Babylonian Square rule for Trapezoid Area and eTCL demo example calculator, numerical analysis

This page is under development. Comments are welcome, but please load any comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks, gold 12dec2018


Introduction

gold Here are some TCL calculations for trapezoid area using both conventional formulas and the Babylonian square law formula. The calculator uses modern units as meters for convenience to modern users and textbook examples. The units are not critical in the calculator internal math and any consistent units and output might be used like dollars to donuts. A number of Babylonian math problems have been found on Babylonian square law formula for trapezoid areas and volumes, ref Friberg.

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 Square Rule for Trapezoid Area

In the TCL calculator, the Babylonian square rule for trapezoid area uses three given's as 1) parallel long_side length meters l, 2) parallel short_side length meters s, and 3) feed no units f. Most available tablets here suggest l > s as a condition where feed => 0 , but exceptions are possible. This paper refers to meters, square meters, and degrees to orient the modern user in problems compatible with modern textbooks. The feed parameter f = (l-s)/h. Rearranging terms, trapezoid height h = (l-s)/f. The Babylonian feed parameter for trapezoids may be defined as the rate or coefficient that for a given unit of height or depth (y), usually cubits, the horizontal distance (x) recedes N1 units. The Babylonian math problems dealt with trapezoids where the long and short parallel sides were either 1) horizontal, eg. 0 degrees) 2) vertical, eg. 90 degrees, or 3) in-between, eg. 45 degrees. If the long and short parallel sides of the trapezoid are horizontal, the feed parameter in the conventional and modern sense is equivalent to horizontal(r1-r2)/height(y) =oc x/y =oc inverse slope =oc cotangent (x/y). If the long and short parallel sides of the trapezoid are vertical, the feed parameter in the conventional and modern sense is equivalent to vertical(r1-r2)/horizontal(x) =oc y/x =oc slope =oc tangent (y/x). Also, the Babylonian square law for trapezoid area was extended to volumes by *width factor w. The trapezoid_area_sq_law gives trapezoidal volume = w*(l**2-s**2)/(2.*f).

Conventional Western Formula for Trapezoid Area

The conventional median is median m1 = .5*(l+s). The conventional trapezoid area is area2 = .5*(l+s)*h. The conventional trapezoidal volume gives vol2 = .5*(l+s)*h*w.

For comparison of the Babylonian Square Rule with conventional formulas, TCL code will include redundant procs, calculation paths, and formulas to compute trapezoid area and volume.


Push Button Operation

For the push buttons, the recommended procedure is push a 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 TCL calculator increments a new testcase number internally, eg. TC(1), TC(2)., TC(3), TC(4) etc. 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.


Table, Math Terms Associated with Babylonian Trapezoids

Table , Math Terms Associated with Babylonian Trapezoids printed in tcl formatmodern equivalent, English comment if any
sag ki.gu : side earth trapezoid
sag ki.ta : side earth its , possessive case upper front of the trapezoid
sag an.na :side upper upper front of the trapezoid
us gi.na : length front true length of the trapezoid
sag gi.na : side front true front of trapezoid
N1 kus i-ku-lu : N1 cubit in eats it (it or man), read left to right feed parameter of the trapezoid noun derived from Sumerian SOV phrase
sag du : side triangle
N1 arakarum : N1 multiply transformation coefficient, usually scale up

Note. These Math Terms cover 4 or 5 cuneiform languages, and 5000 years of history. Math Terms are best understanding of context and math connotation here,not blessed by linguists and cuneiform dictionaries.


