Here is a summary of Tosfos. Hope this proves useful.

TOSFOS DH "l'Tekufos"

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 Eiruvin (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.

Background: 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 (which Tosfos will proceed to explain): Those who count the Molad back to the beginning of the world (i.e.for the purpose of figuring out the year) will need to take away 7 days, 9 hours, and 642 Chalakim (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.).

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.

Question: There is no source for such a statement.

Background to 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.

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 as if the entire year before Rosh Hashanah actually existed.

To figure out the 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 (stated by Tosfos). The Molad of Nisan of Tohu was therefore on Wednesday at 9h 642 ch. The Molad of Tishrei of Tohu is similalry 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."

All the best,

Yaakov Montrose

In addition, here is what we wrote about this issue in response to a previous question:

Dear Avraham,

It is very helpful to learn Shevilei d'Rakia of Rebbi Yisrael Lipshitz, the author of the commentary Tiferes Yisrael on Mishnayos. It is found in Volume 3 (or Mo'ed 1), before Mishnayos Shabbos. See especially sections 34,35 and also 9, 17 and 31. (The beginning of section 34 is different than Tosfos).

I will try to give you enough background information to deal with the concepts mentioned in Tosfos.

(a) Tosfos explains that the difference between Rebbi Yehoshua and Rebbi Eliezer is whether the very first Tekufah (equinox or solstice), i.e. the Tekufah that occurred at the moment in which the sun was created, was in Nisan or Tishrei. This affects the time at which the Tekufah will occur in all future years, as follows.

The solar year is calculated as 365.25 days. Therefore a quarter year is 91.25 days plus 1.5 hours (or 91d 7.5h). The sun was created at the beginning of the night of the 4th day of Creation. That moment was the moment of the Tekufah (equinox). According to Rebbi Yehoshua, this was on Wed. night a few days before the first of Nisan. Half a year later, in Tishrei, the Tekufah occurred 15 hours later in the week than it did in Nisan, or the 15th hour of Wednesday at the end of Elul (which can be written in notation form as d4 h15 - we drop the month when making these calculations for simplicity. The reason the Tekufah of Tishrei is on Wed. is because the Tekufah of Nisan was on Wed., and 182 days - the amount of days in 2 Tekufos - is divisible by 7, so there is no change in the day of the week that the Tekufah occurs two months later.)

According to Rebbi Eliezer, the first Tekufah occurred at the beginning of the night of that same Wed. at the end of Elul, or 15 hours earlier than Rebbi Yehoshua's Tekufah would have occurred. Thus, according to Rebbi Eliezer -- the author of the Mishnah in Rosh Hashanah -- the Tekufah will always be 15 hours earlier than according to Rebbi Yehoshua.

(b) Tosfos then explains when the first Molad occurred, from which every subsequent Molad will be calculated. The Molad occurs every 29 days 12 hours and 793 Chalakim (a Chelek is'1/1080 part of an hour).

Adam was created on the sixth day and was commanded at the beginning of the 9th hour of the day to keep his Mitzvos. Tosfos assumes he immediately saw and sanctified the moon, making it Rosh Chodesh. One cannot see the moon until 6 hours after the Molad, so the Molad must have been 6 hours earlier, or d6 h14 (the 14th hour from the beginning of the night, i.e. the second hour from the beginning of daybreak, on Friday).

However, for the purposes of calculating the Molad, we first extrapolate backwards what the Molad would have been one year before the Tishrei in which the world was created, and this is used as the starting point for calculating the current Molad. Tosfos explains that the reason for this is because the Creation began several days before Rosh Hashanah (Adam was created on Rosh Hashanah), so we add an entire year to our count when counting the number of years since creation (since "part of a year is like a whole year"). This is called 'the year of Tohu.'

To know the day and time at which that earlier Molad 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 438ch earlier than the Molad of Adam's Tishrei (and we drop the first 7 days again, leaving 2d 4h 438ch). This comes out to be d4 h9 ch742 (i.e. the 9th hour of Wed. plus 742 ch).

The Molad of Tishrei of Tohu was 12 x (1d 12h 793ch) earlier than the Molad of Adam's Tishrei. This comes out to d2 h5 ch202, which Tosfos refers to as 'BaHaRaD.'

(c) According to what we have calculated above, at the time of Adam the Tekufah - with regard to which we rule like Rebbi Yehoshua - was d4 h15, while the Molad - with regard to which we rule like Rebbi Eliezer - was d6 h14. This means that the Tekufah occurred 1d 23h earlier than the Molad in Tishrei of that year.

What was the difference in time between the Tekufah and the Molad in the Nisan of Tohu? To calculate this figure, we find the difference in length between half a solar year and half a lunar year, and add that figure to the 1d 23h difference that we have for Tishrei. (Because the lunar year is shorter than the solar year, the Tekufah will precede the Molad in Nisan by a greater amount than in did in the following Tishrei.)

Half a solar year is 365.25/2 days,; half a lunar year is (29.5d 793ch) x 6. The difference between the two is 5d 10h 642ch. Add that figure to the 1d 23hr by which the Tekufah preceded the Molad in Tishrei, and you will find that the Tekufah preceded the Molad in Nisan by 7d 9h 642ch (which Tosfos refers to as ZaT TaRMaB).

Tosfos sums this up as follows: The Molad we calculate from, when determining the current Molad, is BaHaRaD (d2 h5 ch204) of the year of Tohu (the year before Adam). The Tekufah we calculate from is the Tekufah of Nisan of the year of Tohu, 6 months before Adam was created.

(d) Tosfos concludes with a question. The Molad changes by 2d 4h 438ch over 6 months, as we saw above (b). Thus, if the world had been created six months earlier than it was the Molad would be off by more than two days. Why then is there a Machlokes between Rebbi Yehoshua and Rebbi Eliezer about when the first Molad was! 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 - in d6 h14 of which was the first of Tishrei or in d6 h14 which was the first of Nisan !

We hope this made things clearer. We'll be happy to explain more.

All the best,

Reuven Weiner