Sumerian Hollow Cylinder Formula and eTCL Slot Calculator Demo Example, numerical analysis

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gold Here is some eTCL starter code for calculating the volume of a hollow cylinder. Most of the testcases involve replicas or models, using assumptions and rules of thumb.

Neugebauer published a cuneiform math problem on kiln design, which computes the volume of a hollow cylinder. I have modified a slot calculator in eTCL to handle calculations for the hollow cylinder volume and calculate the Sumerian coefficients. At this point, its easier to use modern geometric equations in the eTCl calculator and compute the volume accurately. Then try to understand how the Sumerians developed their geometric coefficients. Although the geometric coefficients were passed through successive cultures as late as 400 BCE, the main compilation of the geometric coefficients was probably during the King Naram-Sin reforms of the Akkadian empire in 2150 BCE.

From clay tablet YBC7997, internal and external volumes of a hollow cylinder is equated to squared ratio of radius1*radius1 over radius2*radius2. Not sure about the accurate math derivation and maybe numerical coincidence, but the tablet appears to be using inner volume equals outer volume times radius1*radius1 over radius2*radius2. Hereafter, the paragraph will use the modern decimal notation, PI (3.14...), and carry extra decimal points from the eTCL calculator, whereas the Sumerians used 3 and round numbers. radius1 would be the radius of the hollow and radius2 would be the radius of the outer cylinder. In the tablet, the circumference of the outer cylinder was 1.5 units and the ratio of the inner radius to outer radius would be 1:4. The diameter of the outer cylinder would be 1.5/PI or 0.4774, and radius2 would be 0.4774/2 or .2387. radius1 would be .2387/4 or 0.0597. The height of the cylinder would be 1 unit. Using conventional formulas the volume of the outer cylinder would be PI*radius2*radius2*height, substituting 3.14*.2387*.2387*1, 0.1790. The conventional volume of the inner cylinder would be PI*radius2*radius2*height, substituting 3.14*.0597*.0597*1, 0.0111. The volume of the hollow cylinder would be outer cylinder minus inner cylinder, 0.1790-0.0111, 0.1678 . In squared proportions, the radius1*radius1 over radius2*radius2 would be (1*1)/(4*4),1/16,0.0625. The Sumerians found the inner cylinder volume (hollow) as (1/16) * outer cylinder volume, (1/16) * 0.1790, 0.0111 in modern notation. Not on the tablet, but it follows that the hollow or outer cylinder volume would be (1-1/16)* outer cylinder volume, (15/16)*0.1790, 0.1678 . In Sumerian base60, the factor would be 1/16 or 3/60+~45/3600.

Pseudocode and Equations

    set inner_cylinder_a=b*(c*c/d*d)_  [*  0.1790[/  [* 1. 1.  ] [* 4. 4.  ]   ]] # 0.0111
    set hollow_cylinder_a=b*(c*c/d*d)_  [*  0.1790[- 1. [/  [* 1. 1.  ] [* 4. 4.  ]]   ]] # 0.1678 
    set inner_cylinder_  [ eval expr [pi]*.0597*.0597*1 ] # 0.0111
    set outer_cylinder_  [ eval expr [pi]*.2387*.2387*1 ] # 0.1790

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 10printed in tcl wiki format
quantity value comment, if any
testcase number:10
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

Screenshots Section

figure 1.

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


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   |&"  

gold This page is copyrighted under the TCL/TK license terms, this license .

Comments Section

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