Updated 2017-10-25 19:12:09 by gold

Babylonian Square Side Rule & Diagonal Procedure Algorithm and eTCL demo example calculator, numerical analysis edit

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

gold Here is some eTCL starter code for Babylonian Square Side Rule & Diagonal Algorithm in calculator shell.

The Babylonian Square Side Rule from clay tablets was loaded into an eTCL calculator shell. The Babylonian Square Side Rule is of historical interest, but the rule is not very accurate and dependent on an initial input or guess. The square side rule with a single trial input becomes less accurate with increasing N. In some math problems, the Babylonians used the Square Side Rule for calculating the diagonals of squares and rectangles. The Babylonians did not use algebra notation, so the reader will have to bear some anachronisms in the eTCL pseudocode. Succesive or iterated math solutions are called algorithms and the Square Side Rule procedure is one of the earliest algorithms documented. The TCL procedures are descendants of this idea.

In the Babylonian problem texts on clay tablets, there are numbers and coefficients which were used in determining geometric dimensions, the amount of materials, and the daily work rates of the workers. One difficulty is determining the effective power of the numbers in base 60. For example, 20 could represent either 20*3600, 20*60, 20, 20/60, 20/3600, or even 1/20. The basic dimensions of geometric figures and final tallies were presented in the Babylonian accounts on clay tablets, but sometimes the calculations were left off the tablet, broken off, or garbled. At least one approach for the modern reader and using modern terminology is to develop the implied algebraic equations from the Babylonian numbers. Then the eTCL calculator can be run over a number of testcases to validate the algebraic equations. In the eTCL code, extra significant digits are retained to check the function errors, but not to imply greater precision in the algorithm. The Babylonians did not use percent estimates, but that seemed the most understandable error rating for the eTCL calculator.

For the first testcase of sqrt 2, the initial trial_square_root entry of 2 gave an error of 6.06 percent. For the second testcase of sqrt 10, the initial trial_square_root entry of 3 gave an error of 0.138 percent. For the third testcase, the initial trial_square_root entry of 9 gave an error of 0.555 percent. An iterative solution with successive trial_square_root entries loaded as 9, 10.05555, and 10.000153437 showed 1.177E-8 percent error in three interations. The formula factor is usually integer 2 in the texts, rarely three. A console program was added below for picking trial_square_root entries.

Pseudocode Section edit

    # using  pseudocode for Babylonian procedure algorithm.
            3 quantities needed
            target number
            trial_square_root,   w.a. guess
            formula factor , usually 2 or 3
            set approximate_root [* [/ 1. $formula_factor ] [+ [* [/ 1. $trial_square_root ] $target_number] $trial_square_root ]   ]
            set side5  $trial_square_root
            set side6 [+ [* [/ 1. $trial_square_root ] $target_number] $trial_square_root ]  
            set side7 [* [/ 1. $formula_factor ]] 
     check approx. root from square side rule with sqrt function in TCL
     ref. errorx procedure        
     check_answer   new area =? desired goal , desired goal reached (yes/no)
     set answers and printout with resulting values

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 1printed in tcl wiki format
quantity value comment, if any
1:testcase_number
2.0 :target number N
2.0 :trial square root
2.0 :function factor
3.0 :answers: intermediate term in formula
4.0 :trial square
1.414 :square root from TCL sqrt function
6.066 :percentage error
1.5 :approximate square root from square side rule

Testcase 2

table 2printed in tcl wiki format
quantity value comment, if any
2:testcase_number
10.0 :target number N
3.0 :trial square root
2.0 :function factor
6.333 :answers: intermediate term in formula
9.0 :trial square
3.162 :square root from TCL sqrt function
0.138 :percentage error
3.166 :approximate square root from square side rule

Testcase 3

table 3printed in tcl wiki format
quantity value comment, if any
3:testcase_number
100.0 :target number N
9.0 :trial square root
2.0 :function factor
20.111 :answers: intermediate term in formula
81.0 :trial square
10.0 :square root from TCL sqrt function
0.555 :percentage error
10.0555 :approximate square root from square side rule

Screenshots Section

figure 1.


References:

  • Daniel F. Mansfield , N.J. Wildberger.
  • Plimpton 322 is Babylonian exact sexagesimal trigonometry.
  • Ref. square side rule
  • Historia Mathematica, August 2017 DOI: 10.1016/j.hm.2017.08.001
  • Newsweek, Babylonian Tablet Could Hold Mathematical
  • Secrets For Today's Researchers, Josh Lowe , 8/25/17
  • Plimpton 322 Tablet and the Babylonian Method of
  • Generating Pythagorean Triples,
  • Abdulrahman A. Abdulaziz, University of Balamand, 2010
  • Seaching for Babylonian Triplets Slot Calculator Example
  • Babylonian trailing edge algorithm and reverse sequence algorithm for reciprocals, eTCL demo example calculator, numerical analysis
  • [Babylonian Field Expansion Procedure Algorithm and eTCL demo example calculator, numerical analysis]
  • [Babylonian False Position Algorithm and eTCL demo example calculator, numerical analysis]
  • 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
  • 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.
  • Wikipedia, Methods of computing square roots,
  • Ref. square side rule also known as Babylonian method or Heron's method
  • Another simple database
  • Richard Suchenwirth
  • editRecord
  • Oneliner's Pie in the Sky, ref. errorx procedure
  • One Liners
  • Category Algorithm
  • Square Root
  • [Babylonian Number Series and eTCL demo example calculator]
  • Brahmagupta Area of Cyclic Quadrilateral and eTCL demo example calculator
  • 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
  • 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
  • Poles and walls in Mesopotamia and Egypt , Melville,
  • www.sciencedirect.com/science/article
  • Pythagorean ‘Rule’ and ‘Theorem’ – Mirror of the Relation Between Babylonian and Greek Mathematics,
  • Jens Høyrup, Roskilde University
  • On Plimpton 322. Pythagorean numbers in Babylonian mathematics, E. M. Bruins
  • PLIMPTON 322: A UNIVERSAL CUNEIFORM TABLE FOR OLD BABYLONIAN
  • MATHEMATICIANS, BUILDERS, SURVEYORS AND TEACHERS
  • Rudolf Hajossy

