Jerusalem Talmud Rosh Hashanah 1:4

משנה: עַל שִׁשָּׁה חֳדָשִׁים הַשְּׁלוּחִין יוֹצְאִין, עַל נִיסָן מִפְּנֵי הַפֶּסַח וְעַל אָב מִפְּנֵי הַתַּעֲנִית וְעַל אֱלוּל מִפְּנֵי רֹאשׁ הַשָּׁנָה וְעַל תִּשְׁרֵי מִפְּנֵי תַקָּנַת הַמּוֹעֲדוֹת וְעַל כִּסְלֵיו מִפְּנֵי חֲנוּכָּה וְעַל אֲדָר מִפְּנֵי הַפּוּרִים. וּכְשֶׁבֵית הַמִּקְדָּשׁ קַייָם יוֹצְאִין אַף עַל אִייָר מִפְּנֵי פֶסַח קָטָן׃ MISHNAH: For six months messengers are going out²⁶²To inform the people about the start of a new lunar month in the absence of a computed calendar. This is a late development which in the Mishnah is placed in Second Temple times whereas in Halakhah 2:1 (Note 39) it is placed later.. For Nisan because of Passover, and for Av because of the fast day²⁶³The 9th of Av, anniversary of the destruction of the Temple; obviously a later development., and for Elul because of New Year’s Day²⁶⁴Since New Year’s Day is also the first of a month, it could not be determined in advance. If the first of Elul was known, the 30th of Elul could be observed everywhere as 1st of Tishre, since traditionally Elul was 29 days (Halakhah 3:1 Note 14, Babli 19b; Shevi`it10:2 Note 41, Sanhedrin1:2 Note 213); a second day had to be observed for the rare possibility that Elul was found to have 30 days., and for Tishre to fix the holidays²⁶⁵To avoid that the fast of the Day of Atonement had to be observed for two days., and for Kislew because of Ḥanukkah, and for Adar because of Purim. And when the Temple existed they also went for Iyar because of the minor Pesaḥ²⁶⁶On the 14th of Iyar; Pesaḥim9:1..

הלכה: וְיֵצְאוּ אַף עַל הָעֲצֶרֶת. עֲצֶרֶת חֲתוּכָה הִיא. אַשְׁכָּחַת אֲמַר. פְּעָמִים חֲמִשָּׁה פְעָמִים שִׁשָּׁה פְעָמִים שִׁבְעָה. שְׁלֵימִים חֲמִשָּׁה. כְּסִדְרָן שִׁשָּׁה. חֲסֵירִין שִׁבְעָה. HALAKHAH: Should they not also go for Pentecost? Pentecost is cut out. You find to say sometimes five, sometimes six, sometimes seven. Full months five, regular six, deficient seven²⁶⁷Since as the Greek name indicates, Pentecost is 50 days after Passover, once Passover is fixed, Pentecost also is fixed. In the pharisaic-rabbinic system, where the count starts on the second day of Passover, Pentecost is the 5th of Sivan if both Nisan and Iyar are 30 days, it is the 6th if one of them (in the computed calendar, Nisan) is 30 days and the other month 29, it is the 7th if both months have 29 days. The lengths of these months are irrelevant for the fixation of Pentecost. Babli 6b; cf. Note 100..

לֹא כְבָר יָֽצְאוּ בָאֱלוּל. אֶלָּא בְגִין מוֹדָעָא דְאִיתְקַדַּשׁ יַרְחָא. אָמַר רִבִּי יְהוֹשֻׁעַ בֶּן לֵוִי. אֲנָא עָרֵיב לְאִילֵּין דְּאָֽזְלִין לִנְמוּרִין דְּלֵית חַד מִינְּהוֹן מַייִת מִי אֲזִל. תַּמָּן חָשִׁין לְצוֹמָא רָבָּה תְּרֵין יוֹמִין. אֲמַר לוֹן רַב חִסְדָּא. לָמָּה אַתֶּם מַכְנִיסִין עֲצְמְכֶם לְמִסְפֵּק הַזֶּה. חֲזָקָה שֶׁאֵין בֵּית דִּין מִתְעַצְּלִין בּוֹ. אָבוּהּ דְּרִבִּי שְׁמוּאֵל בַּר רַב יִצְחָק חָשׁ עַל גַּרְמֵיהּ וְצָם תְּרֵין יוֹמִין. אַפְסַק כָּרוכֵהּ וֹדְמָךְ. Did they not already go for Elul²⁶⁸Therefore it should not be necessary to send messengers for Tishre.? But to announce that the new Moon has been sanctified. Rebbi Joshua ben Levi said, I am a guarantor for those who go to Nemurin that none of them will die when going. There²⁶⁹From here on the paragraph is copied in Ḥallah1:1, Notes 83–86. “There” means Babylonia. It could mean Nimurin, whose location would have to be on the road to Iraq since it was within the jurisdiction of Rav Ḥisda, and R, Joshua ben Levi was encouraging people to spread the news about the correct date of the Day of Atonement, here called the Great Fast. they were concerned about the Great Fast two days. Rav Ḥisda^269aIn A: Ḥizqiah. told them, why do you bring yourselves into this serious doubt? The presumption is that the Court will not tarry in this matter²⁷⁰Even without a computed calendar one may be sure that New Year’s Day is the 30th of Elul and according to all biblical rules only fast one day on Tishre 10.. The father of Rebbi Samuel bar Rav Isaac was concerned for himself and fasted two days. His intestine split and he died.