Question: Why does the Gemara bother with this question, only to establish that the Gemara is according to Rebbi Yishmael? The Gemara already established this a few lines earlier!

Explanation: This is a question.

Observation: Rav Ashi's answer (that there are three Roshei Shanim that everyone agrees on, only counting those starting at the beginning of a month) is only according to Rebbi Meir, as Rebbi Shimon does not count the first of Elul. However, the Mishnah is still Rebbi's opinion based on the Tanaim, and not just the opinion of Rebbi Meir, as Rebbi Meir does not agree regarding the Rosh Hashanah l'Regalim (he holds one Regel is enough).

Observation: The Targum translates this as "the rams alight onto the sheep," which is a modest terminology for their relations. Rashi later (11a, DH v'Oso) explains that when Rebbi Yehoshua Ben Levi states that the world was created in Nisan, this is shown by the fact that animals mate at this time (just as when the world originally needed to start being populated by the various species of animals). However, Rashi here does not mention this fact (NOTE: The Gilyon Hashas points out that Tosfos clearly did not have this text regarding the animals in the Gemara, unlike the text of Rashi.)

Question: How can the Gemara say that grain is only finished and ready to have Ma'aser taken close to the new year of Ma'aser (the first of Tishrei)? Doesn't the Mishnah state in Ma'asros (1:3) that they are ready to have Ma'aser taken when they are one-third ripe?

Answer: Our Gemara is not referring to when Ma'aser could be taken, but rather to when it practically is taken, which is usually towards Tishrei.

Implied Question: Where does it state in our Mishnah that Tishrei is the Rosh Hashanah for the Ma'aser of grain?

Answer: This is stated in our Mishnah through the word "Yerakos," as interpreted by the Gemara later (12a).

Question: How does the Gemara know this to be true?

Answer#1: It could be that the Gemara learns the law of Ma'aser where the Pasuk states "ha'Yotzei ha'Sadeh Shanah Shanah" (Devarim 14:22) from the Pasuk "mei'Reishis ha'Shanah" (Devarim 11:12) which is referring to the Rosh Hashanah of Tishrei (8b).

Answer#2: It could be based on the fact that Bnei Yisrael were not obligated in Ma'aser until they started keeping Shemitah (see Rashi in Kesuvos 25a, DH v'Lo Nischayvu). As the years of Shemitah are based on the first of Tishrei, Ma'aser must also be based on the first of Tishrei.

Explanation: Rav Chisda holds like Rebbi Zeira who says the Rosh Hashanah "l'Shanim" on the first of Tishrei is referring to the Tekufah.

Question: According to this answer, why didn't the Mishnah include that the first of Tishrei is the Rosh Hashanah for the dating of the years of gentile kings in Shtaros?

Answer: Our Mishnah is not talking about the new year's effect on matters which involve gentiles.

Explanation: Rebbi Eliezer and Rebbi Yehoshua argue regarding the Molad and Tekufah. Rebbi Eliezer holds that these start from the first of Tishrei, while Rebbi Yehoshua holds that they start from the first of Nisan (for information regarding what is the Tekufah and Molad, see Sifsei Chachamim on our Tosfos, and in his introduction to Rosh Hashanah at length).

Observation: The Gemara in Eruvin (56a) which states that the Tekufah of Nisan only falls on the "quarters" (6th, 12th, 18th 24th hrs.) of the day must be according to Rebbi Yehoshua.

Note: Being that a Tekufah is 91 days and seven and one half hours, if the Tekufah is counted from Nisan to Nisan one will always end up with 6 extra hours, making the Tekufah fall on the sixth hour. The extra six hours of the next year will mean the Tekufah will fall on the 12th hour, the next year on the 18th, and so on.

Observation: Those who count the Molad back from the beginning of the world (i.e. for the purpose of calculating the monthly Molad) will need to take away 7 days, 9 hours, and 642 Chalakim (Note: Chazal split an hour into 1,080 "Chalakim" - "parts" to an hour, due to the fact that the number 1,080 is divisible by every number from one to ten besides seven, making it easy to split into halves, thirds, etc.). This length of time is referred to by Tosfos as ZaT TaRMaB. (Note: Tosfos will now proceed to explain why.)

