## Introduction edit

gold Here is some eTCL starter code on a calculator for human language root words. The impetus for these calculations was checking some references in Sumerian and cuneiform mathematics. Most of the testcases involve experiments or models, using assumptions and rules of thumb.

The original formulas were developed by Morris Swadesh in the changes of native American languages, published 1950 to 1955. Swadesh published a lexical constant fraction of 0.86 in 1955, restating that 86% of original root words should remain in a source language after 1000 years. In the various papers, Swadesh used selected word lists of 215, 200, and 100 root words. The original formula was N(t) = N(0)*exp(-lambda*t). In eTCL notation, the remaining root words N2= [*kay [/ N1 [exp [/ \$t 1000.]]]], where N1 is the original set of root words, 1000. years for millennium, and t is the time in years. The constant kay equals [* [/ 86. 200.] [exp 1. ]] or 1.16886, where [exp 1. ]] is familiar natural number 2.718. As a rule of thumb, a second language with different roots of 50 percent is not easily understandable from a first or source language, and probably would be considered a second language. There might be 5 percent error in comparison lists of two languages, so the 50% criterion from the rule of thumb can be relaxed somewhat. In modeling a descendent language from a source language, the criterion is postulated that a language with more than 46 percent of root words remaining is a descendent of the source language.

For the eTCL calculator, one enters time in years, number of root words in list at time zero (usually 200), and the remaining root words after time. Using the Swadesh formula and constant of 86, the minimal or predicted number of remaining root words is estimated. The decimal fraction of remaining words would be [/ N2 N1] and the percentage of remaining root words after time would be [* [/ N2 N1] 100.]. The eTCL calculator can report if more than 46 percent of root words is passed as a criterion, suggesting that the second language set (remaining roots) is a descendent language. The eTCL calculator can report if { \$percent >= 46. } { set passfaillogic 1. }, 1 for yes or 0 for no. If the remaining root words after time is not known accurately, one can reload the predicted number as roots2 and push solve a second time for tentative lower bound. For example in testcase 4, the eTCL calculator returned a percentaqe of 60 percent for 120 remaining root words, but less accuracy. Reloading the minimal 85 root words, the eTCL returned 42.5 percent. So the lower/upper bounds on the solution would be 42 to 60 percent.

A related formula that Swadish used was separation time in millennia equals (ln C) / (2* (ln R) ), where C is the fraction of roots words that two languages have in common and R is the Swadesh ratio (0.86). The separation time is an estimate of when two related languages separated from a common source language before present (BP). For example, an early study found that the French and German languages have 33% root words in common or decimal ratio 0.33. The separation in years would be (ln C) / (2* (ln R) ), ln(.33)/(2*ln(.86)) =-1.1086 / (2*-.1508), or 3.675 millennia before present (BP).

A more sophisticated analysis of the F&G separation indicated the common roots of F&G at fraction 0.351 and 35.1 percentage. One error source was the limited volcabulary lists of 200 or so, which introduce a quantization error of about [- 1 [/66 67]] or 1 percent for each word in error. From the 200 word list tallies, each word in error means that the separation formula estimates about 50 years per word in error. Overall, there was about a 15 percent error in the linguistic calculations. Consequently, the time of separation between F&G was between 3400 and 4200 years BP. One historical event that took place near the F&G separation was the volcanic eruption of the island of Thera in 1475 BCE or 3489 BP. Other papers suggest the Minoan eruption of Thera took place near 1628 BCE based on tree rings and near 1613-1660 BCE based on carbon dating. It has been speculated that the volcanic dust, crop failures, and poor grassland yields damaged the ecology of the Middle East and triggered one of the Celtic or Indo-European invasions of Europe.

Continuing with the eTCL calculator on the separation of French and German, one loads 200 root words for time zero and 66 words as remaining roots. The solved ratio would be [/ 66 200] or 0.33 as above for the common roots of French and German. The eTCL notation for years separation would be set separation [* 1000. [/ [log \$newratio ]] [* 2. [log .86 ]]]] years BP. This function for separation was loaded into the calculation routine, but can be pasted into the eTCL console. The fraction 0.33 estimated the years of separation as 3675 before present or 1661 BCE +-200 years. For another bound on the F&G separation, the fraction 0.351 or 70/200 was returned in the separation formula as 3480 years BP or 1466 BCE , +-200 years. While quantization error was not the only error, the quantization error on the two calculations can be estimated as [* 50. [70-66]] or 200 years. This added separation function shows how the calculator can be modified, but the separation function should be handled carefully since the interpretation and derivation of the separation function is a little different.

Ironically, the first estimate for the F&G separation at 1661 BCE +-200 years was in close agreement within the bounds of tree ring estimates and carbon 14 dates. For the first estimate, the reported error of +-200 years was based on the 200 word lists. The analysis might point out that reducing the wordlist from 200 to 100 root words substantially increases the quantization error. The quantization error from a hundred word list would be [- 1 [/ 33 34]] , 3 percent quantization error for each word. Continuing with 4 words in error, the total error on the F&G problem would be 12 percent (+-400 years) using the 100 word list. The conclusion is that human languages change over the years. If there are N1 number of roots in the source language, there will be some smaller fraction N2 left over the years.

