**Hamurabi.tcl Text Adventure Game and TCL Demo Example**

This page is under development. Constructive comments are welcome, but please load any constructive comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks, [gold] 08Jul2020
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<<TOC>>
***Preface***
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[gold] Here are some TCL calculations for. 
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***Introduction*** 
The  
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From the references, . 

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***  Procedures ***
 
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In the 
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**Pseudocode Section**
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            # end of file, pseudocode:
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***Testcases Section***  

In planning any software, it is advisable to gather a number of testcases to check the results of the program. The math for the testcases can be checked by pasting statements in the TCL console. Aside from the TCL calculator display, when one presses the report button on the calculator, one will have console show access to the   capacity  functions (subroutines). 
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**** Testcase 1  , !st right sided trapezoid  ****
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The  
 

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**** Testcase 2  ,  d  ****
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**** Testcase 3 ,   B   ****
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The following scenari 
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**** Testcase 4 ,    ****
 

**** Testcase 5  ,  ****
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The following scenario was used for the _th testcase. 
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**** Testcase 6  ,   e****
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The following scenario was used for the 6th testcase. There is  Astrology software available to derive some testcases from modern numbers. It turns out that the  year 2020 offers an interesting retrograde of Jupiter in astrology. 

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***Screenshots Section***
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****figure 1a.****
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[bablylonian_trapezoid_cal]
****figure 1b. Babylonian Astronomy Trapezoid Area Concept & Formulas ****
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[bablylonian_trapezoid]
****figure 1c. Previous sample of vaporware ****
[Babylonian Square rule for Trapezoid Area png panel] 
****figure 1d., previous  sample vaporware****

[Babylonian Square Side Rule png]
****figure 2.****
[Babylonian Square rule for Trapezoid Area png printout] 
****figure 3.****
[Babylonian Square rule for Trapezoid Area png more concept]
****figure 4.****
[Babylonian Square rule for Trapezoid Area png concept]
****figure 5.****
[Babylonian Combined Market Rates png trapezoidal prism concept diagram 3]
****figure 6.****
[Babylonian Combined Market Rates png concept diagram 2]
****figure 7.****
[Babylonian Square rule for Trapezoid Area png geometry grid overlay]
****figure 8.****
[Babylonian Square rule for Trapezoid Area png geometric figure trapeziod band]
****figure 9.****
[Babylonian Trapezoid Bisection Algorithm and eTCL demo small scale fields png] 
****figure 10.****
[Babylonian Trapezoid Bisection Algorithm large scale fields]
****figure 11.****
[Babylonian Trapezoid Bisection Algorithm and eTCL demo smallest plots png]
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****figure 12.Babylonian_zigzag_function_for_full_moon_eclipses****
[Babylonian_moon_zigzag]
****figure 13.Babylonian_step_function_for_full_moon_eclipses****
[Babylonian_moon2]
****figure 14.Babylonian_step_functions_with_quadratic_zigzags_overlay_for_Jupiter_occultation_of_star****
[Babylonian_Expansion_jupiter]
****figure 15.Babylonian_Expansion_Procedure_120_day_chart ****
[Babylonian_Expansion_Procedure_120_day_chart]
****figure 16.Contrast in 2 Different Prediction Algorithms (meaning,  2 possibilities) ****
[Babylonian_Expansion_Procedure_Jupiter_velocity]
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****figure 16.quadratic_regression ****
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          0               0.2           
         60               0.1583      
        120               0.0250    
======  
[Babylonian_Expansion_quadratic_regression_correction]
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****figure 17.image of Jupiter taken by NASA's Hubble Space Telescope ****
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[Babylonian_Expansion_Procedure_image_Jupiter_NASA]

