Updated 2016-04-27 19:45:47 by gold

## Sumerian Paint & Bitumen Coating and eTCL Slot Calculator Demo Example edit

gold Here is some eTCL starter code for calculating materials for ancient Sumerian painters, waterproofing roof crews, and floor pavement crews. The impetus for these calculations was checking prices and materials in some cuneiform texts and modern replicas. Most of the testcases involve replicas or models, using assumptions and rules of thumb.

Some order of magnitude calculations were made for a trial Sumerian house of 2300 BCE, with many assumptions. The Orchard House was 30.11 meters long, 11.5 meters wide, and walls 3 meters high with a surface area of 30.11*11.5, or 346.6 square meters. As constructed, the front wall was 10.45 meters and back wall was 11.5 meters, presumably the difference of (11.5-10.45) or 1 meter was the threshold or entrance. One side of the house was extended as a garden wall. For the Orchard House, the ratio of wall length to width was 30.33/11.5 , 2.618:1, or rounding 3:1. For an initial cut of the wall volume using hand calculation, the total wall volume would be the sum of the front wall L*H*T (11.5*3*0.33), the rear wall(11.5*3*0.33), side wall (30.11*3*0.33), and second side wall (30.11*3*0.33). The eTCL calculations make the assumption that all bricks are of the "2/3 cubit" burned brick type. The garden wall would enclose 60*60 sq. meters or 4*60 perimeter of square iku, based on modern assumptions. Using an assumption that the wall thickness of the Orchard House was 0.33 meters, the eTCL calculated a wall volume of 77.7 cubic meters. If the "2/3 cubit" burned brick was used, the total number was 8924 bricks. The estimated workforce was 2 foremen (salary of 50 liters), one junior scribe (10 liters), 56 craftsmen (10 liters), and 16 subsistence men (0.5 liter).

While the number of bricks has not observed on any receipt, the estimated workdays for making bricks would be 8924/240 or 37.2 workdays, within the nominal workforce for one day. The estimated workdays for digging clay for bricks would be 77.7/3 or 25.9 workdays, within the nominal workforce for one day. Presumably, the bulk of the bricks were made on site, but the available text doesn't state so. For comparison, the average Sumerian mudbrick house was about 90 sq. meters, and the average room size was 3 meters width by 3.65 meters length,3* 3.65, or about 10.95 square meters. The Sumerian mudbrick houses generally contained rooms of 11 square meters built around a central courtyard. For the Orchard House, the general guidelines suggest about 6 rooms (3*3.6 m.) on both sides of a central courtyard about 5 meters wide.

The bare walls and roof of Orchard House have been raised in 2300 BCE, as a concept. Now, 1) the exterior walls have to be waterproofed with bitumen, 2) the interior walls have to be painted with red, yellow, white, and black, 3) the interior floors have to be paved with bitumen, and 4) roof and beams have to be waterproofed with bitumen. All this will require an eTCL calculator that makes an estimate of a rectangular surface area, calculates the appropriate film thickness or application thickness from a Sumerian coefficient, and then calculates the required paint or bitumen in liters. The price of paint, bitumen, and labor can be estimated from the URIII equivalency lists circa 2300 BCE. The astronomers of Babylon were kind enough to forward weather reports and price lists for their era, circa 1900 BCE.

The Bry formula takes a Sumerian coefficient and estimates the application thickness of yellow paint, bitumen, butter oil etc. The film thickness in centimeters equals (1/coeff.)*0.4977*100. Once the film thickness (T) is estimated, the volume of paint or bitumen is estimated as L*W*T and converted into liters. The Sumerians originally developed this formula for .... water irrigation of barley fields! So after the painters and floor pavers are set to work in the heat of the sun, one can jump into the cool irrigation canal and find some examples of water irrigation for the eTCL calculator.

The customary procedure is find a testcase and make some trial calculations with the handheld calculator, the eTCL green calculator console, or the google expression calculator. Folks with weak eyes or small screens might want to try the large print numerals of the green calculator on gold. Hereafter, eTCL notation will be used, which can be pasted into the green console window.

The first testcase is calculating the thickness and volume of bitumen coating on the floor of an average Sumerian house. The average Sumerian mudbrick house for a free craftsman was about 90 sq. meters or about 3 sars rounded. Taking an average from an excavation report, the square root can be used to develop house length and width for the eTCL calculator, L = [sqrt 90.] and W = [sqrt 90.]. In the crowded cities, most of these rectangular quarters shared common walls and looked somewhat like apartment buildings of one story. Using easy non-prime numbers for the wall lengths, the rectangular house would be length of 12 meters and width of 8 meters. the surface area would be [* 12 8], 96 sq meters, or [/ 96. 32.], exactly 3 sars. With a bitumen coefficient of 16, the Bry formula estimates the thickness of the bitumen floor as [* 0.4977 100. [/ 1. 16. ]] or 3.11 centimeters. The volume of the bitumen would be the area of the floor times the bitumen thickness as [* 12. 100. 8. 100. 3.11 ] or 2.99E6 cc. The bitumen coating would be [* 2.99E6 .001] or 2990 liters. In UrIII, the price of bitumen was 1 silver piece for 2 barig or 120 liter as [/ 2990 120. ] or 24.9 silver pieces. For comparison, the eTCL calculator returns a surface area of 96 sq. meters and a coating volume of 2986 liters.

