**Babylonian Multiplication Algorithm 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] ---- <> [gold] Here is some eTCL starter code for Babylonian Multiplication Algorithm in calculator shell. The Babylonians did not use algebra notation. For examples, algorithm will give product of long side and short front side in meters for product of field area in meters squared. This eTCL calculator was used in study of almost square fields and quasi_cubes. User should be able to multiply terms as 11*10 or 61*60 in entry fields. . ---- **Pseudocode Section** ====== # using pseudocode for Babylonian Multiplication Algorithm # possible problem instances, almost square fields # babylonian multiplication rule a * b = ((a + b)/2)^2 - ((a - b)/2)^2 initialize algorithm_result = 1. term1 = ((a + b)/2) term2 = ((a - b)/2) # babylonian looks up squares in table algorithm_result = tranformed square (term1 ) - square (term2 ) # probably should put algorithm in subroutine or procedure for # transfer to other program check algorithm check_product = expr a * b set answers and printout with resulting values pseudocode: need test cases > small,medium, giant pseudocode: need testcases within range of expected operation. pseudocode: are there any cases too small or large to be solved? ====== ***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 | |& &| 11.0 :|long side meters | |& &| 10.0 :|short front side meters| |& &| 1.0 :|optional side meters| |& &| 1.1000 :|answers: ratio long over short | |& &| 110.25 :|squared term1 | |& &| 0.25 :|squared term2 | |& &| 110.0 :|check product (expr a*b) square meters | |& &| 110.0 :|product from algorithm square meters | |& **** Testcase 2 **** %|table 2|printed in| tcl wiki format|% &| quantity| value| comment, if any|& &| 2:|testcase_number | |& &| 61.0 :|long side meters | |& &| 60.0 :|short front side meters| |& &| 1.0 :|optional meters| |& &| 1.0166 :|answers: ratio long over short | |& &| 3660.25 :|squared term1 | |& &| 0.25 :|squared term2 | |& &| 3660.0 :|check product (expr a*b) square meters | |& &| 3660.0 :|product from algorithm square meters | |& **** Testcase 3 **** %|table 3|printed in| tcl wiki format|% &| quantity| value| comment, if any|& &| 3:|testcase_number | |& &| 3601.0 :|long side meters | |& &| 3600.0 :|short front side meters| |& &| 1.0 :|optional meters| |& &| 1.000277 :|answers: ratio long over short | |& &| 12963600.25 :|squared term1 | |& &| 0.25 :|squared term2 | |& &| 12963600.0 :|check product (expr a*b) square meters | |& &| 12963600.0 :|product from algorithm square meters | |& ---- ***Screenshots Section*** ****figure 1.**** [Babylonian Multiplicatiion Algorithm and example demo eTCL calculator screenshot ] ---- ***References:*** * Mathematical Cuneiform Texts, Neugebauer and Sachs * Extraction of Cube Roots in Babylonian Mathematics, Kazuo Muroi, Centaurus Volume 31, issue 3, 1988 * Babylonian Mathematical Texts II-III Author(s): A. Sachs Source: Journal of Cuneiform Studies, Vol. 6, No. 4 * (1952), pp. 151-156 Published by: The American Schools of Oriental Research * Computing the Cube Root, Ken Turkowski, Apple Computer Technical Report #KT-32 10 February 1998 * Approximating Square Roots and Cube Roots , Ali Ibrahim Hussenom, 2014/11/04 * Aryabhata’s Root Extraction Methods, Abhishek Parakh , Louisiana State University, Aug 31st 2006 * Another Method for Extracting Cube Roots, Brian J. Shelburne, * Dept of Math and Computer, Science Wittenberg University * Jeanette C. Fincke* and Mathieu Ossendrijver* BM 46550 – a Late Babylonian Mathematical Tablet with * Computations of Reciprocal Numbers,Zeitschrift für Assyriologie 2016; 106(2): 185–197 * Interpretation of reverse algorithms in several mesopotamian texts, Christine Proust * 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 * Wikipedia search engine * 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] * [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 ---- **Appendix Code** ***appendix TCL programs and scripts *** ====== # pretty print from autoindent and ased editor # Babylonian Multiplication Algorithm calculator # written on Windows XP on eTCL # working under TCL version 8.5.6 and eTCL 1.0.1 # gold on TCL WIKI, 15jan2017 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 {{} { long side meters :} } lappend names { short front side meters :} lappend names { optional meters : } lappend names { answers: ratio long over short : } lappend names { squared term1 :} lappend names { squared term2 : } lappend names { check product (expr a*b) square meters : } lappend names { product from algorithm square meters :} 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 Multiplication Algorithm from TCL WIKI, written on eTCL " tk_messageBox -title "About" -message $msg } 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 long_side $side1 set short_front_side $side2 set height $side3 # babylonian multiplication rule a * b = ((a + b)/2)^2 - ((a - b)/2)^2 set term1 [* [+ $long_side $short_front_side ] .5 ] set term1 [** $term1 2 ] set term2 [* [- $long_side $short_front_side ] .5 ] set term2 [** $term2 2 ] # babylonian uses table lookup here set algorithm_result [- $term1 $term2 ] set check_product_tcl [expr { $long_side * $short_front_side }] set ratio_long_over_short [/ $long_side $short_front_side ] set side4 $ratio_long_over_short set side5 $term1 set side6 $term2 set side7 $check_product_tcl set side8 $algorithm_result } proc fillup {aa bb cc dd ee ff gg hh} { .frame.entry1 insert 0 "$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 reference_factor flag console show; puts "%|table $testcase_number|printed in| tcl wiki format|% " puts "&| quantity| value| comment, if any|& " puts "&| $testcase_number:|testcase_number | |& " puts "&| $side1 :|long side meters | |&" puts "&| $side2 :|short front side meters| |& " puts "&| $side3 :|optional meters| |& " puts "&| $side4 :|answers: ratio long over short | |&" puts "&| $side5 :|squared term1 | |&" puts "&| $side6 :|squared term2 | |&" puts "&| $side7 :|check product (expr a*b) square meters | |&" puts "&| $side8 :|product from algorithm square meters | |&" } frame .buttons -bg aquamarine4 ::ttk::button .calculator -text "Solve" -command { calculate } ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 11.0 10.0 1.0 1.1 110.25 0.25 110. 110.} ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 61. 60. 1. 1.01 3660.25 0.25 3660. 3660. } ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 3601.0 3600.0 1.0 1.0001 12963600.0 0.25 12963600. 12963600.0 } ::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 Multiplication Algorithm 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 |&" ====== ---- [gold] This page is copyrighted under the TCL/TK license terms, [http://tcl.tk/software/tcltk/license.html%|%this license]. **Comments Section** <> Please place any comments here, Thanks. <> Numerical Analysis | Toys | Calculator | Mathematics| Example| Toys and Games | Games | Application | GUI ---- <> Development | Concept| Algorithm