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***References:***
   * A Geometric Algorithm with Solutions to Quadratic Equations
   * in a Sumerian Juridical Document from Ur III Umma
   * Joran Friberg,  Chalmers University of Technology, Gothenburg, Sweden
   * Geometric        division problems, quadratic equations,        and recursive
   * geometric        algorithms in        Mesopotamia,Joran Friberg  
   * google search engine <Trapezoid area bisection> 
   * Wikipedia search engine <Trapezoid area > 
   * mathworld.wolfram.com, Trapezoid and right trapezoid
   * Mathematical Treasure: Old Babylonian Area Calculation, uses ancient method
   * Frank J. Swetz , Pennsylvania State University
   * Wikipedia, see temple of Edfu, area method used as late as 200 BC in Egypt.
   * [Oneliner's Pie in the Sky] 
   * [One Liners] 
   * [Category Algorithm]
   * [Babylonian Number Series and eTCL demo example calculator]
   * [Brahmagupta Area of Cyclic Quadrilateral and eTCL demo example calculator]
   * [Babylonian Trapezoid Bisection Algorithm and eTCL demo example calculator, numerical analysis]
   * [Chinese Horse Race Problems from Suanshu, DFP, and example eTCL demo calculator, numerical analysis]
   * [Gauss Approximate Number of Primes and eTCL demo example calculator]
   * Land surveying in ancient Mesopotamia, M. A. R. Cooper
   * [Sumerian Approximate Area Quadrilateral and eTCL Slot Calculator Demo Example , numerical analysis] 
   * Thomas G. Edwards, Using the Ancient Method of False Position to Find Solutions
   * Joy B. Easton, rule of double false position
   * Vera Sanford, rule of false position
   * www.britannica.com, topic, mathematics trapezoid
   * [Sumerian Equivalency Values, Ratios, and the Law of Proportions with Demo Example Calculator]
   * [Babylonian Sexagesimal Notation for Math on Clay Tablets in Console Example] 
   * Babylonians Tracked Jupiter With Advanced Tools: Trapezoids, Michael Greshko, news.nationalgeographic.com
   * Geometry in Babylonian Astronomy, Cluster of Excellence Topology, Humboldt University of Berlin
   * Mathieu Ossendrijver: „Ancient Babylonian astronomers calculated Jupiter’s position
   * from the area under a time-velocity graph“, in: Science, January 29, 2016.
   * Late Babylonian Field Plans in the British Museum, books.google.com/books 
   * Karen Rhea Nemet-Nejat
   * Late Babylonian Surface Mensuration Author(s): Marvin A. Powell Source: jstor
   * translation:  trapezoid in two babylonian astronomical cuneiform
   * texts for jupiter (act 813 & act 817) from the seleucid era , 310 BC -75 AD 
   * Otto Neugebauer, Astronomical Cuneiform Texts, 3 Vols. 
   * Lund Humphreys, London, 1955:405,430-31. 
   * DeSegnac, MS 3908 A RE-CONSTRUCTION,  D.A.R. DeSegnac
   * A draft for an essay 
   * DeSegnac, MENTAL COMPUTING OF THREE ARCHAIC
   * MESOPOTAMIAN PUZZLES W 20044, 35, W 20044, 20 & W 20214, essay draft
   * DeSegnac, HARMONY OF NUMBERS I and II,  D.A.R. DeSegnac, A draft for an essay

