40 \cr
 \+&00 &1.33333       &1.42008      
       &1.5           &1.57735       &1.65464       &1.7337
       &1.8163        & \cr
 \+&01 &1.237653      &1.324053     
       &1.402561      &1.477746      &1.552221      &1.627878
       &1.706373      & \cr
 \+&02 &1.187417      &1.272749
       &1.349919      &1.423492      &1.496065      &1.569499
       &1.645393      & \cr
 \+&03 &1.147486      &1.231629
       &1.307493      &1.379587      &1.450471      &1.521965
       &1.595619      & \cr
 \+&04 &1.113248      &1.196158 
       &1.270744      &1.341437      &1.410751      &1.480464
       &1.552075      & \cr
 \+&05 &1.082793      &1.164457
       &1.23779       &1.307136      &1.374962      &1.443
       &1.5127        & \cr
 \vskip.5cm
 \hrule
 \vskip.5cm

 \smallskip
 \smallskip
 \hrule
 \vskip.5cm
 \settabs\+\quad
   &$AB\%$\quad 
   &1.627057$\,$      &1.642596$\,$      &1.658348$\,$
   &1.674328$\,$      &1.980196$\,$      &2$\,$&\cr %sample line
 \+&&&$B$-Only $\%$&&&&& \cr
 \+&$AB\%$&45&46&47&48&49&50 \cr
 \+&00 &1.90453       &1.922958      &1.941696
       &1.960768      &1.980196      &2 &\cr
 \+&01 &1.789387      &1.806694      &1.824274
       &1.842146      &1.860327      &  &\cr
 \+&02 &1.725354      &1.741988      &1.758872
       &1.776022      &1.793457      &  &\cr
 \+&03 &1.67297       &1.689029      &1.705319
       &1.721855      &              &  &\cr
 \+&04 &1.627057      &1.642596      &1.658348
       &1.674328      &              &  &\cr
 \+&05 &1.58547       &1.600524      &1.615776
       &              &              &  &\cr
 \vskip.5cm
 \hrule
 \vskip.5cm

 \centerline{Case I:  Chord joining intersection points lies between
                      the two centers}
 \centerline{\bf Case II:  Chord lies to one side of both centers.}
 \vfill\eject

 \vskip.5cm
 \centerline{TABLE 3}
 \centerline{TWO INTERSECTING CIRCLES PROBLEM}
 \centerline{Error in the Estimated Distance between the two
             centers:  (dest - $d$)}
 \smallskip
 \smallskip
 \hrule
 \vskip.5cm
 {\ee \settabs\+
   &{\ee $AB\%$}
   &{\be \phantom{-}0.003585}\quad&{\be -0.001890}
   &{\be -0.000220}          &{\be \phantom{-}0.002339}
   &{\be \phantom{-}0.005130}&{\be \phantom{-}0.007643}      
   &{\be \phantom{-}0.009292}&{\be \phantom{-}0.009392}
   &{\be \phantom{-}0.006917}&{\be \phantom{-}0.000295}      
   &{\be -0.01170}&\cr %sample line
 \+&&&&&&{\ee $B$}-Only {\ee $\%$}&&
     &&&&&&&& \cr
 \+&$AB\%$&00&05&10&15&20&25&30&35&40&45&50 \cr
 \+&00 &{\be \phantom{-}0.003585 }  &{\be -0.00189}      
       &{\be  -0.00022}             &{\be \phantom{-}0.002339 }    
       &{\be \phantom{-}0.005130}   &{\be \phantom{-}0.007643 }       
       &{\be \phantom{-}0.009292 }  &{\be \phantom{-}0.009392 }      
       &{\be \phantom{-}0.006917}   &{\be \phantom{-}0.000295 }       
       &{\be -0.01170 } \cr
 \+&05 &{\be -0.00796}              &{\be \phantom{-}0.000948 }      
       &{\be -0.00314}              &{\be -0.00465 }   
       &{\be  -0.00442}             &{\be -0.00306 }      
       &{\be  -0.00110}             &{\be \phantom{-}0.000865 }     
       &{\be  \phantom{-}0.002138}  &{\be \phantom{-}0.001824 } 
       & \cr
 \+&10 &{\se  -0.00111}              &-0.00320
       &\phantom{-}0.000151  &\phantom{-}0.000254
       &-0.00029       &-0.00075     
       &-0.00096       &-0.00100
       &-0.00117       &-0.00193
       &  \cr
 \+&15 &{\se -0.00128}&-0.00331
       &-0.00005   &-0.00039
       &-0.00121  &-0.00152       
       &-0.00099    &\phantom{-}0.000469
       &\phantom{-}0.002778    && \cr
 \+&20 &{\se -0.00096}&-0.00263
       &\phantom{-}0.000458 &-0.00030
       &-0.00137       &-0.00157       
       &-0.00037     &\phantom{-}0.002464 
       &\phantom{-}0.007088    && \cr
 \+&25 &{\se -0.00044}     &-0.00166
       &\phantom{-}0.001121&-0.00009
       &-0.00143   &-0.00152    
       &\phantom{-}0.000319    &\phantom{-}0.004538
       &&& \cr
 \+&30 &{\se \phantom{-}0.000144}&-0.00065
       &\phantom{-}0.001711\phantom{-}\phantom{-}&0.000014
       &-0.00158   &-0.00154    
       &\phantom{-}0.001003     &\phantom{-}0.006707
       &&& \cr
 \+&35 &{\se \phantom{-}0.000727}&\phantom{-}0.000235
       &\phantom{-}0.002122       &-0.00005
       &-0.00188      &-0.00163      
       &\phantom{-}0.001762
       &&&& \cr
 \+&40 &{\se \phantom{-}0.001248}&\phantom{-}0.00930
       &\phantom{-}0.002310&-0.00031
       &-0.00229    &-0.00168   
       &\phantom{-}0.002798
       &&&& \cr
 \+&45 &{\se \phantom{-}0.001670} &\phantom{-}0.001372
       &\phantom{-}0.002261       &-0.00074
       &-0.00272      &-0.00151
       &&&&& \cr
 \+&50 &{\se \phantom{-}0.001965}&\phantom{-}0.001534
       &\phantom{-}0.001995&-0.00124
       &-0.00300      &-0.00084
       &&&&& \cr
 \+&55 &{\se \phantom{-}0.002114}&\phantom{-}0.001407
       &\phantom{-}0.001563&-0.00172
       &-0.00292
       &&&&&& \cr
 \+&60 &{\se \phantom{-}0.002102}&\phantom{-}0.001011
       &\phantom{-}0.001052&-0.00195
       &-0.00207
       &&&&&& \cr
 \+&65 &{\se \phantom{-}0.001914}&\phantom{-}0.000389
       &\phantom{-}0.000606&-0.00165
       &&&&&&& \cr
 \+&70 &{\se \phantom{-}0.001542} &-0.00037
       &\phantom{-}0.000606&-0.00165
       &&&&&&& \cr
 \+&75 &{\se \phantom{-}0.000976}&-0.00113
       &\phantom{-}0.001043
       &&&&&&&& \cr
 \+&80 &{\se \phantom{-}0.000210}&-0.00160
       &\phantom{-}0.003186
       &&&&&&&& \cr
 \+&85 &{\se -0.00076}&-0.00124
       &&&&&&&&& \cr
 \+&90 &{\se -0.00194}&\phantom{-}0.001460
       &&&&&&&&& \cr
 \+&95 &{\se -0.00334}
       &&&&&&&&&& \cr
 \+&100&{\se -0.00496}
       &&&&&&&&&& \cr }
 \vskip.5cm
 \hrule
 \vskip.5cm
 \centerline{Actual Distance from Table 1; Estimated Distance from
             Polynomials 1, 2, and 3.}
 \centerline{Polynomial 1}
 \centerline{\bf Polynomial 2}
 \centerline{\sl Polynomial 3}
 \vfill\eject
 \centerline{\bf Los Angeles, 1994--A Spatial Scientific View}
 \smallskip 
 \smallskip
 \centerline{Sandra L. Arlinghaus,}
 \centerline{Institute of Mathematical Geography and
             University of Michigan}
 \smallskip
 \centerline{William C. Arlinghaus,}
 \centerline{Lawrence Technological University}
 \smallskip
 \centerline{Frank Harary,}
 \centerline{New Mexico State University}
 \smallskip
 \centerline{John D. Nystuen,}
 \centerline{University of Michigan}
 \smallskip

    An algorithm discussed by  Maria Hasse  (Hasse, 1961; Harary,
 Norman,  and Cartwright,  1965)  offers  a  method  for  finding
 the  shortest  distance  between  any  two nodes in a network of
 $n$ nodes when given  only  distances  between  adjacent  nodes. 
 The algorithm  is  one  that  focuses  on  structure alone,  and
 it is therefore  spatial.  The  procedure  is  similar  in  form 
 to  that  used  to  multiply  matrices,  given  two $n \times n$
 matrices $A$ and $B$.   To find the entries in their  Hasse-sum, 
 matrix $C$,  take  the minimum of the  row-by-column sums; thus,
 the entry
 $$ 
 c(21) = 
 \hbox{min} \{a(21)+b(12),a(22)+b(22),a(23)+b(32),...,a(2n)+b(n2)\}.
 $$
 The  results  below show  the outcome of applying this tool from
 theoretical spatial science to the real-world:  to one change in
 the Los Angeles freeway pattern following the recent devastating
 earthquake (January 17, 1994).
     
 \noindent{\bf Los Angeles, 1994.}

      When a recent earthquake caused a disastrous  collapse of a
 span  of  the  Santa Monica  freeway,  according  to  television
 reports  the  world's  busiest  freeway  (carrying  an estimated
 300,000 vehicles per weekday),  municipal authorities managed to
 keep the city moving.  They employed a well-balanced combination
 of alternate routing using  intelligent  vehicle highway systems
 (IVHS) in which traffic lights  along surface routes paralleling
 freeways were coordinated in response  to  user demand, together
 with media messages urging people to stay  off  roadways and the
 effective dispersal of information concerning  alternate routes. 
 Outside forecasters of doom predicted massive  gridlock that did
 not occur in regions where alternate routing was available.  

     In  the  analysis below, we test Hasse's algorithm against a
 changed  adjacency  configuration  and  interpret  the  results. 
 Indeed, what would  a  forecaster using the Hasse algorithm have
 predicted in this situation?

     The  map  in  Figure  1  shows a  portion of the Los Angeles
 freeway  system, and nearby major surface arterials, linking Los
 Angeles  International  Airport  (LAX) to the Central City (CC). 
 We  tightened  our  focus  to consider  what  sort of impact the
 partial closing of the Santa Monica freeway might have on travel
 times  to  and  from  the airport  and the downtown region.  The
 routing in Figure 1 is along  freeways, only, that form a square
 envelope around the direct diagonal  route  (that does not exist
 in the real world)  linking LAX to the CC.   Any  rupture  along
 this  circuit  will  completely destroy one of the two  possible
 routes, sending all the traffic along one path only.  Thus, when
 the Santa Monica freeway was ruptured  (Figure 2)--cross-hatched
 area  on Figure  1 -- all the traffic would have been forced due 
 east and then north, if only freeway linkages were employed.