Testcases Section

Testcase 1

table 1printed in tcl format
quantity value comment, if any
1:testcase_number
38.0 :long_side length , meters:
21.0 :short_side length , meters:
1.0625 :feed rating:
1.0 :optional, default=1 , width meters:
16.0 :answers, height calculated, meters :
29.5 :conventional median, (l+s)/2, meters:
1.0625 :feed calculated:
472.0 : volume, width * B. sq law, meters cubed:
472.0 :conventional area formula, .5*(l+s)*h, meters squared:
472.0 :trapezoid area from square rule, meters squared:

Testcase 2

table 2printed in tcl format
quantity value comment, if any
2:testcase_number
18.0 :long_side length , meters:
6.0 :short_side length , meters:
1.330 :feed rating:
1.0 :optional, default=1 , width meters:
9.022 :answers, height calculated, meters :
12.0 :conventional median, (l+s)/2, meters:
1.329 :feed calculated:
108.270 : volume, width * B. sq law, meters cubed:
108.270 :conventional area formula, .5*(l+s)*h, meters squared:
108.270 :trapezoid area from square rule, meters squared:

---

Testcase 3

table 3printed in tcl format
quantity value comment, if any
3:testcase_number
24.0 :long_side length , meters:
20.0 :short_side length , meters:
0.333 :feed rating:
1.0 :optional, default=1 , width meters:
12.012 :answers, height calculated, meters :
22.0 :conventional median, (l+s)/2, meters:
0.333 :feed calculated:
264.264 : volume, width * B. sq law, meters cubed:
264.264 :conventional area formula, .5*(l+s)*h, meters squared:
264.264 :trapezoid area from square rule, meters squared:


Screenshots Section

figure 1.

Babylonian Square rule for Trapezoid Area png panel

figure 2.

Babylonian Square rule for Trapezoid Area png printout

figure 3.

Babylonian Square rule for Trapezoid Area png more concept

figure 4.

Babylonian Square rule for Trapezoid Area png concept

figure 5.

Babylonian Combined Market Rates png trapezoidal prism concept diagram 3

figure 6.

Babylonian Combined Market Rates png concept diagram 2

figure 7.

Babylonian Square rule for Trapezoid Area png geometry grid overlay

figure 8.

Babylonian Square rule for Trapezoid Area png geometric figure trapeziod band


References:

  • Amazing-Traces-of-a-Babylonian-Origin-in-Greek-Mathematics,
  • Joran Friberg. Primary reference
  • google < Babylonian trapezoid Wikipedia >
  • Wikipedia search engine < trapezoid >
  • Wikipedia search engine <mesopotamian unit measurement >
  • Wolfram Alpha search engine < trapezoid >
  • Ancient Babylonian Algorithms, Donald E. Knuth, Stanford University
  • Category Algorithm
  • A Remarkable Collection of Babylonian Mathematical Texts, Joran Friberg, 2007
  • Chalmers University of Technology Gothenburg, Sweden
  • Joran Friberg and andrew george,
  • Six more mathematical cuneiform texts in the Schoyen collection, MS 3895,
  • Amazing Traces of a Babylonian Origin in Greek Mathematics,
  • Joran Friberg
  • A Remarkable Collection of Babylonian Mathematical Texts:
  • Manuscripts in the Schøyen Collection: Cuneiform Texts I},
  • Joran Friberg
  • Babylonian Mathematical Astronomy: Procedure Texts,
  • by Mathieu Ossendrijver, trapezoids in astronomy
  • measures Jupiter motion

Pseudocode & Equations Section

        formulas
        following includes algebraic expressions,
        terms, and notation 
        Western trapezoid formulas
        and trapezoidal prism volume
        following includes algebraic expressions,
        terms, and notation not used on tablets
        side_aa refs top, t. on side
        side_bb refs bottom, t. on side
        side_cc refs height, t. on side
        median_length = .5*(side_aa+side_bb)
        trapezoid_area= 0.5*(side_aa+side_bb)*height
        trapezoid_area= median*height
        trapezoidal_prism_volume=
        t. vol=length*(0.5*(side_aa+side_bb))*height
        median divides equal side lengths
        Equations                                       reference
            1) median = .5*(l+s)                        conventional
            2) trapezoid area = .5*(l+s)*h              conventional
            3) feed = (l-s)/h                           trapezoid_area_sq_law
            4) h = (l-s)/feed                           trapezoid_area_sq_law
            5) trapezoid area = (l**2-s**2)/(2.*feed)   trapezoid_area_sq_law  
            # Also, the B. square rule for trapezoid area
            # was extended to volumes by *width factor. 
            6) trapezoidal volume = .5*(l+s)*h*w              conventional
            7) trapezoidal volume = w*(l**2-s**2)/(2.*feed)   trapezoid_area_sq_law 
         # end of file, pseudocode:

Appendix Code

appendix TCL programs and scripts

        # pretty print from autoindent and ased editor
        # Babylonian Trapezoid Area  V2
        # written on Windows XP on TCL
        # working under TCL version 8.6
        # gold on  TCL Club, 4oct2018
        package require Tk
        package require math::numtheory
        namespace path {::tcl::mathop ::tcl::mathfunc math::numtheory }
        set tcl_precision 17
        frame .frame -relief flat -bg aquamarine4
        pack .frame -side top -fill y -anchor center
        set names {{} {long_side length , meters:} }
        lappend names {short_side length , meters:}
        lappend names {feed rating: }
        lappend names {optional, default=1 , width meters: }
        lappend names {answers, height calculated :}
        lappend names {conventional median, (l+s)/2, meters::}
        lappend names {conventional area formula, .5*(l+s)*h, meters squared: }
        lappend names {trapezoid area from square rule, meters squared:}
        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 Babylonian Trapezoid Area  V2
            from TCL ,
            # gold on  TCL Club, 10aug2018 "
            tk_messageBox -title "About" -message $msg }
        proc self_help {} {
            set msg "Calculator for Babylonian Trapezoid Area V2
            from TCL ,
            # self help listing
            # 3 givens follow.
            1) long_side length meters
            2) short_side length meters
            3) feed, B. parameter related oc modern slant or slope
            4) optional height meters
            The solution from the clay tablets
            is genuine antique method from 1600 BCE
            and different looking from modern math methods.
            # The Babylonians did not use modern algebraic notation,
            # so the reader will have to bear some anachronisms in the TCL code.
            # Comparing conventional formula(s) for trapezoid area
            # to Babylonian square rule for trapezoid area.
            # For comparision, code will include redundant paths & formulas
            # to compute trapezoid area. The calculator uses modern
            # units as meters for convenience to modern users and textbooks.
            # Any convenient and consistent in/output units might be used
            # like inches, feet, nindas, cubits, or dollars to donuts.
            # Equations
            1) median = .5*(l+s)                        conventional
            2) trapezoid area = .5*(l+s)*h              conventional
            3) feed = (l-s)/h                           trapezoid_area_sq_law
            4) h = (l-s)/f                              trapezoid_area_sq_law
            5) trapezoid area = (l**2-s**2)/(2.*feed)   trapezoid_area_sq_law
            # Also, the B. square rule for trapezoid area
            # was extended to volumes by *width.
            6) trapezoidal volume = .5*(l+s)*h*w              conventional
            7) trapezoidal volume = w*(l**2-s**2)/(2.*feed)   trapezoid_area_sq_law
            # 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
            # copyrighted under the TCL/TK 8.6 license terms.
            # Editorial rights 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.
            Try copy and paste following into green screen console
            set answer [* 1. 2. 3. 4. 5. ]
            returns  120
            # gold on  TCL Club, 10aug2018 "
            tk_messageBox -title "About" -message $msg }
        proc trapezoid_solution { long_side short_side feed width} {
            global trapezoid_area height_calc feed_calc median trapezoidal_vol
            set median [* .5 [+ $long_side $short_side ]]
            set height_calc [/ [- $long_side $short_side] $feed ]
            set trapezoid_area [/ [- [* $long_side $long_side] [* $short_side $short_side] ] [* 2. $feed ] ]
            set trapezoidal_vol [* $width [/ [- [* $long_side $long_side] [* $short_side $short_side] ] [* 2. $feed ] ]]
            set feed_calc [ expr {  ($long_side-$short_side)/ $height_calc } ]
            puts "  check  $long_side $short_side $feed height_calc= $height_calc area= $trapezoid_area feed= $feed_calc  "
            return $trapezoid_area  }
        proc calculate {     } {
            global answer2
            global side1 side2 side3 side4 side5
            global side6 side7 side8
            global testcase_number
            global trapezoid_area height_calc feed_calc median trapezoidal_vol
            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 side5 1
            set side6 1
            set side7 1
            set side8 1
            set long_side $side1
            set short_side $side2
            set feed $side3
            set width $side4
            # height_calc calculated from feed and sides,
            # default for width equals 1.
            set trapezoid_area_sq_law [ trapezoid_solution $side1 $side2 $side3 $side4 ]
            set trapezoid_area_conventional [ expr { .5*($long_side+$short_side)* $height_calc }  ]
            set side5 $height_calc
            set side6 $feed_calc
            set side7 $trapezoid_area_conventional
            set side8 $trapezoid_area_sq_law
        }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$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 trapezoid_area height_calc feed_calc median trapezoidal_vol
            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 . " Babylonian Trapezoid Area 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 format|% "
            puts "&| quantity| value| comment, if any|& "
            puts "&| $testcase_number:|testcase_number | |&"
            puts "&| $side1 :|long_side length , meters: |   |&"
            puts "&| $side2 :|short_side length , meters: | |& "
            puts "&| $side3 :|feed rating: | |& "
            puts "&| $side4 :|optional, default=1 , width meters:| |  |&"
            puts "&| $side5 :|answers, height calculated, meters :| |  |&"
            puts "&| $median :|conventional median, (l+s)/2, meters:| |  |&"
            puts "&| $side6 :|feed calculated:| |  |&"
            puts "&| $trapezoidal_vol :| volume, width * B. sq law, meters cubed:| |  |&"
            puts "&| $side7 :|conventional area formula, .5*(l+s)*h, meters squared:| |  |&"
            puts "&| $side8 :|trapezoid area from square rule, meters squared: |  |&"
            