Implied Question: Where does this number come from?

Answer #1: Some say that the moon acted as if it was placed in Niduy after it told Hash-m that it is not appropriate that the moon and the sun should be the same size (see Chulin 60b). It therefore moved backwards this amount of time.

Refutation: There is no source for such a statement.

Answer #2: The world was created on the twenty-fifth of the month (according to Rebbi Eliezer, the twenty-fifth of Elul), six days before the creation of Adam ha'Rishon on the first of the month. Adam ha'Rishon was commanded at the beginning of the ninth hour not to eat from the Eitz ha'Da'as (see Sanhedrin 38b), and he probably was commanded then to be Mekadesh the Chodesh which he must have promptly done.

We can assume that when Adam was Mekadesh the Chodesh the new moon was unable to be seen at least six hours beforehand, as the new moon is always covered up at least six hours before it is seen. This would mean that the Molad was at the beginning of the third hour of daylight on that day, fifteen hours from the start of the previous night. This can be symbolized as "VID - Vav, Yud Daled," meaning that this took place on the sixth day at the end of the fourteenth hour of creation. Since Rosh Hashanah did not occur until Friday, when Adam announced the new month, it is evident that the creation of man occurred before Rosh Hashanah, on the 25th of Elul.

There is something called a "Shanah Shel Tohu" - "year of emptiness" which we use to calculate what the Molad would have been in the year before man was created. This is based on the concept of "Yom Echad ba'Shanah Chashuv Shanah," - "one day of a year is considered a year." Being that the world was created at the "end" of the year, six days before the "Rosh Hashanah" when man was created, we calculate the Molad and Tekufah as if the entire year before Rosh Hashanah actually existed.

Note: To determine the day and time at which the Molad of Nisan in the "Shanah Shel Tohu" would have occurred, we simply subtract for each month 29 days 12 hours and 793 Chalakim. Since we are calculating only the day of the week, and not the day of the month, we can drop the first 28 days (which are divisible by 7), which leaves us (in shorthand notation) 1d 12h 793ch to subtract per month.

We can now calculate the Molad of the Tishrei and Nisan of the year of Tohu. The Molad of Nisan of Tohu was 6 x (1d 12h 793ch) = 9d 4h 438 ch earlier than the Molad of Adam's Tishrei. We drop the first seven days again, leaving 2d 4h 438ch (as stated by Tosfos). [The Molad of Nisan of Tohu was therefore on Wednesday at 9h 742 ch.] The Molad of Tishrei of Tohu is similarly reached by subtracting 12 x (1d 12h 793ch) from the Molad of Adam's Tishrei. This comes out to 2d (Monday) 5h 204 ch, which Tosfos refers to as "BaHaRaD".

We rule like Rebbi Yehoshua regarding the Tekufah, as is implied by the Gemara in Eruvin (56a), and Rebbi Eliezer regarding the Molad. (Note: The Tosfos ha'Rosh explains that we rule like Rebbi Yehoshua regarding the solar-dependent Tekufah, and therefore Tosfos understands that the Tekufah of Rebbi Yehoshua is to be started from Nisan, despite the fact that we follow Rebbi Eliezer regarding the lunar-dependent Molad, and assume that Adam was created on Rosh Hashanah of Tishrei.)

The Tekufah of Tishrei thus occurred on Wednesday of creation at 15 hours, as follows. The length of a Tekufah is 91d 7.5h, and there are two Tekufos between Adam's Tishrei and the preceding Nisan of Shanah Shel Tohu, which is when the sun was created at the beginning of Wednesday night (according to Rebbi Yehoshua's understanding). The length of two Tekufos is 182d 15h, thus the first Tekufah of Adam's Tishrei was 15 hours into Wednesday. We already know that the first Molad was at "VID," Friday at the fourteenth hour (see 2 above). Thus the Tekufah was 1d 23h before the Molad of Adam's Tishrei.

