## Introduction

gold Here are some TCL calculations for the Sumerian Surveyor Area Formula. The Surveyor formula will be described in modern algebraic terms and modern units here. The Sumerian surveyor formula is ( (L1+L2)/2)* (W1+W2)/2) or (L1+L2 +W1+W2)/4 for rectangles and 4 sided figures. This formula is mostly of historical interest in early Sumerian mathematics, since the formula is accurate only on squares or rectangles. The surveyor formula is very inaccurate where angles are no longer 90 degrees, known to fail grossly on rhomboids and some quadrilaterals, and was abandoned or unreported by the Babylonian mathematicians (circa 1500 BCE). Perimeter is included on the calculator as perimeter equals L1+L2+W1+W2. Although the TCL calculator uses modern meters as length measure, any consistent units may be used with stipulations or installed in the calculator code. Most of the testcases involve replicas or models, using assumptions and rules of thumb.

The Sumerian surveyor formula is ( (L1+L2)/2)* (W1+W2)/2) or (L1+L2 +W1+W2)/4 for a rectangle and 4 sided figure. The first testcase was all sides equal 1 meter. For the formula (L1+L2 +W1+W2)/4, substituting, area equals ( 1+1 +1+1)/4, results in 1 square meter. The perimeter is (L1+L2 +W1+W2), substituting, perimeter = ( 1+1 +1+1), 4 meters.

The second testcase was L1=2,L2=2,W1=4,and W2=4. For the formula (L1+L2 +W1+W2)/4, substituting, area equals ( 2+2 +4+4)/4, 12/4, 3 square meters. The perimeter was 2+2 +4+4, 12 meters.

The third testcase was L1=3,L2=3,W1=5,and W2=5. For the formula (L1+L2 +W1+W2)/4, substituting, area equals ( 3+3 +5+5)/4, 16/4, 4 square meters. The perimeter was 3+3 +5+5, 16 meters. --- One problem in math history is dating the invention of math formulas, the use and duration in society, and the interaction between the cuneiform math and other cultures. Evidence for the surveyors formula has been found carved on the Edfu Temple in ancient Egypt.

### Pseudocode and Equations

```    # following statements can be pasted into eTCL console
set testcase_1  [ eval expr  ( 1+1 +1+1)/4 ]  # 1 square meters
set testcase_2  [ eval expr  ( 2+2 +4+4)/4 ]  # 3 square meters
set testcase_3  [ eval expr  ( 3+3 +5+5)/4 ]  # 4 square meters
set testcase_1_perimeter   [ eval expr  ( 1+1 +1+1) ]  # 4 square meters
set testcase_2_perimeter   [ eval expr  ( 2+2 +4+4) ]  # 12 square meters
set testcase_3_perimeter   [ eval expr ( 3+3 +5+5)  ]  # 16 square meters```

### 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
1.0 :length L1 meters meters
1.0 :length L2 meters
1.0 :width W1 meters
1.0 :width W2 meters
1.0 :average length meters
1.0 :average width meters
4.0 :perimeter meters
1.0 :surface area square meters

#### Testcase 2

table 2printed in tcl wiki format
quantity value comment, if any
testcase number:2
2.0 :length L1 meters meters
2.0 :length L2 meters
4.0 :width W1 meters
4.0 :width W2 meters
2.0 :average length meters
4.0 :average width meters
12.0 :perimeter meters
3.0 :surface area square meters

#### Testcase 3

table 3printed in tcl wiki format
quantity value comment, if any
testcase number:3
3.0 :length L1 meters meters
3.0 :length L2 meters
5.0 :width W1 meters
5.0 :width W2 meters
3.0 :average length meters
5.0 :average width meters
16.0 :perimeter meters
4.0 :surface area square meters

### 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, Jöran 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

## Appendix Code

### appendix TCL programs and scripts

```        # pretty print from autoindent and ased editor
# Sumerian Surveyor Formula calculator
# written on Windows XP on eTCL
# working under TCL version 8.5.6 and eTCL 1.0.1
# gold on TCL WIKI, 15may2016
# comment follows from gold, 12Dec2018
# pretty print from autoindent and ased editor
# Sumerian Surveyor Formula Calculator V2
# written on Windows XP on TCL
# working under TCL version 8.6
# Revamping older program from 2016.
# One of my early TCL programs on wiki.
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 L1 meters :} }
lappend names {length L2 meters :}
lappend names {width W1 meters: }
lappend names {width W2 meters :}
lappend names {answers: average length meters:}
lappend names {average width meters: }
lappend names {surface area square meters: }
lappend names {surface area square 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
set msg "Calculator for Sumerian Surveyor Area Formula V2
from TCL ,
# gold on  TCL Club, 12Dec2018  "
tk_messageBox -title "About" -message \$msg }
proc self_help {} {
set msg " Sumerian Surveyor Area Formula V2
from TCL Club ,
# self help listing
# problem, Sumerian Surveyor Area Formula V2
# 4 givens follow.
1) length L1 meters:
2) length L2 meters:
3) width W1 meters:
4) width W2 meters:
# 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.
# This posting, screenshots, and TCL source code is
# Editorial rights and disclaimers
# retained under the TCL/TK license terms
# and will be defended as necessary in court.
Conventional text editor formulas
or  formulas grabbed from internet
screens can be pasted into green console.
# gold on  TCL Club, 12Dec2018 "
tk_messageBox -title "Self_Help" -message \$msg }
proc pi {} {expr acos(-1)}
proc calculate {     } {
global side1 side2 side3 side4 side5
global side6 side7 side8
global testcase_number
global average_length average_width total_surface_area
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 length1 \$side1
set length2 \$side2
set width1 \$side3
set width2 \$side4
set average_length [/ [+ \$length1 \$length2 ] 2. ]
set average_width [/ [+  \$width1 \$width2 ] 2. ]
set total_surface_area [/ [+ \$length1 \$length2 \$width1 \$width2 ] 4. ]
set perimeter [+ \$length1 \$length2 \$width1 \$width2 ]
set side5 \$average_length
set side6 \$average_width
set side7 \$perimeter
set side8 \$total_surface_area
}
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 average_length average_width total_surface_area
console eval {.console config -bg palegreen}
console eval {.console config -font {fixed 20 bold}}
console eval {wm geometry . 40x20}
console eval {wm title . " Sumerian Surveyor Area V2 Report, screen grab and paste from console 2 to texteditor"}
console eval {. configure -background orange -highlightcolor brown -relief raised -border 30}
console show;
puts "%|table \$testcase_number|printed in| tcl wiki format|% "
puts "&| quantity| value| comment, if any|& "
puts "&| testcase number:|\$testcase_number | |&"
puts "&| \$side1 :|length L1 meters meters |   |&"
puts "&| \$side2 :|length L2 meters | |& "
puts "&| \$side3 :|width W1 meters| |& "
puts "&| \$side4 :|width W2 meters| |&"
puts "&| \$side5 :|average length meters | |&"
puts "&| \$side6 :|average width meters |  |&"
puts "&| \$side7 :|perimeter meters|  |&"
puts "&| \$side8 :|surface area square meters|  |&"
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 1.0 1.0 1.0 1.0  1.0 1.0 4.0 1.0}
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 2. 2. 4.0 4.0   2.0  4.0 12.0 3.0 }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 3. 3. 5.0 5.0   3.0 5.0  16.0 4.0 }
::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  .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 Surveyor Area 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   |&"  ```