Biruni Estimate of Earth Diameter Slot Calculator Example
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[gold] Here is an eTCL script on
to estimate the radius or diameter of the earth
using Al Biruni figures and method
from the text Masudi Canon.
An equation for the dip angle and height of
mountain was used in computing the earth's diameter.
The equation is striking in that it involved a
single location versus two locations.
Spelled Al Bīrūnī , Al Beruni, Al Bayrooni (born 5 September 973CE)
In planning any software,
height waS 652+1/20 cubits
radius was 12851359cubits
circumference was 80780039 cubits
arabic cubit ~ 4,933 mm
circumference ~ 25000+2/7 english miles
dip reportly 34 minutes
testcase was height 479 meters,
479 meters is modern survey of Nandian Mountian.
Persian cubit of about 500.1 mm,
dip angle was 30 or 34 minutes,
30/60 degree or 34/60 degrees.
(652.05 cubits * cos (.56666) / (1 - cos(.56666) = ?
Very sensitive equation, cosine would
have to be known to 5th position.
reverse on Biruni's angle constant
652.05/(1-x)=r
652.05*x=r(1-x)
x= r(652.05+r)= 0.99949265 radians
or 0.577156 degrees.
Loading 0.577156 degrees and 652.05 cubits,
calculator is getting 12851427 cubits,
close to Biruni's answer.
For meters, the Nandian survey height
as 479 meters? ... Anyway.
Loading 652.05 cubits *.5 or 326.025 meters and
0.577156 degrees is returning 6425.714 meters
Modern estimate of earth radius at
equator is 6356.75 km.
Biruni overage is about 1.08 percent.
Trying to find some earlier estimates of
earth's circumference from other cultures.
Most of the early constants
were given underived without sources.
Scanty sources and circular reasoning
, no good text reference,
speculates that the Sumerians might
have used 3600 beru as circumference,
with the earth circumference as 3600*10.8 km.
Given ~, the earth radius would be 573 beru
9_33,base 60)
and diameter would be 1146 beru (19_6,base 60).
Still trying to find out if "miles"
on clay tablet BM 92687 are beru or else.
Chinese Han text.
# On sumerian clay tablet, there are 655200
"beru ina sama" summed up, akkadian for "beru in sky."
This factors to 364*60*30.
There are 86400 seconds in a day.
There are 655200 beru in sky over 360 conventional
degrees or 1820 per degree. The apparent
disk of sun or moon was supposed to
be 1000 beru (~to 0.5 degree).
In conventional astronomy, a degree contains
60*60 or 3600 arcseconds. So a beru in sky
may something like a double second (meaning double
arcsecond). In modern constants, a double arcsecond
would be equivalent to 1800/111.3km or 16.17 double
arcseconds per km ( from one latitude degree
equals 111.3 km). A sky triangle of one degree
and an opposite side of 1800 would have a side of
1800/tan(1) or 103137.638. A second and
almost equal side would
be 1800/sin(1) or 103121.93.
considerations for circle, 360*60*60=1296000
364*60*60=1310400 , say 1310400/2=655200.
Arithmetic Classic of the Gnomon and the Circular Paths of
Heaven reported triplet as follows.
60000 li, 24948 km, 15502.2miles
80000 li, 33264 km , reputed altitude of sun,
100000 li ,41580km, reputed distance from measuring stick to sun
Chinese li was 0.4158 km (0.25837 miles).
2* babylonian 12709 =25418
# Chen Zi of first century reported indirect numbers
leading to earth diameter of 911489 li or 378997km.
Chen Zi placed tropic of cancer +23.5 latitude
at 16000 li south
of his location, using modern terms.
Chen Zi placed tropic of capricorn -23.5 latitude
at 135000 li south
of his location, using modern terms.
difference of tropic latitude is
+23.5-(-23.5) or 47 degrees.
Ratio of 47 degree tropic difference to 360
circumference is 47/360.
47/360 =(135000-16000)/x
x =(135000-16000)/(47/360)
x= 119000/(47/360)
x = 911490 li or 911490 li*.415 km/li or 378268 km.
Still trying for original text, but
it seems his li in the text
has a factor of ten involved.
The "li" in the text is about
0.043958 km or 43.958 meters.
The number 119000 is mentioned several times
in the text, and appears to apply to
the distance between the tropics, although
subject to interpretation.
Can only suggest Chen Zi's number for the
earth circumference was 37900. km
and implied constant was 105.27 km per degree,
in modern terms.
# more.
Eratosthenes third century BCE 46250 km
Ptolemy c140CE 28800 km
The giant of Chinese legend Pan Gu (or panku, ancient basin)
held the earth and sky apart at 90000 li (45000 km).
The Egyptians held that the N/S length of
Egypt was 1/50 of the globe circumference.
