## Babylonian Weight Riddle Problems 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 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 12Dec2018

## Introduction

gold Here is some TCL calculations for Babylonian weight riddle problems in calculator shell.

In the cuneiform math problems and coefficient lists on clay tablets, there are coefficient numbers which were used in determining the amount of materials and the daily work rates of the workers. In most cases, the math problem is how the coefficient was used in estimating materials, work rates, and math problems. One difficulty is determining the effective magnitude or power of the number coefficient in the base 60 notation. In cuneiform, numbers in base 60 are written using a relative notation. For example, 20 could represent either 20*3600,20,20/60, 20/3600, or even 1/20. The basic dimensions and final tallies were presented in the cuneiform accounts on clay tablets, but some calculations, some units, and some problem answers (aw shucks!) were left off the tablet. Successive or iterated math solutions are called algorithms and the Babylonian methods are some of the earliest algorithms documented circa 1600 BCE. The TCL procedures are descendants of this idea. The Babylonians did not use algebra notation, decimal notation, or modern units, so the reader will have to bear some anachronisms in the TCL code. At least one approach for the modern reader and using modern terminology is to develop the implied algebraic equations and decimal equivalents from the cuneiform numbers. Then the TCL calculator can be run over a number of testcases to validate the algebraic equations.

## Babylonian Weight Riddle Problems

The Babylonians did not use algebra notation. The answer was given without worked solution, so problem was solved with algebra, ref Neugebauer and Sachs. User should be able to add and subtract terms of linear equation by 60/+7/+11 or 60/-7/-11 in entry fields. .

## Pseudocode Section

```       # using pseudocode for Babylonian weight riddle problems
# possible problem instances
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
60.0 :final weight
7.0 :fraction 1/a
11.0 :fraction 1/b
1.0 :answers: optional
1. :optional
1. :optional
1. :optional
48.125 :initial weight

#### Testcase 2

table 2printed in tcl wiki format
quantity value comment, if any
2:testcase_number
60.0 :final weight
8.0 :fraction 1/a
12.0 :fraction 1/b
1.0 :answers: optional
1. :optional
1. :optional
1. :optional
49.230 :initial weight

#### Testcase 3

table 3printed in tcl wiki format
quantity value comment, if any
3:testcase_number
120.0 :final weight
12.0 :fraction 1/a
15.0 :fraction 1/b
1.0 :answers: optional
1. :optional
1. :optional
1. :optional
103.846 :initial weight

### 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 <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
• 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 Weight Riddle Problems calculator
# written on Windows XP on eTCL
# working under TCL version 8.5.6 and 1.0.1
# gold on TCL WIKI, 25jan2017
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 {{} { final weight  :} }
lappend names { fraction 1/a :}
lappend names { fraction 1/b : }
lappend names { answers: optional : }
lappend names { optional :}
lappend names { optional : }
lappend names { optional : }
lappend names { initial weight   :}
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 Weight Riddle Problems
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 weight \$side1
set fraction1 \$side2
set fraction2 \$side3
# initialize placeholder answer
set result 1.
set term1 [+ 1. [/ 1. \$fraction1 ]]
set term2   [/ 1. \$fraction2 ]
set term3 [+ \$term1 [* \$term2  \$term1] ]
set result [/ \$weight \$term3 ]
set side5 1.
set side6 1.
set side7 1.
set side8 \$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
console show;
puts "%|table \$testcase_number|printed in| tcl wiki format|% "
puts "&| quantity| value| comment, if any|& "
puts "&| \$testcase_number:|testcase_number | |& "
puts "&| \$side1 :|final weight  |   |&"
puts "&| \$side2 :|fraction 1/a  | |& "
puts "&| \$side3 :|fraction 1/b  | |& "
puts "&| \$side4 :|answers: optional| |&"
puts "&| \$side5 :|optional  | |&"
puts "&| \$side6 :|optional |  |&"
puts "&| \$side7 :|optional   |  |&"
puts "&| \$side8 :|initial weight |  |&"
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 60.  7.   11.0 1.  1.  1. 1. 48.0}
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 60.  8.0  12.0  1.  1.  1. 1. 49.0 }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 120. 12.0 15.0 1.  1.  1. 1. 104.0 }
::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 Weight Riddle Problems 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   |&"  ```

## Comments Section

Please place any comments here with your wiki MONIKER and date, Thanks gold 12Dec2018

 Category Numerical Analysis Category Toys Category Calculator Category Mathematics Category Example Toys and Games Category Games Category Application Category GUI

 Category Development Category Concept Category Algorithm