 \midinsert 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad\qquad&LAX\quad
                &4
                &-----
                &{\bf X}
                &-----
                &6
                &-----
                &8
                &-----
                &10
                &-----
                &12
                &-----
                &$\bullet $
                &\quad CC\cr %sample line
 \+& &4& & & &6& &8& &10& &12& &16&CC&\cr
 \+& &$\bullet $&-----&{\bf X}&-----&$\bullet $&-----&$\bullet $
                &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr
 \+& &$\vert $  & & & & & & & & & & & &$\vert $&&\cr
 \+& &$\vert $  & & & & & & & & & & & &$\vert $&&\cr
 \+& &$\vert $  & & & & & & & & & & & &$\vert $&&\cr
 \+&${\phantom{LA}}$3 &$\bullet$&&&&&&&&&&&&$\bullet $&15&\cr
 \+& &$\vert $  & & & & & & & & & & & &$\vert $&&\cr
 \+& &$\vert $  & & & & & & & & & & & &$\vert $&&\cr
 \+& &$\vert $  & & & & & & & & & & & &$\vert $&&\cr
 \+&${\phantom{LA}}$2&$\bullet $&&&&&&&&&&&&$\bullet $&14&\cr
 \+& &$\vert $  & & & & & & & & & & & &$\vert $&&\cr
 \+& &$\vert $  & & & & & & & & & & & &$\vert $&&\cr
 \+& &$\vert $  & & & & & & & & & & & &$\vert $&&\cr
 \+& &$\bullet $&-----&-----&-----&$\bullet $&-----&$\bullet $
                &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr
 \+&LAX &1& & & &5& &7& &9& &11& &13&&\cr
 \vskip.5cm
 \noindent{\bf Figure 1.}  LAX denotes Los Angeles International Airport.
 CC denotes the Central City.  Routes are along major expressways.
 The X indicates the rupture in the Santa Monica freeway
 caused by the January 17, 1994 earthquake.  Consider that all lines,
 whether dashed or solid, represent continuous graphical linkage between
 adjacent nodes.  The only break in the freeway is at the X.
 \vskip.5cm
 \hrule
 \endinsert

 \midinsert
 \vskip.5cm
 \hrule
 \vskip5in
 \noindent{\bf Figure 2.}  Drawing based on a photo, showing damage
 to the Los Angeles freeways, from the {\sl New York Times\/}, Tuesday,
 January 18, 1994.
 \vskip.5cm
 \hrule
 \endinsert

 \topinsert 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad\qquad&LAX\quad
                &4
                &-----
                &{\bf X}
                &-----
                &6
                &-----
                &8
                &-----
                &10
                &-----
                &12
                &-----
                &$\bullet $
                &\quad CC\cr %sample line
 \+& &4& & & &6& &8& &10& &12& &16&CC&\cr
 \+& &$\bullet $&-----&{\bf X}&-----&$\bullet $&-----&$\bullet $
                &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr
 \+& &$\vert $  & & & &$\vert $ &
                &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &18& &20& &22& &24& &$\vert $&&\cr
 \+&${\phantom{LA}}$3 &$\bullet $&-----&-----&-----&$\bullet $
                      &-----&$\bullet $&-----&$\bullet $&-----
                      &$\bullet $&-----&$\bullet $&15&\cr
 \+& &$\vert $  & & & &$\vert $ &
                &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &17& &19& &21& &23& &$\vert $&&\cr
 \+&${\phantom{LA}}$2 &$\bullet $&-----&-----&-----&$\bullet $
                      &-----&$\bullet $&-----&$\bullet $&-----
                      &$\bullet $&-----&$\bullet $&14&\cr
 \+& &$\vert $  & & & &$\vert $ &
                &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\bullet $&-----&-----&-----&$\bullet $&-----&$\bullet $
                &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr
 \+&LAX &1& & & &5& &7& &9& &11& &13&&\cr
 \vskip.5cm

 \noindent{\bf Figure 3.}  Same basic pattern as Figure 1, with surface 
 routes added, and intersections of surface routes added as nodes in the
 graph.  LAX denotes Los Angeles International Airport.
 CC denotes the Central City.  Routes are along major expressways.
 The X indicates the rupture in the Santa Monica freeway
 caused by the January 17, 1994 earthquake.  Consider that all lines,
 whether dashed or solid, represent continuous graphical linkage between
 adjacent nodes.  The only break in the freeway is at the X.
 \vskip.5cm
 \hrule
 \endinsert

     To overcome this apparently disastrous traffic situation, it
 is  natural  to  introduce alternate routes along roads that are
 already  present.   Indeed, in earlier  mathematical  references
 there is  consideration  of this sort of rerouting problem after
 some edges of a network  have  been  deleted (Menger, 1927; Ford
 and Fulkerson, 1962).  One set  of major surface routes is added
 to the map in Figure 1  to offer a  number  of  different routes
 (Figure 3). The matrix $A$ (Figure 4) displays time-distances in
 tabular form across the network shown in Figure 3.  The entry of
 12 in the first row,  second column  indicates  that it takes 12
 quarter-minutes to travel from the node  labelled  1 to the node
 labelled 2.  A zero in this matrix indicates that there are zero
 quarter-minutes required as travel time--thus, zeroes  appear in
 this matrix only along the main diagonal.  Nodes are treated  as
 points within which no travel is possible. An asterisk indicates 
 that  there is no direct linkage between corresponding entries--
 an  asterisk in  the  (1,3)  position indicates that there is no
 single edge of the graph linking node 1 to node 3. All numerical
 entries  are  expressed in  quarter-minutes;  the Pascal program
 (Figure 5), was written to display  integral results.  (Use of a
 spreadsheet is possible but is far more time-consuming.)  Travel
 times were calculated from  distances in the 1993  Rand  McNally
 Road Atlas, assuming (from field experience) an average speed of
 40 mph.

 \topinsert 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad\qquad&LAX\quad
                &4
                &-----
                &{\bf X}
                &-----
                &6
                &-----
                &8
                &-----
                &10
                &-----
                &12
                &-----
                &$\bullet $
                &\quad CC\cr %sample line
 \+& &4& & & &6& &8& &10& &12& &16&CC&\cr
 \+& &$\bullet $&-----&{\bf X}&-----&$\bullet $&-----&$\bullet $
                &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr
 \+& &$\vert $  & & & &$\vert $ &
                &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+&${\phantom{LA}}$3 &$\bullet $&-----&-----&-----&-----
                      &-----&-----&-----&-----&-----
                      &-----&-----&$\bullet $&15&\cr
 \+& &$\vert $  & & & &$\vert $ &
                &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+&${\phantom{LA}}$2 &$\bullet $&-----&-----&-----&-----
                      &-----&-----&-----&-----&-----
                      &-----&-----&$\bullet $&14&\cr
 \+& &$\vert $  & & & &$\vert $ &
                &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\vert $  & & & &$\vert $ & 
                &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr
 \+& &$\bullet $&-----&-----&-----&$\bullet $&-----&$\bullet $
                &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr
 \+&LAX &1& & & &5& &7& &9& &11& &13&&\cr
 \vskip.5cm

 \noindent{\bf Figure 6.}  Same basic pattern as Figure 1, with surface 
 routes added; unlike Figure 2, in this case intersections of
 surface routes are NOT added as nodes in the graph.  The surface
 routes have limited access.  LAX denotes Los Angeles
 International Airport. CC denotes the Central City.  Routes are
 along major expressways.  The X indicates the rupture in the
 Santa Monica freeway caused by the January 17, 1994 earthquake. 
 Consider that all lines, whether dashed or solid, represent
 continuous graphical linkage between adjacent nodes.  The only
 break in the freeway is at the X.
 \vskip.5cm
 \hrule
 \endinsert

     Higher powers of the matrix $A$  count  numbers  of paths of
 longer  length. The matrix $A^2$ counts paths of 2 edges as well
 as those  of one edge.  Thus, in $A^2$  one would expect to find
 an entry indicating the total time to travel from node 1 to node
 3,  as well  as entries  representing travel times across single
 graphical edges from node 1 to node 2 and from node 2 to node 3;
 indeed, as would be hoped the  travel time of 30 quarter-minutes
 from nodes 1 to 3 is the sum of the travel times from 1 to 2 (12
 quarter-minutes) and from 2 to 3 (18 quarter-minutes).  

    The Hasse operator (erroneously referred to as the Hedetniemi 
 operator  in  some  earlier work,  corrected  by  F. Harary  who
 also notes that this procedure may also be present in literature
 earlier than Hasse's 1961 article) always  selects  the shortest
 path if more than one is available.   Other  algorithms  execute 
 similar  calculations;   however,   Floyd's  algorithm  provides 
 easy  display of  lengths  only  (and  not  the  components that
 compose them), while Dijkstra's algorithm is  not  designed  for 
 easy  display  of  results  but  does  permit the  determination
 of  the  actual  position  of  the shortest path.  Both of these
 algorithms require the same number of steps independent  of  the 
 actual  data;  Hasse's  does not -- it stops shorter than  would
 Floyd's or Dijkstra's in many  situations.   Further detail  has
 been published elsewhere  (Harary, Norman, and Cartwright, 1960;
 Arlinghaus, Arlinghaus, and Nystuen, 1990).

     The matrices $A$ through  $A^8$  show  travel  times  across
 paths  of  varying  length  for  the freeway system prior to the
 earthquake (Figure 4a).  The algorithm stops when $A^{n+1}=A^n$;
 in this case,  therefore,  the  last matrix  with new entries is
 $A^8$--the matrix $A^9$  is calculated  to know when to stop the
 iteration.  A different initial matrix is  required  to  capture
 the  linkage  pattern  between  LAX  and  CC  following the 1994
 earthquake (Figure 4b)--the Santa Monica freeway  was  shattered
 between nodes 4 and 6 on the graph in Figure 3.  The matrix  $B$
 in  Figure  6 indicates a new adjacency pattern; in $A$, the 4th
 row, 6th column contained a  value of 30 to represent the direct
 linkage  between  nodes  4 and  6.   The  corresponding entry in
 matrix $B$  is an  asterisk -- that is,  there  is no path, of a
 single  graphical  edge, available  between nodes 4 and 6.  When
 Hasse's algorithm is run using $B$  (Figure 4b),  instead of $A$
 (Figure 4a), as the initial matrix, the iteration  requires  the
 same number of stages; however, some of  the entries are  larger
 in $B$  than  in  $A$,  reflecting  the need for longer paths to
 provide  alternate routes around the earthquake-altered freeway. 
 In the eighth iteration, the $B$-iterate contains entries in the
 (4,6),  (4,8),  (4,10),  (4,12),  and  (4,16) positions that are
 about  30  quarter-minutes  larger  than  are the entries in the
 corresponding eighth A-iterate.  This increase in the structural
 model comes purely from spatial pattern--it does not address the
 natural increase in congestion that one would also expect.  

     The surface route pattern that was introduced  permitted all
 turns at  each of  the surface route intersections; this sort of
 strategy appears  desirable,  but because turns (especially U.S.
 left-hand turns onto two way streets) generally force additional
 slowing of the traffic one might  consider further alteration of
 the structural model.

    Figure 6 shows a modified form of the map in Figure 3; in it,
 the  nodes  17  through  24  have  been omitted.  This graphical
 omission  corresponds  to  the  real-world  notion of preventing
 intersecting traffic  flows within the interchanges.  One way to
 reduce congestion is to  prohibit  all turns.  Another is to use
 traffic  lights  in  a  manner  that  responds  to  the  traffic
 itself,  rather  that to  estimates of traffic.  The  structural
 model  in  Figure  6  represents  this  sort  of  approach;  the 
 north-south route from node 5 to node 6  does not intersect  any
 of the east-west surface vehicular flows.  

      Figure  7a  shows  the initial matrix $C$ representing this
 particular    structural    model    that   permits   restricted  
 pre-earthquake travel  across  surface routes.   Figure 7b shows
 the initial matrix $D$ representing the model with  the  rupture
 in  the  Santa  Monica  freeway.  When Hasse's algorithm is run, 
 there are clearly once  again a  number  of locations that stand
 out:  the (4,6) entry, for example, goes from 30 quarter-minutes
 to 114 quarter-minutes in this  case.  Indeed, there is not even
 any path available of  length less than 5 for this entry:  there
 is an asterisk in this position in $D^4$ --the only asterisk for
 this entry in the $C$ iteration sequence, with the bridge in, is
 in  the  first  matrix.  The  last entries to come into the  $D$ 
 sequence  iteration  are (4,8) and (4,9)--this situation tallies 
 with the  relationships  shown on the map  in Figure 3.  Traffic 
 engineers might  choose  this  latter  model during times of the
 day  when  volumes are not  high  at  the  nodes  showing  large 
 increases,  or  some  other  strategy  that  responds to traffic
 history.

      The path structure from node 1 (LAX) to node 16 (CC) is not 
 altered;  the Santa  Monica  freeway  was not the shortest route
 from  LAX  to  CC  although  its  closure  no  doubt adds to the
 congestion  along  shorter  routes.   Most of the entries in the
 fourth row of $D^7$   to the  right of, and including, the sixth
 column  show  increases  in  time -- some  only  slight and some
 substantial.  Only  the  fourth  row and the fourth  column show
 altered  time  patterns,  pre-  and  post-earthquake --  Hasse's 
 algorithm  shows  that  the  underlying spatial structure of the
 road  network  is sufficient to provide alternate routing to and
 from  LAX  to  CC and between many of the intervening locations. 
 This finding  matches what has apparently happened in the actual
 post-earthquake environment.

 \noindent{\bf Policy Implications}

      In order to turn the elegant theoretical tool of Hasse into
 one a traffic engineer might actually employ, there are a number
 of  policy  implications  to consider -- policy changes  can put
 real-world teeth into theory.
 
 \item{1.}
     No  turns  except  onto  expressways  means  maximum   flow;
 however,  this strategy is awkward for those living in the area. 
 Indeed,  even if it  is assumed  that people can  turn off  onto
 minor  streets  but  cannot  turn  at major intersections, these
 local turns cause a lower average speed.

 \item{2.}
     Permit right hand turns only --not too disruptive of flow so
 speed is  maintained.   The algorithm still holds,  even with an
 asymmetric matrix.

 \item{3.}
     Permit  all  turns  --  there  are  a  number of engineering
 strategies that  might  have corresponding structural components
 in a graphical model.   Left hand turns slow the system.  Insert
 different average speeds or times on the edges of the structural
 model.

 \item{4.}  Use one-way streets--this strategy equalizes left and
 right turns; it, too, produces asymmetric adjacency matrices.
 \vfill\eject

 \noindent{\bf References}

 \ref Arlinghaus, S., Arlinghaus, W., and Nystuen, J.  1990. 
 The  Hedetniemi matrix sum:  an algorithm for shortest path
 and shortest distance.  {\sl Geographical Analysis\/}, Vol.
 22, No. 4, 351-360.

 \ref Ford, L. R., Jr., and Fulkerson, D. R.  1962.
 {\sl Flows in networks\/}. 
 Princeton, N.J.:  Princeton University Press.

