## Introduction

gold Here is TCL calculations for the sowing rate of barley and wheat in ancient Sumerian fields. The impetus for these calculations was checking grain sowing rates in a cuneiform text,The Farmer's Instructions or Farmer's Almanac. Most of the testcases involve replicas or models, using assumptions and rules of thumb.

In the cuneiform math problems and coefficient lists on clay tablets, there are coefficient numbers which were used in determining the amount of materials and the daily work rates of the workers. In most cases, the math problem is how the coefficient was used in estimating materials, work rates, and math problems. One difficulty is determining the effective magnitude or power of the number coefficient in the base 60 notation. In cuneiform, numbers in base 60 are written using a relative notation. For example, 20 could represent either 20*3600,20,20/60, 20/3600, or even 1/20. The basic dimensions and final tallies were presented in the cuneiform accounts on clay tablets, but some calculations, some units, and some problem answers (aw shucks!) were left off the tablet. Successive or iterated math solutions are called algorithms and the Babylonian methods are some of the earliest algorithms documented circa 1600 BCE. The TCL procedures are descendants of this idea. The Babylonians did not use algebra notation, decimal notation, or modern units, so the reader will have to bear some anachronisms in the TCL code. At least one approach for the modern reader and using modern terminology is to develop the implied algebraic equations and decimal equivalents from the cuneiform numbers. Then the TCL calculator can be run over a number of testcases to validate the algebraic equations. Babylonian Cubic Equations into TCL

### Farmer's Instructions

From a paraphrased translation of "The Farmer's Instructions" , 3 measures will be spent on each 18 iku area. The number 3 is a bare number 3 without units and refers to the amount of seed on a standard surface area of 18 iku. An iku is equivalent to 3.6E3 square meters and 18 iku is equivalent to 1 bur or 6.48 hectares. From this and other texts, the Sumerians reckoned seed or grain quantities in volume units. The unitless 3 could represent either 3*3600,3,3/60, 3/3600, or even 1/3 times some unknown volume unit. The most likely unit is some multiple of the sila, which is equivalent to a liter. The task here is evaluate the unitless 3 in terms of Sumerian or other sowing rates. However, the research would like to find the seeding rate represented by unitless 3 and compare the seeding rates mentioned in other cuneiform texts.

"The Farmer's Instructions" also has a discussion on seed and farrows, which can be used to derive one possible interpretation of the sowing rate in the customary silas per bur. The key phrases are 8 farrows in a ninda and 1 volume gin of seed per ninda. A ninda is length unit of 12 cubits or 6 meters. A gin of seed is about 1/60 of a sila or liter. The numbers set up as a small square plot of 12*12 cubits , 144 square cubits, or 36 square meters, requiring 8 gin or 8/60 sila of seed for 36 square meters. There are 8 farrows in the plot with each farrow of width 12 cubits/8 farrows or 1.5 cubits. A Sumerian bur is 6.48E4 square meters or 6.48 hectares. Setting proportions of (8/60)/36= volume_seed/6.48E4, volume_seed = {(8/60)*6.48E4)} /36 , 240 sila per bur.

### Testcases

From other texts, the sowing rate in Sumer and the Mesopotanian countries ranged from 240 to 360 silas per bur, but 270 silas per bur is closest to an often standard sowing rate. Using proportions, the standard sowing rate reduces as 270 silas per bur, 15 sila per iku, or 9 volume gin per small square plot of 144 square cubits. For comparison, the most quoted modern European rate for barley is 100 kilograms per hectare. The eTCL calculator will convert the silas per bur into kilograms per hectare using the density of barley (0.62 kg/liter) or the density of wheat (0.78 kg/liter). Note that the density of barley and other grains changes with the seed moisture content or humidity, so comparison with kilograms of seed before standard moisture content (1920 etc) can introduce a 2 to 5 percent error. Likewise, some modern countries use different standards of moisture content in their agricultural reports.

The reported rate was 10080 silas over a surface area of 672 iku or 241.9 hectares. The hand calculation was (10080/672)*(18 iku / 1 bur) =270 silas per bur and (10080/672)*(18 iku / 1 bur)*(0.68 kg/l)*(1 bur/6.48 hectares)= 28.4 kg/hectare. The eTCL calculator reported 270 silas per bur and 28.9 kg/hectare. Ref. CDLI entry: P011010 on VAT 12656 by Robert Englund.

