Updated 2016-06-13 19:09:36 by gold

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

This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER in your comment with the same courtesy that I will give you. Its very hard to reply intelligibly without some background of the correspondent. Thanks,gold

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 . The factor would be (1*1)/(4*4), 1/16.

Using the eTCL calculator, one can compare the results of the eTCL calculator with the Sumerian math problem on kiln design. The discussion follows closely the translation by Neugebauer and uses converted metric units and decimal notation. Extra decimal points are kept to check the accuracy of the eTCL calculator, not to imply greater accuracy on the tablet figures. The eTCL calculator was loaded as outer diameter = 2.86 meters, circumference = 9 meters, kiln height = 6 meters, outer radius1=4, inner core radius2 = 1. The eTCL calculator returns 2.40909 decimal cubic meters for the inner core. Using 3 for PI, the tablet calculations show 8/60+26/3600+15/21600 sars or 2.5424 decimal cubic meters for the inner core. The clay tablet has a (2.5424/2.40909)-1, or 5.55336 percent overshoot error, mostly due to using 3 for PI.

Neugebauer reported the coefficient for clay tablets as 10 in several coefficient lists. A tentative calculation can be made assuming the 10 coefficient means the area for recommended tablet size. The tablet surface area might be 10/21600 sar which converts to 0.0148 square meters, 148 square centimeters. No shape information provided, but a square tablet of roughly 12 cm on a side would fit the assumed definition. A math problem or specific text references needed to confirm this definition, but at least some tablets in the Sumerian era have roughly 148 sq.cm. surface area.

From the Neugebauer and Robson papers, the thickness of a log is 4_48_00. The terminology is complex, but the coefficient is normally used in cylinder volume problems as 4/60 + 48/3600 in base60, converting to 0.08 in decimal notation. The thickness of a log is equivalent to 1/(4*PI) and used is some math problems in lieu of 3 for PI. The common Sumerian formula for circular area was circumference squared divided by 12, C*C/12 or C*C*(1/12). Using the log coefficient, the formula for circular area used C*C*(4/60 + 48/3600). The volume of a "log" was C*C*(4/60 + 48/3600)*height. Some problems averaged the top and bottom end of the log figure as (.5*(C1+C2))*(4/60 + 48/3600)*height. So 4/60 + 48/3600 representing a form of 3.125 is a little different and closer approach to the modern PI (3.14...). The error in using 3 for PI is undershoot 4.5 percent error and the error of 3.125 is undershoot 0.52 percent error.

Problems in Cylinder Volume Tablets

In some math problems for volumes of cylinders and cones, there are a number of constants and math operations that are difficult to understand (by me!). These include the imla coefficient of 18, calling for the square of area or cube of volume at the start of some problems, and the use of seeds or seed numbers at the start of a volume problem.

The Robson paper cited an imla coefficient of 18, which has been associated with circumference calculations in math problems. Imla means mud depth or mud vertical distance in Sumerian and possibly may be a factor like the log coefficient combined with 3 for PI and some other integer. The imla coefficient is 18 is reciprocal of daily excavation task as 18 workdays per volume sar. 1/imla has decimal notation is 0.05555 sar volume of excavation, base60 3_20, 3/60+20/3600. The daily excavation task appears as 3_20 volume sars per workday in coefficient lists and Babylonian math problems. As example, a small canal needs an excavation of 5 sars, (5*18), or 90 cubic meters. 5 sars times 18 workdays per sar volume calls for 90 workers working one day or one worker digging 90 workdays. The imla coefficient is intended to save base60 division from 5/0_3_20 and gives an answer by multiplication, probably from a product lookup table.
set reciprocal_excavation_task  [ eval expr  1./18. ] # decimal notation 0.05555555555555555
set decimal_excavation_task  [ eval expr  (3./60.)+(20./3600.)] # decimal notation 0.05555555555555555
set solution_task [ eval expr  5.*18.] # decimal answer is 90. workdays
set alternate_solution_task [ eval expr  5./.05555 ] # decimal answer is 90.0090 workdays
# decimal points added to check eTCL calculators, not to imply greater accuracy than the clay text problems.

In other texts, seed or seed measure is sometimes used to refer to seed cubits or seed area (converted from square nindans). But not sure this is the reference in the math problem.

Using 3 for PI, a possible Babylonian formula for right circular cone or heap might be volume_cone= (1/3)*((area*area)/12)*height, where the area is the area of the base circle. An alternate formula might be volume_cone= ((area*area)/36)*height. Along similar lines, a half cone might be half__cone_volume = (1/2)*((area*area)/36)*height, and rearranging terms, half__cone_volume = ((area*area)/18)*height. Since taxes were half the harvest under King Shulgi, the half cone formula might have been used to collect harvest taxes.