The second testcase is calculating the thickness and volume of bitumen coating on the floor of the Orchard House. The Orchard House was 30.11 meters long by 11.5 meters wide. The surface area would be [* 30.11 11.5] or 346.265 square meters. The generic Bry formula is (1/coeff.)*0.4977*100. With a bitumen coefficient of 15, the Bry formula estimates the thickness of the bitumen floor as [* 0.4977 100. [/ 1. 15. ]] or 3.317 centimeters. In cubic centimeters, the volume of the bitumen would be the area of the floor times the bitumen thickness as [* 30.11 100. 11.5 100. 3.317] or 1.14E7 cc. Converting into liters, the bitumen coating would be [* 1.14E7 .001] or 1.14E4 liters. In UrIII, the price of bitumen was 1 silver piece for 2 barig or 120 liter as [/ 3317 300.] or 27.6 silver pieces. For comparison, the eTCL calculator returns a surface area of 346 sq. meters and a coating volume of 1.1E4 liters.

The third testcase is calculating the thickness and volume of bitumen coating on the floor of an average small room of L/H 3.65/3 meters. The surface area is [* 3.65 3.] or 10.95 sq. meters. The bitumen coating is [* 0.4977 100. [/ 1. 15]] or 3.3 cm. For comparison, the eTCL calculator returns a surface area of 10.95 sq. meters and a coating thickness of 3.318 cm, and a coating volume of 363 liters.

The fourth testcase is calculating the thickness and volume of yellow orpiment paint on an interior wall of the Orchard House. Orpiment paint is a yellow pigment that probably is a mixture of pigment and butter oil. The interior wall was 3.65 meters long by 3 meters height. The surface area would be [* 3.65 3.] or 10.95 square meters. With a orpiment coefficient of 54, the Bry formula estimates the thickness of the bitumen floor as [* 0.4977 100. [/ 1. 54]] or 0.92 centimeters. In cubic centimeters, the volume of the orpiment paint would be the area of the floor times the bitumen thickness as [* 3.65 100. 3. 100. 0.92] or 1.0E5 cc. Converting into liters, the orpiment coating would be [* 1.0E5 .001] or 100 liters. In UrIII, the price of orpiment was 1 silver piece for 600(?) manas or 298 kilograms. However, the eTCL calculator is fixed at the value of bitumen, giving [/ 100 120.] or 0.83 silver pieces. For comparison, the eTCL calculator returns a surface area of 10.95 sq. meters and a coating thickness of 0.92 cm, and a coating volume of 363 liters.

The fifth testcase is spreading tallow on a wooden awning placed in the courtyard to shade the gatekeeper. Tallow or butter oil was used for waterproofing and preserving wood as a cheaper substitute for bitumen. The awning spans the courtyard over a 5 meters width and 3 meters depth. The surface area would be [* 5. 3.]or 15 square meters. With a tallow coefficient of 20, the Bry formula estimates the thickness of the bitumen floor as [* 0.4977 100. [/ 1. 20]] or 2.49 cm. The eTCL calculator returns a thickness of 2.4885 cm for tallow and suggests 373 liters will be needed for the awning.