 \ref Harary, F., Norman R., and Cartwright, D.  1965. 
 {\sl Structural Models:   An Introduction to the Theory of
 Directed Graphs\/}.  New York:  Wiley.

 \ref Hasse, Maria.  1961.  Uber die Behandlung graphentheorischer
 Probleme unter Verwendung der Matrizenrechnung,
 {\sl Wiss. Z. Techn. Univer. Dresden\/}, {\bf 10},
 1313-1316.

 \ref Menger, K. Zur allgemeinen Kurventheorie. 1927.  
 {\sl Fund. Math.\/}, {\bf 10}, 96-115.  
 \vfill\eject

 \centerline{\bf Figures containing tables}
 \smallskip 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&* &* &18&* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* 
&* \cr
 \+&12&00&18&* &* &* &* &* &* &* &* &* &* &* &* &* &6 &* &* &* &* &* &* 
&* \cr
 \+&* &18&00&24&* &* &* &* &* &* &* &* &* &* &* &* &* &18&* &* &* &* &* 
&* \cr
 \+&* &* &24&00&* &30&* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* 
&* \cr
 \+&18&* &* &* &00&* &12&* & *&* &* &* &* &* &* &* &12&* &* &* &* &* &* 
&* \cr
 \+&* &* &* &30&* &00&* &12&* &* &* &* &* &* &* &* &* &21&* &* &* &* &* 
&* \cr
 \+&* &* &* &* &12&* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* &* &* &* 
&* \cr
 \+&* &* &* &* &* &12&* &00&* &9 &* &* &* &* &* &* &* &* &* &21&* &* &* 
&* \cr
 \+&* &* &* &* &* &* &6 &* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* &* 
&* \cr
 \+&* &* &* &* &* &* &* &9 &* &00&* &6 &* &* &* &* &* &* &* &* &* &21&* 
&* \cr
 \+&* &* &* &* &* &* &* &* &6 &* &00&* &6 &* &* &* &* &* &* &* &* &* 
&15&* \cr
 \+&* &* &* &* &* &* &* &* &* &6 &* &00&* &* &* &6 &* &* &* &* &* &* &* 
&21\cr
 \+&* &* &* &* &* &* &* &* &* &* &6 &* &00&12&* &* &* &* &* &* &* &* &* 
&* \cr
 \+&* &* &* &* &* &* &* &* &* &* &* &* &12&00&12&* &* &* &* &* &* &* &3 
&* \cr
 \+&* &* &* &* &* &* &* &* &* &* &* &* &* &12&00&21&* &* &* &* &* &* &* 
&3 \cr
 \+&* &* &* &* &* &* &* &* &* &* &* &6 &* &* &21&00&* &* &* &* &* &* &* 
&* \cr
 \+&* &6 &* &* &12&* &* &* &* &* &* &* &* &* &* &* &00&12&9 &* &* &* &* 
&* \cr
 \+&* &* &18&* &* &21&* &* &* &* &* &* &* &* &* &* &12&00&* &9 &* &* &* 
&* \cr
 \+&* &* &* &* &* &* &15&* &* &* &* &* &* &* &* &* &9 &* &00&12&9 &* &* 
&* \cr
 \+&* &* &* &* &* &* &* &21&* &* &* &* &* &* &* &* &* &9 &12&00&* &9 &* 
&* \cr
 \+&* &* &* &* &* &* &* &* &15&* &* &* &* &* &* &* &* &* &9 &* &00&12&6 
&* \cr
 \+&* &* &* &* &* &* &* &* &* &21&* &* &* &* &* &* &* &* &* &9 &12&00&* 
&12\cr
 \+&* &* &* &* &* &* &* &* &* &* &15&* &* &3 &* &* &* &* &* &* &6 &* 
&00&12\cr
 \+&* &* &* &* &* &* &* &* &* &* &* &21&* &* &3 &* &* &* &* &* &* 
&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4a.i  This is the initial matrix, $A$}. 
 Figure 4a contains a set of nine tables (i to ix) illustrating
 the use of Hasse's algorithm on part of the LA freeway/surface
 route system, shown in Figure 3, prior
 to the earthquake of January 17, 1994.  Travel times are in
 one-quarter minutes.  An asterisk indicates that the travel time
 between locations is too large to enter the matrix.  A double-zero
 indicates an entry of 0.
 \vskip.5cm
 \hrule
 \vfill\eject
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&* &18&* &30&* &* &* &* &* &* &* &* &* &18&* &* &* &* &* &* 
&* \cr
 \+&12&00&18&42&18&* &* &* &* &* &* &* &* &* &* &* &6 &18&15&* &* &* &* 
&* \cr
 \+&30&18&00&24&* &39&* &* &* &* &* &* &* &* &* &* &24&18&* &27&* &* &* 
&* \cr
 \+&* &42&24&00&* &30&* &42&* &* &* &* &* &* &* &* &* &42&* &* &* &* &* 
&* \cr
 \+&18&18&* &* &00&* &12&* &18&* &* &* &* &* &* &* &12&24&21&* &* &* &* 
&* \cr
 \+&* &* &39&30&* &00&* &12&* &21&* &* &* &* &* &* &33&21&* &30&* &* &* 
&* \cr
 \+&30&* &* &* &12&* &00&* &6 &* &12&* &* &* &* &* &24&* &15&27&21&* &* 
&* \cr
 \+&* &* &* &42&* &12&* &00&* &9 &* &15&* &* &* &* &* &30&33&21&* &30&* 
&* \cr
 \+&* &* &* &* &18&* &6 &* &00&* &6 &* &12&* &* &* &* &* &21&* 
&15&27&21&* \cr
 \+&* &* &* &* &* &21&* &9 &* &00&* &6 &* &* &* &12&* &* &* &30&33&21&* 
&27\cr
 \+&* &* &* &* &* &* &12&* &6 &* &00&* &6 &18&* &* &* &* &* &* &21&* 
&15&27\cr
 \+&* &* &* &* &* &* &* &15&* &6 &* &00&* &* &24&6 &* &* &* &* &* 
&27&33&21\cr
 \+&* &* &* &* &* &* &* &* &12&* &6 &* &00&12&24&* &* &* &* &* &* &* 
&15&* \cr
 \+&* &* &* &* &* &* &* &* &* &* &18&* &12&00&12&33&* &* &* &* &9 &* &3 
&15\cr
 \+&* &* &* &* &* &* &* &* &* &* &* &24&24&12&00&21&* &* &* &* &* 
&15&15&3 \cr
 \+&* &* &* &* &* &* &* &* &* &12&* &6 &* &33&21&00&* &* &* &* &* &* &* 
&24\cr
 \+&18&6 &24&* &12&33&24&* &* &* &* &* &* &* &* &* &00&12&9 &21&18&* &* 
&* \cr
 \+&* &18&18&42&24&21&* &30&* &* &* &* &* &* &* &* &12&00&21&9 &* &18&* 
&* \cr
 \+&* &15&* &* &21&* &15&33&21&* &* &* &* &* &* &* &9 &21&00&12&9 
&21&15&* \cr
 \+&* &* &27&* &* &30&27&21&* &30&* &* &* &* &* &* &21&9 &12&00&21&9 &* 
&21\cr
 \+&* &* &* &* &* &* &21&* &15&33&21&* &* &9 &* &* &18&* &9 &21&00&12&6 
&18\cr
 \+&* &* &* &* &* &* &* &30&27&21&* &27&* &* &15&* &* &18&21&9 
&12&00&18&12\cr
 \+&* &* &* &* &* &* &* &* &21&* &15&33&15&3 &15&* &* &* &15&* &6 
&18&00&12\cr
 \+&* &* &* &* &* &* &* &* &* &27&27&21&* &15&3 &24&* &* &* 
&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4a.ii.  This is the power 2 matrix, $A^2$}. 
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&* &30&* &36&* &* &* &* &* &* &* &18&30&27&* &* &* &* 
&* \cr
 \+&12&00&18&42&18&39&30&* &* &* &* &* &* &* &* &* &6 &18&15&27&24&* &* 
&* \cr
 \+&30&18&00&24&36&39&* &48&* &* &* &* &* &* &* &* &24&18&33&27&* &36&* 
&* \cr
 \+&54&42&24&00&* &30&* &42&* &51&* &* &* &* &* &* &48&42&* &51&* &* &* 
&* \cr
 \+&18&18&36&* &00&45&12&* &18&* &24&* &* &* &* &* &12&24&21&33&30&* &* 
&* \cr
 \+&* &39&39&30&45&00&* &12&* &21&* &27&* &* &* &* &33&21&42&30&* &39&* 
&* \cr
 \+&30&30&* &* &12&* &00&48&6 &* &12&* &18&* &* &* 
&24&36&15&27&21&33&27&* \cr
 \+&* &* &48&42&* &12&48&00&* &9 &* &15&* &* &* &21&42&30&33&21&42&30&* 
&36\cr
 \+&36&* &* &* &18&* &6 &* &00&48&6 &* &12&24&* &* &30&* 
&21&33&15&27&21&33\cr
 \+&* &* &* &51&* &21&* &9 &48&00&* &6 &* &* &30&12&* 
&39&42&30&33&21&39&27\cr
 \+&* &* &* &* &24&* &12&* &6 &* &00&48&6 &18&30&* &* &* &27&* 
&21&33&15&27\cr
 \+&* &* &* &* &* &27&* &15&* &6 &48&00&* &36&24&6 &* &* &* 
&36&39&27&33&21\cr
 \+&* &* &* &* &* &* &18&* &12&* &6 &* &00&12&24&45&* &* &* &* &21&* 
&15&27\cr
 \+&* &* &* &* &* &* &* &* &24&* &18&36&12&00&12&33&* &* &18&* &9 &21&3 
&15\cr
 \+&* &* &* &* &* &* &* &* &* &30&30&24&24&12&00&21&* &* &* 
&24&21&15&15&3 \cr
 \+&* &* &* &* &* &* &* &21&* &12&* &6 &45&33&21&00&* &* &* &* &* 
&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&* &* &* &* &* &* &* &00&12&9 
&21&18&30&24&* \cr
 \+&30&18&18&42&24&21&36&30&* &39&* &* &* &* &* &* &12&00&21&9 &30&18&* 
&30\cr
 \+&27&15&33&* &21&42&15&33&21&42&27&* &* &18&* &* &9 &21&00&12&9 
&21&15&27\cr
 \+&* &27&27&51&33&30&27&21&33&30&* &36&* &* &24&* &21&9 &12&00&21&9 
&27&21\cr
 \+&* &24&* &* &30&* &21&42&15&33&21&39&21&9 &21&* &18&30&9 &21&00&12&6 
&18\cr
 \+&* &* &36&* &* &39&33&30&27&21&* &27&* &21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&* &* &* &* &* &* &27&* &21&39&15&33&15&3 &15&36&24&* &15&27&6 
&18&00&12\cr
 \+&* &* &* &* &* &* &* &36&33&27&27&21&27&15&3 &24&* 
&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4a.iii  This is the power 3 matrix, $A^3$}.
 \vskip.5cm
 \hrule
 \vfill\eject 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&* &36&* &42&* &* &* &* &* &18&30&27&39&36&* &* 
&* \cr
 \+&12&00&18&42&18&39&30&48&36&* &* &* &* &* &* &* &6 
&18&15&27&24&36&30&* \cr
 \+&30&18&00&24&36&39&48&48&* &57&* &* &* &* &* &* &24&18&33&27&42&36&* 
&48\cr
 \+&54&42&24&00&60&30&* &42&* &51&* &57&* &* &* &* &48&42&57&51&* &60&* 
&* \cr
 \+&18&18&36&60&00&45&12&54&18&* &24&* &30&* &* &* 
&12&24&21&33&30&42&36&* \cr
 \+&51&39&39&30&45&00&57&12&* &21&* &27&* &* &* &33&33&21&42&30&51&39&* 
&48\cr
 \+&30&30&48&* &12&57&00&48&6 &54&12&* &18&30&* &* 
&24&36&15&27&21&33&27&39\cr
 \+&* &48&48&42&54&12&48&00&54&9 &* &15&* &* 
&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&* &* &18&* &6 &54&00&48&6 &54&12&24&36&* 
&30&42&21&33&15&27&21&33\cr
 \+&* &* &57&51&* &21&54&9 &48&00&54&6 &* 
&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&* &* &* &24&* &12&* &6 &54&00&48&6 &18&30&51&36&* 
&27&39&21&33&15&27\cr
 \+&* &* &* &57&* &27&* &15&54&6 &48&00&48&36&24&6 &* 
&45&48&36&39&27&33&21\cr
 \+&* &* &* &* &30&* &18&* &12&* &6 &48&00&12&24&45&* &* &30&* 
&21&33&15&27\cr
 \+&* &* &* &* &* &* &30&* &24&42&18&36&12&00&12&33&27&* &18&30&9 &21&3 
&15\cr
 \+&* &* &* &* &* &* &* &39&36&30&30&24&24&12&00&21&* 
&33&30&24&21&15&15&3 \cr
 \+&* &* &* &* &* &33&* &21&* &12&51&6 &45&33&21&00&* &* &* 
&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&* &* &27&* &* &00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&* &45&* &* &33&* &12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&* &9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&* &30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&* &30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&* &36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&* &30&* &* &36&* &27&48&21&39&15&33&15&3 &15&36&24&* &15&27&6 
&18&00&12\cr
 \+&* &* &48&* &* &48&39&36&33&27&27&21&27&15&3 &24&* 
&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4a.iv  This is the power 4 matrix, $A^4$}.
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&* &42&* &48&* &* &* 
&18&30&27&39&36&48&42&* \cr
 \+&12&00&18&42&18&39&30&48&36&57&42&* &* &33&* &* &6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&* &63&* &* &51&* 
&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&30&72&42&* &51&* &57&* &* &* &63&48&42&57&51&66&60&* 
&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&* &30&39&* &* 
&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&30&45&00&57&12&63&21&* &27&* &* 
&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&* 
&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&42&54&12&48&00&54&9 &60&15&* 
&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&* &18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&* &57&57&51&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&* &* &24&* &12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&* &* &63&57&* &27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&* &* &* &30&* &18&* &12&54&6 &48&00&12&24&45&39&* &30&* 
&21&33&15&27\cr
 \+&* &33&* &* &39&* &30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&* &* &51&* &* 
&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&* &* &* &63&* &33&* &21&57&12&51&6 &45&33&21&00&* 
&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&* &00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&* &39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&* &36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&* &42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4a.v  This is the power 5 matrix, $A^5$}.
 \vskip.5cm
 \hrule
 \vfill\eject
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&69&42&* &48&45&* &* 
&18&30&27&39&36&48&42&54\cr
 \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&* &6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&60&63&* 
&51&51&69&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&30&72&42&78&51&* &57&* &* 
&75&63&48&42&57&51&66&60&72&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&* 
&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&30&45&00&57&12&63&21&69&27&* 
&60&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 
&54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&42&54&12&48&00&54&9 
&60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&78&18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&69&57&57&51&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&60&* &24&69&12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&* &63&63&57&69&27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&45&* &* &30&* &18&63&12&54&6 
&48&00&12&24&45&39&51&30&42&21&33&15&27\cr
 \+&45&33&51&* &39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&* 
&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&* &* &69&63&* &33&63&21&57&12&51&6 
&45&33&21&00&60&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4a.vi  This is the power 6 matrix, $A^6$}.
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&* 
&18&30&27&39&36&48&42&54\cr
 \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&30&72&42&78&51&84&57&* 
&75&75&63&48&42&57&51&66&60&72&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&30&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 
&54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&42&54&12&48&00&54&9 
&60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&78&18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&69&57&57&51&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&75&63&63&57&69&27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&45&63&* &30&72&18&63&12&54&6 
&48&00&12&24&45&39&51&30&42&21&33&15&27\cr
 \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&* &66&69&63&72&33&63&21&57&12&51&6 
&45&33&21&00&60&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4a.vii  This is the power 7 matrix, $A^7$}.
 \vskip.5cm
 \hrule
 \vfill\eject 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&78&18&30&27&39&36&48&42&54\cr
 \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&30&72&42&78&51&84&57&87&75&75&63&48&42&57&51&66&60&72&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&30&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 
&54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&42&54&12&48&00&54&9 
&60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&78&18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&69&57&57&51&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&75&63&63&57&69&27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&45&63&87&30&72&18&63&12&54&6 
&48&00&12&24&45&39&51&30&42&21&33&15&27\cr
 \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&78&66&69&63&72&33&63&21&57&12&51&6 
&45&33&21&00&60&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4a.viii  This is the power 8 matrix, $A^8$}.
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&78&18&30&27&39&36&48&42&54\cr
 \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&30&72&42&78&51&84&57&87&75&75&63&48&42&57&51&66&60&72&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&30&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 
&54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&42&54&12&48&00&54&9 
&60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&78&18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&69&57&57&51&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&75&63&63&57&69&27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&45&63&87&30&72&18&63&12&54&6 
&48&00&12&24&45&39&51&30&42&21&33&15&27\cr
 \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&78&66&69&63&72&33&63&21&57&12&51&6 
&45&33&21&00&60&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4a.ix  This is the power 9 matrix, $A^9$}.
 It is identical to the matrix in Figure 4a.viii and so the algorithm
 terminates. 
 \vskip.5cm
 \hrule
 \vfill\eject