The Romans use 75 roman miles as eq to
one degree of latitude.
The Mogul era "Ain" states the earth’s circumference was
8000 farsakh(1600CE).
In some texts of Aryabhatiya,
the diameter of the earth is mentioned as 3300 yojanams.
Brahmagupta ( 7th c) wrote that the
circumference of the earth was 36,000 km, in modern units.
Sumeria 38880 km? 3600 beru?
Eratosthenes 46250 km 15.6 percent high
ptolemy 28800km 15.6 precent low
aristotle 48000 km 19.9 percent high
chen zi 37826.8 km 5.7 percent short
Biruni 39834.7 km 0.4 percent short
Brahmagupta 36,000 km 11.1 per cent short
Roman (2nd c) 39933 km 0.187 percent short*
Arabia/India 35840 11 percent short **
* meaning 75 roman miles/deg * 360. deg. *1.479 scale km/r. mile
** sailor's marks , 224 isba * 8 zams per isba * 20 km per zams
arabic zam = sanscrit zama = Muldive Dhevehi dama ref. sumerian dana???
lateen day's sail will raise pole star one isba.
***Screenshots Section***
'''Comments Section'''
Please place any comments here, Thanks.
References:
[http://www.highbeam.com/doc/1G2-2830900460.html]
[http://en.wikipedia.org/wiki/Biruni]
The history of cartography, Volume 2
By John Brian Harley, David Woodward
****Appendix Code****
appendix TCL programs and scripts
*************
*FIRST VERSION
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# Biruni Estimate of Earth Diameter
# written on Windowws XP on eTCL
# working under TCL version 8.5.6 and eTCL 1.0.1
# gold on TCL WIKI , 8aug2010
frame .frame -relief flat -bg aquamarine4
pack .frame -side top -fill y -anchor center
set names {{} {not_used:} {not_used:} {not_used:} {dip angle:} {height:} {answer km:} { km/1000:}}
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 "Calculator for Biruni Earth Diameter.
from TCL WIKI,
written on eTCL "
tk_messageBox -title "About" -message $msg
}
proc pi {} {expr acos(-1)}
proc interlinear { xx1 yy1 xx3 yy3 xx2 } {
return [expr { ((($xx2-$xx1)*($yy3-$yy1))/($xx3-$xx1))+ $yy1 } ] ;}
proc pol { xx1 yy1 xx3 yy3 xx2 } {
return [expr { ((($xx2-$xx1)*($yy3-$yy1))/($xx3-$xx1))+ $yy1 } ] ;}
proc errorx {aa bb} {expr { $aa > $bb ? (($aa*1.)/$bb -1.)*100. : (($bb*1.)/$aa -1.)*100.}}
proc strat5 { xx1 } {
global side1 side2 side3
global side4 side5 side6 side7
set ss side1
set tt $side2
set xxx [ expr { $side4*[pi]/180. } ]
# set side4 [ expr { $side5*cos($side4)/(1.-cos($side4))} ]
# set side5 [ expr { $side5*cos($side4)/(1.-cos($side4))} ]
set side6 [ expr { 1.*$side5*cos($xxx)/(1.-1.*cos($xxx))} ]
set side7 [ expr { 1.*$side5*cos($xxx)/(1.-1.*cos($xxx))} ]
set side7 [ expr { (1.*$side7)*.001 } ]
return $side7
}
proc calculate { } {
global colorwarning
global colorback
global answer2 answer3
global side1 side2 side3 side4 side5 side6 side7
set answer2 5
set answer2 [ strat5 $side1 ]
set side7 $answer2
}
proc fillup {aa bb cc dd ee ff gg} {
.frame.entry1 insert 0 "$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 {} {
console show;
puts "
The interpolation function takes
two know points on a line and
solves for an intermediate point.
The points are xx1,yy1 xx2,yy2 and xx3,?yy3?
The input order of the five items
is xx1 yy1 xx2 yy2 xx3
and solving for ?yy3?.
The interpolation function loaded as
proc pol. User should be able to write
pol 50. 1000. 200. 1200. 150.
and save answer (1133.3) on console."
}
frame .buttons -bg aquamarine4
::ttk::button .calculator -text "Solve" -command { calculate }
::ttk::button .test2 -text "Testcase1" -command {clearx;fillup 10. 1. 1. .577156 652.05 12.85E6 12.85E4}
::ttk::button .test3 -text "Testcase2" -command {clearx;fillup 100. 1. 1. .577156 479 9.44E5 9.44E3}
::ttk::button .test4 -text "Testcase3" -command {clearx;fillup 1000. 1. 2. .56666 652.05 13.33E6 13.33E4}
::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
bind . <Motion> {wm title . "Biruni Earth Diameter"}
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