### Possible Broadcast Seeding Rates

It is possible that some cuneiform tablets are using broadcast rates, since the date for the invention of the seed drill is not known and probably not all regions used the seed drill. The available UrIII documentation suggests that the ancient seed drill was used for rates between 240 and 360 silas per bur. One advantage of the seed drill in ancient times was using less seed in sowing a field by hand broadcasting. For example, some Neo-Babylonian math problems with the seeding measure terminology assume a much higher (~2*x) rate of seeding than the assessed seed drill. For comparison with the ancient seed drill rates, modern broadcast rates range from 50 to 130 kilograms barley per hectare, which is equivalent to 520 to 1360 silas barley per bur. In any case, the eTCL calculator does not assume use of the seed drill and works equally well with broadcast seed.

The following seeding rate comes from a definition in a Neo-Babylonian math problem, ref. MCT. The reported rate was 33.333 liters over a surface area of 100*100 cubits, 10,000 square cubits, or 2500 square meters. The land conversion is 2.5E3/3.6E3, or 0.6944 iku. The eTCL calculator reported 864 silas per bur and 82.7 kg/hectare. Note that the 2500 sq. meters is very close to the Roman iugerum area unit of 2518 sq. meters or 0.2518 hectares, considered a day's plowing in Roman times.

The following two seeding rates come from the seeding measure terminology and definitions, used in Neo-Babylonian math problems. The first seeding rate was 33.333 liters over a surface area of 10,800 square cubits or 2700 sq. meters. The land conversion is 2.7E3/3.6E3, or 0.75 iku. The eTCL calculator reported 800 silas per bur and 76.5 kg/hectare. The second seeding rate was 33.333 liters over a surface area of 12,000 square cubits or 3000 sq. meters. The land conversion is 3.0E4/3.6E3 or 0.833 iku. The eTCL calculator reported 720 silas per bur and 68.9 kg/hectare. Ref. Apsamikku in Neo-Babylonian Mathematics, Eleanor Robson

Calculating Jemdet Nasr barley harvests in the Late Uruk II capacity system, Damerow and Englund suggested circa 25 liters per basic grain unit and 15 units per bur. The gist is that 15 units*25 liters per unit is 375 liters per bur. The result of 375 liters per bur falls on the high side of the (later) standard sowing rates between 240 and 360 liters of seed grain per bur (UrIII, 2300 BCE). The eTCL calculator converted 375 silas per bur to 35.9 kilograms barley per hectare.

Assuming ten days to plow a bur, a tentative Jemdet Nasr grain budget would be \$total_grain =~ seed_grain( \$seeding_rate times * \$surface_area 1 bur) + 4_oxen(4 ox *(8 l/d)*10 d)+3_men_plowing_team(3 men*(2 l/d)*10 d)+porters((\$total_grain/25)*1 liters). The budget implies a rule of thumb that \$total_grain =~ 2*\$seeding_grain_rate*\$surface_area or \$surface_area =~ \$total_grain/(2*\$seeding_grain_rate). Further, \$seeding_grain_rate =~ \$total_grain /(2*\$surface_area ).

### Seeding Area Problem in NeoBabylonian Style

Here is a model seeding area problem in NeoBabylonian style. A reference seeding rate is defined from reference area equals seeds and the areas of a house are totaled in seeds. The reference area is a square of 12 cubits on a side and is equivalent to 75 seeds. There are 6 sections in the house, 5 sections or rooms of 25 seeds each and one section or central courtyard of 55 seeds. The central courtyard may be treated as 2 regular rooms of 25 seeds, if convenient. The area of house totals 180 seeds. Not sure if these units in the translation are regular cubit (0.4977 meter) or seed cubits (1 meter), but the principles are the same.

The first issue would be finding the seeding rate and its reciprocal. The seeding rate is 75/(12*12), decimal 0.5208 seeds per square cubit. The reciprocal seeding rate is (12*12)/75, decimal 1.92 seeds per square cubit. The Babylonian would calculate the base 60 version of the seeding rate ( eg. 30/60 + discounted 75/3600 ) and look up the reciprocal in a table, finding 60/30 or 2. The total area of the house is 180*((12*12)/75)), 345.6 sq cubits or 86.4 sq. meters. Each regular room is 25*((12*12)/75)), 48 sq cubits or 12 sq. meters. The central courtyard is 55*((12*12)/75)), 15.6 sq cubits or 26.4 sq. meters. Note that the solutions are using the reciprocal of the seeding rate, which the Babylonians would do to avoid division in the main assessment.

