Islamic calendar

The Islamic Calendar ( Arabic التقويم الهجري, DMG at-taqwīm al-hiǧrī orالتقويم الإسلامي at-taqwīm al-islāmī ) is a purely lunar calendar . Its calendar years consist of 12 lunar months of 29 or 30 days and are 354 or 355 (on average about 354 ¹ ⁄ ₃ ) days long, i.e. 10 to 12 days shorter than the 365 or 366 (about 365 ¹ ⁄ ₄ ) days long solar years of the Christian calendar. As a result, 33 years of the Islamic calendar correspond to about 32 years of the Christian calendar. During this period, the dates of the Islamic year move once through a solar year. From the perspective of a solar calendar , the fasting month of Ramadan begins10 to 12 days earlier than the previous year.

Today's date according to the
Islamic calendar :

---

8. Muharram 1442
(= August 27, 2020)
[ update ]

The Islamic calendar begins with the year of the emigration ( hijra ) of the prophet Mohammed from Mecca to Medina. According to the Christian calendar , that was in the year 622. According to the Islamic calendar, years are usually given in western languages ​​with the abbreviation AH (= Anno Hegirae ) or in German with d. H. (= [in the year] of the Hejra ) provided.

The Islamic calendar and the Islamic time reckoning are mainly used for religious purposes today. Ramadan and the other twelve lunar months of the calendar begin in many Islamic countries on the day on the evening of which, for the first time after the new moon, a thin crescent moon is observed as a new light event by a religious authority. Due to the dependence on the geographical longitude , the date can be different in different places. There are also calendars (“cyclical” calendars) that are based on projections for the occurrence of the new light - also location-specific.

Since a pure lunar calendar is useless for agriculture, the use of solar calendars has been preserved everywhere in the Islamic world : since 1927, the Gregorian (solar) calendar has been used in everyday and economic life in almost all Islamic countries.

Islamic Calendar
Linden Museum , Stuttgart

history

The old Arabic calendar as the background of the Islamic calendar

The ancient Arabic month names that were adopted into Islam
No.	Surname
1	Muharram
2	Safari
3	Rabīʿ al-auwal
4th	Rabīʿ ath-thānī
5	Jumādā l-ūlā
6th	Jumādā th-thāniya
7th	Radschab
8th	Shaʿbān
9	Ramadan
10	Schauwāl
11	Dhū l-Qaʿda
12	Dhū l-Hijah

The Islamic calendar with its month names goes back to the old Arabic calendar, which was a lunisolar calendar and was widespread in large parts of the Arabian Peninsula . The year started in autumn, probably mid to late September. It consisted of twelve lunar months, which were counted from new moon to new moon as in today's Islamic calendar; every two or three years, however, a thirteenth month was added so that the beginning of the year could be maintained in autumn. This leap month was called nasīʾ , “shift”, because it shifted the first month of the new year. With this system, the old Arabic calendar was similar to the Jewish calendar .

Some of the month names in the ancient Arabic calendar refer to the seasons. For example, the Arabic word Rabīʿ , which occurs in the names of the third and fourth months, means "spring". Al-Biruni , who wrote a work on the different times, explains this with the fact that a lot of rain fell in ancient Arabia and the flowers were in bloom during this time.

The calendar was mainly used to calculate the dates of pilgrimages and markets. Four months of the year were considered sacred: the month of Rajab because of taking place at this time umra , the month of Dhu al-Hijjah and the two months preceding and following due to take place at this time Hajj . During these four months there was a general duty of peace (cf. Sura 9:36 ). The markets of ʿUkāz and Dhū l-Madschāz were also calculated according to this calendar.

According to al-Azraqī , in early times the responsibility for the calculation lay with members of the Kinda tribe because they “usually provided the kings of the Arabs”. Then that privilege passed to the Kināna tribe . In the period immediately before Islam, it belonged to the Banū Fuqaim clan, who belonged to the Kināna. At the end of the Hajj, the man in charge of the intercalation stood on the square in front of the Kaaba and gave a speech in rhyming prose in which he announced whether or not a leap month would be inserted before the start of the new year.

The Islamic Calendar Reform

After the Muslim conquest of Mecca in 630, the old Arabic calendar was reformed: the leap month nasīʾ was abolished and a purely lunar calendar was created in this way. The exact timing of the abolition is not clear. Ibn Ishāq mentions them both for the pilgrimage led by Abū Bakr in 631 and for Muhammad's farewell pilgrimage in 632.