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** More References **
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   * The Marduk Star Nēbiru, Immanuel Freedman,Cuneiform Digital Library Bulletin 2015:3, 8 November 2015
   * Computing planetary positions - a tutorial with worked examples, webpage By Paul Schlyter, Stockholm, Sweden
   * New perspective on the Exaltation of Planets, Blog of Kiril Stoychev, Bulgarian Astrological Association, December    
   * 20, 2015
   * Probability for Lunar Occultation, for maximum distance from the ecliptic, note by dotancohen,  Apr 7 2014, 
   * Mathematics  Stack Exchange
   * Babylonian Mathematical Astronomy, Chapter,  January 2015, Mathieu Ossendrijver, Freie Universität Berlin
   * Babylonian Market Predictions, Chapter on April 2019,  Mathieu Ossendrijver, Freie Universität Berlin
   * BM 32339+32407+32645 - New Evidence for Late Babylonian Astrology. 
   * Chapter on September 2018, Mathieu Ossendrijver, 
   * Freie Universität Berlin
   * Ancient Babylonians took first steps to calculus, article  By Ron Cowen, Science magazine, 29Jan2016
   * Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph,
   * Article by Mathieu Ossendrijver, Science magazine, 29Jan2016
   * Babylonian mathematical astronomy procedure texts, bibliography 
   * Babylonian astronomy,  Wikipedia, collected 2020
   * Babylonian and Indian Astronomy: Early Connections, Subhash Kak, February 17, 2003
   * Electrical & Computer Engineering, Louisiana State University, Baton Rouge, LA
   * [Babylonian Trapezoid Bisection Algorithm and eTCL demo example calculator, numerical analysis]
   * [Babylonian Field Expansion Procedure Algorithm and example demo eTCL calculator, numerical analysis]
   * Article by Mathieu Ossendrijver, mathematical terminology in Babylonian astronomical texts, 2012
   * [Babylonian False Position Algorithm and eTCL demo example calculator, numerical analysis]
   * [Babylonian Combined Market Rates and eTCL demo example calculator, numerical analysis]
   * [Babylonian Brothers Inheritance Problems Algorithm and eTCL demo example calculator, numerical analysis]
   * [Babylonian Cubic Equation Problem and eTCL demo example calculator, numerical analysis]
   * [Sumerian Base 60 conversion and eTCL demo example calculator, numerical analysis]
   * [Aryabhat Sum of Squares and Cubes and eTCL demo example calculator, numerical analysis]
   * [Sumerian Approximate Area Quadrilateral and eTCL Slot Calculator Demo Example , numerical analysis]
   * Article by Mathieu Ossendrijver, Babylonian-computational-astronomy-print.pdf, 2012
   * Babylonian Astronomy in Context, 
   * Keeping the Watch: Babylonian Astronomical Diaries and More, Mathieu Ossendrijver
   * Humboldt University Berlin, April 3 2018
   * Exploring Jupiter: The Astrological Key to Progress, Prosperity & Potential,
   * 310 Pages , 1996  by Stephen Arroyo
   * A Primer on Logarithms by Shailesh Shirali,  Mathematical Marvels Universities Press
   * Studies on the Ancient Exact Sciences in Honor of Lis Brack-Bernsen, eds John M. Steele
   * Mathieu Ossendrijver
   * A study of Babylonian planetary theory, subtitle  The outer planets, by Teije de Jong1
   * date 31 August 2018
   * Under One Sky: Astronomy and Mathematics in the Ancient Near East
   * Author(s): John M. Steele, Annette Imhausen, Publisher: Ugarit-Verlag, Year: 2002
   * Rising Time Schemes in Babylonian Astronomy, Author(s): John M. Steele, Publisher: Springer, Year: 2017
   * Observation, theory and practice in late Babylonian astronomy: Some preliminary observations, Author(s): Steele J.
   * Ancient astronomy and celestial divination, Author(s): N M Swerdlow
   * The Babylonian theory of the planets, Author(s): Noel M. Swerdlow,
   * Publisher: Princeton University Press, Year: 1998
   * Administrative Timekeeping in Ancient Mesopotamia, Robert Englund,
   * Journal of the Economic and Social History of the Orient 31 (1988) 121-185
   * Bisectable Trapezia in Babylonian Mathematics, Lis Brack Bernsen,Olaf Schmidt 
   * First published: April 1990 
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***Table, Math Terms Associated with Babylonian Trapezoids ***
%|Table , Math Terms Associated with Babylonian Trapezoids  |printed in tcl format|modern equivalent, English| comment if any|% 
&| sag ki.gu :| side earth   |  trapezoid |   |&
&| sag ki.ta :| side earth its , possessive case | upper front of the trapezoid | |  |&
&| sag an.na  :|side upper  | upper front of the trapezoid  | ||& 
&|us gi.na :| length front  | true length of the trapezoid||& 
&| sag gi.na :| side front| true front of trapezoid |  |&
&| N1 kus i-ku-lu :| N1 cubit in eats it (it or man),  read left to right | feed parameter of the trapezoid | noun derived from Sumerian SOV phrase   |&
&| sag du :| side | triangle |  |&
&|N1 arakarum :| N1 multiply | transformation coefficient, usually scale up |  |&
Note. These Math Terms cover 4 or 5 cuneiform languages, and 5000 years of history. Math Terms are best understanding of  context  and math connotation here,not blessed by linguists and cuneiform dictionaries. 
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**Appendix Code**

