## Introduction

gold Here is some eTCL starter code for calculating daily pay of ancient Sumerian porters. The impetus for these calculations was checking mud brick construction and porter pay in some math problems, translation texts, and modern replicas. Most of the testcases involve replicas or models, using assumptions and rules of thumb.

In the Sumerian coefficient lists on clay tablets, there are coefficients which were used in determining the daily work rates of the construction workers and porters. In most cases, the math problem is how the coefficient was used in estimating materials and work rates. One difficulty is determining the effective power of the coefficient in base 60. For example, 20 could represent either 20*3600,20,20/60, 20/3600, or even 1/20. The work teams and final tallies were presented in the Sumerian accounts on clay tablets, but the calculations were left off the tablet. At least one approach for the modern reader using modern terminology and decimal notation is to develop the implied algebraic equations from the Sumerian numbers. Then the eTCL calculator can be run over a number of testcases to validate the algebraic equations.

For the first testcase from tablet YBC 4673, Neugebauer and Robson translated a math problem for a porter carrying 5 sixties of bricks for a distance of 30 rods (180 meters). The standard pay presented in the math problem was 9 sixties of bricks over 30 rods for pay of 10 liters of grain. Pay should be calculated in proportion to a standard pay, given on this tablet occasion. However, some tablets are broken off, obscured, or otherwise leave off the pay standard. The base60 answer on the tablet was 5+1/2+3/60+(1/3)*(1/60) or converting pay of 5.5555 decimal liters of grain. Setting proportions and using decimal notation, new_pay*9.*30.=10.*5.*30., new_pay=(10.*5.*30.)/(9.*30.), and new_pay=5.5555 liters of grain. A Sumerian rod equals 6 meters, but the units cancel out in the proportions equation, so merely enter consistent units in the eTCL calculator. Extra decimals are kept to check the accuracy of the eTCL calculator, not to imply higher accuracy off the numbers from the clay tablet.

The task standard in the first testcase can be used to estimate a trial days task of porting. Setting proportions for a days wage of 2 liters grain, 2liters/X=10liters/9, X=(2*9)/10, X= 1.8 sixties of bricks. rounding 2 sixties of bricks. Setting proportions for a days wage of 1 liters grain, 1liters/X=10liters/9, X=(1*9)/10, X= 0.9 sixties of bricks. rounding 1 sixties of bricks.