Most published coefficient lists are lists of terse numbers and short phrases. There was one corrupted coefficient list that had phrases of tasks for the students or future scribes. As corrupted as this tablet is, the tablet adds a lot for modern eyes. It is difficult to establish the prospective use of the coefficients, apart from the math problems. The available tasks were
```  construct a brick mould inside a brick mould.
construct a river (inside) a river with an earth slope.
remove a governors' river (from) a river.
construct a canal.
construct a circle inside a circle.
divide a circle into two equal halves.
construct a circle inside a brick mould.
A brick mould inside a circle.        ```

Sumerian phrases for coefficients were either 1) formula "number sa noun (or phrase)", 2) formula " number noun (or phrase)", or 3) formula "number igigub noun (or phrase)". The first formula "number sa noun" is not actual value of noun but a proportion between different units. "number sa noun" was typically used to transform sar area to another units. whether volume or number of objects. For example, one sar area to fixed number of bricks as sar area transformed to (objects), brick sar of 12*60 or 720 bricks. In this context, sa meant not "of", but "in connection with" . The phrase "number noun" was used to mean that the noun was the value of number. The second formula "number noun" suggests the value in direct relationship for following noun. The third formula "number igigub noun" suggests the value in reciprocal relationship for following noun. Igigub means "I see it " in Sumerian.

The third coefficient formula was "number igigub use" as "20/60 igigub use". 60/20 was reciprocal use to avoid division. There were tables for multiplication of numbers with simple prime factors (2,3,5). In the math problems, division was often limited to simple numbers from 1 to 9, which could be memorized. From the coefficient lists and proportions, there were four possible variants of proportions or equations that could be used to solve the problem. For the igigub phrases, the targeted relationship in the math problems used the reciprocal 60/20. One says targeted, because some math problems give the all three numbers, including solution. From the text with left out lines and phrases, it is often difficult to determine the intent of the math problem.

From the coefficient lists, known coatings range in thickness from orpiment at 0.92 cm to one instance of bitumen at 4.2 cm. In the tables below, various substances are listed with numbers from the coefficient lists, but not all substances may use the Bry equation.

### Pseudocode and Equations

```  Bry formula thickness = (1/coeff.)*0.4977*100
volume = [* length width thickness] # meters,meters,centimeters
volume in cubic centimeters = [* length 100. width 100. thickness]
Sumerian price in silver = [/ liters 120.]
Sumerian price in liters grain = [* silver 300.]
# 1 silver piece = 1 gur = 300 liters of grain
liters = [* volume_in_cubic_cm .001]
price? = raw materials + labor  +  profit
price? = raw materials + heat process
price? = raw materials + labor
average price per unit  = revenue  / units sold```

### Table 1, Bitumen Coefficients and Prices

Bitumen Coefficients and Prices
includes prices & equivalencies from different regions and eras
coefficient transliterated english possible decimal /fraction
15 igi.gub.esiri.e coefficient refined pitch 15/60
16 igi.gub.esiri coefficient raw pitch 16/60
15 ssa esiri coefficient pitch 15/60
12 ssa esiri coefficient (refined?) pitch 12/60
10 45 06 ssa ina ki-ri-im coefficient (refined?) pitch 0.1794
15 ssa esiri ? coefficient (raw?) pitch 15/60
15 ssa esiri-e-aIt coefficient refined pitch 15/60
2.5 barig (of) esir-e-a price wet pitch 2.5 barig for 1 shekel , URIII
10 gu (of) esir-had price dry pitch 10 gu for 1 shekel , URIII
12 gu (of) esir-had price dry pitch 12 gu for 1 shekel, URIII
4 ban (of) esir-had price dry pitch 4 ban for 1 shekel, Babylon 1900 BC

Note: esir-e-a (literally, bitumen watery) measured by barrels ( or jars) of 60 liters eq. esir-had (literally, bitumen dry) measured by weights of 30 kilogram (manload). Any coefficient calculation would have to account for units of wet volume (SE system S* for wet capacity) or dry weight (EN system E)

### Table 2, Pigment and Dye Coefficients, includes prices & equivalencies

Paint, Pigment, and Dye Coefficients table printed in tcl wiki format
includes prices & equivalencies from different regions and eras
quantity coef. film thickness cm comment, if any
orpiment 540.922 yellow pigment
madder 202.488 reddish dye from plant, possible ringer?
gypsum 11_084.52??? white pigment?, possible ringer?
tallow 20 2.488 also, butter oil
water 481.03687 used in irrigation

### Example of Sumerian reciprocal use (igigub)

Proportions & 4 variant equations derived from reciprocal or igigub comment
20/60 listed as igigub (intended use as reciprocal) in coefficient list
20/60 = 4 sar / 12 gur direct relationship from proportions
60/20 not in table. Lists of common reciprocals (base 60) were available. Does appear as whole number (3) in math problems as intermediate step.
(60/20) = 12 gur / 4 sar reciprocal relationship, invert proportions above
12 gur * (20/60) = 4 sar not targeted relationship, seeking answer of gurs
12 gur / (60/20) = 4 sar not targeted relationship, seeking answer of gurs
4 sar / (20/60) = 12 gur not used commonly. Division avoided in math problems.
4 sar * (60/20) = 12 gur targeted relationship, commonly used in math problems as solution of problem.