 \centerline{\bf Figures containing tables}
 \smallskip 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&* &* &18&* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* 
&* \cr
 \+&12&00&18&* &* &* &* &* &* &* &* &* &* &* &* &* &6 &* &* &* &* &* &* 
&* \cr
 \+&* &18&00&24&* &* &* &* &* &* &* &* &* &* &* &* &* &18&* &* &* &* &* 
&* \cr
 \+&* &* &24&00&* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* 
&* \cr
 \+&18&* &* &* &00&* &12&* & *&* &* &* &* &* &* &* &12&* &* &* &* &* &* 
&* \cr
 \+&* &* &* &* &* &00&* &12&* &* &* &* &* &* &* &* &* &21&* &* &* &* &* 
&* \cr
 \+&* &* &* &* &12&* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* &* &* &* 
&* \cr
 \+&* &* &* &* &* &12&* &00&* &9 &* &* &* &* &* &* &* &* &* &21&* &* &* 
&* \cr
 \+&* &* &* &* &* &* &6 &* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* &* 
&* \cr
 \+&* &* &* &* &* &* &* &9 &* &00&* &6 &* &* &* &* &* &* &* &* &* &21&* 
&* \cr
 \+&* &* &* &* &* &* &* &* &6 &* &00&* &6 &* &* &* &* &* &* &* &* &* 
&15&* \cr
 \+&* &* &* &* &* &* &* &* &* &6 &* &00&* &* &* &6 &* &* &* &* &* &* &* 
&21\cr
 \+&* &* &* &* &* &* &* &* &* &* &6 &* &00&12&* &* &* &* &* &* &* &* &* 
&* \cr
 \+&* &* &* &* &* &* &* &* &* &* &* &* &12&00&12&* &* &* &* &* &* &* &3 
&* \cr
 \+&* &* &* &* &* &* &* &* &* &* &* &* &* &12&00&21&* &* &* &* &* &* &* 
&3 \cr
 \+&* &* &* &* &* &* &* &* &* &* &* &6 &* &* &21&00&* &* &* &* &* &* &* 
&* \cr
 \+&* &6 &* &* &12&* &* &* &* &* &* &* &* &* &* &* &00&12&9 &* &* &* &* 
&* \cr
 \+&* &* &18&* &* &21&* &* &* &* &* &* &* &* &* &* &12&00&* &9 &* &* &* 
&* \cr
 \+&* &* &* &* &* &* &15&* &* &* &* &* &* &* &* &* &9 &* &00&12&9 &* &* 
&* \cr
 \+&* &* &* &* &* &* &* &21&* &* &* &* &* &* &* &* &* &9 &12&00&* &9 &* 
&* \cr
 \+&* &* &* &* &* &* &* &* &15&* &* &* &* &* &* &* &* &* &9 &* &00&12&6 
&* \cr
 \+&* &* &* &* &* &* &* &* &* &21&* &* &* &* &* &* &* &* &* &9 &12&00&* 
&12\cr
 \+&* &* &* &* &* &* &* &* &* &* &15&* &* &3 &* &* &* &* &* &* &6 &* 
&00&12\cr
 \+&* &* &* &* &* &* &* &* &* &* &* &21&* &* &3 &* &* &* &* &* &* 
&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4b.i  This is the initial matrix, $B$}. 
 Figure 4b contains a set of nine tables (i to ix) illustrating
 the use of Hasse's algorithm on the LA freeway system following
 the earthquake of January 17, 1994.  Travel times are in
 one-quarter minutes.  An asterisk indicates that the travel time
 between locations is too large to enter the matrix.  A double-zero
 indicates an entry of 0.
 \vskip.5cm
 \hrule
 \vfill\eject
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&* &18&* &30&* &* &* &* &* &* &* &* &* &18&* &* &* &* &* &* 
&* \cr
 \+&12&00&18&42&18&* &* &* &* &* &* &* &* &* &* &* &6 &18&15&* &* &* &* 
&* \cr
 \+&30&18&00&24&* &39&* &* &* &* &* &* &* &* &* &* &24&18&* &27&* &* &* 
&* \cr
 \+&* &42&24&00&* &* &* &* &* &* &* &* &* &* &* &* &* &42&* &* &* &* &* 
&* \cr
 \+&18&18&* &* &00&* &12&* &18&* &* &* &* &* &* &* &12&24&21&* &* &* &* 
&* \cr
 \+&* &* &39&* &* &00&* &12&* &21&* &* &* &* &* &* &33&21&* &30&* &* &* 
&* \cr
 \+&30&* &* &* &12&* &00&* &6 &* &12&* &* &* &* &* &24&* &15&27&21&* &* 
&* \cr
 \+&* &* &* &* &* &12&* &00&* &9 &* &15&* &* &* &* &* &30&33&21&* &30&* 
&* \cr
 \+&* &* &* &* &18&* &6 &* &00&* &6 &* &12&* &* &* &* &* &21&* 
&15&27&21&* \cr
 \+&* &* &* &* &* &21&* &9 &* &00&* &6 &* &* &* &12&* &* &* &30&33&21&* 
&27\cr
 \+&* &* &* &* &* &* &12&* &6 &* &00&* &6 &18&* &* &* &* &* &* &21&* 
&15&27\cr
 \+&* &* &* &* &* &* &* &15&* &6 &* &00&* &* &24&6 &* &* &* &* &* 
&27&33&21\cr
 \+&* &* &* &* &* &* &* &* &12&* &6 &* &00&12&24&* &* &* &* &* &* &* 
&15&* \cr
 \+&* &* &* &* &* &* &* &* &* &* &18&* &12&00&12&33&* &* &* &* &9 &* &3 
&15\cr
 \+&* &* &* &* &* &* &* &* &* &* &* &24&24&12&00&21&* &* &* &* &* 
&15&15&3 \cr
 \+&* &* &* &* &* &* &* &* &* &12&* &6 &* &33&21&00&* &* &* &* &* &* &* 
&24\cr
 \+&18&6 &24&* &12&33&24&* &* &* &* &* &* &* &* &* &00&12&9 &21&18&* &* 
&* \cr
 \+&* &18&18&42&24&21&* &30&* &* &* &* &* &* &* &* &12&00&21&9 &* &18&* 
&* \cr
 \+&* &15&* &* &21&* &15&33&21&* &* &* &* &* &* &* &9 &21&00&12&9 
&21&15&* \cr
 \+&* &* &27&* &* &30&27&21&* &30&* &* &* &* &* &* &21&9 &12&00&21&9 &* 
&21\cr
 \+&* &* &* &* &* &* &21&* &15&33&21&* &* &9 &* &* &18&* &9 &21&00&12&6 
&18\cr
 \+&* &* &* &* &* &* &* &30&27&21&* &27&* &* &15&* &* &18&21&9 
&12&00&18&12\cr
 \+&* &* &* &* &* &* &* &* &21&* &15&33&15&3 &15&* &* &* &15&* &6 
&18&00&12\cr
 \+&* &* &* &* &* &* &* &* &* &27&27&21&* &15&3 &24&* &* &* 
&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4b.ii.  This is the power 2 matrix, $B^2$}. 
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&* &30&* &36&* &* &* &* &* &* &* &18&30&27&* &* &* &* 
&* \cr
 \+&12&00&18&42&18&39&30&* &* &* &* &* &* &* &* &* &6 &18&15&27&24&* &* 
&* \cr
 \+&30&18&00&24&36&39&* &48&* &* &* &* &* &* &* &* &24&18&33&27&* &36&* 
&* \cr
 \+&54&42&24&00&* &63&* &* &* &* &* &* &* &* &* &* &48&42&* &51&* &* &* 
&* \cr
 \+&18&18&36&* &00&45&12&* &18&* &24&* &* &* &* &* &12&24&21&33&30&* &* 
&* \cr
 \+&* &39&39&63&45&00&* &12&* &21&* &27&* &* &* &* &33&21&42&30&* &39&* 
&* \cr
 \+&30&30&* &* &12&* &00&48&6 &* &12&* &18&* &* &* 
&24&36&15&27&21&33&27&* \cr
 \+&* &* &48&* &* &12&48&00&* &9 &* &15&* &* &* &21&42&30&33&21&42&30&* 
&36\cr
 \+&36&* &* &* &18&* &6 &* &00&48&6 &* &12&24&* &* &30&* 
&21&33&15&27&21&33\cr
 \+&* &* &* &* &* &21&* &9 &48&00&* &6 &* &* &30&12&* 
&39&42&30&33&21&39&27\cr
 \+&* &* &* &* &24&* &12&* &6 &* &00&48&6 &18&30&* &* &* &27&* 
&21&33&15&27\cr
 \+&* &* &* &* &* &27&* &15&* &6 &48&00&* &36&24&6 &* &* &* 
&36&39&27&33&21\cr
 \+&* &* &* &* &* &* &18&* &12&* &6 &* &00&12&24&45&* &* &* &* &21&* 
&15&27\cr
 \+&* &* &* &* &* &* &* &* &24&* &18&36&12&00&12&33&* &* &18&* &9 &21&3 
&15\cr
 \+&* &* &* &* &* &* &* &* &* &30&30&24&24&12&00&21&* &* &* 
&24&21&15&15&3 \cr
 \+&* &* &* &* &* &* &* &21&* &12&* &6 &45&33&21&00&* &* &* &* &* 
&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&* &* &* &* &* &* &* &00&12&9 
&21&18&30&24&* \cr
 \+&30&18&18&42&24&21&36&30&* &39&* &* &* &* &* &* &12&00&21&9 &30&18&* 
&30\cr
 \+&27&15&33&* &21&42&15&33&21&42&27&* &* &18&* &* &9 &21&00&12&9 
&21&15&27\cr
 \+&* &27&27&51&33&30&27&21&33&30&* &36&* &* &24&* &21&9 &12&00&21&9 
&27&21\cr
 \+&* &24&* &* &30&* &21&42&15&33&21&39&21&9 &21&* &18&30&9 &21&00&12&6 
&18\cr
 \+&* &* &36&* &* &39&33&30&27&21&* &27&* &21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&* &* &* &* &* &* &27&* &21&39&15&33&15&3 &15&36&24&* &15&27&6 
&18&00&12\cr
 \+&* &* &* &* &* &* &* &36&33&27&27&21&27&15&3 &24&* 
&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4b.iii  This is the power 3 matrix, $B^3$}.
 \vskip.5cm
 \hrule
 \vfill\eject 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&* &36&* &42&* &* &* &* &* &18&30&27&39&36&* &* 
&* \cr
 \+&12&00&18&42&18&39&30&48&36&* &* &* &* &* &* &* &6 
&18&15&27&24&36&30&* \cr
 \+&30&18&00&24&36&39&48&48&* &57&* &* &* &* &* &* &24&18&33&27&42&36&* 
&48\cr
 \+&54&42&24&00&60&63&* &72&* &* &* &* &* &* &* &* &48&42&57&51&* &60&* 
&* \cr
 \+&18&18&36&60&00&45&12&54&18&* &24&* &30&* &* &* 
&12&24&21&33&30&42&36&* \cr
 \+&51&39&39&63&45&00&57&12&* &21&* &27&* &* &* &33&33&21&42&30&51&39&* 
&48\cr
 \+&30&30&48&* &12&57&00&48&6 &54&12&* &18&30&* &* 
&24&36&15&27&21&33&27&39\cr
 \+&* &48&48&72&54&12&48&00&54&9 &* &15&* &* 
&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&* &* &18&* &6 &54&00&48&6 &54&12&24&36&* 
&30&42&21&33&15&27&21&33\cr
 \+&* &* &57&* &* &21&54&9 &48&00&54&6 &* 
&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&* &* &* &24&* &12&* &6 &54&00&48&6 &18&30&51&36&* 
&27&39&21&33&15&27\cr
 \+&* &* &* &* &* &27&* &15&54&6 &48&00&48&36&24&6 &* 
&45&48&36&39&27&33&21\cr
 \+&* &* &* &* &30&* &18&* &12&* &6 &48&00&12&24&45&* &* &30&* 
&21&33&15&27\cr
 \+&* &* &* &* &* &* &30&* &24&42&18&36&12&00&12&33&27&* &18&30&9 &21&3 
&15\cr
 \+&* &* &* &* &* &* &* &39&36&30&30&24&24&12&00&21&* 
&33&30&24&21&15&15&3 \cr
 \+&* &* &* &* &* &33&* &21&* &12&51&6 &45&33&21&00&* &* &* 
&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&* &* &27&* &* &00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&* &45&* &* &33&* &12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&* &9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&* &30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&* &30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&* &36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&* &30&* &* &36&* &27&48&21&39&15&33&15&3 &15&36&24&* &15&27&6 
&18&00&12\cr
 \+&* &* &48&* &* &48&39&36&33&27&27&21&27&15&3 &24&* 
&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4b.iv  This is the power 4 matrix, $B^4$}.
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&* &42&* &48&* &* &* 
&18&30&27&39&36&48&42&* \cr
 \+&12&00&18&42&18&39&30&48&36&57&42&* &* &33&* &* &6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&* &63&* &* &51&* 
&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&63&72&72&* &81&* &* &* &* &* &* &48&42&57&51&66&60&* 
&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&* &30&39&* &* 
&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&63&45&00&57&12&63&21&* &27&* &* 
&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&* 
&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&72&54&12&48&00&54&9 &60&15&* 
&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&* &18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&* &57&57&81&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&* &* &24&* &12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&* &* &63&* &* &27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&* &* &* &30&* &18&* &12&54&6 &48&00&12&24&45&39&* &30&* 
&21&33&15&27\cr
 \+&* &33&* &* &39&* &30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&* &* &51&* &* 
&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&* &* &* &* &* &33&* &21&57&12&51&6 &45&33&21&00&* 
&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&* &00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&* &39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&* &36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&* &42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4b.v  This is the power 5 matrix, $B^5$}.
 \vskip.5cm
 \hrule
 \vfill\eject
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&69&42&* &48&45&* &* 
&18&30&27&39&36&48&42&54\cr
 \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&* &6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&60&63&* 
&51&51&69&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&63&72&72&78&81&* &87&* &* &75&* 
&48&42&57&51&66&60&72&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&* 
&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&63&45&00&57&12&63&21&69&27&* 
&60&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 
&54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&72&54&12&48&00&54&9 
&60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&78&18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&69&57&57&81&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&60&* &24&69&12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&* &63&63&87&69&27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&45&* &* &30&* &18&63&12&54&6 
&48&00&12&24&45&39&51&30&42&21&33&15&27\cr
 \+&45&33&51&* &39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&* 
&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&* &* &69&* &* &33&63&21&57&12&51&6 
&45&33&21&00&60&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4b.vi  This is the power 6 matrix, $B^6$}.
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&* 
&18&30&27&39&36&48&42&54\cr
 \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&63&72&72&78&81&84&87&* 
&75&75&93&48&42&57&51&66&60&72&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&63&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 
&54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&72&54&12&48&00&54&9 
&60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&78&18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&69&57&57&81&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&75&63&63&87&69&27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&45&63&* &30&72&18&63&12&54&6 
&48&00&12&24&45&39&51&30&42&21&33&15&27\cr
 \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&* &66&69&93&72&33&63&21&57&12&51&6 
&45&33&21&00&60&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4b.vii  This is the power 7 matrix, $B^7$}.
 \vskip.5cm
 \hrule
 \vfill\eject 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&78&18&30&27&39&36&48&42&54\cr
 \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&63&72&72&78&81&84&87&87&75&75&93&48&42&57&51&66&60&72&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&63&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 
&54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&72&54&12&48&00&54&9 
&60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&78&18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&69&57&57&81&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&75&63&63&87&69&27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&45&63&87&30&72&18&63&12&54&6 
&48&00&12&24&45&39&51&30&42&21&33&15&27\cr
 \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&78&66&69&93&72&33&63&21&57&12&51&6 
&45&33&21&00&60&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4b.viii  This is the power 8 matrix, $B^8$}.
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&78&18&30&27&39&36&48&42&54\cr
 \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 
&18&15&27&24&36&30&42\cr
 \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr
 \+&54&42&24&00&60&63&72&72&78&81&84&87&87&75&75&93&48&42&57&51&66&60&72&72\cr
 \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr
 \+&51&39&39&63&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr
 \+&30&30&48&72&12&57&00&48&6 
&54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr
 \+&60&48&48&72&54&12&48&00&54&9 
&60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr
 \+&36&36&54&78&18&63&6 &54&00&48&6 
&54&12&24&36&57&30&42&21&33&15&27&21&33\cr
 \+&69&57&57&81&63&21&54&9 &48&00&54&6 
&54&42&30&12&51&39&42&30&33&21&39&27\cr
 \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 
&18&30&51&36&48&27&39&21&33&15&27\cr
 \+&75&63&63&87&69&27&60&15&54&6 &48&00&48&36&24&6 
&57&45&48&36&39&27&33&21\cr
 \+&48&45&63&87&30&72&18&63&12&54&6 
&48&00&12&24&45&39&51&30&42&21&33&15&27\cr
 \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 
&15\cr
 \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr
 \+&78&66&69&93&72&33&63&21&57&12&51&6 
&45&33&21&00&60&51&51&42&42&33&36&24\cr
 \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 
&21&18&30&24&36\cr
 \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 
&30&18&36&30\cr
 \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 
&21&15&27\cr
 \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 
&27&21\cr
 \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 
&18\cr
 \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 
&12&00&18&12\cr
 \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 
&18&00&12\cr
 \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 
&24&36&30&27&21&18&12&12&00\cr
 \vskip.5cm
 \noindent{\bf Figure 4b.ix  This is the power 9 matrix, $B^9$}.
 It is identical to the matrix in Figure 4b.viii and so the algorithm
 terminates. 
 \vskip.5cm
 \hrule
 \vfill\eject