CONCLUSION OF ANSWER #2: To determine the difference in time between the Tekufah and Molad of Nisan in the year of Tohu, we can simply calculate the difference between half a solar year (2 x 91d 7.5h = 182d 15h) and half a lunar year (6 x 29d 12h 793 ch = 177d 4h 438ch). We then add that number to the difference of 1d 23h that we already have found. The above equation shows a difference of 5d 10h 642ch plus 1d 23h, or a total difference of 7d 9h 642ch (referred to by Tosfos as ZaT TaRMaB). This is the amount of time by which the Tekufah of Nisan preceded the Molad in the Shanah Shel Tohu.

Since today we count the years of the world (to determine the Molad) from the Tekufah of Tishrei of Shenas Tohu, as previously explained (f:3), we must subtract ZaT TaRMaB to arrive at the correct time of the Molad from the Nisan of the Shenas Tohu.

Question: The Molad changes by 2d 4h 438 ch over six months, as we saw above (f:5). Thus, if the Molad would have occurred six months earlier than it actually did, the Molad would be off by more than two days. Why, then, is there an argument between Rebbi Yehoshua and Rebbi Eliezer over when the first Molad occurred? It should be simple enough to determine by sight the time of the true Molad today, within two days, and to extrapolate from that when the first Molad must have been on Friday (d:6) at the end of the fourteenth hour of either Nisan or Tishrei?

Rashi's Explanation: What does the Gemara mean by "Chag she'ha'Chodesh Miskaseh Bo" - "the holiday that that the month is covered in it? At this time, the moon is not seen to those far away from it, meaning in the morning to those in the west and in the evening to those in the east, as it is very small right after it becomes a new moon.

Proof: Rabeinu Tam supports this explanation from a Midrash (see Sifsei Chachamim for the explanation of the Midrash at length).

Rabeinu Tam's Explanation: "Miskaseh" refers to "preparation," meaning that Rosh Hashanah is the Yom Tov that the month is ready for. A similar definition of "Keseh" is found in the Pasuk "l'Yom ha'Keseh" (see Rashi in Mishlei 7:20, DH l'Yom ha'Keseh).

R. Meshulam's Explanation #1: This alludes to the Chatas of Rosh Chodesh, which is "covered up," meaning that it is not brought, on Rosh Hashanah.

Question: Rabeinu Tam asked that from the Tosefta and from a Tefilah instituted by Rebbi Eliezer ha'Kalir it seems clear that the Chatas of Rosh Chodesh is brought on Rosh Hashanah!

Answer: Rabeinu Meshulam explained that he didn't mean that the Chatas is not brought, just that it is not mentioned in the Pasuk "Milvad Olas ha'Chodesh u'Minchasah" (Bamidbar 29:6) which mentions the other Korbanos of Rosh Chodesh brought on Rosh Hashanah.

R. Meshulam's Explanation #2: The Korban Musaf of Rosh Chodesh is "hidden," meaning not mentioned, in the Tefilah of Musaf.

Rabeinu Tam argued on this, and mentioned the Korban Musaf of Rosh Chodesh in Musaf.

Observation: According to the opinions (Rav Ashi, Rav Chisda) that the Rosh Hashanah of Regalim in the Mishnah does not really mean that the Rosh Hashanah for Regalim is on the first of the month, but rather that it is in the middle of this month (l'Regel she'Bo, see 4a), one could have answered our Gemara's question in a similar fashion (reading of Tosfos according to the Maharsha).

Observation: As the Rabanan don't hold that the beginning of Yovel is Kodesh, they do not need a Pasuk to teach that we do not add more than the rest of the year to Yovel. If Yovel begins late, it is logical that it does not end later than it should (as opposed to Shabbos etc. which begins early and ends late).

Question: On the contrary, they should require a special teaching because the Torah states Yovel is a "Shanah" - "year." Doesn't the simple explanation of "year" mean a full year?

Answer: Being that the Torah also states that specifically "the fiftieth year" is holy (Yovel), this redefines our understanding of what a year is in this case.