As can be seen from the Qur'anic verse ( Sura 9:37 ), which refers to this event, the interposition of the Nasīʾ was judged as pagan behavior and human intervention in the divine world order. One of the results of the calendar reform was that the Kināna , who previously had the right to determine the leap months and thus also determine the movement of goods and military activities in the Arabian Peninsula, lost much of their power, as did the traditional cycle of markets in the Arabian Peninsula Peninsula disappeared. But it was even more important that in this way the pure lunar year with its 354 days became the basis for the Islamic calendar and henceforth served to determine the religious festivals.

The Islamic calendar has adopted the old Arabic month names unchanged. However, the names lost their original seasonal meaning because they wandered through the year from now on.

The continuation of solar calendars in Islamic countries

Advantages of the Islamic lunar calendar were the simple determination of the beginning of the month and the uniform duration of months and years. The lunar year, however, is not practicable for an economic system based on agriculture that is dependent on the seasons. In particular, it is difficult to set dates on which the annual taxes were to be paid. Therefore, in almost all countries where Islam has taken root, a solar calendar was continued:

• in the Maghreb the Julian calendar , later the Gregorian , with Roman month names (yanayir, fibrayir, maris, abril etc.),
• also in Maschrek , but with ancient oriental month names (kanun ath-thani, schubat, adhar, nisan , ayar, haziran, tammuz , ab, aylul, tischrin al-awwal, tischrin ath-thani, kanun al-awwal),
• in Egypt the Coptic calendar ,
• in the Iranian area the old Iranian solar calendar , which however has taken the year of the hijra as its starting point,
• in the Ottoman Empire at times the Rumi calendar , a variant of the Julian calendar with Hijra year counting and the beginning of the year first in September, later in March.

However, this resulted in synchronization problems. In the Ottoman Empire, for example, income, mostly from agriculture and often in kind, was collected on dates according to the sun-based Rumi calendar. The expenses, such as the salaries of officials, soldiers, etc., however, were due according to the Islamic calendar. To synchronize the income and expenditure year, the Hijra year count has also been extended to the years of the Rumi calendar. But because the solar years were longer than the lunar years, there were discontinuities in the year counting of the Rumi calendar, ie there were expenditure years to which no income year could be assigned. The endeavor of the Ottoman administration to avoid the invoicing inconveniences associated with such discontinuities resulted in the aforementioned bringing forward of the beginning of the year of the Rumi calendar from September to March.

In everyday and economic life, however, the Gregorian calendar has been used in almost all Islamic countries since the early 20th century . In the area of ​​the Ottoman Empire , the lunar calendar was officially replaced by the Gregorian calendar on March 1, 1917.

For the months of the Gregorian or Julian calendar, the well-known European month names are used in some parts of the Arab world, while the rest of the time use oriental month names. The French names are in use in Iran. In addition, the Iranian solar calendar officially applies .

The lunar calendar

The Islamic calendar is a lunar calendar . The months are lunar months with a duration of 30 or 29 days. The calendar is strictly based on astronomical observation: A new month begins with new light , that is the moment after the new moon , when the waxing crescent moon can be observed again for the first time in the evening after sunset.

Twelve months make up a lunar year. At 354 or 355 days, this is 10 to 12 days shorter than a solar year. In contrast to lunisolar calendars such as the Jewish calendar or the Chinese calendar, there is no compensation in the Islamic calendar through leap months. Therefore, the beginning of the year is shifted backwards by 10 to 12 days compared to the Gregorian calendar. 33 lunar years correspond to about 32 solar years.

How the calendar works

Since the onset of the new light is unstable and difficult to predict, a calendar with alternating months of 30 and 29 days and the resulting 354 day common year are used. This includes an occasional leap day to adjust the calendar lunar year to the astronomical period of 12 synodic months of about 29.53059 days each.

In a 30-year calendar cycle, a day is added eleven times to the last calendar month. If the year divided by 30 has a remainder of 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, or 29, it is a leap year . The last cycle began in 1411 AH, the next one begins in 1441 AH.

The calendar cycle length is 30 × 354 + 11 = 10631 days . The astronomical cycle length is 360 × 29.53059 = 10631.0124 days. The deviation of 0.0124 days in 30 years does not add up to one day until around 2500 years.