 \centerline{\bf Figure for Hasse algorithm in Pascal}

 \noindent program hasse(input,output);

 \noindent const max=999999;

 \qquad\qquad\qquad\qquad n=24;
 
 \noindent type hed=array[1..n,1..n] of integer;

 \noindent var a:array[1..n] of hed;

 \qquad done:boolean;

 \qquad i,j,k,num:integer;

 \noindent procedure print(matrix:hed);

 \noindent begin
 
 \qquad for i:=i to n do

 \qquad\qquad begin

 \qquad\qquad for j:=1 to n do

 \qquad\qquad\qquad if matrix[i,j]=max then write (' *')

 \qquad\qquad\qquad\qquad else write(matrix[i,j]:4);

 \qquad\qquad writeln

 \qquad\qquad end

 \noindent end;

 \noindent procedure hedsum(power, init:hed;var next:hed;var flag:boolean);

 \noindent var row,col,min,middle,temp:integer;

 \noindent begin

 \qquad flag:=true;

 \qquad for row:=1 to n do

 \qquad for col:=1 to n do

 \qquad begin

 \qquad\qquad min:=power[row,col];

 \qquad\qquad for middle:=1 to n do

 \qquad\qquad begin

 \qquad\qquad\qquad temp:=power[row,middle]+init[middle,col];

 \qquad\qquad\qquad if temppower[row,col] then flag:=false;

 \qquad end

 \noindent end;

 \noindent $\{$main program$\}$  %remark--$\{$ is { and $\}$ is }

 \noindent begin

 \qquad for i:=1 to n do for j:=1 to n do a[1][i,j]:=max;

 \qquad for i:=1 to n do a[1][i,i]:=0;

 \qquad repeat

 \qquad\qquad readln(i,j,num);

 \qquad\qquad a[1][i,j]:=num;

 \qquad\qquad a[1][j,i]:=num;

 \qquad until eof;

 \qquad page; writeln('this is the initial matrix');writeln;

 \qquad print(a[1]);

 \qquad k:=0;

 \qquad repeat

 \qquad\qquad k:=k+1;

 \qquad\qquad hedsum(a[k],a[1],a[k+1],done);
 \qquad\qquad page; writeln('this is power',k+1:5); writeln;

 \qquad\qquad print(a[k+1]);

 \qquad until (done) or (k=n-1);

 \qquad writeln;

 \qquad writeln('the number of steps was', k:5)

 \noindent end.