### Table 1 : Swadesh Constants and Separation of F&G

Swadesh Constants and Separation of F&G table printed in tcl wiki format
quantity value comment, if any
first estimate F&G separation 1661 BCE first cut, used common root fraction 0.33, +-200 years
second estimate F&G separation 1466 BCE used common root fraction 0.351
Minoan volcanic eruption of Thera 1475 BCE used chronology of Egyptian Kinglist primarily
Minoan volcanic eruption of Thera 1618 BCE used carbon dating from Thera and Knossos palace
Minoan volcanic eruption of Thera 1620-1660 BCE used Canadian tree rings and Californian bristle pines
Possibly more than one volcanic eruption of Thera 1450-1660 BCE in period of interest.
Swadesh-200 fraction 0.81 decimal used with list of 200 root words, Swadesh original use
Swadesh-200 constant 81 percentage used with list of 200 root words, Swadesh original use
Swadesh-200 lambda 0.19 1.-lambda(0.19)=0.81, [exp [*-1. \$lambda \$t]] Swadesh original use
Lees-200 fraction 0.805 decimal used with list of 200 root words, Lees revised use of 0.805 +- 0.0176 with 90% confidence
Lees-200 constant 80.5 percentage used with list of 200 root words, Lees revised use
Lees-200 lambda 0.195 used with list of 200 root words, Lees revised use
Swadesh-100 fraction 0.86 decimal used with list of 100 root words, Swadesh revised use
Swadesh-100 constant 86 percent used with list of 100 root words, Swadesh revised use
Swadesh-100 lambda 0.14 1.-lambda(0.14)=0.86, [exp [*-1. \$lambda \$t]]Swadesh revised use
years of separation, ref. Swadesh [* 1000. [/ [log \$newratio ]] [* 2. [log .86 ]] years BP root words common to F&G over source
years of separation, ref. Swadesh [- [* 1000. [/ [log \$newratio ]] [* 2. [log .86 ]] \$current_year ] years BCE years separation rated to BCE