Calculation of the beginning of the month

The following table shows the lengths of the individual months and the beginning of the month calculated according to this cyclical calendar for the Islamic years 1436 to 1438. The date is highlighted in bold if the month Dhū l-Hiddscha has 30 days as a leap month.

month	Number of days	Gregorian date of the beginning of the month
		1436		1437		1438		1439
Muharram	30th	October 25, 2014		15th October 2015		3rd October 2016		22nd September 2017
Safari	29	November 24, 2014		November 14, 2015		November 2, 2016		October 22, 2017
Rabīʿ al-awwal	30th	23rd December 2014		December 13, 2015		December 1, 2016		20th November 2017
Rabīʿ ath-thānī	29	January 22, 2015		January 12, 2016		December 31, 2016		20th December 2017
Jumada l-ula	30th	February 20, 2015		February 10, 2016		29th January 2017		January 18, 2018
Jumādā th-thāniya	29	March 22, 2015		March 11, 2016		February 28, 2017		17th February 2018
Radschab	30th	April 20, 2015		April 9, 2016		29th March 2017		18th March 2018
Shaʿbān	29	May 20, 2015		May 9, 2016		April 28, 2017		17th April 2018
Ramadan	30th	June 18, 2015		June 7, 2016		May 27, 2017		May 16, 2018
Shawwal	29	July 18, 2015		July 7, 2016		June 26, 2017		June 15, 2018
Dhu l-qaʿda	30th	August 16, 2015		5th August 2016		July 25, 2017		July 14, 2018
Dhū l-Hijah	29 (30)	15th September 2015		4th September 2016		August 24, 2017		13th August 2018

Islamic holidays

Of the Islamic feast days , the month of fasting Ramadān and the month of pilgrimage Dhū l-iddscha have particularly great religious significance. The festival of sacrifice ( ʿĪd al-aḍḥā ) on the 10th of Dhū l-hiddscha is the highest festival for Muslims, the second highest is the festival of breaking the fast ( ʿĪd al-fitr ) at the end of Ramadān. In the first ten days of the month of Muḥarram , the Shiites celebrate their highest funeral festival , the Ashura rites, in which they commemorate the death of Imam Husain ibn Ali at the battle of Karbala with processions, passion plays and flagellation .

For religious purposes in most Islamic countries it is not astronomical calculation that is decisive, but observation of the moon. The new month begins when the crescent moon ( hilāl ) is visible again ( new light ) after a new moon . If a sighting is not possible due to the weather, it ends after the 30th day. This form of observation is particularly important for determining the beginning and end of the Ramadan month.

However, this means that the calculated start of the month can differ from the actual start of the month. Therefore, in printed calendars, dates on Islamic festivals are often noted: "The actual date may differ by 1 to 2 days". Corresponding postponements of celebrations can have unpleasant consequences, as the following newspaper note shows:

"RIAD, January 16, 2005 (dpa). One of the highlights of the Islamic pilgrimage, the festival of sacrifice, has surprisingly been brought forward by a day by religious scholars. ... The decision came as a surprise to the approximately two million pilgrims who are preparing for the climax of the Hajj. Thus the pilgrimage ritual on Mount Arafat is celebrated on Wednesday and the festival of sacrifice on Thursday. This decision led to chaos in Egypt, as many had already bought train tickets to be with their relatives for the festival. "

The Hijri Year Count

The starting point for the Islamic calendar is the 1st Muharram of the year in which Mohammed emigrated from Mecca to Medina. This is July 16 622. According to the Arabic word for emigration, hijra , is this era as Hijri referred -Zeitrechnung. Years after this calendar are marked with AH for "anno hegirae" (year of the hijra).

According to the astronomer al-Biruni (973-1048), who lived 400 years after the reported event, the Islamic Hidschrī calendar was introduced by the second caliph ʿUmar ibn al-Chattāb in 638 of the Christian era . The earliest evidence of the census, which begins in 622, is an inscription by Muawiyas I in Greek, which speaks of the "era of the Arabs". This has led to the assumption that it was only later Islamic tradition that constructed the reference point “emigration from Mecca”. Most of the old evidence leaves open whether the census refers to the "era of the Arabs" or the hijra.