 \noindent {\bf Figure 5.}  Computer program, written in Pascal, of
 W. C. Arlinghaus; originally presented on a poster by Arlinghaus,
 Arlinghaus, and Nystuen, ``Elements of Geometric Routing Theory--II"
 Association of American Geographers, National Meetings, Toronto,
 Ontario, April 1990.   
 \vfill\eject
 \centerline{\bf Figures containing tables}
 \smallskip 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&* &* &18&* &* &* &* &* &* &* &* &* &* &*  \cr
 \+&12&00&18&* &* &* &* &* &* &* &* &* &* &33&* &*  \cr
 \+&* &18&00&24&* &* &* &* &* &* &* &* &* &* &45&*  \cr
 \+&* &* &24&00&* &30&* &* &* &* &* &* &* &* &* &*  \cr
 \+&18&* &* &* &00&42&12&* & *&* &* &* &* &* &* &*  \cr
 \+&* &* &* &30&42&00&* &12&* &* &* &* &* &* &* &*  \cr
 \+&* &* &* &* &12&* &00&45&6 &* &* &* &* &* &* &*  \cr
 \+&* &* &* &* &* &12&45&00&* &9 &* &* &* &* &* &*  \cr
 \+&* &* &* &* &* &* &6 &* &00&45&6 &* &* &* &* &*  \cr
 \+&* &* &* &* &* &* &* &9 &45&00&* &6 &* &* &* &*  \cr
 \+&* &* &* &* &* &* &* &* &6 &* &00&45&6 &* &* &*  \cr
 \+&* &* &* &* &* &* &* &* &* &6 &45&00&* &* &* &6  \cr
 \+&* &* &* &* &* &* &* &* &* &* &6 &* &00&12&* &*  \cr
 \+&* &33&* &* &* &* &* &* &* &* &* &* &12&00&12&*  \cr
 \+&* &* &45&* &* &* &* &* &* &* &* &* &* &12&00&21 \cr
 \+&* &* &* &* &* &* &* &* &* &* &* &6 &* &* &21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7a.i  This is the initial matrix, $C$}. 
 Figure 7a contains a set of seven tables (i to vii) illustrating
 the use of Hasse's algorithm on the LA freeway system and the 
 limited access surface route network (Figure 6) prior
 to the earthquake of January 17, 1994.  Travel times are in
 one-quarter minutes.  An asterisk indicates that the travel time
 between locations is too large to enter the matrix.  A double-zero
 indicates an entry of 0.
 \vskip.5cm
 \hrule
 \vfill\eject
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&* &18&60&30&* &* &* &* &* &* &45&* &*  \cr
 \+&12&00&18&42&30&* &* &* &* &* &* &* &45&33&45&*  \cr
 \+&30&18&00&24&* &54&* &* &* &* &* &* &* &51&45&66 \cr
 \+&* &42&24&00&72&30&* &36&* &* &* &* &* &* &69&*  \cr
 \+&18&30&* &72&00&42&12&48&18&* &* &* &* &* &* &*  \cr
 \+&60&* &54&30&42&00&51& 6&* &15&* &* &* &* &* &*  \cr
 \+&30&* &* &* &12&51&00&45&6 &51&12&* &* &* &* &*  \cr
 \+&* &* &* &36&48& 6&45&00&51&9 &* &15&* &* &* &*  \cr
 \+&* &* &* &* &18&* &6 &51&00&45&6 &51&12&* &* &*  \cr
 \+&* &* &* &* &* &15&51&9 &45&00&51&6 &* &* &* &12 \cr
 \+&* &* &* &* &* &* &12&* &6 &51&00&45&6 &18&* &51 \cr
 \+&* &* &* &* &* &* &* &15&51&6 &45&00&51&* &24&6  \cr
 \+&* &45&* &* &* &* &* &* &12&* &6 &51&00&12&24&*  \cr
 \+&45&33&51&* &* &* &* &* &* &* &18&* &12&00&12&33 \cr
 \+&* &45&45&69&* &* &* &* &* &* &* &27&24&12&00&21 \cr
 \+&* &* &66&* &* &* &* &* &* &12&51&6 &* &33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7a.ii.  This is the power 2 matrix, $C^2$}. 
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&60&30&66&36&* &* &* &57&45&57&*  \cr
 \+&12&00&18&42&30&72&42&* &* &* &51&* &45&33&45&66 \cr
 \+&30&18&00&24&48&54&* &60&* &* &* &72&63&51&45&66 \cr
 \+&54&42&24&00&72&30&81&36&* &45&* &* &* &75&69&90 \cr
 \+&18&30&48&72&00&42&12&48&18&57&24&* &* &63&* &*  \cr
 \+&60&72&54&30&42&00&51& 6&57&15&* &21&* &* &99&*  \cr
 \+&30&42&* &81&12&51&00&45&6 &51&12&57&18&* &* &*  \cr
 \+&66&* &60&36&48& 6&45&00&51&9 &57&15&* &* &* &21 \cr
 \+&36&* &* &* &18&57&15&51&00&45&6 &51&12&24&* &57 \cr
 \+&* &* &* &45&57&15&51&9 &45&00&51&6 &57&* &33&12 \cr
 \+&* &51&* &* &24&* &12&57&6 &51&00&45&6 &18&30&51 \cr
 \+&* &* &72&* &* &21&57&15&51&6 &45&00&51&39&27&6  \cr
 \+&57&45&63&* &* &* &18&* &12&57&6 &51&00&12&24&45 \cr
 \+&45&33&51&75&63&* &* &* &24&* &18&39&12&00&12&33 \cr
 \+&57&45&45&69&* &99&* &* &* &33&30&27&24&12&00&21 \cr
 \+&* &66&66&90&* &* &* &21&57&12&51&6 &45&33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7a.iii  This is the power 3 matrix, $C^3$}.
 \vskip.5cm
 \hrule
 \vfill\eject 
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&60&30&66&36&75&42&* &57&45&57&78 \cr
 \+&12&00&18&42&30&72&42&78&48&* &51&72&45&33&45&66 \cr
 \+&30&18&00&24&48&54&60&60&* &69&69&72&63&51&45&66 \cr
 \+&54&42&24&00&72&30&81&36&87&45&* &51&87&75&69&90 \cr
 \+&18&30&48&72&00&42&12&48&18&57&24&63&30&63&75&*  \cr
 \+&60&72&54&30&42&00&51& 6&57&15&63&21&* &105&99&27\cr
 \+&30&42&60&81&12&51&00&45&6 &51&12&57&18&30&* &63 \cr
 \+&66&78&60&36&48& 6&45&00&51&9 &57&15&63&* &42&21 \cr
 \+&36&48&* &87&18&57&15&51&00&45&6 &51&12&24&36&57 \cr
 \+&75&* &69&45&57&15&51&9 &45&00&51&6 &57&45&33&12 \cr
 \+&42&51&69&* &24&63&12&57&6 &51&00&45&6 &18&30&51 \cr
 \+&* &72&72&51&63&21&57&15&51&6 &45&00&51&39&27&6  \cr
 \+&57&45&63&87&30&* &18&63&12&57&6 &51&00&12&24&45 \cr
 \+&45&33&51&75&63&105&30&* &24&45&18&39&12&00&12&33\cr
 \+&57&45&45&69&75&99&* &42&36&33&30&27&24&12&00&21 \cr
 \+&78&66&66&90&* &27&63&21&57&12&51&6 &45&33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7a.iv  This is the power 4 matrix, $C^4$}.
 \vskip.5cm
 \hrule
 \vfill\eject 
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&60&30&66&36&75&42&81&48&45&57&78 \cr
 \+&12&00&18&42&30&72&42&78&48&78&51&72&45&33&45&66 \cr
 \+&30&18&00&24&48&54&60&60&66&69&69&72&63&51&45&66 \cr
 \+&54&42&24&00&72&30&81&36&87&45&93&51&87&75&69&57 \cr
 \+&18&30&48&72&00&42&12&48&18&57&24&63&30&63&75&69 \cr
 \+&60&72&54&30&42&00&51& 6&57&15&63&21&69&105&48&27\cr
 \+&30&42&60&81&12&51&00&45&6 &51&12&57&18&30&42&63 \cr
 \+&66&78&60&36&48& 6&45&00&51&9 &57&15&63&54&42&21 \cr
 \+&36&48&66&87&18&57&15&51&00&45&6 &51&12&24&36&57 \cr
 \+&75&78&69&45&57&15&51&9 &45&00&51&6 &57&45&33&12 \cr
 \+&42&51&69&93&24&63&12&57&6 &51&00&45&6 &18&30&51 \cr
 \+&81&72&72&51&63&21&57&15&51&6 &45&00&51&39&27&6  \cr
 \+&48&45&63&87&30&69&18&63&12&57&6 &51&00&12&24&45 \cr
 \+&45&33&51&75&63&105&30&54&24&45&18&39&12&00&12&33\cr
 \+&57&45&45&69&75&48&42&42&36&33&30&27&24&12&00&21 \cr
 \+&78&66&66&57&69&27&63&21&57&12&51&6 &45&33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7a.v  This is the power 5 matrix, $C^5$}.
 \vskip.5cm
 \hrule
 \vfill\eject  
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&60&30&66&36&75&42&81&48&45&57&78 \cr
 \+&12&00&18&42&30&72&42&78&48&78&51&72&45&33&45&66 \cr
 \+&30&18&00&24&48&54&60&60&66&69&69&72&63&51&45&66 \cr
 \+&54&42&24&00&72&30&81&36&87&45&93&51&87&75&69&57 \cr
 \+&18&30&48&72&00&42&12&48&18&57&24&63&30&42&54&69 \cr
 \+&60&72&54&30&42&00&51& 6&57&15&63&21&69&60&48&27\cr
 \+&30&42&60&81&12&51&00&45&6 &51&12&57&18&30&42&63 \cr
 \+&66&78&60&36&48& 6&45&00&51&9 &57&15&63&54&42&21 \cr
 \+&36&48&66&87&18&57& 6&51&00&45&6 &51&12&24&36&57 \cr
 \+&75&78&69&45&57&15&51&9 &45&00&51&6 &57&45&33&12 \cr
 \+&42&51&69&93&24&63&12&57&6 &51&00&45&6 &18&30&51 \cr
 \+&81&72&72&51&63&21&57&15&51&6 &45&00&51&39&27&6  \cr
 \+&48&45&63&87&30&69&18&63&12&57&6 &51&00&12&24&45 \cr
 \+&45&33&51&75&42&60&30&54&24&45&18&39&12&00&12&33\cr
 \+&57&45&45&69&54&48&42&42&36&33&30&27&24&12&00&21 \cr
 \+&78&66&66&57&69&27&63&21&57&12&51&6 &45&33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7a.vi  This is the power 6 matrix, $C^6$}.
 \vskip.5cm
 \hrule
 \vfill\eject  
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&60&30&66&36&75&42&81&48&45&57&78 \cr
 \+&12&00&18&42&30&72&42&78&48&78&51&72&45&33&45&66 \cr
 \+&30&18&00&24&48&54&60&60&66&69&69&72&63&51&45&66 \cr
 \+&54&42&24&00&72&30&81&36&87&45&93&51&87&75&69&57 \cr
 \+&18&30&48&72&00&42&12&48&18&57&24&63&30&42&54&69 \cr
 \+&60&72&54&30&42&00&51& 6&57&15&63&21&69&60&48&27\cr
 \+&30&42&60&81&12&51&00&45&6 &51&12&57&18&30&42&63 \cr
 \+&66&78&60&36&48& 6&45&00&51&9 &57&15&63&54&42&21 \cr
 \+&36&48&66&87&18&57& 6&51&00&45&6 &51&12&24&36&57 \cr
 \+&75&78&69&45&57&15&51&9 &45&00&51&6 &57&45&33&12 \cr
 \+&42&51&69&93&24&63&12&57&6 &51&00&45&6 &18&30&51 \cr
 \+&81&72&72&51&63&21&57&15&51&6 &45&00&51&39&27&6  \cr
 \+&48&45&63&87&30&69&18&63&12&57&6 &51&00&12&24&45 \cr
 \+&45&33&51&75&42&60&30&54&24&45&18&39&12&00&12&33\cr
 \+&57&45&45&69&54&48&42&42&36&33&30&27&24&12&00&21 \cr
 \+&78&66&66&57&69&27&63&21&57&12&51&6 &45&33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7a.vii  This is the power 7 matrix, $C^7$}.
 The algorithm terminates in six steps; this matrix is identical
 to $A^6$.
 \vskip.5cm
 \hrule
 \vfill\eject 
 \centerline{\bf Figures containing tables}
 \smallskip 
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&* &* &18&* &* &* &* &* &* &* &* &* &* &*  \cr
 \+&12&00&18&* &* &* &* &* &* &* &* &* &* &33&* &*  \cr
 \+&* &18&00&24&* &* &* &* &* &* &* &* &* &* &45&*  \cr
 \+&* &* &24&00&* &* &* &* &* &* &* &* &* &* &* &*  \cr
 \+&18&* &* &* &00&42&12&* & *&* &* &* &* &* &* &*  \cr
 \+&* &* &* &* &42&00&* & 6&* &* &* &* &* &* &* &*  \cr
 \+&* &* &* &* &12&* &00&45&6 &* &* &* &* &* &* &*  \cr
 \+&* &* &* &* &* & 6&45&00&* &9 &* &* &* &* &* &*  \cr
 \+&* &* &* &* &* &* &6 &* &00&45&6 &* &* &* &* &*  \cr
 \+&* &* &* &* &* &* &* &9 &45&00&* &6 &* &* &* &*  \cr
 \+&* &* &* &* &* &* &* &* &6 &* &00&45&6 &* &* &*  \cr
 \+&* &* &* &* &* &* &* &* &* &6 &45&00&* &* &* &6  \cr
 \+&* &* &* &* &* &* &* &* &* &* &6 &* &00&12&* &*  \cr
 \+&* &33&* &* &* &* &* &* &* &* &* &* &12&00&12&*  \cr
 \+&* &* &45&* &* &* &* &* &* &* &* &* &* &12&00&21 \cr
 \+&* &* &* &* &* &* &* &* &* &* &* &6 &* &* &21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7b.i  This is the initial matrix, $D$}. 
 Figure 7b contains a set of seven tables (i to vii) illustrating
 the use of Hasse's algorithm on the LA freeway system and the 
 limited access surface route network (Figure 6) following
 the earthquake of January 17, 1994.  Travel times are in
 one-quarter minutes.  An asterisk indicates that the travel time
 between locations is too large to enter the matrix.  A double-zero
 indicates an entry of 0.
 \vskip.5cm
 \hrule
 \vfill\eject
 \vskip.5cm
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&* &18&60&30&* &* &* &* &* &* &45&* &*  \cr
 \+&12&00&18&42&30&* &* &* &* &* &* &* &45&33&45&*  \cr
 \+&30&18&00&24&* &* &* &* &* &* &* &* &* &51&45&66 \cr
 \+&* &42&24&00&* &* &* &* &* &* &* &* &* &* &69&*  \cr
 \+&18&30&* &* &00&42&12&48&18&* &* &* &* &* &* &*  \cr
 \+&60&* &* &* &42&00&51& 6&* &15&* &* &* &* &* &*  \cr
 \+&30&* &* &* &12&51&00&45&6 &51&12&* &* &* &* &*  \cr
 \+&* &* &* &* &48& 6&45&00&51&9 &* &15&* &* &* &*  \cr
 \+&* &* &* &* &18&* &6 &51&00&45&6 &51&12&* &* &*  \cr
 \+&* &* &* &* &* &15&51&9 &45&00&51&6 &* &* &* &12 \cr
 \+&* &* &* &* &* &* &12&* &6 &51&00&45&6 &18&* &51 \cr
 \+&* &* &* &* &* &* &* &15&51&6 &45&00&51&* &27&6  \cr
 \+&* &45&* &* &* &* &* &* &12&* &6 &51&00&12&24&*  \cr
 \+&45&33&51&* &* &* &* &* &* &* &18&* &12&00&12&33 \cr
 \+&* &45&45&69&* &* &* &* &* &* &* &27&24&12&00&21 \cr
 \+&* &* &66&* &* &* &* &* &* &12&51&6 &* &33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7b.ii.  This is the power 2 matrix, $D^2$}. 
 \vskip.5cm
 \hrule
 \vfill\eject
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&60&30&66&36&* &* &* &57&45&57&*  \cr
 \+&12&00&18&42&30&72&42&* &* &* &51&* &45&33&45&66 \cr
 \+&30&18&00&24&48&* &* &* &* &* &* &72&63&51&45&66 \cr
 \+&54&42&24&00&* &* &* &* &* &* &* &* &* &75&69&90 \cr
 \+&18&30&48&* &00&42&12&48&18&57&24&* &* &63&* &*  \cr
 \+&60&72&* &* &42&00&51& 6&57&15&* &21&* &* &* &*  \cr
 \+&30&42&* &* &12&51&00&45&6 &51&12&57&18&* &* &*  \cr
 \+&66&* &* &* &48& 6&45&00&51&9 &57&15&* &* &* &21 \cr
 \+&36&* &* &* &18&57& 6&51&00&45&6 &51&12&24&* &57 \cr
 \+&* &* &* &* &57&15&51&9 &45&00&51&6 &57&* &33&12 \cr
 \+&* &51&* &* &24&* &12&57&6 &51&00&45&6 &18&30&51 \cr
 \+&* &* &72&* &* &21&57&15&51&6 &45&00&51&39&27&6  \cr
 \+&57&45&63&* &* &* &18&* &12&57&6 &51&00&12&24&45 \cr
 \+&45&33&51&75&63&* &* &* &24&* &18&39&12&00&12&33 \cr
 \+&57&45&45&69&* &* &* &* &* &33&30&27&24&12&00&21 \cr
 \+&* &66&66&90&* &* &* &21&57&12&51&6 &45&33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7b.iii  This is the power 3 matrix, $D^3$}.
 \vskip.5cm
 \hrule
 \vfill\eject 
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54&18&60 &30&66&36&75&42&* &57&45 &57&78 \cr
 \+&12&00&18&42&30&72 &42&78&48&* &51&72&45&33 &45&66 \cr
 \+&30&18&00&24&48&90 &60&* &* &78&69&72&63&51 &45&66 \cr
 \+&54&42&24&00&72&*  &* &* &* &* &* &96&87&75 &69&90 \cr
 \+&18&30&48&72&00&42 &12&48&18&57&24&63&30&63 &75&*  \cr
 \+&60&72&90&* &42&00 &51& 6&57&15&63&21&* &105&* &27 \cr
 \+&30&42&60&* &12&51 &00&45&6 &51&12&57&18&30 &* &63 \cr
 \+&66&78&* &* &48& 6 &45&00&51&9 &57&15&63&*  &42&21 \cr
 \+&36&48&* &* &18&57 & 6&51&00&45&6 &51&12&24 &36&57 \cr
 \+&75&* &78&* &57&15 &51&9 &45&00&51&6 &57&45 &33&12 \cr
 \+&42&51&69&* &24&63 &12&57&6 &51&00&45&6 &18 &30&51 \cr
 \+&* &72&72&96&63&21 &57&15&51&6 &45&00&51&39 &27&6  \cr
 \+&57&45&63&87&30&*  &18&63&12&57&6 &51&00&12 &24&45 \cr
 \+&45&33&51&75&63&105&30&* &24&45&18&39&12&00 &12&33 \cr
 \+&57&45&45&69&75&*  &* &42&36&33&30&27&24&12 &00&21 \cr
 \+&78&66&66&90&* &27 &63&21&57&12&51&6 &45&33 &21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7b.iv  This is the power 4 matrix, $D^4$}.
 \vskip.5cm
 \hrule
 \vfill\eject 
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54 &18&60 &30&66&36&75 &42&81&48&45&57&78 \cr
 \+&12&00&18&42 &30&72 &42&78&48&78 &51&72&45&33&45&66 \cr
 \+&30&18&00&24 &48&90 &60&87&66&78 &69&72&63&51&45&66 \cr
 \+&54&42&24&00 &72&114&84&* &* &102&93&96&87&75&69&90 \cr
 \+&18&30&48&72 &00&42 &12&48&18&57 &24&63&30&42&75&69 \cr
 \+&60&72&90&114&42&00 &51& 6&57&15 &63&21&69&105&48&27\cr
 \+&30&42&60&84 &12&51 &00&45&6 &51 &12&57&18&30&42&63 \cr
 \+&66&78&87&*  &48& 6 &45&00&51&9  &57&15&63&54&42&21 \cr
 \+&36&48&66&*  &18&57 &6 &51&00&45 &6 &51&12&24&36&57 \cr
 \+&75&78&78&102&57&15 &51&9 &45&00 &51&6 &57&45&33&12 \cr
 \+&42&51&69&93 &24&63 &12&57&6 &51 &00&45&6 &18&30&51 \cr
 \+&81&72&72&96 &63&21 &57&15&51&6  &45&00&51&39&27&6  \cr
 \+&48&45&63&87 &30&69 &18&63&12&57 &6 &51&00&12&24&45 \cr
 \+&45&33&51&75 &42&105&30&54&24&45 &18&39&12&00&12&33 \cr
 \+&57&45&45&69 &75&48 &42&42&36&33 &30&27&24&12&00&21 \cr
 \+&78&66&66&90 &69&27 &63&21&57&12 &51&6 &45&33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7b.v  This is the power 5 matrix, $D^5$}.
 \vskip.5cm
 \hrule
 \vfill\eject  
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54 &18&60 &30&66 &36&75 &42&81&48&45&57&78 \cr
 \+&12&00&18&42 &30&72 &42&78 &48&78 &51&72&45&33&45&66 \cr
 \+&30&18&00&24 &48&90 &60&87 &66&78 &69&72&63&51&45&66 \cr
 \+&54&42&24&00 &72&114&84&111&90&102&93&96&87&75&69&90 \cr
 \+&18&30&48&72 &00&42 &12&48 &18&57 &24&63&30&42&54&69 \cr
 \+&60&72&90&114&42&00 &51& 6 &57&15 &63&21&69&60&48&27 \cr
 \+&30&42&60&84 &12&51 &00&45 &6 &51 &12&57&18&30&42&63 \cr
 \+&66&78&87&111&48& 6 &45&00 &51&9  &57&15&63&54&42&21 \cr
 \+&36&48&66&90 &18&57 & 6&51 &00&45 &6 &51&12&24&36&57 \cr
 \+&75&78&78&102&57&15 &51&9  &45&00 &51&6 &57&45&33&12 \cr
 \+&42&51&69&93 &24&63 &12&57 &6 &51 &00&45&6 &18&30&51 \cr
 \+&81&72&72&96 &63&21 &57&15 &51&6  &45&00&51&39&27&6  \cr
 \+&48&45&63&87 &30&69 &18&63 &12&57 &6 &51&00&12&24&45 \cr
 \+&45&33&51&75 &42&60 &30&54 &24&45 &18&39&12&00&12&33 \cr
 \+&57&45&45&69 &54&48 &42&42 &36&33 &30&27&24&12&00&21 \cr
 \+&78&66&66&90 &69&27 &63&21 &57&12 &51&6 &45&33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7b.vi  This is the power 6 matrix, $D^6$}.
 \vskip.5cm
 \hrule
 \vfill\eject  
 \hrule
 \vskip.5cm
 \settabs\+\qquad\qquad\qquad
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$
   &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line
 \+&00&12&30&54 &18&60 &30&66 &36&75 &42&81&48&45&57&78 \cr
 \+&12&00&18&42 &30&72 &42&78 &48&78 &51&72&45&33&45&66 \cr
 \+&30&18&00&24 &48&90 &60&87 &66&78 &69&72&63&51&45&66 \cr
 \+&54&42&24&00 &72&114&84&111&90&102&93&96&87&75&69&90 \cr
 \+&18&30&48&72 &00&42 &12&48 &18&57 &24&63&30&42&54&69 \cr
 \+&60&72&90&114&42&00 &51& 6 &57&15 &63&21&69&60&48&27\cr
 \+&30&42&60&84 &12&51 &00&45 &6 &51 &12&57&18&30&42&63 \cr
 \+&66&78&87&111&48& 6 &45&00 &51&9  &57&15&63&54&42&21 \cr
 \+&36&48&66&90 &18&57 & 6&51 &00&45 &6 &51&12&24&36&57 \cr
 \+&75&78&78&102&57&15 &51&9  &45&00 &51&6 &57&45&33&12 \cr
 \+&42&51&69&93 &24&63 &12&57 &6 &51 &00&45&6 &18&30&51 \cr
 \+&81&72&72&96 &63&21 &57&15 &51&6  &45&00&51&39&27&6  \cr
 \+&48&45&63&87 &30&69 &18&63 &12&57 &6 &51&00&12&24&45 \cr
 \+&45&33&51&75 &42&60 &30&54 &24&45 &18&39&12&00&12&33 \cr
 \+&57&45&45&69 &54&48 &42&42 &36&33 &30&27&24&12&00&21 \cr
 \+&78&66&66&90 &69&27 &63&21 &57&12 &51&6 &45&33&21&00 \cr
 \vskip.5cm
 \noindent{\bf Figure 7b.vii  This is the power 7 matrix, $D^7$}.
 The iteration terminates after 6 steps; this matrix is
 identical to $D^6$.
 \vskip.5cm
 \hrule
 \vfill\eject  


