**Babylonian Number Series and eTCL demo example calculator, numerical analysis  **

This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER  in your comment with the same courtesy that I will give you. Its very hard to reply intelligibly without some background of the correspondent. Thanks,[gold]
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<<TOC>>

[gold] Here is some eTCL starter code for Babylonian number series.

# following statements can be pasted into eTCL console
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 set project 1.0
 set sum_of_squares [ expr 1*1+2*2+3*3+4*4+5*5+6*6+7*7+8*8+9*9+10*10 ] # answer 385
 set sum_of_squares [ expr 5*5+6*6+7*7+8*8+9*9+10*10 ] # answer 355
 set sum_of_squares [ expr 2*2+3*3+4*4+5*5 ]   # answer 54
 set sum_of_squares [ expr 1*1+2*2+3*3+4*4+5*5+6*6+7*7+8*8+9*9+10*10+11*11+12*12+13*13+14*14+15*15 ] # answer 1240
 set sum_of_2_powers [ expr 1**1+2**2+3**2+4**2+5**2+6**2+7**2+8**2+9**2+10**2]   
======

One tablet gives a method for finding the sum of a ladder (geometric series?). From 1 to 10, sum the powers (literally the "ladder") of 2. The last term you add is 8_32 or decimal 512. Subtract 1 from 8_32, obtaining 8_31. Add 8_31 to 8_32, obtaining the answer 17_3 or decimal 1023.  modern formula = 2**n+2**(n-1). There is uncertaincy in the math term "ladder or stair", since 512 is 2**9 (rather than 2**10 as 1024). 

One tablet gives a method for finding the sum of a quadratic series. Add the squares from 1 to 10, the last term is 100. The answer is 6_25 in base 60 or decimal 385. The modern formula is (1/3+( 2/3 )*n)*k

For the first testcase, the first term is 1, final term is 100, sum is 385. For the second  testcase, the final term is 100.For the third  testcase, the final term is 55.

One tablet showed a method for finding an approximate square root, using sqrt(a**2 + b) ~=~ a + b/(2*a). As the eTCL console program developed, a**2 was the highest square below the rational number N1, and b was N1-a**2. The error in the square root was about 0.38 percent. The square root method was modified for the cube root using b/(3*a)  and the error in the cube root was about 8 percent.
======
  set sqrt_error [* 100. [- 1. [/ 3.1666666666666665 3.1622776601683795]] ]  # 0.3879257199 percent error
  set cubert_error [* 100. [- 1. [/ 2.3333333333333335 2.154434690031884]]]  # 8.30 percent error
  # program finds highest square below rational number, for initial guess of square root
  proc list2 { aa bb} { for {set i 1} {[* [+ $i 1] [+ $i 1]  ]<=[- $bb 1]} {incr i} {set j 1 };return $i}
  list2 1 100    #answer 9  ,  9*9 = 81 < 100
  # program finds highest cube below rational number, for initial guess of cube root
  proc list3 { aa bb} { for {set i 1} {[* [+ $i 1] [+ $i 1]  [+ $i 1] ]<=[- $bb 1]} {incr i} {set j 1 };return $i}
  [list3 1 100]    #answer 4  ,   4*4*4=64 < 100
  # list squares 
  proc listsq { aa bb} { for {set i 1} {$i<=$bb} {incr i} {lappend booboo [ * $i  $i 1. ] };return $booboo}
  [listsq 1 5]
  #answer 1.0 4.0 9.0 16.0 25.0
  proc listnumbers { aa bb} { for {set i 1} {$i<=$bb} {incr i} {lappend booboo [ expr 1.* $i] };return $booboo}
  #returns list of integer numbers from aa to bb as reals with decimals, usage listnumbers 1 5
  [listnumbers 1 5] 
  # answer is 1.0 2.0 3.0 4.0 5.0 
  proc sumsquares5 x {expr {$x<2? 1: $x*$x+[sumsquares5 [incr x -1]]}}
  # usage sumsquares5 2 # answer 5
  proc sum_ladder x {expr {$x<1? 1: 2**$x+[sum_ladder [incr x -1]]}}
  # usage sum_ladder 2  # answer 7 = 2**0+2**1+2**2
======  
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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). 

**** Testcase 1 ****
%|table 1|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 1:|testcase_number | |&
&| 0.0 :|start|   |&
&| 10.0 :|end   | |& 
&| 1.0 :|optional | |& 
&| 1.0 :|optional | |&
&| 0.0 :|answers: initial term | |&
&| 100 :|final term   |  |&
&| 385 :|sum  |  |&
&| 385 :|sum |  |&
 
**** Testcase 2 ****
%|table 2|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 2:|testcase_number | |&
&| 2.0 :|start|   |&
&| 5.0 :|end   | |& 
&| 1.0 :|optional | |& 
&| 1.0 :|optional | |&
&| 1.0 :|answers: initial term | |&
&| 25 :|final term   |  |&
&| 54 :|sum  |  |&
&| 54 :|sum |  |&

**** Testcase 3 ****
%|table 3|printed in| tcl wiki format|% 
&| quantity| value| comment, if any|& 
&| 3:|testcase_number | |&
&| 0.0 :|start|   |&
&| 15.0 :|end   | |& 
&| 1.0 :|optional | |& 
&| 1.0 :|optional | |&
&| 1.0 :|answers: initial term | |&
&| 225 :|final term   |  |&
&| 1240 :|sum  |  |&
&| 1240 :|sum |  |&

