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Astronomical Gearing
Al-Biruni's mechanical calendar
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Instruments
carrying gearings of toothed wheels (cogwheels)
were used in the Late Antike or Byzantine period
as calendrical devices, some of them displaying
the phase (age) of the Moon, its position in the
zodiac, and the position of the Sun in the
zodiac.
A calendrical device with eight cogwheels was described by the Persian Muslim scholar and polymath al-Biruni (978-1048).  The rotating Moon disc: 59 days rotation period, corresponding to 29.5 days for the synodic month. The rotation period of the wheel (V, 39 cogs) for the sidereal month:  The rotation period of blue wheel (VIII, 60 cogs):  The rotation period of the red wheel (VI, 49 cogs) for the year: 
|
| Full Moon |
JD | Diff days |
Long. Moon |
Diff Long. | Long. Sun | Diff Long. |
| 1996 Jan 05 UT
22:08 |
2450088.369 | - | 104.80° | - |
284,86° | - |
| 2001 Jan 09 UT 20:24 | 2451919.350 | 1830.98 | 109.65° | 4.85° | 289,65° | 4.80° |
| 2006 Jan 14 UT 09:48 | 2453749.908 | 1830.56 | 114.08° | 4.44° | 294,09° | 4.43° |
| 2011 Jan 19 UT 21:21 | 2455581.390 | 1831.48 | 119.45° | 5.36° | 299,45° | 5.37° |
| 2016 Jan 24 UT 01:46 | 2457411.573 | 1830.18 | 123.49° | 4.04° | 303,49° | 4.04° |
| 2021 Jan 28 UT 19:16 | 2459243.303 | 1831.73 | 129,09° | 5.60° | 309,09° | 5.60° |
Al-Biruni's Moon Box:
from Wiedemann
Counting the cogs of al-Biruni's drawing above:
| wheel |
I |
II |
III |
IV |
V |
VI |
VII |
VIII |
| cogs | 7 | 10 | 20 | 23 | 40 | 48 | 59 | 58 |
In his paper, Wiedemann mentions the drawing to be erroneous:
"Die hier mitgeteilte Figur 2 ist der Leydener Handschrift entnommen;
ihr entspricht diejenige in der Berliner Handschrift,
beide sind nicht ganz korrekt" (p. 8)
My applet is using these numbers of cogs:
| wheel |
I |
II |
III |
IV |
V |
VI |
VII |
VIII |
| cogs | 7 | 10 | 19 | 25 | 39 | 49 | 59 | 60 |
| The applet is also computing the apparent
longitude of the Sun and the Moon, using algorithms of
Jean
Meeus, taking into account the difference ∆T between Dynamical Time and Universal Time from MICA. Comparing the applet's results and JPL ephemerides for the Sun the mean absolute error is only 0.00004° = 0,15'', Moon the mean absolute error is only 0.002° = 0,72''. Selecting a date from the menu the position of the moon is set to the mean position of that date, remaining constant up to the next date:  The
applet's moon is
moving by 13.187°
per day, while the
mean
longitude
increases by
13.176°. The
difference goes up
to 3.8° per year:
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Al
Biruni's Moon Calendar (YouTube) Abū Rayḥān al-Bīrūnī (Wikipedia) Hill, al-Hassan:
Engeneering in Arabic-Islamic Civilization Abu
Arrayhan
Muhammad
ibn
Ahmad
al-Biruni
(MacTutor History of Mathematics archive) The
Science of al-Biruni (A. C.
Sparavigna) Antikythera
Mechanism
Sun
and
Moon
Flash Animation The
Antikythera
Mechanism
I:
Java-animation
of The Sun-Moon Assembly The
Antikythera Mechanism
(age-of-the-sage.org) HORIZONS
Web-Interface (ephemerides for solar-system
bodies, JPL) |
| Books,
Articles |
| J.V. Field, D.R. Hill, M.T. Wright:
Byzantine and Arabic Mathematical Gearing, Science
Museum, London 1985. Jo Marchant: Decoding the Heavens, Solving the Mystery of the World's First Computer; William Heinemann, London 2008. Jo Marchant: Die Entschlüsselung des Himmels, Der erste Computer - ein 2000 Jahre altes Rätsel wird gelöst; Rowohlt 2011. Wiedemann, Eilhard: Ein Instrument, das die Bewegung von Sonne und Mond darstellt; Der Islam; Band 4, Heft 1, 1913; pp 5-13. Hill, D. R.: Al-Biruni's Mechanical Calendar; Annals of Science; Vol 42, Issue 2, 1985; pp 139-163. (Summary) Jean Meeus: Astronomical Algoritms, Willmann-Bell, First English Edition 1991. MICA (Multiyear Interactive Computer Almanac) by U. S. Naval Observatory |
Updated:
2023, Oct 04