Appendix Code edit

appendix TCL programs and scripts

        # pretty print from autoindent and ased editor
        # Babylonian Square Side Rule calculator
        # written on Windows XP on eTCL
        # working under TCL version 8.5.6 and eTCL 1.0.1
        # gold on TCL WIKI, 2oct2017
        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 {{} {target number N :} }
        lappend names {trial square root:}
        lappend names {formula factor (usually 2 ) : }
        lappend names {answers: intermediate term in formula }
        lappend names {trial square :}
        lappend names {square root from TCL sqrt function : }
        lappend names {percentage error: }
        lappend names {approximate square root from square side rule :} 
        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 Square Side Rule
            from TCL WIKI,
            written on eTCL "
            tk_messageBox -title "About" -message $msg } 
       proc ::tcl::mathfunc::precision {precision float}  {
            #  tcl:wiki:Floating-point formatting, [AM]
            set x [ format "%#.5g" $float ]
            return $x
           }   
            #proc errorx always returns a positive error. 
            #Normally assume $aa is human estimate,
            #assume $bb is divinely exact.
       proc errorx  {aa bb} {expr { $aa > $bb ?   (($aa*1.)/$bb -1.)*100. : (($bb*1.)/$aa -1.)*100.}}
       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 target_number $side1
            set trial_square_root $side2
            set formula_factor $side3
       set approximate_root [* [/ 1. $formula_factor ] [+ [* [/ 1. $trial_square_root ] $target_number] $trial_square_root ]   ]
            set side4 [+ [* [/ 1. $trial_square_root ] $target_number] $trial_square_root ]  
            set side5 [* $trial_square_root $trial_square_root]
            set side6 [sqrt $target_number ]  
            set side7 [ errorx $approximate_root [sqrt  $target_number ]   ]
            set side8 $approximate_root
                    }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$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 answer2
            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 :|target number N  |   |&"
            puts "&| $side2 :|trial square root | |& "  
            puts "&| $side3 :|formula factor (usually 2 )| |& "
            puts "&| $side4 :|answers: intermediate term in formula| |&"
            puts "&| $side5 :|trial square | |&"
            puts "&| $side6 :|square root from TCL sqrt function |  |&"
            puts "&| $side7 :|percentage error |  |&"
            puts "&| $side8 :|approximate square root from square side rule |  |&" 
            }
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { set side8 0 ; calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 2.  2.0  2.0 3.0  4.0   1.414    6.066  1.5}
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 10.0 3.0 2.0 6.333 9.0   3.162 0.138     3.166}
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 100.0 9.0 2.0  20.111  81.0    10.0  0.5555  10.0555}
        ::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 Square Side Rule Calculator"          

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 for Newton's Method

see SQUARE ROOT on wiki
        # gold on TCL WIKI, 2oct2017
        # console program for square root
        # additional verbose Newton's Method
        package require Tk
        package require math::numtheory
        set tcl_precision 17
             console show
             namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
         proc square_root_function { number_for_root  } {   
             set counter 0
             set epsilon .0001
             while { $counter < 50.  } {
             if { [* $counter $counter 1. ]   > [* $number_for_root 1.] } {break}            
             incr counter 
             }   
             set  square_root_estimate $counter
             while {1} {
             set keeper $square_root_estimate        
             set starter $square_root_estimate
             set remainder   [* $starter $starter  1. ]  
             set remainder [- $number_for_root [* $starter $starter  1. ] ]
             set  square_root_estimate  [+ $starter [/ $remainder [* 2. $starter ]]]
             if {abs($keeper - $square_root_estimate) < $epsilon} break
                   }
             return $square_root_estimate
                }     
            puts " [ square_root_function 10. ] "

console program for brackets of root

                # pretty print from autoindent and ased editor
                # console program for token multiplication and square root
                # additional verbose  
                # working under TCL version 8.5.6 and eTCL 1.0.1
                # program written on Windows XP on eTCL
                # gold on TCL WIKI, 10Mar2017
                package require Tk
                package require math::numtheory
                namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
                set tcl_precision 17
                console show
             global keeper target_number keep_under
             set keeper 0
             proc square_root_functionx { number_for_root  } {  
             global keeper target_number keep_under
             set counter 0
             set epsilon .0001
             while { $counter < 1000.  } {
             set keeper [* $counter $counter 1. ] 
             set target_number $number_for_root
             if { [* $counter $counter 1. ]   > [* $number_for_root 1.] } {break}  
             set keep_under $counter         
             incr counter 
             }   
             return $counter}
            puts " function gives [ square_root_functionx 10. ] ,  root for $target_number between $keep_under and  [ square_root_functionx 10. ] , max square $keeper "
            #returns positive integers under and over root.
            # function gives 4 ,  root for 10. between 3 and  4 , max square 16.0 

gold This page is copyrighted under the TCL/TK license terms, this license.

Comments Section edit

Please place any comments here, Thanks.