        }
        frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 38.0 21.0 1.0625   1.  16. 1.0625  472.  472.}
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 18.0 6.0 1.33      1.   1.0  1.333    108.  108. }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 24.0 20. 0.333     1. 1.0  0.333   264. 264. }
        ::ttk::button .clearallx -text clear -command {clearx }
        ::ttk::button .about -text about -command {about}
        ::ttk::button .self_help -text self_help -command {self_help}
        ::ttk::button .console2 -text report -command { reportx }
        ::ttk::button .exit -text exit -command {exit}
        pack .calculator  -in .buttons -side top -padx 10 -pady 5
        pack  .clearallx .console2 .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 . "Babylonian Trapezoid Area  V2"
        # gold on TCL Club, 4oct2018
        # This posting, screenshots, and TCL source code is
        # copyrighted under the TCL/TK 8.6 license terms.
        # Editorial rights retained under the TCL/TK license terms
        # and will be defended as necessary in court.
        # end of file

appendix, procs for Tablet VAT_7848

        # Tablet VAT_7848, published in MCT, Seleucid t. circa 300 BCE
        # numbers from trapezoid problem in VAT_7848
        # VAT_7848 uses version of Pythagorean theorem
        # Numbers for isoceles trapezoid, not sure if formula
        # works for all cases. ?cases of acute and right angle traps?
        # both slant sides r3=30
        # short parallel side s= 14
        # long parallel side l= 50
        # height h=24
        # trapezoid_area = 768
        # check formulas for trapezoid
        set trapezoid_height2 [ expr { sqrt ( 30**2- ((50-14)/2.)**2 ) } ]
        # square terms = expr sqrt(576), height = 24
        set trapezoid_area2 [ expr 24.*((50-14)/2.) ] # answer = 768.
        # develop TCL procs
        proc trapezoid_height { l s r3  }  {return  [ expr { sqrt ( $r3**2- (($l-$s)/2.)**2 ) } ] }
        proc trapezoid_area {s l h} {return [  expr { $h*(($l+$s)/2.) } ] }
        trapezoid_height 30. 50. 14. # answer = 24
        trapezoid_area 50. 14. 24.  # answer = 768
        # end of file