In terms of work baskets, Hoeyrup set some tentative dimensions for conical baskets, which can be examined in terms of surface area. The small and large workbasket can be treated as treated as cones or truncated cones in the eTCL calculators. If the small and large basket have the same height and small end diameter for carrying on the head, then the difference of small and large baskets might be the large or open end diameters. It has been noted that some Egyptian wall murals show conical baskets for carrying mud and some Babylonian work standards refer to a hod(?) for carrying mortar. Not absolute proof, but the ratio of the reed packs to make the small and large baskets fairly bracket the calculated ratio of the cone surface areas.
set surface_area_large_basket [ eval expr   2.*[pi]*(1./3.)*(4./5.)+  2.*[pi]*(1./3.)*(1./3.)   ]
set surface_area_small_basket [ eval expr   2.*[pi]*(1./4.)*(4./5.)+  2.*[pi]*(1./4.)*(1./4.)   ]
set reed_pack_ratio [eval expr (3.+1./3.)/(4.+1./2.) ]  #0.7407407407407408
set mud_baskets_surface_area_reduction_ratio [ eval expr  (1./4.)/(1./3.) ]  #  ratio=0.75
set mud_baskets_surface_area_ratio  [ eval expr  ([pi]*(1./4.)*(4./5.))/([pi]*(1./3.)*(4./5.) ) ] 
#  ratio=0.75,h=4/5 cubit, r_l=1/3 cubit,r_s=1/4 cubit, vol_large_basket = 11.1 liters, vol_small_basket= 6.25 liters

Thinking out loud, many of the listings for Babylonian math exercises are variants of similar problems and use the same numbers over and over. Its conceivable to pick special numbers to simplify the math. For example, 1.5 nindians is a common circumference and 0.5 nindians (~1.5/PI) is a common diameter in the circle problems. Suppose a cylinder volume as (C*C*H)/(1/4*3) was defined with the area equals height, then a cube could be introduced and substituted as H=C into the problem as (C*C*C)*(1/4*3).

Pseudocode and Equations

    set volume_bowl  [eval expr (5./60.)*(5./60.)*(4.*60.+48.) ]  #  2 sila in Babylonian calculation
    set conventional_vol? [eval expr 2.*((.5*.5/(3.*4.))*.09) ] # 0.00375 liters per m*m
    set conv_circle [eval expr 1.5*1.5*(1/12.) ] # text answer 11/60+15/3600, decimal 0.1875
    set strange_formula [eval expr $area*$area*(4.*60.+48.)]  # possible sila per kus*kus factor
    45 carrying baskets (45*25=1125kg) per 1us (360 meters) per workday
    1_15 talents dirt (1_15=1/60+15/3600=37.5 kg) for 1 dana (1 dana=beru=10.8 km) per workday
    6 bricks (b=8.5 minas>>6*8.5*0.4977kg,25.3827) for beru 10.8 kilometers per day
    large basket (dusu)  =2_13_20 volume shekel=1/27 volume shekel=13.33 kilograms=11.1 liters
    small basket = 1_4 volume shekel=1/48 volume shekel=7.5 kilograms=6.25 liters
    1 volume shekel = 300 liters = 12 talents =12*30=360 kilograms
    1 talent =12 volume shekels 

Pseudocode and Equations

    # following statements can be pasted successively into eTCL console
    proc pi {} {expr acos(-1)}  
    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
    set overshoot_error [eval expr (2.5424/2.40909)-1. ] # 0.05533624729669717
    set error_log_coefficient [eval expr (1.-3.125/[pi])*100.] # undershoot 0.52 percent error
    set error_3_for_PI_coefficient [eval expr (1.-3.0/[pi])*100.] # undershoot 4.5 percent error
    set log_coefficient_reciprocal [ eval expr (1./4.*[pi]) ] # 0.7853981633974483
    set imla_reciprocal_PI_coefficient [ eval expr (1./6.*[pi]) ] # 0.5235987755982988

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.

figure 2.

Concept drawing for Sumerian kiln math problem


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, Joran 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
  • The design of Babylonian waterclocks : Astronomical and experimental evidence,Steele
  • BRICKS AND MUD IN METRO-MATHEMATICAL CUNEIFORM TEXTS, Joran Friberg
  • Jens Hoeyrup, Peter Damerow-Changing Views on Ancient Near Eastern Mathematics-Reimer (2001)

Appendix Code edit

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 edit

Please place any comments here, Thanks.