Version 27 of Babylonian Shadow Length & Angles and eTCL Slot Calculator Demo Example, numerical analysis

Babylonian Shadow Lengths & Angles and eTCL Slot Calculator Demo Example

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 in your comment with the same courtesy that I will give you. Its very hard to reply intelligibly without some background of the correspondent. Thanks,gold

• Babylonian Shadow Lengths & Angles and eTCL Slot Calculator Demo Example
• Gradient Coefficients
• Shadow Lengths
• Calculator Use
  • Pseudocode and Equations
  • Table 1,Babylonian Day Length Circa 600 BCE
  • Table 2,Babylonian Noon Shadows, hand calculator
  • Testcases Section
    • Testcase 1
    • Testcase 2
    • Testcase 3
  • Screenshots Section
    • figure 1.
    • figure 2.
    • figure 3.
    • figure 4.
    • figure 5.
  • References:
• Appendix Code
  • appendix TCL programs and scripts
  • Pushbutton Operation
• Comments Section

gold Here is some eTCL starter code for calculating gradient and tangent angles from the Old Babylonian coefficient tables. The impetus for these calculations was the gnomon instructions tablets. Most of the testcases involve replicas or models, using assumptions and rules of thumb.

The eTCL calculator is estimating the gradient and tangent angles from integer ratios, especially those from the coefficient tables, Babylonian shadow length tables (Mul Apin), and gnomon instructions tablets. In pseudocode, each angle is $kay*atan($N1/$N2), where N1&N2 are integers and N1 > N2. Some analysts believe the Babylonian gnomon was 1 cubit high ( 0.5 meter ) in the Mul Apin, but the length of the gnomon is not explicitly stated or obvious in the available texts. As an intermediate step, the eTCL calculator can be used to calculate a table of tangent angles.

Gradient Coefficients

In the Old Babylonian coefficient tables, the 7 readable gradients in a set are 1/7, 1/6, 1/5, 1/4, 1/3, 1/2, and 2/3. There are unreadable lines before and after the set of readable gradients. On the same tablet, there are readable lines that have 1/1 sides for a square. In other texts, there is a math problem that calls for a 1 cubit vertical decline in a cubit horizontal distance (1/1), equivalent to a tangent angle of 45 degrees. So its probably safe to add the 1/1 ratio at equivalent 45 degrees to the table of standard gradients

Shadow Lengths

There are several tablets that deal with the predictions for the length of the gnomon shadow over the day, including the Mul Apin and the gnomon instructions tablets. For the lunar eclipse predictions, the Babylonian astronomers reported the eclipses or potential eclipses as time periods after sunrise or time periods before sunset. The duration of day or the day1/day2 ratio could be used in a corrective factor as <main eclipse prediction> + kay*<corrective factor>. From the Mul Apin, the corrective ratio 1:1 was used for shortest day over longest day for the year (near Babylon). Another corrective ratio 2:3 has been found in later LB texts. Using the day/night ratios, the Babylonian astronomers developed a piecewise solution to a nonlinear problem. Although not completely understood, some elements of the gnomon instructions tablets appear to be followup on the day/night duration predictions and the eclipse correction factor.

In the gnomon instructions tablets, there are a number of terms that need resolution. There is handuhhu of the Cancer (month 4) and handuhhu of the Capricorn (month 10). In other contexts, handuhhu has meant lock part or rooster spur. Also, handuhhu same was used for the boundary? pin? of the heavens. The LB. handuhhu shares root words with Su. he-nun (plenty), Su. he-en-du (path), Su. hen-zer (bronze pin), and Akk. he-gal (point? great). In astrology terms, arrow, Su. kak-si-di, or Su. gag.si.sa was used to designate month 5 (or star Sirius). Another astrologic term in the gnomon instructions, mar-du or mar-tu was used to designate month 5 as Perseus. Its possible that handuhhu or mar-tu refers to the border or end of month 4. In later astronomy definitions, the constellation Cancer or month 4 extended 30 degrees, and if the center of Cancer is the center of the gnomon shadow chart, the bounds of Cancer might extend plus or minus 15 degrees of center.

Lets look at cuneiform terms for shadow and possible etymology in the gnomon instructions texts. The LB. word translated as shadow ( nis-mi ) already carries the connotation of tool. The Sumerian verb was nissu...lal (shadow ... reach or extend). The Su. nissu was a compound word written as nis-mi ( nis=tool, mi=substitute). Another variant word for shadow was Su. gissu or Akk. sillu.

The gnomon instructions tablets are partly broken and the late Babylonian terminology is so different from the other OB. astronomy and math texts, it is difficult to follow the internal narrative. One approach is study the initial instructions on setting the gnomon up and the final chart construction for shadow lengths, up to the broken off portion. If the bounds or limits of the final chart can be determined from modern calculations, then perhaps the internal narrative will make more sense. The initial instructions are to brick up on the left and right of the na (stone gnomon?) which suggested building a temporary support.. For the final chart construction, the "2 kus sag" suggests drawing on a wooden board 2 cubits on a side. From the shadow predictions in the Mul Apin, the Babylonian astronomers appear to be concentrating on measuring the midmorning shadow and the noon shadow in cubits.

