Greg Killian: A Rosh Chodesh (New Moon) - Shulchan Aruch
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 Shulchan Aruch (Orach Chayim 417).
 Chullin 60b
 There are two basic reasons for reciting Hallel. First, we recite Hallel on festival days in order to praise HaShem as we celebrate His festival (see RAMBAN, Shoresh 1, who understands that reciting Hallel is part of the mitzva of Simchat Yom Tov). Second, we recite Hallel in order to commemorate a miraculous salvation from danger.
 Shulchan Aruch 418:1
 Shulchan Aruch, Orach Chayim 419:1-2, Mishna Berurah, Orach Chayim 418:2, Ben Ish Chai, Vayikra 10, 1 Samuel 20:24
 Bamidbar 10:10, Pesachim 77a and Shavuot 10a, Taanit 29a, Leviticus 23:4 and Rashi ad loc.
 The Tur in Hilchot Rosh Chodesh
 Chullin 60b
 Near relatives being disqualified from offering evidence together.
 I.e., found by the Beth din to be unreliable.
 Ex. XII, 1, 2.
 Even if you are near relatives.
 The communal leaders, to sanctify the month on the strength of it. Nothing, however, is implied about relatives.
 Lit., ‘parts’ (sc. of one hour), 73/1080 X 60 m == 4 m 3 1/3 sec. The new moon, therefore, could not be seen on the twenty-ninth day.
 As a funeral oration would not be delivered on New Moon, which was regarded as a holy day.
 Midrash Sod Halbbur. On the Mystery of the New Moon.
 Rosh HaShana 1 and 2.
 The commencement of the month was dated from the time when the earliest visible appearance of the new moon was reported to the Sanhedrin. If this happened on the 30th day of the current month, that month was considered to have ended on the preceding 29th day, and was called deficient. But if no announcement was made on the 30th day, that day was reckoned to the current month, which was then called full, and the ensuing day was considered the first of the next month.
 The ‘calculation’ as to which and how many months were to be intercalated. It was an established rule that no year should consist of less than four nor more than eight full months.
 The proclamation by formal ‘sanctification’ of the new moon on the thirtieth day.
 The thirtieth day.
 I.e., it is patent to all that the next day is the new moon, as no month exceeds 30 days.
 An Acronym for: “Torah a Neviim a Ketuvim” which literally means the “Law the Prophets and the Writings”, which are the names used by Yeshua and the Apostolic Writers for the Old Testament.
 The thirtieth day is known as the day of prolongation ( ruchg ouh ) as it is the day which is added to make the preceding month full (v. supra p. 21, n. 7). In the case of the two Adars the thirtieth day of each is sanctified as the New Moon of the next month.
 I.e., that the Beth din is Jerusalem fixed the New Moon of Adar II on the thirtieth day of the first Adar, the thirtieth day always being regarded as the ‘proper time’ of New Moon.
 This section is an excerpt from “The Book of Our Heritage” , by Eliyahu Kitov.
 Shulchan Aruch Orach Chaim 426:1 Rema, Magen Avraham 1.
Ramban citing Sefer HaBahir
From the Jewish Encyclopedia
 The American Heritage Dictionary of the English Language
 The American Heritage Dictionary of the English Language.
 I.e., on the thirtieth day of the outgoing month.
 ‘the day of the prolongation’. V. supra, p. 81, n. 1.
 On account of Sabbath.
 Through having drunk too much on Sabbath, and become intoxicated (Rashi).
 It was customary to abstain from work on New Moon (v. Tosaf. s.v. ouan ). In this case the thirtieth day would always he kept as New Moon from doubt, and if the actual day fixed was the thirty-first, there would be two days New Moon.
 Because other people might have seen the new moon.
 Because it could not be proved that they had not seen it (Rashi). R. Hananel: Provided they had seen a semblance of the new moon].
 Heb. rucg lit., ‘taking across’: the word used for the prolonging of the year and the month.
 This was a Baraitha made up of enigmatic sentences like the one which follows.
 I.e., that there should be no appearance of the old moon in this period, viz., after the closing of the twenty-ninth day; otherwise New Moon cannot be proclaimed on the thirtieth.
 Because if the conjunction is calculated to have been after midday and they claim to have seen the new moon before nightfall, they are not telling the truth.
 Which would imply that in Babylon the new moon is not visible till eighteen hours after its birth (Rashi).
 Which would imply that in Israel the new moon is visible six hours after its birth (Rashi).
 Lev. XXIII, 32, in connection with fasting on the Day of Atonement. This shows that the day follows the night in reference to the festivals.
 Ex. XII, 18, in connection with eating unleavened bread on Passover. This shows that the festivals end at even.
 Lit., ‘the interpretation of exegeses’.
 According to R. Johanan, the ‘night’ referred to is on the same footing as the night of the Day of Atonement which commences at nightfall. But according to Resh Lakish, it is on a par with the first night of Passover, which, in relation to the Paschal lamb, was a continuation of the afternoon before. Hence Resh Lakish holds that even if the old moon was seen in the early part of the evening, the next day may still be declared New Moon.