Since there are multiples of 6 and 12 in the reciprocal, factoring and approximate solutions could estimate the seeding rate reciprocal as (12*12)/72, (12*12)/(6*12), 12/6, reduction 2. The regular room would be (25)*(2), approximately 50 sq cubits and 4 percent off. The regular rooms here are compatible with the Orchard House postings in Sumerian Construction Rates, Sumerian Workcrew & Payroll, and Sumerian Paint & Bitumen Coating.

The seeding measure problems look peculiar to modern eyes. The early math used different units for length, width, and depth ( LWD)measures. The early multiplication of different units and measures tended to degenerate into unit conversion exercises. The seeding measure problems are thought to hop over some of cumbersome mixed unit multiplication, addition, and division for surface areas. Only the later Neo-Babylonian math used square cubits for area for some problems, which seems a more sensible measure of surface area. Some Neo-Babylonian property or barley field planning maps are bizarre sums of triangles, rectangles, and quadrilaterals. Possibly, the seeding measures make property settlements easier like elder brother gets 135 seeds (3/4) and younger brother get 45 seeds (1/4) of the 180 seed house and fields.

### Push Button Operation

For the push buttons in the eTCL calculator, 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. Aside from the TCL calculator display, when one presses the report button on the calculator, one will have console show access to the functions (subroutines).

Samples of Sumerian Seed Drill Rates & from Theory assumes seed drill,regular farrows per 6 meter strip and barley seed tcl wiki format
silas per bur No. farrows per ninda (6 meter strip) farrow width centimeters barley kg/ha comment, if any
2408 75 22.96 earliest and most doc'ed, Farmer's Instructions,Umma and Girsu UrIII, Fara tablets, & Babylon etc
2709 66.625.83Fara ED IIIa
30010 60 28.70 Umma and Girsu during the Ur III period
33011 54.5 31.57 break point for 11 rows in the math theory, not reported in the cuneiform texts
36012 5034.44 Umma and Girsu during the Ur III period
> 360 rates > 360 silas per bur, maybe suspect broadcast rates???
390 13 46.1 37.3 next break point for 13 rows in the math theory, not reported in the cuneiform texts

