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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 a clay tablet, 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 2*PI*radius2*radius2*height, substituting 2*3.14*.2387*.2387*1, 0.3578. The conventional volume of the inner cylinder would be 2*PI*radius2*radius2*height, substituting 2*3.14*.0597*.0597*1, 0.0224. The volume of the hollow cylinder would be outer cylinder minus inner cylinder, 0.3578-0.0224, 0,3354. 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) * 0.3578, 0.0224 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)* 0.3578, 0.3354. In Sumerian base60, the factor would be 1/16 or 3/60+45/3600.
set inner_cylinder_a=b*(c*c/d*d)_ [* 0.3578 [/ [* 1. 1. ] [* 4. 4. ] ]] # 0.0223625 set hollow_cylinder_a=b*(c*c/d*d)_ [* 0.3578 [- 1. [/ [* 1. 1. ] [* 4. 4. ]] ]] # 0.3354375 set inner_cylinder_ [ eval expr 2*[pi]*.0597*.0597*1 ] # 0.0224 set outer_cylinder_ [ eval expr 2*[pi]*.2387*.2387*1 ] # 0.3578
price? = raw materials + labor + profit price? = raw materials + heat process price? = raw materials + labor average price per unit = revenue / units sold
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).
table 1 | printed in | tcl wiki format |
---|---|---|
quantity | value | comment, if any |
testcase number: | 1 |
table 2 | printed in | tcl wiki format |
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quantity | value | comment, if any |
testcase number: | 2 |
table 3 | printed in | tcl wiki format |
---|---|---|
quantity | value | comment, if any |
testcase number: | 3 |
# 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 outer circumference meters :} lappend names {cylinder height meters: } lappend names {optional, cylinder ratio nom no units :} lappend names {optional, cylinder ratio dennom no units :} lappend names {answers: 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 lot order size from TCL WIKI, written on eTCL " tk_messageBox -title "About" -message $msg } proc calculate { } { global answer2 global side1 side2 side3 side4 side5 global side6 side7 side8 global discounted_inventory 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 cost_order $side1 set number_sales $side2 set holding_cost $side3 set quantity [/ [* $cost_order $number_sales ] $holding_cost] set quantity [sqrt $quantity ] set side6 $quantity set order_cost [* $quantity $cost_order ] set side7 $order_cost set interval $side4 set shelf_life_per_interval [/ $quantity $number_sales ] set shelf_life_days [* $shelf_life_per_interval $interval ] set side8 $shelf_life_days set inventory_trial_value $side5 set discounted_inventory [* $inventory_trial_value $quantity ] } proc fillup {aa bb cc dd ee ff gg hh} { .frame.entry1 insert 0 "$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 discounted_inventory 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 diameters meters | |&" puts "&| $side2 :|cylinder outer circumference meters | |& " puts "&| $side3 :|cylinder outer circumference meters | |& " puts "&| $side4 :|cylinder ratio nom no units | usually 1 |&" puts "&|$side5 :|cylinder outer volume cubic meters | usually 4 |&" puts "&| $side6:|answers: 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 10.0 50.0 5.0 7.0 1.0 10.0 100. 1.4} ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 100.0 1000.0 10.0 7.0 1.0 100. 1.E4 0.70 } ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 10.00 100.0 10.0 7.0 1.0 10.0 100. 0.70 } ::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 "
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 .
Please place any comments here, Thanks.