appendix, Calculations for another Tablet DPA34

            # Tablet DPA34, pub Friberg,
            # Old Akkadian period (c. 2340-2200 BC)
            # example: field expansion procedure (algorithm)
            # example: trapezoid field problem
            # probably task was calc area and converting
            # to regular “almost round” units of area.
            # school problem  on near trap. field
            # dimensions in 3 lines as follows.
            # 26 nindas north 110 nindas east
            # 58 nindas south 106? nindas west
            # (given area) 2 bur 1 ese 3 iku (cuneiform units of area)
            # note: answer and solution method not supplied on tablet.
            # Answer above equivalent to
            # expr 2*1800+1*6*100+3*100 = 4500 nindas
            # Not on the tablet, but a check answer could be
            # computed from the approximate quadrilateral formula.
            # Opposite sides are a&c,b&d, but formula not exact
            # condition {a b c d}   > 0
            proc  approx_quadrilateral_area { a b c d } {
                return [ expr {  (($a+$c)/2.)*(($b+$d)/2.) } ]
            }
            set field area [ approx_quadrilateral_area 26 110 58 106  ]
            # proc answer 4536??? decimal nindas, problems in agreement
            # online internet calc answer = 4480.86
            # online cyclic quadrilateral = 4480.8570
            # Strangely, average expr (4536+4480)/2=4508.0 is
            # right on the money.
            # end of file

appendix, Calculations for another Tablet W20044

            # Tablet W20044, pub CDLI, Friberg & D.A.R. DeSegnac,
            # Archaic period (c. (3350 to 3200 BCE)
            # example: field expansion procedure (algorithm)
            # example: trapezoid (algorithm)
            # probably task was calculating area and converting
            # to regular “almost round” units of area.
            # school problem  on near trap. field
            # dimensions in 4 slots as follows.
            # 75 nindas (north?, | stroke)
            # 100 nindas (east?, - stroke)
            # 60 nindas (south?, | stroke )
            # 120 nindas (west?, - stroke)
            # Not on the tablet,
            # ( area book answer) is 7425 square nindas
            # 7425 square nindas conv. 4 burs  and 2 iku (units of area)
            # area  = 4*1800+2*100 = 7400 square nindas
            # Not on the tablet, but a check answer could be
            # computed from the cyclic quadrilateral formula.
            set field_area1 [ approx_quadrilateral_area 26 110 58 106  ]
            # proc answer 7325.978 decimal nindas, 4 burs.
            # quadrilateral broken into 2 right triangles.
            # ajacent sides used
            set field_area2 expr 60*100*.5+120*75*.5
            # 7500.0 = = 4 burs 3 iku.
            approximate quad = expr ((75.+60.)/2.)* ((100.+120.)/2.)
            #    7425.0, gives 4 burs 2 iku rounded
            # 60*60 sq. nindas  = 2 bur, 1 bur = 1800 sq ninda
            # Friberg shows area from approx. formula was
            # expr {7425-4*1800} # or  225 sq. n. over a whole number in 4 burs.
            # 7425/(4*1800.) = 1.03125 decimal over whole number (1).
            # Field expansion procedure algorithm as square of (side+1/60),
            # (side+1/60 fraction)+(side+1/60 fraction)=
            expr { (1+1/60.)*(1+1/60) } = 1.0336 decimal
            # is close to result.
            # Also, a sqrt of expr (4*1800)**.5 = 84.852 decimal or 84+51/60
            # test square of same area = 84.8*84.8=7191 sq nindas
            # test square of same area, 84.8, on side, 
            # expr { (84.8+84.8*1/60.)*(84.8+84.8*1/60.) }= 7432.73
             average long side = expr (120+100)/2. = 110.0
             average long side = expr (60+75)/2. =  67.5
             average ratio of average sides expr ((120+100)/2.)/( (60+75)/2.)=1.629
             The golden mean is expr (1. +sqrt(5))/2 = 1.61803
             short avg over long avg is expr ((60+75)/2.)/ ((120+100)/2.)= 0.6136
             expr 1./((1. +sqrt(5))/2.)=   0.61803
             This is the earliest documented example of Golden Mean that
             has been found, in Archaic period (c. (3350 to 3200 BCE).
             In base_60 and rounded to near 1/60 (B. regular number), the Golden Mean
             or Golden Ratio would be expr { 1+ .61803*60} or   1+37/60. 
             The Golden Ratio Conjugate = expr {1/1.61803} = .61803 or 37/60.
            # end of file