***appendix TCL programs and scripts ***
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======
       # pretty print from autoindent and ased editor
        # Babylonian Astronomy Trapezoid Area  V2
        # written on Windows 10 on TCL
        # working under TCL version 8.6
        # gold on  TCL Club, 4Jul2020
        package require Tk
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        frame .frame -relief flat -bg aquamarine4
        pack .frame -side top -fill y -anchor center
        set names {{} {side height1 , degrees per day:} }
        lappend names {side height2 , degrees per day:}
        lappend names {base line length1, days: }
        lappend names {base line length2, days: }
        lappend names { answers, bisection velocity_c degrees per day:}
        lappend names { bisection, half area displacement degrees:}
        lappend names { bisection_time_c  days:}
        lappend names { area displacement degrees:}
        foreach i {1 2 3 4 5 6 7 8} {
            label .frame.label$i -text [lindex $names $i] -anchor e
            entry .frame.entry$i -width 35 -textvariable side$i
            grid .frame.label$i .frame.entry$i -sticky ew -pady 2 -padx 1 }
        proc about {} {
            set msg "Calculator for Babylonian Astronomy Trapezoid Area  V2
            from TCL ,
            # gold on  TCL Club, 10Jul2020 "
            tk_messageBox -title "About" -message $msg }
        proc self_help {} {
            set msg "Calculator for Babylonian Astronomy Trapezoid Area V2
            from TCL ,
            # self help listing
            # 4 givens follow.
            1) side height1 , degrees per day
            2) side height2 , degrees per day
            3) base line length1, days 
            4) base line length2, days
            The solution from the clay tablets
            is genuine antique method from 1600 BCE
            and different looking from modern math methods.
            # The Babylonians did not use modern algebraic notation,
            # so the reader will have to bear some anachronisms in the TCL code.
            # Comparing conventional formula(s) for trapezoid area
            # to Babylonian square rule for trapezoid area.
            # For comparison, code will include redundant paths & formulas
            # to compute trapezoid area. The calculator uses modern
            # units  for convenience to modern users and textbooks.
            # Any convenient and consistent in/output units might be used
            # like inches, feet, nindas, cubits, or dollars to donuts.
            # Equations
            1) median = .5*(l+s)                        conventional
            2) trapezoid area = .5*(l+s)*h              conventional
            3) feed = (l-s)/h                           trapezoid_area_sq_law
            4) h = (l-s)/f                              trapezoid_area_sq_law
            5) trapezoid area = (l**2-s**2)/(2.*feed)   trapezoid_area_sq_law
            # Also, the B. square rule for trapezoid area
            # was extended to volumes by *width.
            6) trapezoidal volume = .5*(l+s)*h*w              conventional
            7) trapezoidal volume = w*(l**2-s**2)/(2.*feed)   trapezoid_area_sq_law
            # Recommended procedure is push testcase and fill frame,
            # change first three entries etc, push solve,
            # and then push report. Report allows copy and paste
            # from console to conventional texteditor. For testcases
            # testcase number is internal to the calculator and
            # will not be printed until the report button is pushed
            # for the current result numbers.
            # This posting, screenshots, and TCL source code is
            # copyrighted under the TCL/TK 8.6 license terms.
            # Editorial rights retained under the TCL/TK license terms
            # and will be defended as necessary in court.
            Conventional text editor formulas or  grabbed from internet
            screens can be pasted into green console.
            Try copy and paste following into green screen console
            set answer [* 1. 2. 3. 4. 5. ]
            returns  120
            # gold on  TCL Club, 10jul2020 "
            tk_messageBox -title "About" -message $msg }
        # following proc is  not used now
        proc trapezoid_solution { long_side short_side feed width} {
            global trapezoid_area height_calc feed_calc median trapezoidal_vol
            set median [* .5 [+ $long_side $short_side ]]
            set height_calc [/ [- $long_side $short_side] $feed ]
            set trapezoid_area [/ [- [* $long_side $long_side] [* $short_side $short_side] ] [* 2. $feed ] ]
            set trapezoidal_vol [* $width [/ [- [* $long_side $long_side] [* $short_side $short_side] ] [* 2. $feed ] ]]
            set feed_calc [ expr {  ($long_side-$short_side)/ $height_calc } ]
            puts "  check  $long_side $short_side $feed height_calc= $height_calc area= $trapezoid_area feed= $feed_calc  "
            return $trapezoid_area  }
        proc calculate {     } {
            global answer2
            global side1 side2 side3 side4 side5
            global side6 side7 side8
            global testcase_number
            incr testcase_number
            set side1 [* $side1 1. ]
            set side2 [* $side2 1. ]
            set side3 [* $side3 1. ]
            set side4 [* $side4 1. ]
            set side5 [* $side5 1. ]
            set side6 [* $side6 1. ]
            set side7 [* $side7 1. ]
            set side8 [* $side8 1. ]
            set side5 1.
            set side6 1.
            set side7 1.
            set side8 1.
            set velocity_0 $side1
            set velocity_1 $side2
            set base_line [ expr { abs (  $side4 - $side3 )} ]
            set calculation_1 [ expr { .5 * ( $side1 + $side2 ) * 60. } ]
            set bisection_1 [ expr { .5 * $calculation_1 } ]
            set bisection_velocity_c [ expr {  ((($velocity_0)**2 +($velocity_1)**2)/2.)**.5 }]
            set bisection_time_c [ expr { $base_line * (  $velocity_0 -  $bisection_velocity_c  ) /  ( $velocity_0 -  $velocity_1  )  }]
            set side5 $bisection_velocity_c
            set side6 $bisection_1
            set side7 $bisection_time_c
            set side8 $calculation_1
        }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$aa"
            .frame.entry2 insert 0 "$bb"
            .frame.entry3 insert 0 "$cc"
            .frame.entry4 insert 0 "$dd"
            .frame.entry5 insert 0 "$ee"
            .frame.entry6 insert 0 "$ff"
            .frame.entry7 insert 0 "$gg"
            .frame.entry8 insert 0 "$hh"
        }
        proc clearx {} {
            foreach i {1 2 3 4 5 6 7 8 } {
                .frame.entry$i delete 0 end } }
        proc reportx {} {
            global side1 side2 side3 side4 side5
            global side6 side7 side8
            global testcase_number
            console show;
            console eval {.console config -bg palegreen}
            console eval {.console config -font {fixed 20 bold}}
            console eval {wm geometry . 40x20}
            console eval {wm title . " Babylonian Astronomy Trapezoid  V2 Report , screen grab and paste from console 2 to texteditor"}
            console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
            puts "%|table $testcase_number|printed in| tcl format|% "
            puts "&| quantity| value| comment, if any|& "
            puts "&| $testcase_number:|testcase_number | |&"
            puts "&| $side1 :|side height1 , degrees per day: |   |&"
            puts "&| $side2 :|side height2 , degrees per day: | |& "
            puts "&| $side3 :|base line length1, days: | |& "
            puts "&| $side4 :|base line length2, days: | |&"
            puts "&| $side5 :|answers, bisection velocity_c degrees per day:| |  |&"
            puts "&| $side6 :|bisection, half area displacement degrees:| |  |&"
            puts "&| $side7 :|bisection_time_c days:| |  |&"
            puts "&| $side8 :|area displacement degrees:| |  |&"
            