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***Screenshots Section***
****figure 1.****
[Babylonian Number Series and eTCL demo example calculator screenshot 2]
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***References:***
   *   goggle "Babylonian Number cuneiform Series"
   * [Oneliner's Pie in the Sky] 
   * [One Liners] 
   * google <  Babylonian Interest Rates Wikipedia >
   * Ancient Babylonian Algorithms, Donald E. Knuth, Stanford University 
   * The oldest example of compound interest in Sumer, seventh power of four thirds (4/3)**7 ,  Kazuo MUROI  
   * Interest, Price, and Profit: An Overview of Mathematical Economics in YBC 46981,
   * Robert Middeke-Conlin Christine Proust,  CNRS and Université Paris Diderot
   * Compound Interest Doubling Time Rule: Extensions and Examples from Antiquities,
   * Saad Taha Bakir,  ISSN: 2241 - 1968 (print), 2241 – 195X (online) 
   * Two Sumerian Words of Fractions in Babylonian Mathematics: igi-n-gál and igi-te-en , Kazuo MUROI 
   * O. Neugebauer and A. Sachs, Mathematical Cuneiform Texts, American Oriental Society, Series 29. New Haven, 1945. 
   * K. Muroi, Interest Calculation in Babylonian Mathematics:
   * New Interpretations of VAT 8521 and VAT 8528, Historia Scientiarum, 39, (1990), 29-34
   * O. Nuegebauer, The Exact Sciences in Antiquity, Second edition, Dover Publication Inc., New York, 1969

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**Appendix Code**

***appendix TCL programs and scripts ***

======
        # pretty print from autoindent and ased editor
        # Babylonian Number Series calculator
        # written on Windows XP on eTCL
        # working under TCL version 8.5.6 and eTCL 1.0.1
        # gold on TCL WIKI, 15dec2016
        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 {{} {start :} }
        lappend names {end :}
        lappend names {optional : }
        lappend names {optional : }
        lappend names {answers: initial term :}
        lappend names {final term: }
        lappend names {sum: }
        lappend names {sum:} 
        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 Number Series 
            from TCL WIKI,
            written on eTCL "
            tk_messageBox -title "About" -message $msg } 
             proc sum_function { start end } { 
             global side1 side2 side3 side4 side5
             global side6 side7 side8   
             set sum 0
             set counter 1
             set start [int $start ]
             set counter [int $start ]
             set end [int $end ]
             while { $counter < 1E6  } {
             set sum [+ $sum [* $counter $counter ]]
             if { $counter   < $start } {set side5 [* $counter $counter ] } 
             set side6 [* $counter $counter ]
             if { $counter   > [- $end 1. ] } { break}              
             incr counter 
             }
             return $sum  }  
       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 start $side1  
            set end $side2 
            set sum_of_squares [sum_function $start $end ]
            set side7 $sum_of_squares
            set side8 $sum_of_squares
                    }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$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;
            puts "%|table $testcase_number|printed in| tcl wiki format|% "
            puts "&| quantity| value| comment, if any|& "
            puts "&| $testcase_number:|testcase_number | |&"
            puts "&| $side1 :|start|   |&"
            puts "&| $side2 :|end   | |& "  
            puts "&| $side3 :|optional | |& "
            puts "&| $side4 :|optional | |&"
            puts "&| $side5 :|answers: initial term | |&"
            puts "&| $side6 :|final term   |  |&"
            puts "&| $side7 :|sum  |  |&"
            puts "&| $side8 :|sum |  |&"                     
            }
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 0.0 10.  1.0 1.0  0.0  100. 385 385}
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 2.0 5.0 1.0 1.0  1.0  25. 55. 55. }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 0.0 15. 1.0 1.0  1.0   225 1240 1240 }
        ::ttk::button .clearallx -text clear -command {clearx }
        ::ttk::button .about -text about -command {about}
        ::ttk::button .cons -text report -command { reportx }
        ::ttk::button .exit -text exit -command {exit}
        pack .calculator  -in .buttons -side top -padx 10 -pady 5
        pack  .clearallx .cons .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 Number Series Calculator"      
======
**Console program for approximate roots using Babylonian method**
======
                  # autoindent from ased editor
                  # console program for babylonian algorithm for roots.
                  # follows tablet formulas but not very accurate  
                  # written on Windows XP on eTCL
                  # working under TCL version 8.5.6 and eTCL 1.0.1
                  # TCL WIKI , 12dec2016
             console show
             package require math::numtheory
             #namespace path {math::numtheory}
             namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
             set tcl_precision 17
         proc square_root_function { number_for_root  } {   
             set counter 0
             while { $counter < 50.  } {
             if { [* $counter $counter 1. ]   > [* $number_for_root 1.] } {break}            
             incr counter 
             }      
             set starter [- $counter 1  ]
             set remainder   [* $starter $starter 1. ]  
             set remainder [- $number_for_root [* $starter $starter 1. ] ]
             set  square_root_estimate  [+ $starter [/ $remainder [* 2. $starter ]]]
              }  
         proc cube_root_function { number_for_root  } {   
             set counter 0
             while { $counter < 50.  } {
             if { [* $counter $counter $counter  1. ]   > [* $number_for_root 1.] } {break}            
             incr counter 
                     }      
             set starter [- $counter 1  ]
             set remainder   [* $starter $starter $starter 1. ]  
             set remainder [- $number_for_root [* $starter $starter $starter 1. ] ]
             set  square_root_estimate  [+ $starter [/ $remainder [* 3. $starter ]]]
              }  
             puts " [square_root_function 10 ] "
             puts " [cube_root_function 10 ] "
======
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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   |&"  
======

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[gold] This page is copyrighted under the TCL/TK license terms, [http://tcl.tk/software/tcltk/license.html%|%this license]. 

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