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1.Origin of Islamic Calendar - Part 1
2. Related Verses of Holy Quran and Hadiths - Appendixi
3. Facts about the moon - Appendix 
4 Difference of Opnions in Interpreting Hadiths - Part 2
5. Fuqaha Positions on Regional and Global Sighting 
‘Ikhtilaf vs. Ittihad al-Matali

Verses of Holy Quran and Hadiths Relating to the Crescent sighting for the purposes of calendaring.

Holy Quran regards moon and sun merely as sign to determine dates and does not require us to see  the moon to begin and end fasting. But the Hadiths, which indicates doing so show a method to those people who were not conversant with modern Calendars and watches and their basic principle is to follow a definite time schedule. Just as we use watches for prayer and iftar time instead of the methods described in Hadith we can use Islamic Hijri calendar, as is the practice in Egypt and many other Islamic countries and Boharas in Pakistan and India. Al Azhar University approved that practice.

The word "moon" appear 86 time(s) in 68 hadith(s) in Bukhari translation.
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The word "moon" appear 30 time(s) in 28 verse(s) in Quran in Yusuf Ali translation.
 

Verses of Holy Quran From translation by YUSUFALI. 

002.185 Ramadhan is the (month) in which was sent down the Qur'an, as a guide to mankind, also clear (Signs) for guidance and judgment (Between right and wrong). So every one of you who is present (at his home) during that month should spend it in fasting, but if any one is ill, or on a journey, the prescribed period (Should be made up) by days later. Allah intends every facility for you; He does not want to put to difficulties. (He wants you) to complete the prescribed period, and to glorify Him in that He has guided you; and perchance ye shall be grateful.

002.189 They ask thee concerning the New Moons. Say: They are but signs to mark fixed periods of time in (the affairs of) men, and for Pilgrimage. It is no virtue if ye enter your houses from the back: It is virtue if ye fear Allah. Enter houses
through the proper doors: And fear Allah: That ye may prosper. 

006.096. He it is that cleaveth the day-break (from the dark): He makes the night for rest and tranquility, and the sun and moon for the reckoning (of time): Such is the judgment and ordering of (Him), the Exalted in Power, the Omniscient.

009.036  The number of months in the sight of Allah is twelve (in a year)- so ordained by Him the day He created the heavens and the earth; of them four are sacred: that is the straight usage. So wrong not yourselves therein, and fight the Pagans all together as they fight you all together. But know that Allah is with those who restrain themselves. 

009.037 Verily the transposing (of a prohibited month) is an addition to Unbelief: the Unbelievers are led to wrong thereby: for they make it lawful one year, and forbidden another year, in order to adjust the number of months forbidden by Allah and make such forbidden ones lawful. The evil of their course seems pleasing to them. But Allah guideth not those who reject Faith. 

010.005 It is He Who made the sun to be a shining glory and the moon to be a light (of beauty), and measured out stages for her; that ye might know the number of years and the count (of time). Nowise did Allah create this but in truth and righteousness. (Thus) doth He explain His Signs in detail, for those who understand. 

14.33: and maketh the sun and moon, constant in their course, to be of service unto you, and hath made of service unto you the night and the day. 

036.037  And a Sign for them is the Night: We withdraw therefrom the Day, and behold they are plunged in darkness; 

036.038 And the sun runs his course for a period determined for him: that is the decree of (Him), the Exalted in Might, the All-Knowing. 

036.039 And the Moon,- We have measured for her mansions (to traverse) till she returns like the old (and withered) lower part of a date-stalk. 

036.040  It is not permitted to the Sun to catch up the Moon, nor can the Night outstrip the Day: Each (just) swims along in (its own) orbit (according to Law). 

054.001 The Hour (of Judgment) is nigh, and the moon is cleft asunder. 

055.005  The sun and the moon follow courses (exactly) computed; 
 
 

Hadiths from Sahih Bukhari & Muwatta Imam Malik

Volume 3, Book 31, Number 131: Narrated Abdullah bin Umar: 
Allah's Apostle said, "The month (can be) 29 nights (i.e. days), and do not fast till you see the moon, and if the sky is overcast, then complete Sha'ban as thirty days." 

Volume 3, Book 31, Number 132: Narrated Ibn 'Umar: 
The Prophet said, "The month is like this and this," (at the same time he showed the fingers of both his hands thrice) and left out one thumb on the third time. 

Volume 3, Book 31, Number 133: Narrated Abu Huraira: 
The Prophet or Abu-l-Qasim said, "Start fasting on seeing the crescent (of Ramadan), and give up fasting on seeing the crescent (of Shawwal), and if the sky is overcast (and you cannot see it), complete thirty days of Sha'ban." 

Volume 3, Book 31, Number 134: Narrated Um Salama: 
The Prophet vowed to keep aloof from his wives for a period of one month, and after the completion of 29 days he went either in the morning or in the afternoon to his wives. Someone said to him "You vowed that you would not go to your wives for one month." He replied, "The month is of 29 days." 

Volume 3, Book 31, Number 135: Narrated Anas: 
Allah's Apostle vowed to keep aloof from his wives for one month, and he had dislocation of his leg. So, he stayed in a Mashruba for 29 nights and then came down. Some people said, "O Allah's Apostle! You vowed to stay aloof for one month," He replied, "The month is of 29 days." 