 \centerline{\bf 5.  SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE}

 \centerline{\bf  BACK ISSUES OF {\sl SOLSTICE\/} ON A GOPHER}

 \noindent {\sl Solstice\/} is available on a GOPHER from the
 Department of Mathematics at Arizona State University:
      PI.LA.ASU.EDU port 70 

 \centerline{\bf  BACK ISSUES OF {\sl SOLSTICE\/} AVAILABLE ON FTP}
  
 \noindent This section shows the exact set of commands that  work 
 to  download {\sl Solstice\/} on  The  University  of  Michigan's 
 Xerox  9700.   Because different universities will have different
 installations  of {\TeX},  this  is  only a rough guideline which
 {\sl might\/} be of use to the reader. (BACK   ISSUES   AVAILABLE
 using anonymous ftp to open um.cc.umich.edu, account  GCFS;  type
 cd GCFS after  entering system;  then type ls to get a directory;
 then type get solstice.190 (for example) and download it or  read
 it according to local constraints.) Back issues will be available
 on this account; this account is ONLY for back issues;  to  write
 Solstice,  send   e-mail   to   Solstice@UMICHUM.bitnet   or   to
 Solstice@um.cc.umich.edu .   Issues  from  this  one  forward are
 available on FTP on account IEVG (substitute IEVG for GCFS above).

 First  step  is  to  concatenate  the  files  you  received   via
 bitnet/internet.   Simply  piece  them together in your computer,
 one  after  another,  in  the  order  in which they are numbered,
 starting with the number, ``1."

 The  files  you  have received are ASCII files;  the concatenated
 file  is  used  to  form  the  .tex file from which the .dvi file
 (device  independent)  file is formed.  They should run, possibly
 with a few harmless ``vboxes" over or under.
 \noindent
 ASSUME YOU HAVE SIGNED ON AND ARE AT THE SYSTEM PROMPT, \#.
 \smallskip
  \# create -t.tex
 
 \# percent-sign t from pc c:backslash words backslash
    solstice.tex to mts -t.tex char notab
 
     (this command sends my file, solstice.tex, which I did as
      a WordStar (subdirectory, ``words") ASCII file to the
      mainframe)
 
 \# run *tex par=-t.tex
 
     (there may be some underfull (or certain over) boxes that
      generally  cause  no  problem;  there should be no other
      ``error"  messages  in  the  typesetting--the  files you
      receive were already tested.)

