Sod Ha’iber
Understanding the Hebrew Calendar

The Rambam writes in Kiddush HaChodesh:

הלבנה נסתרת בכל חדש ואינה נראת כמו שני ימים או פחות או יותר מעט. כמו יום אחד קודם שתדבק בשמש בסוף החדש וכמו יום אחד אחר שתדבק בשמש ותראה במערב בערב.

The moon is hidden every month, a day before the conjunction and a day after the conjunction, after which it is seen in the evening in the west. 

That sighting was what the Torah was referring to in Parshat Bo.

החדש הזה לכם ראש חדשים ראשון הוא לכם לחדשי השנה.

This month shall be for you the beginning of the months, it shall be for you the first of the months of the year. “החדש” refers to the renewal of the moon, that sliver of the visible moon in the above picture.

Implicit in the Rambam’s statement is that from the time of the conjunction until the time one can first see the New Moon is 24 hours and the total time from when the moon disappears (Old Moon) until it reappears is 48 hours.  This seams to conflict with the Talmud Rosh HaShanah 20b:

There is good reason for the Rambam to differ with 

From where do the extra 12 hours come?  Pesachim 94b:

חכמי ישראל אומרים ביום חמה מהלכת למטה מן הרקיע ובלילה למעלה מן הרקיע.

The Sages of Israel maintain: The sun travels beneath the sky by day and above the sky at night.

After the moon in Eretz Yisrael disappears in the West it takes 12 hours above the Rakia to return to be seen in the East.

וחכמי אומות העולם אומרים ביום חמה מהלכת למטה מן הרקיע ובלילה למטה מן הקרקע אמר רבי ונראין דבריהן מדברינו.

The Sages of the nations of the world maintain: It travels beneath the sky by day and below the earth at night. Said Rebbi: And their view is preferable to ours.

The path the moon takes is key to understanding the timing of the moon. From Rosh HaShanah 25a: 

אמר להם רבן גמליאל לחכמים: כך מקובלני מבית אבי אבא: פעמים שבא בארוכה, ופעמים שבא בקצרה. 

Rabban Gamliel said to the Sages, “This I have received from the house of my father’s fathers – [The moon] sometimes takes a long path and sometimes takes a short path.”

We always last see the moon in the East just before sunrise and will next see the moon in the West just after sunset. The moon must disappear within at least a total 36 hour period. This implies that the moon will be seen about 18 hours after the conjunction. Often the moon takes a longer path and the moon disappears for about 60 hours or even about 84 hours.

Twenty-four hours from the Talmud and forty-eight hours from the Rambam, which is right.

Certainly the moon that R’ Simlai’s father saw and the moon that the Rambam saw is the same moon we see today.

It seems to me that R’ Simlai’s observation is right in principle though wrong in it’s conclusion.

The problem is a moon which we know does not generate light but reflects light. Each month as the moon approaches the sun both the moon reflects less sunlight and the ambient light of the sun makes it impossible to see it. 

https://www.staff.science.uu.nl/~gent0113/islam/1438/1438g_01a.pdf

24 hours or 48 hours

As if a dispute of 24 hour or 48 hours is not enough let me add what a decerning observer should be able to ascertain today. First play this video of the moon as it disappears at month’s end and reappears to begin the new month.

We can see in the video the shape of the moon blocking out the stars and the sun but in reality, from earth the moon vanishes without trace for more than 24 hours. The midpoint between when the Old Moon is last seen and the New Moon is first seen from the perspective of the video is the conjunction, when the earth, moon and sun, in that order, are approximately in a straight line. But we terrestrials last see the Old Moon just before sunrise [in the East] and first see the new moon just after sunset [in the West]. This is because close to the time of the conjunction any presence of light from the sun will obliverate the presence of the moon.

Until the sixteenth century with the arrival of Copernicus followed by Kepler, Galileo and Newton

The commentaries note that Rosh HaShanah 20b states that the moon cannot be seen for twenty-four hours every month. Aruch HaShulchan (Hilchot Kiddush HaChodesh, sec. 88) explains that there is no contradiction to the Rambam's statements here. There are twenty-four hours every month in which it is impossible to see the moon. There is in addition, however, approximately one day every month when the probability of seeing the moon is very low.

The Rambam’s view was Geocentric, centered on the earth, which he shared with Aristotle and Ptolemy. In today’s world we would be pressed not to think like Copernicus did with a sun centered model, heliocentric. A heliocentric model would have symmetry between the Old Moon and the New Moon as was seen in the video. 

This is hard to conceptualise abstractly so I would like to take you on a time machine trip to the actual event that has been the cornerstone of the Hebrew Calendar, the moon Hillel ben R’ Yehuda, aka Hillel II, saw in 358 CE. Here I will discuss the physical manifestation of the moon, on another page I will discuss the historical aspect.

If you would like to see the actual path the moon took from the conjunction until it appeared as a New Moon use this link - Path of Moon Tishrei 358 CE.

The Molad of Tishrei 358 CE according to the Gregorian calendar system was

What really happened in 358