Volume 3, Book 31, Number 136: Narrated Abu Bakra: 
The Prophet said, "The two months of 'Id i.e. Ramadan and Dhul-Hijja, do not decrease (in superiority)." 

Volume 3, Book 31, Number 137: Narrated Ibn 'Umar: 
The Prophet said, "We are an illiterate nation; we neither write, nor know accounts. The month is like this and this, i.e. sometimes of 29 days and sometimes of thirty days." 

Volume 3, Book 31, Number 138: Narrated Abu Huraira: 
The Prophet said, "None of you should fast a day or two before the month of Ramadan unless he has the habit of   fasting (Nawafil) (and if his fasting coincides with that day) then he can fast that day." 

Book 18, Number 18.1.2: 
Yahya related to me from Malik from Abdullah ibn Dinar from Abdullah ibn Umar that the Messenger of Allah, may Allah bless him and grant him peace, said, "A month has twenty-nine days in it. Do not start the fast or break it until you see the new moon. If the new moon is obscured from you, then work out (when it should be)." 

Book 18, Number 18.1.3: 
Yahya related to me from Malik from Thawr ibn Zayd ad-Dili from Abdullah ibn Abbas that the Messenger of Allah, may Allah bless him and grant him peace, once mentioned Ramadan and said, "Do not start the fast or break it until you see the new moon. If the new moon is obscured from you, then complete a full thirty days." 

Book 18, Number 18.1.4: 
Yahya related to me from Malik that he had heard that once in the time of Uthman ibn Affan the new moon had been seen in the afternoon and Uthman did not break his fast until evening had come and the sun had set. Yahya said that he had heard Malik say that some one who sees the new moon of Ramadan when he is on his
own should start the fast and not break it if he knows that that day is part of Ramadan. He added, "Some one who sees the new moon of Shawwal when he is on his own does not break the fast, because people suspect the reliability of someone among them who breaks the fast. Such people should say, when they sight the new moon,  'We have seen the new moon.' Whoever sees the new moon of Shawwal during the day should not break his fast but should continue fasting for the rest of that day. This is because it is really the new moon of the night that is
coming ." 

Yahya said that he heard Malik say, "If people are fasting on the day of Fitr thinking that it is still Ramadan and then definite evidence comes to them that the new moon of Ramadan had been seen one day before they began to  fast and that they are now into the thirty-first day, then they should break the fast on that day at whatever time the news comes to them. However, they do not pray the id prayer if they hear the news after the sun has begun to decline." 

Section: Making the Intention to Fast Before Dawn 
Book 18, Number 18.2.5: 
Yahya related to me from Malik from Nafi that Abdullah ibn Umar used to say, "Only some one who makes the intention to fast before dawn (actually) fasts." 
Yahya related to me from Malik from Ibn Shihab that A'isha and Hafsa, the wives of the Prophet, may Allah bless him and grant him peace, also said that. 

Section: Being Quick to Break the Fast 
Book 18, Number 18.3.6: 
Yahya related to me from Malik from Abu Hazim ibn Dinar from Sahl ibn Sad as Saidi that the Messenger of Allah, may Allah bless him and grant him peace, said, "People will remain in good as long as they are quick to  break the fast." 

Book 18, Number 18.1.3: 

Yahya related to me from Malik from Thawr ibn Zayd ad-Dili from Abdullah ibn Abbas that the Messenger of Allah, may Allah bless him and grant him peace, once mentioned Ramadan and said, "Do not start the fast or     break it until you see the new moon. If the new moon is obscured from you, then complete a full thirty days." 

Book 18, Number 18.1.4: 

Yahya related to me from Malik that he had heard that once in the time of Uthman ibn Affan the new moon had been seen in the afternoon and Uthman did not break his fast until evening had come and the sun had set. 

Yahya said that he had heard Malik say that some one who sees the new moon of Ramadan when he is on his own should start the fast and not break it if he knows that that day is part of Ramadan. He added, "Some one  who sees the new moon of Shawwal when he is on his own does not break the fast, because people suspect the reliability of someone among them who breaks the fast. Such people should say, when they sight the new moon, 'We have seen the new moon.' Whoever sees the new moon of Shawwal during the day should not break his fast but should continue fasting for the rest of that day. This is because it is really the new moon of the night that is  coming ." 

Yahya said that he heard Malik say, "If people are fasting on the day of Fitr thinking that it is still Ramadan and then definite evidence comes to them that the new moon of Ramadan had been seen one day before they began to fast and that they are now into the thirty-first day, then they should break the fast on that day at whatever time the news comes to them. However, they do not pray the id prayer if they hear the news after the sun has begun to decline." 

Abu'Umayr ibn Anas reported from his paternal uncles among the Ansaar who said: It was cloudy and we could not see the new moon of Shawwaal, so we started the day fasting, then a caravan came at the end of the day and told the Messenger of Allaah (peace and blessings of Allaah be upon him) that they had seen the new moon of Shawwaal the day before, so he told the people to stop fasting, and they went out to pray the Eid prayer the next day.
 

Part 1
 Part 2