In modern terms, the shadow lengths and time slice of the morning study would be limited by the dimensions of the "2 kus sag" chart. The midmorning shadow would at tangent angle of atan (assumed gmonon/shadow), atan (1 cubit/1 cubit), atan (1/1), 45 degrees. The understanding is that the texts define midmorning by a constant shadow length of 1 cubit, not using time. The time (after sunrise) that the sun reaches the midmorning locus varies throughout the year. At Babylon, the noon shadow from a 1 cubit gnomon varies from 9/60 to 88/60 cubits over the year. The tangent angle at noon would range from atan(1/(9/60)) to atan (1/(88/60)) , 48 to 33.86 degrees in modern terms. Assuming full scale drawing on the shadows, the 2 kus on a side would limit the "2 kus sag" chart to a certain window of shadow sizes and tangent angles. The tangent angles in the 2 kus window would be limited to atan(1/2), above 26.5 degrees. An alternate possibility is to subtract 1 cubit from the "morning lengths", which some of the gnomon instructions imply, and the resultant limit would be atan(1/(1+2), atan(1/3), above 18.4 degrees. Using the earth rotation of 15 degrees per hour, the time slice of a window ref 26.5 degrees would be 2*(90-26.5)/15, decimal 8.4 hours. The time slice of a window of ref 18.4 degrees would be 2*(90-18.4)/15, decimal 9.5 hours.

Using the modern formulas, some trial calculations for the shadow lengths at noon can be made with the hand calculator. These will not have the accuracy of the large computer programs, but can provide some rounded peg points for the shadow charts for a gmonon of 1 cubit length. Babylon was at 33/13/59N, decimal degrees 32.3305N and 44/22/00E, decimal degrees 44.3666E. The axial tilt or obliquity is 23.44 at present; the tilt in Babylonian times was about 23.75 degrees (500 BCE). The noon shadow of the winter solstice (B. month 10) was 1 cubit * tan(32.3305-23.75), 1*tan(8.5805), rounded decimal 0.15 cubits, or 9/60 cubits. The noon shadow of the spring equinox (B. month 1 ) and fall equinox (B. month 7 ) was 1 cubit * tan(32.3305), rounded decimal 0.6329 cubits, or 38/60 cubits. The noon shadow of the summer solstice (B. month 4 ) was 1 cubit * tan(32.3305+23.75), 1*tan(56.0805), rounded decimal 1.487 cubits, or 89/60 cubits.

In reference to shadow lengths in the Mul Apin and the gmonon instructions texts, there is uncertaincy in the obliquity partly from the modeling of the obliquity and partly in that most tablets are undated. Where the tablet date or measurement date is uncertain, the obliquity can not be modeled beyond a certain point. An error of 1-(23.44/23.75) or rounding 2 percent could be attributed to obliquity modeling. Also the Babylonians usually measured a quantity to 1) fractions of the nearest 1/60 and 2) fractions regular in base 60, which introduced a quantization error. For example, rounding odd 37/60 cubits to even and regular 38/60 cubits would have an error of 1-(37/60)/(38/60) or rounding 3 percent error.

Calculator Use

For the eTCL calculator, the length of any gmonon is set as the master length in centimeters. Using proportions, the gmonon length is multiplied by the reciprocal shadow ratio at Babylon and gives the calculated shadow length. For example, loading 100 centimeters and (1/2) gives 100*(1/(1/2)), 100*2, or 200 c. The units are passed through unchanged, so the master length could be in inches or feet with the same units returned on the shadow.

Pseudocode and Equations

  area = [* length width ] # square meters

Table 1,Babylonian Day Length Circa 600 BCE

Table 2,Babylonian Noon Shadows, hand calculator

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

Testcase 2

Testcase 3

Screenshots Section

figure 1.

figure 2.

figure 3.

figure 4.

figure 5.

