Introduction

gold Here is some eTCL starter code for Babylonian cube root rule of eight algorithm in calculator shell.

The Babylonian tablets VAT8587 and YBC6295 with cube root problems have been analysed by Sachs and Muroi. The tablets call for a integer square root solution, but after some manipulation and circular agruments, use a second known integer square root to find the answer. These cube root problems are of historic interest, and there are linguistic and procedural elements in these cube root problems which may aid in understanding other tablets. As best understood, the cube root rule of eight algorithm was loaded into an eTCL 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 and some units were left off the tablet. 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 eTCL calculator can be run over a number of testcases to validate the algebraic equations.

Using modern notation, the cube root rule of eight is that multiplying by eight under the radical has the same value as multiplying the radical by two. In the algorithm, multiplying by eight under the radical and multiplying by half (.5) outside the radical gives the same initial value.

Another tool was factorization or splitting a number into a set of factors whose product equaled a positive integer. This section included some terminology that the Babylonians did not use. A number could be broken down into factors, then take reciprocal of each factor. Then the product of the reciprocal factors was the reciprocal of the original number. For example, take the number 24 which factors into (6)*(4) =24. Then take the reciprocals (1/6)*(1/4) or (0.25)*(0.1666). The product of (0.25)*(0.1666) is 0.041665, the reciprocal of the original number. As a check, 0.041665 * 24 = 1. The factorization could be extended to square roots and cube roots. For example on square roots, factor the number 36 as 9*4. Taking the sqrt(9) *sqrt(4) or 3*2 is 6, which is the square root of the original number 36. For example on cube roots, factor the number 64 as 8*8. Taking the crt(8) * crt(8) or 2*2 is 4, which is the cube root of the original number 64. Also the Babylonians had an operation called doubling. Take a number and its reciprocal, then double the number. The reciprocal of the doubled number is the half of the reciprocal of the first number. For example, take the number 24 and its reciprocal 0.041665 from above, 2*24 is 48, and (1/2)*(1/24) is the reciprocal 0.020833 of the doubled number. Check 0.020833*48 equals 1. However, at least on most clay tablets, the format was operation N1, N1 operation N2, or operation N2 on individual lines. There is little overt evidence for operation on multiple numbers in single line as operation N1 N2 N3, like a series notation, expr {N1*N2*N2}, or ::tcl::mathop:: <* N1 N2 N3>.

The Babylonian tablets VAT8587 and YBC6295 have lines like operation N1 N2, here N2 was a carry over value or extra column2. The instructions place N2 as next or "under" the operation N1, which may be saving N2 as a second set of factors. The trailing edge reciprocal algorithm on this wiki has this feature also. The Babylonian tablets VAT8587 and YBC6295 are demonstrating factorization with the rule of eight. Certainly, factorization could reduce a large cube_root to an easier value to handle.

Pseudocode Section

# using pseudocode for Babylonian cube root rule of eight algorithm
# possible problem instances include, given n ,
# find sqrt(n) from rule of eight and second known sqrt
target_number = supplied value
# cube root rule of eight algorithm
# initilize place holder
set target_number 1.
B. 7.30 notation = 7/60+30/3600= decimal 0.125,  reciprocal of 0.125 = 8
B. 30 notation = 30/60 = 1/2 = decimal 0.5
cube root 0.125 = decimal 0.5
set term2 (8 * (target_number - 0.125))
# assuming integer cube table lookup here
set term3 [** \$term2 [/ 1. 3. ]]
issue  whether operation is  target_number - 0.125 or target_number * 0.125
if target_number * 0.125 is correct,
equivalent to (attempt?)  removing factor of 8 from target number
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
27.0 :target number
1.0 :optional
1.0 :optional
1. :optional
1. :optional
1. :optional
2.995363207447546 :cube root

Testcase 2

table 2printed in tcl wiki format
quantity value comment, if any
2:testcase_number
64.0 :target number
1.0 :optional
1.0 :optional
1. :optional
1. :optional
1. :optional
3.9973941360702776 :cube root

Testcase 3

table 3printed in tcl wiki format
quantity value comment, if any
3:testcase_number
125.0 :target number
1.0 :optional
1.0 :optional
1. :optional
1. :optional
1. :optional
4.9983327774689297 :cube root

Screenshots Section

figure 1. References:

• 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 Cube Root Algorithm 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 {{} { target number  :} }
lappend names { optional :}
lappend names { optional : }
lappend names { answers: optional : }
lappend names { optional :}
lappend names { optional : }
lappend names { optional : }
lappend names { cube root :}
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 Cube Root Algorithm 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 target_number \$side1
# initialize placeholder answer
set cube_root 1.
set term1 [- \$target_number 0.125 ]
set term2 [* \$term1 8. ]
# assuming table look up here
set term3 [** \$term2 [/ 1. 3. ]]
set cube_root [* \$term3 [/ 30. 60.] ]
set side5 1.
set side6 1.
set side7 1.
set side8 \$cube_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 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 :|target number  |   |&"
puts "&| \$side2 :|optional  | |& "
puts "&| \$side3 :|optional  | |& "
puts "&| \$side4 :|answers: optional| |&"
puts "&| \$side5 :|optional  | |&"
puts "&| \$side6 :|optional |  |&"
puts "&| \$side7 :|optional   |  |&"
puts "&| \$side8 :|cube root |  |&"
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 27.  1.   1.0 1.  1.  1. 1. 3.0}
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 64.  1.0  1.0  1.  1.  1. 1. 4.0 }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 125. 1.0  1.0 1.  1.  1. 1. 5.0 }
::ttk::button .clearallx -text clear -command {clearx }
::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 Cube Root Algorithm 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   |&"