Sumerian Porters Pay Formula and eTCL Slot Calculator Demo Example, numerical analysis

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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, Robson translated a math problem for a porter carrying 5 sixties of bricks for a distance of 30 rods. 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.

# 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

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
0.4774 :cylinder outer diameter meters
1.0 :cylinder outer height meters
1.0 :cylinder ratio numerator no unitsusually 1
4.0 :cylinder ratio denominator no units usually 4
1.4997963328237671 :answers:cylinder outer circumference meters
0.2387:radius meters
31.41592653589793 :cylinder lateral surface area square meters
125.66370614359172 : cylinder total surface area square meters
0.1790006923225166: cylinder overall volume cubic meters
0.011187543270157288 : cylinder inner volume cubic meters
0.16781314905235933 :cylinder outer volume cubic meters

Testcase 2

table 2printed in tcl wiki format
quantity value comment, if any
testcase number:2
0.4774 :cylinder outer diameter meters
10.0 :cylinder outer height meters
1.0 :cylinder ratio numerator no unitsusually 1
4.0 :cylinder ratio denominator no units usually 4
1.4997963328237671 :answers:cylinder outer circumference meters
0.2387:radius meters
314.1592653589793 :cylinder lateral surface area square meters
408.4070449666731 : cylinder total surface area square meters
1.790006923225166: cylinder overall volume cubic meters
0.11187543270157288 : cylinder inner volume cubic meters
1.6781314905235931 :cylinder outer volume cubic meters

Testcase 3

table 3printed in tcl wiki format
quantity value comment, if any
testcase number:3
10.0 :cylinder outer diameter meters
10.0 :cylinder outer height meters
1.0 :cylinder ratio numerator no unitsusually 1
4.0 :cylinder ratio denominator no units usually 4
31.41592653589793 :answers:cylinder outer circumference meters
5.0:radius meters
314.1592653589793 :cylinder lateral surface area square meters
408.4070449666731 : cylinder total surface area square meters
785.3981633974483: cylinder overall volume cubic meters
49.08738521234052 : cylinder inner volume cubic meters
736.3107781851078 :cylinder outer volume cubic meters

Testcase 4

table 4printed in tcl wiki format
quantity value comment, if any
testcase number:4
0.5 :cylinder outer diameter meters
0.3 :cylinder outer height meters
1.0 :cylinder ratio numerator no unitsusually 1
4.0 :cylinder ratio denominator no units usually 4
1.5707963267948966 :answers:cylinder outer circumference meters
0.25:radius meters
9.42477796076938 :cylinder lateral surface area square meters
103.67255756846316 : cylinder total surface area square meters
0.05890486225480862: cylinder overall volume cubic meters
0.003681553890925539 : cylinder inner volume cubic meters
0.05522330836388308 :cylinder outer volume cubic meters

Testcase 5

table 5printed in tcl wiki format
quantity value comment, if any
testcase number:5 ref Neugebauer kiln problem
2.86 :cylinder outer diameter meters
8.984954989266807 :cylinder outer circumference meters
6.0 :cylinder outer height meters
1.0 :cylinder ratio numerator no units usually 1
4.0 :cylinder ratio denominator no units usually 4
1.43:answers: radius meters
53.90972993560084 :cylinder outer lateral surface area square meters
60.33397275292661 : cylinder outer lateral surface area square meters
38.5454569039546: cylinder overall volume cubic meters
2.4090910564971626 : cylinder inner volume cubic meters
36.13636584745744 :cylinder outer volume cubic meters

Screenshots Section

figure 1.

Sumerian Hollow Cylinder Formula and eTCL Slot Calculator Demo Example screen.png

figure 2.

Concept drawing for Sumerian kiln math problem

Sumerian Hollow Cylinder Formula and eTCL Slot Calculator Demo


References:

  • 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 Hollow Cylinder Formula 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 {{} {cylinder outer diameter meters:} }
        lappend names {cylinder height meters: }
        lappend names {optional, cylinder ratio numerator no units :}
        lappend names {optional, cylinder ratio denominator no units :}
        lappend names {answers: cylinder outer circumference meters :}
        lappend names {cylinder overall volume cubic meters: }
        lappend names {cylinder inner volume cubic meters: }
        lappend names {cylinder outer volume cubic 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 Sumerian Hollow Cylinder 
            from TCL WIKI,
            written on eTCL "
            tk_messageBox -title "About" -message $msg } 
       proc pi {} {expr acos(-1)}       
       proc calculate {     } {
            global answer2
            global side1 side2 side3 side4 side5
            global side6 side7 side8
            global testcase_number
            global lateral_surface total_surface_area radius
            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 diameter  $side1  
            set overall_circumference $side5
            set height $side2 
            set numerator $side3
            set denominator $side4 
            set overall_circumference [* [pi] $diameter ]
            set radius  [/ $diameter 2. ]
            set ratio_factor [/ [* $numerator $numerator ] [* $denominator $denominator ]]
            set overall_volume [* [pi] $radius $radius  $height ]
            set inner_volume [* $ratio_factor $overall_volume ]
            set outer_volume [* [- 1. $ratio_factor ] $overall_volume ]
            set lateral_surface  [* 2. [pi] $height [+ $denominator $numerator] ]
            set tsa2 [+ $lateral_surface [expr {2.*[pi]*($denominator*$denominator- $numerator*$numerator)} ]]
            set total_surface_area $tsa2
            set side5 $overall_circumference
            set side6 $overall_volume
            set side7 $inner_volume
            set side8 $outer_volume
                    }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$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
            global lateral_surface total_surface_area radius
            console show;
            puts "%|table $testcase_number|printed in| tcl wiki format|% "
            puts "&| quantity| value| comment, if any|& "
            puts "&| testcase number:|$testcase_number | |&"
            puts "&| $side1 :|cylinder outer diameter meters |   |&"
            puts "&| $side2 :|cylinder outer height meters | |& "
            puts "&| $side3 :|cylinder ratio numerator no units|usually 1 |& "
            puts "&| $side4 :|cylinder ratio denominator no units | usually 4  |&"
            puts "&| $side5 :|answers:cylinder outer circumference meters |  |&"
            puts "&| $radius:|radius meters|  |&"
            puts "&| $lateral_surface  :|cylinder lateral surface area square meters|  |&" 
            puts "&| $total_surface_area  :| cylinder total surface area square meters|  |&" 
            puts "&| $side6:| cylinder overall volume cubic meters |  |&"
            puts "&| $side7 :| cylinder inner volume cubic meters|  |&"
            puts "&| $side8  :|cylinder outer volume cubic meters|  |&" 
            }
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup .4774 1.0 1.0 4.0  1.5 0.3578 0.0224 0.3354}
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup .4774 10.0 1.0 4.0 1.5 3.58 0.223 3.35 }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 10. 10.0 1.0   4.0 31.4 1570.0  98.1 1472.6 }
        ::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 . "Sumerian Hollow Cylinder Formula 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   |&"  

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Comments Section

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