        }
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 0.2 0.1583 0.  60.  0.1803 5.3745  28.2601  10.75 }
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 0.1583 0.0250  60.  120.  5.5 2.74949  20.2450  5.5 }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 0.2 0.0250 0.  120.  0.1425 3.375  39.4135  6.75  }
        ::ttk::button .clearallx -text clear -command {clearx }
        ::ttk::button .about -text about -command {about}
        ::ttk::button .self_help -text self_help -command {self_help}
        ::ttk::button .console2 -text report -command { reportx }
        ::ttk::button .exit -text exit -command {exit}
        pack .calculator  -in .buttons -side top -padx 10 -pady 5
        pack  .clearallx .console2 .self_help .about .exit .test4 .test3 .test2   -side bottom -in .buttons
        grid .frame .buttons -sticky ns -pady {0 10}
        . configure -background aquamarine4 -highlightcolor brown -relief raised -border 30
        wm title . "Babylonian Astronomy Trapezoid Area  V2"
        # gold on TCL Club, 8Jul2020
        # This posting, screenshots, and TCL source code is
        # copyrighted under the TCL/TK 8.6 license terms.
        # Editorial rights retained under the TCL/TK license terms
        # and will be defended as necessary in court.
        # end of file
======

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*** Pushbutton Operation***

For the push buttons, the recommended procedure is push testcase and fill frame, change first three entries etc, push solve, and then push report. Report allows copy and paste from console. 