### Pseudocode Section

```    pseudocode can be developed from rules of thumb.
pseudocode: enter time in years, number of remaining root words
pseudocode: output fraction of (remaining root words) over (root words at time zero)
pseudocode: ouput remaining root words as fraction
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

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

ancient languagetable printed in tcl wiki format
quantity value comment, if any
testcase number 1
time in years at time zero : 1000.
root words at time zero: 200.
root words after time (t2-t0) : 86.
answers: min. remaining root words fm formula :85.0
ratio remaining root words : 0.43
remaining root words percentage: 43.0
language descendent criterion > 46% (0 or 1) : 0. logic variable, 1 for yes, 0 for no

#### Testcase 2

ancient language 2 printed in tcl wiki format
quantity value comment, if any
testcase number 2
time in years at time zero : 1000.
root words at time zero: 200.
root words after time (t2-t0) : 120.
answers: min. remaining root words fm formula :85.0
ratio remaining root words : 0.6
remaining root words percentage: 60.0
language descendent criterion > 46% (0 or 1) : 1. logic variable, 1 for yes, 0 for no

#### Testcase 3

ancient language 3 table printed in tcl wiki format
quantity value comment, if any
testcase number 3
time in years at time zero : 1000.
root words at time zero: 200.
root words after time (t2-t0) : 160.
answers: min. remaining root words fm formula :85.0
ratio remaining root words : 0.8
remaining root words percentage: 80.0
language descendent criterion > 46% (0 or 1) : 1. logic variable, 1 for yes, 0 for no

#### Testcase 4

ancient language and lower bound calculation printed in tcl wiki format
quantity value comment, if any
testcase number 4
time in years at time zero : 1000.
root words at time zero: 200.
root words after time (t2-t0) : 120.
answers: min. remaining root words fm formula :85.0
ratio remaining root words : 0.6
remaining root words percentage: 60.0
minimal roots 85 reloaded to establish lower bound ****** ******
time in years at time zero : 1000.
root words at time zero: 200.
root words after time (t2-t0) : 85.
answers: min. remaining root words fm formula :85.0
ratio remaining root words : 0.425
remaining root words percentage: 42.5
language descendent criterion > 46% (0 or 1) : 0. logic variable, 1 for yes, 0 for no
percentge lower to upper bounds :42% to 60%

#### Testcase 5

First estimate, years of separation between F&G table printed in tcl wiki format
quantity value comment, if any
testcase number 5
root words at time zero: 200.
root words after time (t2-t0) : 66.
ratio remaining root words : 0.33
remaining root words percentage: 33.0
language descendent criterion > 46% (0 or 1) : 0. logic variable, 1 for yes, 0 for no
year separation before present BP : 3675 error +- 200 years
year separation BCE :1661.379error +- 200 years
conclusion: possibly Minoan eruption of Thera volcano near 1613-1660 BCE, based on carbon dating example needs check

#### Testcase 6

Second estimate, years of separation between F&Gtable printed in tcl wiki format
quantity value comment, if any
testcase number 6
root words at time zero: 200.
root words after time (t2-t0) : 70.
answers: min. remaining root words fm formula :86.0
ratio remaining root words : 0.35
remaining root words percentage: 35.0
language descendent criterion > 46% (0 or 1) : 0. logic variable, 1 for yes, 0 for no
year separation before present BP : 3480 error +- 200 years
year separation BCE : 1466 error +- 200 years

#### Testcase 7

Separation between latin(200CE) and spanish(2014CE)table printed in tcl wiki format
quantity value comment, if any
testcase number 7
root words at time zero: 200.
root words after time (t2-t0) : 131. from 200*0.655
ratio remaining root words : 0.655
remaining root words percentage: 65.5
language descendent criterion > 46% (0 or 1) : 1. logic variable, 1 for yes, 0 for no
years separation before 2014CE: 1402.705 error +- 200 years
year of separation CE : 611CE error +- 200 years
conclusion: possibly Muslim invasion of 711CE example needs check

#### Testcase 8

Separation between Greek Koine(250BCE) and Cypriot(2014CE) printed in tcl wiki format
quantity value comment, if any
testcase number 8
root words at time zero: 200.
root words after time (t2-t0) : 135.
answers: min. remaining root words fm formula :86.0
ratio remaining root words : 0.675
remaining root words percentage: 67.5
language descendent criterion > 46% (0 or 1) : 1. logic variable, 1 for yes, 0 for no
year separation before present BP : 1303 error +- 200 years
year separation CE : 711 CE error +- 200 years
conclusion: possibly Arab raids on Cyprus, 700-800CE example needs check

#### Testcase 9

Separation between Sung Dynasty Chinese(950CE) and Mandarin(2014)table printed in tcl wiki format
quantity value comment, if any
testcase number 9
root words at time zero: 200.
root words after time (t2-t0) : 159
answers: min. remaining root words fm formula :86.0
ratio remaining root words : 0.795
remaining root words percentage: 79.5
language descendent criterion > 46% (0 or 1) : 1. logic variable, 1 for yes, 0 for no
year separation before present BP : 760 error +- 200 years
year separation CE : 1253 error +- 200 years
conclusion: possibly Genghis Khan invests Peking (1214CE) example needs check

### References:

• Lexicostatistics, google search
• ABC's of Lexicostatistics, Sarah C. Gudschinsa
• Thera [1]
• Classificatiion of the Frisian Dialects, Petra Novotna & Vaclav Balzek
• PLoS One. 2013; 8(8): e71004.
• Environmental Roots of the Late Bronze Age Crisis
• Michael D. Petraglia, Editor, well worth reading
• Toward a phylogenetic chronology of ancient Gaulish, Celtic, and Indo-European
• Peter Forster and Alfred Toth

## Appendix Code edit

### appendix TCL programs and scripts

```        # pretty print from autoindent and ased editor
# human language root words 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 {{} { time years :} }
lappend names {   root words at time zero :}
lappend names { root words remaining after t years: }
lappend names { answers: minimum remaining root words from formula :}
lappend names { ratio remaining roots over roots1(t=0):}
lappend names { remaining roots percentage :}
lappend names { language descendent criterion > 46% (0 or 1) :}
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
grid .frame.label\$i .frame.entry\$i -sticky ew -pady 2 -padx 1 }
proc about {} {
set msg " Human Language Root Words Calculator
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 years \$side1
set roots1 \$side2
set roots2 \$side3
set exponent [/ \$years 1000.]
set kay  [* [/ 86. 200. ] [exp 1. ]]
set calcroots2 [/ [* \$roots1 \$kay ] [exp 1 ] ]
set newratio [/ \$calcroots2 \$roots1 ]
set newratio [/ \$roots2 \$roots1 ]
set percentx [* \$newratio 100. ]
set passfaillogic 0.
if { \$percentx >= 46. } { set passfaillogic 1. }
set calcroots2 [* [int \$calcroots2 ] 1. ]
set side4 \$calcroots2
set side5 \$newratio
set side6 \$percentx
set side7 \$passfaillogic
return \$side7 \$passfaillogic
}
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 "&| time in years at time zero :| \$side1 ||&"
puts "&| root words at time zero:| \$side2 ||& "
puts "&| root words after time (t2-t0) :| \$side3 ||& "
puts "&| answers: min. remaining root words fm formula :|\$side4 ||&"
puts "&| ratio remaining root words :| \$side5 ||& "
puts "&| remaining root words percentage:| \$side6 ||&"
puts "&| language descendent criterion > 46% (0 or 1) :| \$side7 ||&"
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate   }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 1000. 200. 86.  85.0   0.43  43.   0.  }
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 1000. 200. 120.  85.0   0.6  60.   1.  }
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 1000. 200. 160.  85.0   0.8  80.   1.  }
::ttk::button .clearallx -text clear -command {clearx
}
::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 . "Human Language Root Words 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   |&"  ```