Beginning of the year

The following table contains the year beginnings of the Islamic years 1436 to 1471 calculated according to the cyclical calendar. The leap years are highlighted:

islamic year	greg. date		islamic year	greg. date		islamic year	greg. date
1436	October 25, 2014		1448	June 17, 2026		1460	February 6, 2038
1437	15th October 2015		1449	June 6, 2027		1461	January 26, 2039
1438	3rd October 2016		1450	May 25, 2028		1462	January 16, 2040
1439	22nd September 2017		1451	May 15, 2029		1463	January 4, 2041
1440	September 12, 2018		1452	May 4, 2030		1464	December 24, 2041
1441	1st September 2019		1453	April 23, 2031		1465	December 14, 2042
1442	20th August 2020		1454	April 12, 2032		1466	December 3, 2043
1443	August 10, 2021		1455	April 1, 2033		1467	November 22, 2044
1444	July 30, 2022		1456	March 21, 2034		1468	November 11, 2045
1445	July 19, 2023		1457	March 11, 2035		1469	October 31, 2046
1446	July 8, 2024		1458	February 28, 2036		1470	October 21, 2047
1447	June 27, 2025		1459	February 17, 2037		1471	October 9, 2048

conversion

With the following approximation formulas, Islamic years can be converted into Gregorian and vice versa:

${\displaystyle C\approx H\cdot {\frac {32}{33}}+622}$

${\displaystyle H\approx {\frac {33}{32}}\cdot (C-622)}$

Here C denotes the year according to the Christian calendar, H the year according to the Hijra calendar.

Conversion of a Gregorian date into the Islamic calendar

A conversion method was described by Christian Ludwig Ideler and will be presented using a numerical example.

The Gregorian date May 27, 2017 is to be transformed into the Islamic calendar. First, the Gregorian date is converted into the corresponding date in the Julian calendar ( conversion between the Julian and Gregorian calendar ). In 2017, 13 days must be subtracted from the Gregorian date, so that May 14, 2017 in the Julian calendar results. So 2016 was a full Julian year. 2016 years correspond to 504 complete Julian switching cycles (2016: 4 = 504, without remainder) without remaining years.

504 Julian leap periods correspond to 504 × 1461 days = 736 344 days.

In the following switching cycle, no further years have completely expired, so 0 × 365 days = 0 days.

In 2017, May 14th is the 134th day, so there are still 134 days to consider.

In total:

Switching cycles	504 × 1461 days	736 344 days
complete years of new switching cycle	0 × 365 days	0 days
May 14th	134 days	134 days
		736 478 days

From January 1, 1 AD in the Julian calendar to May 14, 2017 (July) 736,478 days have passed.

The number 227 016 (difference in days between the epoch of the Hijra (July 16, 622 AD (July)) and January 1, 1 AD (July)) is now subtracted from this number. It surrender

509 462 days.

From 509 462 days, the number of completely completed Islamic switching cycles of 30 Islamic years each (10 631 days per Islamic switching cycle) and the excess days are determined:

509 462: 10 631 = 47 remainder 9805. Thus 47 complete Islamic switching cycles have expired and 9805 days are redundant. These 9805 days correspond to 27 Islamic years of 9568 days and 237 days are left in the 28th year of the Islamic switching cycle (see Table 2). These 237 days correspond to the 1st Ramadan (see Table 1).

Thus, May 27, 2017 (greg.) Corresponds to the 1st Ramadan 1438 AH (47 × 30 + 28 = 1438).

The following two auxiliary tables, as they are given in a similar way at Ideler, prove to be helpful:

Table 1:

month	Length in Days	summed up Days
Muharram	30th	30th
Safari	29	59
Rabīʿ al-awwal	30th	89
Rabīʿ ath-thānī	29	118
Jumada l-ula	30th	148
Jumādā th-thāniya	29	177
Radschab	30th	207
Shaʿbān	29	236
Ramadan	30th	266
Shawwal	29	295
Dhu l-qaʿda	30th	325
Dhū l-Hijah	29 (30)	354 (355)

Table 2:

year (Leap year *)	summed up Days		year	summed up Days
1	354		16 *	5670
2 *	709		17th	6024
3	1063		18 *	6379
4th	1417		19th	6733
5 *	1772		20th	7087
6th	2126		21 *	7442
7 *	2481		22nd	7796
8th	2835		23	8150
9	3189		24 *	8505
10 *	3544		25th	8859
11	3898		26 *	9214
12	4252		27	9568
13 *	4607		28	9922
14th	4961		29 *	10 277
15th	5315		30th	10 631