For  testcases in a computer session, the eTCL calculator increments a new testcase number internally, eg. TC(1), TC(2) , TC(3) , TC(N). The testcase number is internal to the calculator and will not be printed until the report button is pushed for the current result numbers. The current result numbers will be cleared on the next solve button.   The command { calculate; reportx  } or { calculate ; reportx; clearx  } can be added or changed to report automatically. Another wrinkle would be to print out the current text,  delimiters,  and numbers in a TCL wiki style table as

======
  puts " %| testcase $testcase_number | value| units |comment |%"
  puts " &| volume| $volume| cubic meters |based on length $side1 and width $side2   |&"  
======
**Console program under test. **
***   Expansion subroutine ***

[gold] - 2020-07-5
 

======
 
             # console program written on Windows 10
             # working under TCL version 8.6  
             # gold moniker on TCL WIKI , 7jul2020
             console show
             package require math::numtheory
             namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
             set tcl_precision 17
             proc expansion_procedure_2 { side1 side2 side3 side4 side5 epsilon } { 
             set counter 1
             set token1 $side1
             set token2 $side2
             set token3 $side3
             set epsilon [/ 1. $side5]
             puts " %| Table of Natural Log and Round_off  |% "
             puts " %| counter | value | rounded value | ln |  round off | round off  |% "
             while { $counter < 30.  } {
             #if { [abs [- $side4 [* $token1 $token2 $token3 1. ] ] ]  < $epsilon } {break;} 
             #if { [- $side4 [* $token1 $token2 $token3 1. ] ]   > 0 } {set correction_fraction [* 1. [/ 1. $side5]  ]} 
             #if { [- $side4 [* $token1 $token2 $token3 1. ] ]   < 0 } {set correction_fraction [* -1. [/ 1. $side5]  ]}        
             #set correction_fraction [- $side3 [* $token1 $token2  1. ] ]
             incr counter  
             set token1 [ expr ($token1 + .1 ) ]
             set token2 [ expr (log ($token1)) ]
             set round_off [format %7.2f $token2 ]
             set value_rounded [format %7.2f $token1 ]
             puts " &| $counter | $token1 | $value_rounded | $token2 |  $round_off | $round_off |& "
               }   }
             set side8 [ expansion_procedure_2   1.0 1.0 1.0 1000. 10. 1.   ]
             # printout
======
**printout**
======
 
======

 