### Pseudocode and Equations

```     #pseudocode can be developed from rules of thumb.
#pseudocode: some problems can be solved by proportions (rule of three), to some order of magnitude
#pseudocode: enter quantity1,  quantity2, quantity3 and expected output (quantity4) for testcases.
#pseudocode: enter time in years, number of remaining items
#pseudocode: output fraction of (remaining items) over (items at time zero)
#pseudocode: ouput remaining items as fraction or percent
#pseudocode: output fraction of (quantity4 ) over ( quantity1 at time zero)
#pseudocode: output fraction of (quantity2) * (quantity3 ) over (quantity1 at time zero)
#pseudocode: outputs should be in compatible units.
#pseudocode: rules of thumb can be 3 to 15 percent off, partly since g..in g..out.
#pseudocode: need test cases > small,medium, giant
#pseudocode: need testcases within range of expected operation.
#pseudocode: are there any cases too small or large to be solved?```

### Testcases Section

#### Testcase 1

table 1printed in tcl wiki format
quantity value comment, if any
testcase number:1
240.0 :seed volume silas
18.0 :field surface area iku
0.62 :seed density kilograms per liter
240.0 :seed volume liters
0.0648 :answers: surface area square kilometers
6.48 :surface area hectares
240.0 :seed silas per bur
22.962 :seed kilograms per hectare

#### Testcase 2

table 2printed in tcl wiki format
quantity value comment, if any
testcase number:2
270.0 :seed volume silas
18.0 :field surface area iku
0.62 :seed density kilograms per liter
270.0 :seed volume liters
0.0648 :answers: surface area square kilometers
6.48 :surface area hectares
270.0 :seed silas per bur
25.833 :seed kilograms per hectare

#### Testcase 3

table 3printed in tcl wiki format
quantity value comment, if any
testcase number:3
360.0 :seed volume silas
18.0 :field surface area iku
0.78 :seed density kilograms per liter
360.0 :seed volume liters
0.0648 :answers: surface area square kilometers
6.48 :surface area hectares
360.0 :seed silas per bur
43.333 :seed kilograms per hectare

#### Testcase 4

table 4 printed in tcl wiki format
quanity value comment, if any
testcase number 4
Ur , ED I, first era 2800 BCE
tell area hectares 21 ha (50 acres)
labor force from other sources 2000 men 500 women
animals from other sources 10000 in herds or pens 3000 slaughtered every year
arable land in sq kilometers 60
district (tell) diameter meters 517.
answers: arable land hectares 6000.0
number of families 1000.0
men (or women) 2000.0
total inhabitants 6000.0
inhabitants per hectare 250.
children 4000.0
4.87E4 : sq. meters, total floorspace family dwellingsall one story, proportional premodern pop.
1.33E4: family dwellings, not include public bldg. all one story, proportional premodern pop.
36.5 : sq. meters, average family dwelling, not include public bldg. all one story, proportional premodern pop.
8.1 : sq. meters, average person floorspace, not include public bldg. proportional premodern pop.
12440: hectares grazing land proportional premodern pop.
5266:hectares grain proportional premodern pop., prob. >90% barley
453: hectares vegetable crops proportional premodern pop.
10000: caprivines proportional premodern pop.
750: plowoxen, includes private and state oxen proportional premodern pop.
2000: bovines including plowoxen proportional premodern pop.
10: 10 days nominal, extra ox rations for plowing one bur farmers instructions
6 : liters working ration per ox per day
1.5,3.6,6 : liters animal rations, fat. sheep(1.5),fat. pig(3.6),ox(6)
>>>>>> running final segment
quantity value comment, if any
testcase number:5
347200.0 :total liters (silas) on cited surface area
16660.0 :field surface area iku
0.62 :seed density kilograms per liter
0.0 :convert seed kilograms per hectare (supersedes if not zero)
59.976 :answers: surface area square kilometers
5997.6 :surface area hectares
375.126 :seed silas per bur
35.891 :seed kilograms per hectare

### References:

• The Farmer's Instructions or Farmer's Almanac, ETCSL translation
• Agriculture of the Old Babylonian Period, Gerlinde Mauer,1983
• Alleged Counting with Sexagesimal Place Value Numbers, Joran Friberg ,Chalmers University of Technology, Gothenburg, Sweden,Jun2005
• The Long Career of a Favorite Figure: The apsamikku in Neo-Babylonian Mathematics,Eleanor Robson, University of Cambridge,2007
• Sumerian Harvest Time, By: Miguel Civil, University of Pennsylvania Museum of Archaeology and Anthropology, Jul1963
• Benno Landsberger, major translator of Farmer's Instructions , OLZ , 1929
• Amaranth, Quinoa, Ragi, Tef, and Niger: Tiny Seeds of Ancient History and Modern Interest , R.G. Robinson, Station Bulleti,SB-2949 Agricultural Experiment Station University of Minnesota 1986
• The Ancient Grains Emmer, Einkorn and Spelt, Julie Dawson, Handout January 8, 2013, extension.org and organic_production
• Code of Ur-Nammu, Wikipedia. seeding rate coded into law!

## Appendix Code

### appendix TCL programs and scripts

