New moon (again—it happens every month)

Last month I was confuzzled at the lack of new moon—according to the ephemeris I was using, the moon was past new moon and yet setting earlier than the sun. I couldn't understand how this could be: surely new moon is when the moon has "passed" the sun and is now setting after the sun. To be setting later than the sun, surely it must rising before the sun, and therefore at the end of the lunar month, not the beginning.

Well, it ain't necessarily so, and today there was a new moon at 11:37, and yet the moon still will set earlier (15:52) than the sun (16:10). Eventually, after much head-scratching, I figured out how this can be. See, when the moon passes the sun on its path around the sky, only occasionally does it cross the sun directly—this is when we get solar eclipses. Normally it's either above the sun, or below the sun, relative to the plane of the ecliptic; and given that the sun sets at an angle, this means that the moon can go below the horizon earlier than the sun:

In the above diagram, the moon on Day 1 has already passed the sun, and the crescent is facing to the right, not to the left, yet as moon and sun slide along the direction of the yellow line below the horizon, we can see that the moon has already set, when the sun is still half up.

There we go; another one of life's minor mysteries solved. Which only leaves me to observe the new moon tomorrow, weather permitting, some time between 16:11 (sunset) and 17:10 (moonset).