appendix, Calculations for another Tablet W 19408_76+

         # Tablet W 19408_76+, pub CDLI, Friberg, & D.A.R. DeSegnac, 
         # Archaic period (c. (3350 to 3200 BCE) 
         # example: trapezoid (algorithms)
         # probably task was calculating area and converting 
         # to regular “almost round” units of area.
         # school problem  on near trap. field
         # dimensions in 4 slots as follows.
         # units 60*60 square nindas  = 2 bur, 1 bur = 1800 sq ninda
         # 1280 nindas from expr { 2*600+1*60+2*10 }   (   |   slash ,  north? )
         # 990  nindas expr { 1*600+6*60+3*10 }   (   ____  slash ,  east? )
         # 520 nindas from   expr { 8*60+4*10 }   (   | slash ,  south?)
         # 1410 from   expr { 2*600+ 3*60+3*10 }   (  ____    slash   west?)
         # Area answer not on the tablet, but a check answer could be
         # computed from the cyclic quadrilateral formula.
         # ( area round number) is 1080000.0 square nindas or 600 burs
         # t. area 996142.580 sq nindas from cyclic_quadrilateral_area 1280 990 520 1410
         # c.q. approx 996142/1800 or 553 burs, prob. intended 600 burs.
         # area = 1080000.0 approx_quadrilateral_area 1280 990 520 1410
         # expr {600*1800}   =   1080000 square nindas
         # ratio of averages for opposite sides.
         # 1.333 or 80/60  from expr { ( .5* (990+1410))/(.5*(1280+ 520)) }  
         # Note. 600 burs was relatively huge and unrealistic grain field
         # for a city state. Each plow team was assigned 3 burs to plow
         # over ten day period.
         # The field of 600 burs would require expr {600/3} or 200 plow teams.
         # The city of Umma in the classic URIII period required not more than
         # 60 plow teams for 233.28 sq km of state land.
         # Equivalent square side was expr {sqrt (1080000)} or 1039 nindas
         # end of file

appendix, Golden Rectangle Dimensions for even numbered area units (burs)

                    golden rectangle ratio for 2 bur units
                    short_side_S*long_side_L = 2 bur in golden rectangle ratio
                    (1*S)*(1.61803*S)=expr 2*1800 # 4 bur
                    1.61803*(S**2)=3600 sq nindas
                    S**2= expr { 3600./1.61803 }
                    set S  [expr { sqrt (2224.927) }]
                    S=47.169,  L = 76.321 from expr { 1.61803*47.169}
                    check expr {47.169*76.321 } = 3599.985 square nindas

                    golden rectangle ratio for 4 bur units
                    short_side_S*long_side_L = 4 bur in golden rectangle ratio
                    (1*S)*(1.61803*S)=expr 4*1800 # 4 bur
                    1.61803*(S**2)=7200 sq nindas
                    S**2= expr { 7200./1.61803 }
                    set S  [expr { sqrt 4449.855 }]
                    S=66.7,  L = 107.922 from expr { 1.61803*66.7}
                    check expr {66.7*107.922 } = 7198.3974 square nindas

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