References:

• Babylonian Astronomy, LIS BRACK-BERNSEN AND JOHN M. STEELE†
• Celestial Measurement in Babylonian Astronomy J. M. Steele, 13 Jun 2007
• Lunar Eclipse Astronomy,Kristian Peder Moesgaard,August 2011
• Eclipse Predictions and Earth's Rotation[L1 ]
• Delta T (ΔT) and Universal Time[L2 ]
• Shadow-Length Schemes in Babylonian Astronomy,J. M. Steele, SCIAMVS 14 (2013), 3-39,revised 2012
• Brack-Bernsen, L. and Hunger, H., 2002, “TU 11: A Collection of Rules for the
• Prediction of Lunar Phases and of Month Lengths”, SCIAMVS 3, 3–90
• Late Babylonian procedure texts for gnomons ,Mathieu Ossendrijver – 12 february 2014
• Astronomical Instruments In Ancient India, Shekher Narveker,June, 2007,esp.shanku, gnomon

Appendix Code

appendix TCL programs and scripts

        # pretty print from autoindent and ased editor
        # sun angle calculator
        # written on windows XP on eTCL
        # working under TCL version 8.5.6 and eTCL 1.0.1
        # gold on TCL WIKI , 14jun2014
        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 {{} {master length centimeters:} }
        lappend names {N1  integer:}
        lappend names {N2  integer: }
        lappend names {angle from N1/N2 ratio degrees: }
        lappend names {answer: angle 2 degrees}
        lappend names {angle 3 degrees:}
        lappend names {angle 4 degrees: }
        lappend names {angle 5 degrees: }
        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 sun angles 
            from TCL WIKI,
            written on eTCL "
            tk_messageBox -title "About" -message $msg }
        proc pi {} {expr 1.*acos(-1)}  
        proc radianstodegconst {} {return [/ 180. [pi] ] }
        proc xradianstodegconst {} {return [/ [pi] 180. ] }
        proc degz {} {return [/ 180. [pi]  ]}
        proc degx {aa} {return [ expr { [degz]*atan($aa) }  ]}
            proc calculate {     } {
            global answer2
            global side1 side2 side3 side4 side5
            global side6 side7 side8 testcase_number
            global length0 shadow 
            incr testcase_number
            set $side1 [* $side1 1. ]
            set $side2 [* $side2 1. ]
            set $side3 [* $side3 1. ]
            set $side4 [* $side4 1. ]
            set N1 $side2
            set N2 $side3
            set length0 $side1
            set shadow [* $length0 [/ 1. [/ $N1 $N2 ] ] ]
            set term1 [/ 1. 7. ]
            set term2 [/ 1. 6. ]
            set term3 [/ 1. 5. ]
            set term4 [/ 1. 4. ]
            set term5 [/ 1. 3. ]
            set term7 [/ $N1 $N2 ]
            set angle1 [ degx $term7 ]
            set angle2 [ degx $term2 ]
            set angle3 [ degx $term3 ]
            set angle4 [ degx $term4 ]
            set angle5 [ degx $term5 ]
            set side4 $angle1
            set side5 $angle2
            set side6 $angle3  
            set side7 $angle4 
            set side8 $angle5 
            }
        proc fillup {aa bb cc dd ee ff gg hh} {
            .frame.entry1 insert 0 "$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 testcase_number 
            global length0 shadow 
            console show;
            puts "%|table| printed in|TCL WIKI format |% "
            puts "&|testcase number:| $testcase_number| |& "
            puts "&|shadow 1,  master length:| $side1| |& "
            puts "&|shadow N1 integer: |$side2| |& "
            puts "&|shadow N2 integer: |$side3| |& "
            puts "&|alternate angle from N1/N2 ratio: | $side4| |& "
          puts "&|gmonon length, centimeters: |$length0| |& "
            puts "&|gmonon shadow, centimeters: |$shadow| |& " 
            puts "&|angle 2, degrees: |$side5| |& "
            puts "&|angle 3, degrees: |$side6| |& "
            puts "&|angle 4, degrees: |$side7| |& "
            puts "&|angle 5, degrees: |$side8| |& "
            puts "&| 1/7 ratio | [ degx [/ 1. 7. ] ]|degrees | |& "
            puts "&| 1/6 ratio | [ degx [/ 1. 6. ] ]|degrees | |& "
            puts "&| 1/5 ratio | [ degx [/ 1. 5. ] ]|degrees | |& "
            puts "&| 1/4 ratio | [ degx [/ 1. 4. ] ]|degrees | |& "
            puts "&| 1/3 ratio | [ degx [/ 1. 3. ] ]|degrees | |& "
            puts "&| 1/2 ratio | [ degx [/ 1. 2. ] ]|degrees | |& "
            puts "&| 2/3 ratio | [ degx [/ 2. 3. ] ]|degrees | |& "
            puts "&| 1/1 ratio | [ degx [/ 1. 1. ] ]|degrees | |& "    
       }
         frame .buttons -bg aquamarine4
        ::ttk::button .calculator -text "Solve" -command { calculate   }
        ::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 100. 1. 2.  26.8 9.5  11.3 14.0 18.4 }
        ::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 40. 1.  9.   6.34 9.5  11.3 14.0 18.4 }
        ::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 100. 1. 12.  4.76 9.5  11.3 14.0 18.4 }
        ::ttk::button .clearallx -text clear -command {clearx }
        ::ttk::button .about -text about -command about
        ::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 .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 . "Sun Angle 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   |&"  

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Comments Section