Conversion of an Islamic date into the Julian or Gregorian calendar

The reverse of the Ideler method described above can be used for this conversion. The procedure will again be explained using an example. Ibn Yunus observed a solar eclipse in Kahira (Cairo) on 29 Shawwal 367 AH, based on the epoch of the Hijra on July 15, 622 AD (July). If one chooses July 16, 622 AD (July) for the epoch of the Hijra, the observation took place on the 28th Shawwal 367 AH. On what date in the Julian calendar did this eclipse take place?

First, the completely past Islamic years are divided by 30. This results in: 366: 30 = 12, remainder 6. Thus, 12 complete switching cycles of 30 years (10 631 days) and then an additional 6 individual years have passed since the Hijra era.

The following number of days has passed since the hijra:

Switching cycles	12 × 10 631	127 572
complete years in the new Switching cycle	6 years	2126 (Tab. 2)
completely expired 9 months	9 months	266 (Tab. 1)
Days in the Shawwal	28	28
total		129 992

In addition to these 129 992 days, the number of 227 016 days (difference in days between the epoch of the Hijra (July 16, 622 AD (July)) and January 1, AD 1 (Jul. )) can be added. This gives 357 008 days since January 1, 1 AD (July).

From these 357 008 days, division by 1461 days (Julian switching cycle of 1 × 366 days and 3 × 365 days = 1461 days) results in 244 completely expired Julian switching cycles of 4 years each and the remainder 524 days. These 244 switching cycles correspond to 976 Julian years (244 × 4). 524 days correspond to a common Julian year and 159 days. In total, there are 977 expired Julian years and 159 additional days. The searched date is the 159th day in the 978th ​​Julian year (common year), i.e. June 8th 978 AD (July). On that day, an annular solar eclipse took place in northern Africa, which was partially visible in Kahira.

The weekdays

The week begins - as is common in Judaism and in some western countries - on Sunday and ends on Saturday . So Friday is the weekly public holiday, but not the last day of the week. The change from one day to the next always takes place at sunset . With the exception of Friday, the days of the week in Arabic do not have names, but are counted starting with Sunday:

German	Arabic	Transcription	translation
Sunday	يوم الأحد	Yaum al-aḥad	"First day"
Monday	يوم الإثنين	Yaum al-iṯnain	"Second day"
Tuesday	يوم الثلاثاء	Yaum aṯ-ṯulāṯāʾ	"Third day"
Wednesday	يوم الأَرْبعاء	Yaum al-arbiʿāʾ	"Fourth day"
Thursday	يوم الخَمِيس	Yaum al-ḫamīs	"Fifth day"
Friday	يوم الجُمْعَة	Yaum al-ǧumʿa	"Day of the meeting (for Friday prayer )"
Saturday	يوم السَّبْت	Yaum as-sabt	"Seventh day", " Sabbath "

See also

• List of calendar systems

The Iranian Hijri-Shamsi calendar

In addition to the time calculation according to the Islamic lunar calendar, a time calculation according to solar years is also common in Iran and Afghanistan , which also counts the years since the year of the hijra ( Iranian calendar ). The Arabic name for this era is hijri shamsi ("sun hijra "). To differentiate, the counting of the Islamic lunar years is also called hijri qamari (" lunar hijra "). Since there are around 33 lunar years for every 32 solar years, the Iranian solar calendar is currently (2012) about 42 years behind the Islamic lunar calendar.

The hijri-shamsi year 1395 began on March 20, 2016 ; it ended on March 20, 2017. The hijri-shamsi year 1396 began on March 21, 2017.

literature

• Michael Bonner: "Time Has Come Full Circle": Markets, Fairs and the Calendar in Arabia before Islam , in: Asad Q. Ahmed, Behnam Sadeghi and Michael Bonner (Eds.): The Islamic scholarly tradition: studies in history, law, and thought in honor of Professor Michael Allan Cook . Suffering a.]: Brill 2011. pp. 15-47.
• GSP Freeman-Grenville: The Islamic and Christian Calendars, AD 622-22222 (AH 1-1650). Garnet, Reading, 1995.