### 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

Sumerian math problem to metric table printed in tcl wiki format
nearly average Sumerian house
quantity value comment, if any
testcase number 3
length meters: 12.
width meters: 8.
Sumerian paint/coating coefficient: 15.0
answers: total surface area m*m : 96.0
paint/bitumen film thickness cm: 3.3179thickness of floor coating
paint volume liters: 3185.279
value in silver : 26.543

#### Testcase 2

Orchard House table printed in tcl wiki format
princely quality house built for the lugal mighty or great man
quantity value comment, if any
testcase number 2
length meters: 30.11
width meters: 11.5
Sumerian paint/coating coefficient: 15.0
answers: total surface area m*m : 346.265
paint/bitumen film thickness cm: 3.317thickness of floor coating
paint volume liters: 11489.072
value in silver : 95.742

#### Testcase 3

average room in Orchard House table printed in tcl wiki format
quantity value comment, if any
testcase number 3
length meters: 3.65
width meters: 3.0
Sumerian paint/coating coefficient: 15.
answers: total surface area m*m : 10.95
paint/bitumen film thickness cm: 3.317thickness of floor coating
paint volume liters: 363.320
value in silver : 3.027

#### Testcase 4

yellow paint on wall of average (small) roomtable printed in tcl wiki format
quantity value comment, if any
testcase number 2
length meters: 3.65 considered average room
width meters: 3.considered average room
Sumerian paint/coating coefficient: 54.0thiness film known in coef. texts
answers: total surface area m*m : 10.95
paint/bitumen film thickness cm: 0.921 thiness film known in coef. texts
paint volume liters: 100.922
value in silver : 0.8410

### References:

• the Bitumen Works, Sumerian Coefficients at the Bitumen Works and eTCL Slot Calculator Demo Example edit
• Cities of the Ancient World: [1]
• major paper in understandable prose,Equivalency Values and the Command Economy
• Robert Englund, UCLA [cdli.ucla.edu/staff/englund/publications/englund2012a.pdf]
• Ur III Tablets in the Valdosta State University, search on cdli
• Cuneiform Digital Library Journal, search on Equivalency Values
• Ur III Equivalency Values[2]
• Especially, the Ur III Equivalency Values for esir a and esir had sections.
• The Sumerian keywords -bi, esir, and had search on the cdli
• Mathematical Coefficients of Bitumen, Paul BRY NABU(01-2002)7, in French

## Appendix Code edit

### appendix TCL programs and scripts

```        # pretty print from autoindent and ased editor
# Sumerian paint calculator
# written on Windows XP on eTCL
# working under TCL version 8.5.6 and eTCL 1.0.1
# gold on TCL WIKI , 2may2014
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 {{} { length meters:} }
lappend names { width meters:}
lappend names { coefficient : }
lappend names {answers: total surface area m*m: }
lappend names { paint/bitumen film thickness cm:}
lappend names { paint volume liters:}
lappend names { value in silver:}
foreach i {1 2 3 4 5 6 7} {
label .frame.label\$i -text [lindex \$names \$i] -anchor e
entry .frame.entry\$i -width 35 -textvariable side\$i
set msg "Calculator for Sumerian Paint & Bitumen Coating
from TCL WIKI,
written on eTCL "
tk_messageBox -title "About" -message \$msg }
proc calculate {     } {
global side1 side2 side3 side4 side5
global side6 side7 testcase_number
incr testcase_number
set length   \$side1
set width  \$side2
set coefficient \$side3
set side4 [* \$side1 \$side2 ]
set totsurfacearea  \$side4
set thickness [* 0.4977 100. [/ 1. \$side3 ]]
set side5 \$thickness
set volpaint  [* \$side1 100. \$side2 100. \$thickness]
set volpaint  [* \$volpaint .001]
set side6 \$volpaint
set silver [* 1. [/ \$volpaint 120.]]
set side7 \$silver
}
proc fillup {aa bb cc dd ee ff gg} {
.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"}
proc clearx {} {
foreach i {1 2 3 4 5 6 7} {
.frame.entry\$i delete 0 end } }
proc reportx {} {
global side1 side2 side3 side4 side5
global side6 side7 testcase_number
console show;
puts "%| table |printed in| tcl wiki format|% "
puts "&| quantity| value| comment, if any|& "
puts "&| testcase number| \$testcase_number||& "
puts "&| length meters:| \$side1 ||&"
puts "&| width meters:| \$side2||& "
puts "&| Sumerian paint/coating coefficient:| \$side3||& "
puts "&| answers: total surface area m*m :| \$side4 ||&"
puts "&| paint/bitumen film thickness cm:| \$side5||& "
puts "&| paint volume liters:| \$side6 ||&"
puts "&| value in silver :| \$side7 ||&"
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 12. 8. 15.0  96.1 3.318 3185.  26.5 }
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 30.11 11.5 15.0  346.2 1150.0 3250.  38. }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 3.65 3.0 15. 10.9 3.3 363.3   1.2}
::ttk::button .clearallx -text clear -command {clearx }
::ttk::button .cons -text report -command { reportx }
::ttk::button .exit -text exit -command {exit}
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 Paint & Bitumen Coating 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 (which 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   |&"  ```