 \# run *dvixer par=-t.dvi
 
 \# control *print* onesided
 
 \# run *pagepr scards=-t.xer, par=paper=plain
 \vfill\eject


 \centerline{\bf 6.  SOLSTICE--INDEX, VOLUMES I, II, III, IV}
 \smallskip
 \noindent{\bf Volume IV, Number 2, Winter, 1993}
 \smallskip
 \noindent {\bf 1.}  Welcome to New Readers and Thank You Notes.
 \smallskip
 \noindent {\bf 2.} Press clippings, summary.
 \smallskip
 \noindent {\bf 3.}  Article
 \smallskip
 Villages in Transition:  Elevated Risk of Micronutrient Deficiency.
 \smallskip
 William D. Drake, S. Pak, I. Tarwotjo, Muhilal, J. Gorstein,
 R. Tilden.
 \smallskip
 Abstract; Moving from Traditional to Modern Village Life:  Risks
 during Transition; Testing for Elevated Risks in Transition Villages;
 Testing for Risk Overlap within the Health Sector; Conclusions and
 Policy Implications.
 \noindent {\bf 4.}  Downloading of Solstice
 \smallskip
 \noindent {\bf 5.}  Index to Volumes I (1990), II (1991), III (1992),
 and IV.1 (1993) of Solstice.
 \smallskip
 \noindent {\bf 6.}  Other Publications of IMaGe
 \smallskip
 \noindent {\bf 7.}  Selected recent publications of interest
 involving Solstice Board members, and some goings on about Ann Arbor.
 %___________________________________________________________________
 %___________________________________________________________________
 \smallskip
 \noindent{\bf Volume IV, Number 1, Summer, 1993}
 \smallskip
 \noindent {\bf 1.}  Welcome to New Readers.
 \smallskip
 \noindent {\bf 2.} Press clippings, summary.
 \smallskip
 \noindent {\bf 3.}  Goings on about Ann Arbor--ESRI and IMaGe Gift
 \smallskip
 \noindent {\bf 4.}  Articles
 \smallskip

 Electronic Journals:  Observations Based on Actual Trials,
 1987-Present, by Sandra L. Arlinghaus and Richard H. Zander.

 Headings:
     Abstract; Content issues; Production issues; Archival issues;
     References.  
 \smallskip

 Wilderness As Place, by John D. Nystuen.

 Headings:
     Visual paradoxes; Wilderness defined; Conflict or synthesis;
     Wilderness as place; Suggested readings; Sources; Visual
     illusion authors
 \smallskip

 The Earth Isn't Flat.  And It Isn't Round Either:  Some Significant
     and Little Known Effects of the Earth's Ellipsoidal Shape,
     by Frank E. Barmore.
 reprinted from the {\sl Wisconsin Geographer\/}.

 Headings:  
     Abstract; Introduction; The Qibla problem; The geographic
     center; The center of population; Appendix; References.
 \smallskip

 Microcell Hex-nets? by Sandra L. Arlinghaus

 Headings:
     Introduction; Lattices; Microcell hex-nets; References.
 \smallskip

 Sum Graphs and Geographic Information, by Sandra L. Arlinghaus,
     William C. Arlinghaus, Frank Harary.

 Headings:
     Abstract; Sum graphs; Sum graph unification:  construction;
     Cartographic application of sum graph unification; Sum graph
     unification:  theory; Logarithmic sum graphs; Reversed sum
     graphs; Augmented reversed logarithmic sum graphs; Cartographic
     application of ARL sum graphs; Summary
 \smallskip

 \noindent{\bf 5.}  Downloading of {\sl Solstice\/}. 
 \smallskip

 \noindent{\bf 6.} Index.
 \smallskip

 \noindent{\bf 7.}  Other publications of IMaGe.
 %----------------------------------------------------------------
 %----------------------------------------------------------------
 \smallskip
 \noindent {\bf Volume III, Number 2, Winter, 1992}
 \smallskip

 \noindent {\bf 1.}  A Word of Welcome from A to U.
 \smallskip

 \noindent {\bf 2.}  Press clippings--summary.
 \smallskip

 \noindent {\bf 3.}  Reprints:
 \smallskip

 \noindent {\bf A.} 
 What Are Mathematical Models and What Should They Be?
 by Frank Harary, reprinted from {\sl Biometrie - Praximetrie\/}. 
 \smallskip \noindent {\sl
 1.  What Are They?  2.  Two Worlds:  Abstract and Empirical
 3.  Two Worlds:  Two Levels  4.  Two Levels:  Derviation and
 Selection  5.  Research Schema  6.  Sketches of Discovery
 7.  What Should They Be?
 \/}
 \smallskip

 \noindent {\bf B.}  Where Are We?  Comments on the Concept of
 Center of Population, by Frank E. Barmore, reprinted from
 {\sl The Wisconsin Geographer\/}.
 \smallskip \noindent {\sl
 1.  Introduction  2.  Preliminary Remarks  3.  Census Bureau
 Center of Population Formul{\ae}  4.  Census Bureau Center of
 Population Description  5.  Agreement Between Description and
 Formul{\ae}  6.  Proposed Definition of the Center of 
 Population  7.  Summary  8.  Appendix A  9.  Appendix B
 10.  References
 \/}
 \smallskip

 \noindent {\bf 4.}  Article:
 \smallskip
 The Pelt of the Earth:  An Essay on Reactive Diffusion,
 by Sandra L. Arlinghaus and John D. Nystuen.
 \smallskip \noindent {\sl
 1.  Pattern Formation:  Global Views  2.  Pattern Formation:
 Local Views  3.  References Cited  4.  Literature of Apparent
 Related Interest.
 \/}
 \smallskip

 \noindent {\bf 5.}  Feature
 Meet new{\sl Solstice\/} Board Member William D. Drake;
 comments on course in Transition Theory and listing of
 student-produced monograph.
 \smallskip

 \noindent {\bf 6.} Downloading of Solstice.
 \smallskip

 \noindent {\bf 7.} Index to Solstice.
 \smallskip

 \noindent {\bf 8.} Other Publications of IMaGe.
 \smallskip
 %----------------------------------------------------------------
 %----------------------------------------------------------------
 \noindent {\bf Volume III, Number 1, Summer, 1992}
 \smallskip

 \noindent{\bf 1.  ARTICLES.}

 \smallskip\noindent
 {\bf Harry L. Stern}. 
 \smallskip\noindent
 {\bf Computing Areas of Regions With Discretely Defined Boundaries}.
 \smallskip\noindent
 1. Introduction 2. General Formulation 3. The Plane 4.  The Sphere
 5.  Numerical Example and Remarks.  Appendix--Fortran Program.
 \smallskip

 \noindent{\bf 2.  NOTE }

 \smallskip\noindent
 {\bf Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg}.  
 \smallskip\noindent
 {\bf  The Quadratic World of Kinematic Waves}
 \smallskip

 \noindent{\bf 3.  SOFTWARE REVIEW}
 \smallskip
 RangeMapper$^{\hbox{TM}}$  ---  version 1.4.
 Created  by {\bf Kenelm W. Philip},  Tundra Vole Software,
 Fairbanks, Alaska.  Program and Manual by  {\bf Kenelm W. Philip}.
 \smallskip
 Reviewed by {\bf Yung-Jaan Lee}, University of Michigan.
 \smallskip

 \noindent{\bf 4.  PRESS CLIPPINGS}
 \smallskip

 \noindent{\bf 5.  INDEX to Volumes I (1990) and II (1991) of
             {\sl Solstice}.}
 \smallskip
 %----------------------------------------------------------------
 %----------------------------------------------------------------
 \noindent {\bf Volume II, Number 2, Winter, 1991}
 \smallskip 

 \noindent 1.  REPRINT

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

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

 \noindent 2.  ARTICLE

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

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

 \noindent 3.  FEATURES

 Press clipping; Word Search Puzzle; Software Briefs.
 \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

 \noindent 2.  FEATURES

 \item{i}  Construction Zone --- The logistic curve.
 \item{ii.} Educational feature --- Lectures on ``Spatial Theory"
 \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

 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

 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

 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.
 \smallskip

 \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 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

 William C. Arlinghaus, {\sl Groups, Graphs, and God}
 \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 --- Crossword puzzle on spices.

 \item{v.}   Solution to crossword puzzle.
 \smallskip

 \noindent 4.  SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE
 \smallskip

 \vfill\eject


 \centerline{\bf 7.  OTHER PUBLICATIONS OF IMaGe} 

 \centerline{\bf MONOGRAPH SERIES}
 \centerline{Scholarly Monographs--Original Material, refereed}

 Prices exclusive of shipping and handling;
 payable in U.S. funds on a U.S. bank, only.
 All monographs are \$15.95, except \#12 which is \$39.95.
 
 Monographs are printed by Gryphon Publishing

 1.  Sandra L. Arlinghaus and John D. Nystuen.  Mathematical
 Geography and Global Art:  the Mathematics of  David Barr's
 ``Four Corners Project,'' 1986. 
  
 2.  Sandra L. Arlinghaus.  Down the Mail Tubes:  the Pressured
 Postal Era, 1853-1984, 1986. 
  
 3.  Sandra L. Arlinghaus.   Essays on Mathematical Geography,
 1986.

 4.  Robert F. Austin, A Historical Gazetteer of Southeast Asia,
 1986.  
 
 5.  Sandra L. Arlinghaus, Essays on Mathematical Geography--II,
 1987.  
 
 6.  Pierre Hanjoul, Hubert Beguin, and Jean-Claude Thill,
 Theoretical Market Areas Under Euclidean Distance, 1988. 
 (English language text; Abstracts written in French and
 in English.) 
  
 7.  Keith J. Tinkler, Editor, Nystuen---Dacey Nodal Analysis,
 1988.

 8.  James W. Fonseca, The Urban Rank--size Hierarchy: 
 A Mathematical Interpretation, 1989.

 9.  Sandra L. Arlinghaus,  An Atlas of Steiner Networks, 1989.

 10.  Daniel A. Griffith, Simulating $K=3$ Christaller Central
 Place Structures:  An Algorithm Using A Constant Elasticity of
 Substitution Consumption Function, 1989. 
 
 11.  Sandra L. Arlinghaus and John D. Nystuen,
      Environmental Effects on Bus Durability, 1990.  

 12.  Daniel A. Griffith, Editor.
 Spatial Statistics:  Past, Present, and Future,  1990. 
 
 13.  Sandra L. Arlinghaus, Editor.  Solstice --- I,  1990. 
 
 14.  Sandra L. Arlinghaus, Essays on Mathematical Geography
 --- III, 1991.
 
 15.  Sandra L. Arlinghaus, Editor, Solstice --- II, 1991.
 
 16.  Sandra L. Arlinghaus, Editor, Solstice --- III, 1992.

 17.  Sandra L. Arlinghaus, Editor, Solstice --- IV, 1993.
 %----------------------------------------------------------------
 %----------------------------------------------------------------
 \smallskip
 DISCUSSION PAPERS--ORIGINAL
 Editor, Daniel A. Griffith
 Professor of Geography
 Syracuse University

 1.  Spatial Regression Analysis on the PC:
 Spatial Statistics Using Minitab.  1989.  
 Cost:  \$12.95, hardcopy.
 %----------------------------------------------------------------
 %----------------------------------------------------------------
 \smallskip
 DISCUSSION PAPERS--REPRINTS
 Editor of MICMG Series, John D. Nystuen
 Professor of Geography and Urban Planning
 The University of Michigan

 1.  Reprint of the Papers of the Michigan InterUniversity
 Community of Mathematical Geographers. 
 Editor, John D. Nystuen.
 Cost:  \$39.95, hardcopy.
 
 Contents--original editor:  John D. Nystuen.
 
 1.  Arthur Getis, ``Temporal land use pattern analysis with the
 use of nearest neighbor and quadrat methods."  July, 1963
 
 2.  Marc Anderson, ``A working bibliography of mathematical
 geography."  September, 1963.
 
 3.  William Bunge, ``Patterns of location."  February, 1964.

 4.  Michael F. Dacey, ``Imperfections in the uniform plane."
 June, 1964.
 
 5.  Robert S. Yuill, A simulation study of barrier effects
 in spatial diffusion problems."  April, 1965.
 
 6.  William Warntz, ``A note on surfaces and paths and
 applications to geographical problems."  May, 1965.
 
 7.  Stig Nordbeck, ``The law of allometric growth."
 June, 1965.
 
 8.  Waldo R. Tobler, ``Numerical map generalization;"
 and Waldo R. Tobler, ``Notes on the analysis of geographical
 distributions."  January, 1966.
 
 9.  Peter R. Gould, ``On mental maps."  September, 1966.
 
 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.
 
 11. E. Casetti and R. K. Semple, ``A method for the
 stepwise separation of spatial trends."  April, 1968.
 
 12.  W. Bunge, R. Guyot, A. Karlin, R. Martin, W. Pattison,
 W. Tobler, S. Toulmin, and W. Warntz, ``The philosophy of maps."
 June, 1968.
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 Reprints of out-of-print textbooks.
 
 1.  Allen K. Philbrick.  This Human World.
 
 2.  John F. Kolars and John D. Nystuen.  Human Geography. 
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