```        # TCL source code follows
# pretty print from autoindent and ased editor
# Sumerian Seeding Rate Calculator V2
# written on Windows XP on TCL
# working under TCL version 8.6
# gold on TCL Club, 10Dec2018
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 {{} {seed volume silas ,sila = liter :} }
lappend names {field surface area iku :}
lappend names {seed density kilograms per liter :}
lappend names {optional: seed volume liters :}
lappend names {answers: surface area square kilometers  :}
lappend names {surface area hectares   : }
lappend names {seed silas per bur: }
lappend names {seed kilograms per hectare :}
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 Seeding Rate V2
from TCL,
# gold on TCL Club, 10Dec2018 "
tk_messageBox -title "About" -message \$msg }
proc self_help {} {
set msg " Sumerian Seeding Rate V2
from TCL ,
# self help listing
# problem, Sumerian Seeding Rate V2
# 4 givens follow.
1) seed volume silas (sila = liter)
2) field surface area iku
3) seed density kilograms per liter
4) Optional: seed volume liters
# 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 to conventional texteditor. For testcases
# testcase number is internal to the calculator and
# will not be printed until the report button is pushed
# for the current result numbers.
# >>> copyright notice <<<
# This posting, screenshots, and TCL source code is
# copyrighted under the TCL/TK license terms.
# Editorial rights and disclaimers
# retained under the TCL/TK license terms
# and will be defended as necessary in court.
Conventional text editor formulas or  grabbed from internet
screens can be pasted into green console.
# gold on  TCL Club, 12Dec2018 "
tk_messageBox -title "Self_Help" -message \$msg }
proc break_flag_routine {     } {
global side1 side2 side3 side4 side5
global side6 side7 side8
global spares
global testcase_number
set t9 \$testcase_number
set product_entries [* \$side1 \$side2 \$side3 \$side4 \$side5 \$side6 \$side7 \$side8]
if { \$product_entries < 0.0 } { puts " warning flag! negative numbers detected in product entries ref. tc\$t9" }
foreach item { 1 2 3 4 5 6 7 8 9 10 } {
set error\$item 0 }
#{-?\d*(\.\d+)?}  regsub -all -- {[^0-9.-]} \$input "" number
if { \$side1 < 0.0 } { set side1 .000001 ; set error1 1  }
if { \$side2 < 0.0 } { set side2 .000001 ; set error1 1  }
if { \$side3 < 0.0 } { set side3 .000001 ; set error1 1  }
if { \$side4 < 0.0 } { set side4 .000001 ; set error1 1  }
if { \$side5 < 0.0 } { set side5 .000001 ; set error1 1  }
if { \$side6 < 0.0 } { set side6 .000001 ; set error1 1  }
if { \$side7 < 0.0 } { set side7 .000001 ; set error1 1  }
if { \$side8 < 0.0 } { set side8 .000001 ; set error1 1  }
if { \$error1 == 1 } { puts " warning flag! negative numbers detected, defaulted to positive entries ref. tc\$t9" }
foreach item { 1 2 3 4 5 6 7 8 9 10 } {
set error\$item 0 }
return 1 }
proc calculate {     } {
global side1 side2 side3 side4 side5
global side6 side7 side8
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. ]
break_flag_routine
set barley_density 0.62
set wheat_density 0.78
set grain_density \$side3
set side4 \$side1
set side5 [/ [* \$side2 3600. 1. ] 1E6 ]
set side6 [/ [* \$side2 3600. 1. ] 1E4 ]
set side7 [* [/ \$side1 \$side2 ] 18.]
#set side8 [/ [* [* [/ \$side1 \$side2 ] 18.] \$grain_density] 1. ]
set side8 [/ [* [* [/ \$side1 \$side2 ] 18.] \$grain_density] 6.48 ]
}
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
console show;
console eval {.console config -bg palegreen}
console eval {.console config -font {fixed 20 bold}}
console eval {wm geometry . 40x20}
console eval {wm title . "Sumerian Seeding Rate V2 report  , screen grab and paste from console 2 to texteditor"}
console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
puts "%|table \$testcase_number|printed in| tcl wiki format|% "
puts "&| quantity| value| comment, if any|& "
puts "&| testcase number:|\$testcase_number | |&"
puts "&| \$side1 :|seed volume silas ,sila = liter |   |&"
puts "&| \$side2 :|field surface area iku  | |& "
puts "&| \$side3 :|seed density kilograms per liter| |& "
puts "&| \$side4 :|seed volume liters |  |&"
puts "&| \$side5 :|answers: surface area square kilometers  |  |&"
puts "&| \$side6 :|surface area hectares   |  |&"
puts "&| \$side7 :|seed silas per bur |  |&"
puts "&| \$side8 :|seed kilograms per hectare   |  |&"
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 240.0 18.0 0.62 240.0 0.0648 6.48 240.0  22.9}
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 270.0 18.0 0.62 270.0 0.0648 6.48 270.0  25.8 }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 360.0 18.0 0.78 360.0 0.0648 6.48 360.0 43.33 }
::ttk::button .clearallx -text clear -command {clearx }
::ttk::button .self_help -text self_help -command { self_help }
::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 .self_help .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 Seeding Rate Calculator V2"
```