Somewhat off topic for this blog, but I hope it will help those following daf yomi who are getting to the topic of astronomy and cross-examining witnesses to determine Rosh Chodesh (Rosh haShanah, mostly around 23b-24a).
Really, the statements about the appearance of the moon can be deduced from two things you already know.
1- The moon rises one less time each month than does the sun
The missing moon right before the molad isn’t that the moon just takes a day off. It’s that the moon goes a smidgeon slower across the sky than does the sun.
(You were probably taught that the solar system revolves around the sun, and we could describe the whole thing that way. But it’s unnecessary for the gemara. For those who insist: The moon is moving in the same direction as the earth’s spin, so that at the end of exactly 24 hours we are in the same place in our spin, but the moon is a little ahead. Until we get to seeing the moon in exactly the same spot, it will be more than 24 hours. As I said, this parenthetic confuses you, you don’t need it.)
Because the moon crosses the sky more slowly than the sun, on Rosh Chodesh it will be slightly behind the sun. Moonrise will be slightly after sunrise, and at the end of the day the moon will only be out a little after sunset and be done. The next day, it will be somewhat more behind sunrise. At a half-moon, the moon is rising at noon and setting at midnight. And finally a little before the new moon it will be nearly a day behind the sun, and therefore be slightly ahead of the next sunrise. The moon will rise slightly before dawn, and set slightly before sunset.
Aside: Notice that this means the moon is out during the day as often as at night. We just tend to ignore that pale white-and-blue daytime moon seen close to sunrise or to sunset, depending on the time of month. I have no idea why we teach children otherwise. In terms of the chumash, we are told that Hashem created the moon “lememsheles balaylah — to rule at night”. We aren’t actually told it’s only out at night; only that its role at night is somehow similar (whatever is meant by “ruling”) as the sun’s is during the day.
2- The moon’s light is a reflection of sunlight
Which means that the side of the moon that is lit will always be the one closer to the sun. When there is a partial moon, the “belly” of the lit side will be toward the sun. Or as the gemara puts it, the “horns” of the moon will always point away from the sun.
And you will only see the moon when the sun is either below the horizon or close enough to sunrise or sunset for it to appear dim enough to not wash out the moon’s white-and-blue daytime appearance. Which means that whenever you see the moon, it is close to the middle of the sky than the sun is. And therefore the lit side of the moon will be “down” closer to the horizon, and the “horns” pointing “up” toward the middle of the sky.
Sighting the end of last month: The moon has fallen so far behind the sun that it’s leading the next day’s sun. So it will rise slightly before dawn. Since the moon is ahead of the sun in its east-to-west path, the moon will be to the sun’s west. (The sun will be at the east or south-east horizon, the moon slightly above the horizon.) The thin sliver of moon will be the sun-ward, i.e. eastward side — toward the horizon, and the “horns” will point west, toward the middle of the sky.
Sighting the molad: The moon just started trailing the sun, so it rises slightly after sunrise and set slightly after sunset. The most likely time to see it, given the brightness of the sun, would be in the evening. Since it’s behind of the sun in its path, the moon will be to its east. (The sun will be at the west or south-west horizon, the moon slightly above the horizon.) The thin sliver of moon will be the sun-ward, i.e. westward side — toward the horizon, and the “horns” will point east, which again is toward the middle of the sky.
The moon during the day is blue and white, and is only out during times of change — sunrise or sunset. How does this relate to the comparison of the Jewish People to the moon?