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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 et al 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 ] # 0.3578price? = 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 |
|---|---|---|
| quantity | value | comment, if any |
| testcase number: | 2 |
| table 3 | printed in | tcl wiki format |
|---|---|---|
| quantity | value | comment, if any |
| testcase number: | 3 |
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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