For the second testcase from tablet YBC 4673, Robson translated a math problem asking for number of porters for 39000 bricks over standard distance (30)? The given answer was 60+12 porters. Setting proportions from the previous standard, 1 man / 9 sixties = X/39000 bricks, X = (1.*39000.)/(9.*60), X = 72 porters. The grain wages on the tablet were obscured. Setting proportions from the previous standard, 10 liters / 9 sixties = X/39000 bricks, X = (10.*39000.)/(9.*60.), X = 722 liters. Checking previous answer, if one porter receives pay of 10 liters, then a project of 72 workers should receive 72*10, 720 liters grain. For the second testcase in the eTCL calculator, since brick units cancel out in the calculator, one can either load total bricks as 540 and 39000 bricks, or 650 and 9 sixties of bricks. The eTCL calculator returns 72.2 worker days and 72.22 liters grain.

```# following statements can be pasted into eTCL console
set silas_convert_from_tablet_decimal [ eval expr 5.+1./2.+3./60.+(1./3.)*(1./60.) ] # 5.5555 liters of  grain
set silas_calculation_decimal  [ eval expr (10.*5.*30.)/(9.*30.)]  # 5.5555 liters of grain
set numbers_porters [ eval expr (39000.)/(9.*60.)]  # 72 porters
set project_grain_porters [eval expr (10.*39000.)/(9.*60.)] # 722 liters```

For the third testcase from standards of carrying baskets of mud, one can develop a math problem for a short day like testcase 1 for bricks. The standard pay for porting mud was 2 liters grain to carry 45 baskets of mud for 1 us unit (360 meters ) per workday. Each basket was 25 kilograms of mud and totaled 45*25 or 1125 kilograms. For the work of a half day, the baskets would be 45/2, rounding 23 baskets of mud. An equivalent formula would be new_pay*(standard_load*standard_distance)=(standard_pay*worked_load*worked_distance) , new_pay=(standard_pay*worked_load*worked_distance)/(standard_load*standard_distance). Substituting, new_pay=(2.*23.*360.)/(45.*360.), rounded 1 liter grain. Substituting, new_pay=(2.*23.*360.)/(45.*360.), rounded1 liter grain. For the third testcase in the eTCL calculator, the calculator returns 0.511 worker days and 1.022 liters grain.

`set  new_pay [eval expr (2.*23.*360.)/(45.*360.)] # 1.0222 liters grain`

### 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
testcase number:1
30.0 :porters distance meters
30.0 :porters standard distance kilometers usually
1.0 :porters day integer or fraction usually
1. :porters day integer or fraction
5.555555555555555 :porters pay liters grain
0.09259259259259259 :porters pay silver pieces

#### Testcase 2

table 2printed in tcl wiki format
quantity value comment, if any
testcase number:1
30.0 :porters distance meters
30.0 :porters standard distance kilometers
10.0 :porters standard pay liters usually 2
72.22222222222221 :porters day integer or fraction
722.2222222222222 :porters pay liters grain
12.037037037037036 :porters pay silver pieces

#### Testcase 3

table 3printed in tcl wiki format
quantity value comment, if any
testcase number:3
360.0 :porters distance meters
360.0 :porters standard distance kilometers usually
2.0 :porters standard pay liters usually 2
0.5111111111111111 :porters day integer or fraction usually 1
1.0222222222222221 :porters pay liters grain
0.017037037037037035 :porters pay silver pieces

### References:

• Joran Friberg (2001) Bricks and mud in metro-mathematical cuneiform texts
• A remarkable collection of Babylonian mathematical texts By Joran Friberg
• Eleanor Robson, Mesopotamian Mathematics, 2100-1600 BC (Oxford, 1999)
• Robson, Eleanor, Mesopotamian Mathematics, 2100-1600BCE,Oxford 1999
• Horowitz, Wayne, Late Babylonian Tablet CBS1766, Hebrew University
• Steele, J.M. Celestial Measurement in Bablylonian Astronomy,Annals of Science,2007
• Mathematical Cuneiform Texts, Neugebauer and A. Sachs, American Oriental Society, 1945
• Friberg 1987-90:555,Firberg on tablet BM15285
• Eclipse Prediction and the Length of the Saros in
• Babylonian Astronomy LIS BRACK-BERNSEN∗AND JOHN M. STEELE
• Celestial Measurement in Babylonian Astronomy, J. M. STEELE, University of Durham
• Amazing Traces of a Babylonian Origin in Greek Mathematics, Jöran Friberg and Joachim Marzahn
• The area and the side i added: some old Babylonian geometry, duncan j. Melville
• Sumerian Circular Segment Coefficients and Calculator Demo Example
• Sumerian Coefficients in the Pottery Factory and Calculator Demo Example
• Sumerian Pottery Vessel & Clay Mass and eTCL Slot Calculator Demo Example , numerical analysis
• Mathematics hidden behind the two coefficients of Babylonian geometry, kazuo muroi
• Especially oven/kiln problem in YBC7997, area of ring annulus complements bullseye figure.
• Sumerian Barge & Cargo Calculator and eTCL Slot Calculator Demo Example, numerical analysis
• Sumerian Coefficients at the Weavers Factory and eTCL Slot Calculator Demo Example
• Sumerian Construction Rates and eTCL Slot Calculator Demo Example
• Sumerian Workcrew & Payroll and eTCL Slot Calculator Demo Example, numerical analysis

## Appendix Code

### appendix TCL programs and scripts

```        # pretty print from autoindent and ased editor
# Sumerian Porters Pay calculator
# written on Windows XP on eTCL
# working under TCL version 8.5.6 and eTCL 1.0.1
# gold on TCL WIKI, 5may2016
package require Tk
namespace path {::tcl::mathop ::tcl::mathfunc}
frame .frame -relief flat -bg aquamarine4
pack .frame -side top -fill y -anchor center
set names {{} {porters load quantity:} }
lappend names {porters distance meters :}
lappend names {optional, porters standard load kilograms: }
lappend names {optional, porters standard distance meters:}
lappend names {optional, porters standard pay liters:}
lappend names {answers:  porters day integer or fraction: }
lappend names {porters pay liters grain: }
lappend names {porters pay silver pieces :}
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 Sumerian Porters Pay
from TCL WIKI,
written on eTCL "
tk_messageBox -title "About" -message \$msg }
proc pi {} {expr acos(-1)}
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 worked_distance \$side2
set standard_distance \$side4
set standard_pay  \$side5
set day_fraction 1.
set side7 [/ [* \$standard_pay  \$worked_load \$worked_distance] [* \$standard_load \$standard_distance ] ]
set grain_pay \$side7
set day_fraction [/ \$grain_pay \$standard_pay ]
set side6 \$day_fraction
set silver [* [/ \$grain_pay 2. ] [/ 1. 30. ]]
set side8 \$silver       }
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 :|porters load kilograms|   |&"
puts "&| \$side2 :|porters distance meters| |& "
puts "&| \$side3 :|porters standard load kilograms| |& "
puts "&| \$side4 :|porters standard distance kilometers |      |&"
puts "&| \$side5 :|porters standard pay liters  | usually 2  |&"
puts "&| \$side6 :|porters day integer or fraction |  |&"
puts "&| \$side7 :|porters pay liters grain|  |&"
puts "&| \$side8 :|porters pay silver pieces|  |&"
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 5. 30.0 9.0 30.0  10.0 1.0 5.5 0.09}
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 650. 30.0 9.0 30.0   10.0  1.0 5.5 0.09 }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 23. 360.0 45.0 360.0  2.0 0.51  1.02 0.017 }
::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 . "Sumerian Porters Pay 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   |&"  ```