Updated 2017-02-18 20:00:06 by gold

## Babylonian Multiplication Algorithm and eTCL demo example calculator, numerical analysis edit

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 edit

```       # 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 1printed 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 2printed 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 3printed 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

### 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 edit

### 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
set msg "Calculator for Babylonian Multiplication Algorithm
from TCL WIKI,
written on eTCL "
tk_messageBox -title "About" -message \$msg }
proc calculate {     } {
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 .cons -text report -command { reportx }
::ttk::button .exit -text exit -command {exit}
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   |&"  ```