======
                  # pretty print from autoindent and ased editor
                  # Babylonian  Expansion Procedure Algorithm Calculator V2
                  # console program written on Windows 10  
                  # working under TCL version 8.6  
                  # TCL WIKI , 2jul2020
             console show
             package require math::numtheory
             namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
             set tcl_precision 17
             proc expansion_procedure2 { side1 side2 side3 side4 side5 epsilon } { 
             set counter 1
             set token1 $side1
             set token2 $side2
             set token3 $side3
             set saver1 .00001
             set saver2 .00001
             set saver3 .00001
             set epsilon [/ 1. $side5]
             while { $counter < 1000.  } {
             if { [abs [- $side4 [* $token1 $token2 $token3 1. ] ] ]  < $epsilon } {break;} 
             if { [- $side4 [* $token1 $token2 $token3 1. ] ]   > 0 } {set correction_fraction [* 1. [/ 1. $side5]  ]} 
             if { [- $side4 [* $token1 $token2 $token3 1. ] ]   < 0 } {set correction_fraction [* -1. [/ 1. $side5]  ]}        
             #set correction_fraction [- $side3 [* $token1 $token2  1. ] ]
             set token1 [+ $token1 $correction_fraction ]
             set token2 [+ $token2 $correction_fraction ] 
             set token3 [+ $token3 $correction_fraction ] 
             incr counter  
             puts " &| token1 | $token1 | token2 | $token2 | token3 | $token3 | product | [* $token1 $token2 $token3 ] | correction | $correction_fraction |& "
               }   }
             set side8 [ expansion_procedure_2   9.6 9.6 9.6 1000. 60. .15   ]
             # printout follows
======
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** Table of Natural Log and Round_off **
----
%| Table of Natural Log and Round_off  |% 
%|table |printed in| tcl wiki format|% 
%| quantity| value| comment, if any|% 
%| quantity| value| comment, if any|% 
%| counter | value | rounded value | ln |  round off | round off  |% 
&| 2 | 1.1000000000000001 |    1.10 | 0.095310179804324935 |     0.10 |    0.10 |& 
&| 3 | 1.2000000000000002 |    1.20 | 0.18232155679395479 |     0.18 |    0.18 |& 
&| 4 | 1.3000000000000003 |    1.30 | 0.26236426446749128 |     0.26 |    0.26 |& 
&| 5 | 1.4000000000000004 |    1.40 | 0.33647223662121317 |     0.34 |    0.34 |& 
&| 6 | 1.5000000000000004 |    1.50 | 0.40546510810816466 |     0.41 |    0.41 |& 
&| 7 | 1.6000000000000005 |    1.60 | 0.47000362924573591 |     0.47 |    0.47 |& 
&| 8 | 1.7000000000000006 |    1.70 | 0.53062825106217071 |     0.53 |    0.53 |& 
&| 9 | 1.8000000000000007 |    1.80 | 0.58778666490211939 |     0.59 |    0.59 |& 
&| 10 | 1.9000000000000008 |    1.90 | 0.64185388617239525 |     0.64 |    0.64 |& 
&| 11 | 2.0000000000000009 |    2.00 | 0.69314718055994573 |     0.69 |    0.69 |& 
&| 12 | 2.100000000000001 |    2.10 | 0.74193734472937778 |     0.74 |    0.74 |& 
&| 13 | 2.2000000000000011 |    2.20 | 0.78845736036427061 |     0.79 |    0.79 |& 
&| 14 | 2.3000000000000012 |    2.30 | 0.83290912293510455 |     0.83 |    0.83 |& 
&| 15 | 2.4000000000000012 |    2.40 | 0.87546873735390041 |     0.88 |    0.88 |& 
&| 16 | 2.5000000000000013 |    2.50 | 0.91629073187415555 |     0.92 |    0.92 |& 
&| 17 | 2.6000000000000014 |    2.60 | 0.9555114450274369 |     0.96 |    0.96 |& 
&| 18 | 2.7000000000000015 |    2.70 | 0.993251773010284 |     0.99 |    0.99 |& 
&| 19 | 2.8000000000000016 |    2.80 | 1.0296194171811588 |     1.03 |    1.03 |& 
&| 20 | 2.9000000000000017 |    2.90 | 1.0647107369924289 |     1.06 |    1.06 |& 
&| 21 | 3.0000000000000018 |    3.00 | 1.0986122886681102 |     1.10 |    1.10 |& 
&| 22 | 3.1000000000000019 |    3.10 | 1.1314021114911013 |     1.13 |    1.13 |& 
&| 23 | 3.200000000000002 |    3.20 | 1.1631508098056815 |     1.16 |    1.16 |& 
&| 24 | 3.300000000000002 |    3.30 | 1.1939224684724352 |     1.19 |    1.19 |& 
&| 25 | 3.4000000000000021 |    3.40 | 1.2237754316221163 |     1.22 |    1.22 |& 
&| 26 | 3.5000000000000022 |    3.50 | 1.2527629684953687 |     1.25 |    1.25 |& 
&| 27 | 3.6000000000000023 |    3.60 | 1.2809338454620649 |     1.28 |    1.28 |& 
&| 28 | 3.7000000000000024 |    3.70 | 1.3083328196501793 |     1.31 |    1.31 |& 
&| 29 | 3.8000000000000025 |    3.80 | 1.3350010667323406 |     1.34 |    1.34 |& 
&| 30 | 3.9000000000000026 |    3.90 | 1.3609765531356015 |     1.36 |    1.36 |& 

 
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