Web links

Commons : Islamic Calendar  - collection of pictures, videos and audio files

• Online converter from the Asia-Orient Institute at the University of Zurich
• Islamic-Western Calendar Converter (Based on the Arithmetical or Tabular Calendar) on the website of the University of Utrecht
• The Umm al-Qura Calendar of Saudi Arabia on the website of the University of Utrecht
• The Islamic Calendar of Turkey on the Utrecht University website
• Nikolaus A. Bär: The Islamic calendar , problems of Islamic dating , the beginning of the month in today's Islamic calendar and the times of prayer in Islam . Article and converter at nabkal.de
• German Islamic Scientific Committee on New Moons (DIWAN): Islamic basis and arithmetic rules for determining the lunar months at islam.de

Individual evidence

1. ↑ The prophet Mohamet had forbidden the switching on of months to adapt to the solar year as "desecration of what God has sanctified" . Cf. Nikolaus A. Bär: Prehistory and origin of the Islamic calendar: The ban on switching. nabkal.de
2. ↑ Nikolaus A. Bär: The Islamic Calendar: The year in the Islamic calendar. nabkal.de
3. ↑ Nikolaus A. Bär: Prehistory and origin of the Islamic calendar: The final design of the Islamic calendar. nabkal.de
4. ↑ Meyer's Large Conversation Lexicon. A reference book of general knowledge. Sixth, completely revised and enlarged edition. Leipzig / Vienna, 1905–1909, keyword D ( at woerterbuchnetz.de )
5. ↑ The Koran does not contain how to proceed. According to Islamic law, forecasting is actually forbidden, the traditional observation method has been declared legally binding. Cf. Nikolaus A. Bär: The Islamic calendar: the year in the Islamic calendar, the month. nabkal.de
6. ↑ Bonner: "'Time Has Come Full Circle.'" 2011, pp. 15, 28.
7. ↑ See A. Moberg: nasīʾ. In: The Encyclopaedia of Islam. New Edition . Volume VII, p. 977.
8. ↑ Bonner: "'Time Has Come Full Circle.'" 2011, p. 44.
9. ↑ Bonner: "'Time Has Come Full Circle'". 2011, p. 22.
10. ↑ Bonner: "'Time Has Come Full Circle'". 2011, pp. 23-28.
11. ↑ Bonner: "'Time Has Come Full Circle'". 2011, p. 37f.
12. ↑ Cf. Ibn Hischām: Kitāb Sīrat Rasūl Allāh. From d. Hs. On Berlin, Leipzig, Gotha a. Leyden ed. by Ferdinand Wüstenfeld. 2 volumes Göttingen 1858-59, p. 923 fu 968. Available online at http://archive.org/stream/p2daslebenmuhamm01ibnhuoft#page/104/mode/2up
13. ↑ Bonner: "Time Has Come Full Circle" . 2011, p. 31.
14. ^ Cf. Feroz Ahmad: The Making of Modern Turkey . London / New York 1993, p. 80.
15. ↑ Norbert Lüdtke: Outdoor: Tips, tricks & tricks for adventurers on a world tour Peter Meyer Verlag 2010, p. 165
16. ↑ The corresponding calculation rules developed in the 9th century AD are handed down in a text by the Arab astronomer and mathematician Maslama al-Majriti (10th / 11th centuries AD). Cf. Nikolaus A. Bär: The cyclical Islamic calendar. nabkal.de
17. ^ Friedrich Karl Ginzel: Handbook of the mathematical and technical chronology. Volume 1: Calendar of the Babylonians, Egyptians, Mohammedans, Persians, Indians, Southeast Asians, Chinese, Japanese and Central Americans. Leipzig 1906, p. 255.
18. ↑ FAZ January 17, 2005.
19. ^ Ludwig Ideler: Handbook of mathematical and technical chronology . Volume 2, Berlin 1826, p. 493.
20. ^ Ludwig Ideler: Handbook of mathematical and technical chronology . Volume 2, Berlin 1826, pp. 479 and 481.
21. ^ ^{A b} Ludwig Ideler: Handbook of mathematical and technical chronology . Volume 2, Berlin 1826, p. 489.
22. ^ Theodor von Oppolzer: Canon of the darkness . (= LII. Volume of the memoranda of the mathematical and natural science class of the Imperial Academy of Sciences ). Vienna 1887, p. 208 and sheet no. 104.