You could have stayed up from sunset to sunrise but you would not have seen it on January 30. The moon was close to being right between the earth and sun. The sunlight side of the moon always faces the sun, not the earth. The dark, unilluminated side of the moon was facing the earth that night.
Try looking for the moon at sunset on January 31st. If you are an avid amateur astronomer with correct optics you might be able to find the moon following the sun just after sunset. The 0.2% illuminated moon sets 70 minutes after the sun. Technically you may catch a glimpse of the moon "with considerable difficulty," as they used to say in the mid 1950's science magazines.
Sunset on the first day of February will display a 7% illuminated moon. The very thin crescent will be visible for almost two hours after sunset. This is the crescent everyone can see from a typical suburban backyard. It should appear above the sun set point.
It is historically interesting to begin a month on the first day of that month with the appearance of the crescent moon. If we trace the origin of the word "moon”, we find it means "to measure.” What does the moon measure? It measures time. The word "month" comes from the word "moon."
If you lived in Rome more than two thousand years ago, you would not be surprised seeing the first day of the month coincident with a crescent moon because that was the traditional start of the month, with the first appearance of the crescent moon in the evening sky. An official would go about town, stand on their version of a soapbox to announce how many days to the first evening crescent. He would "call out" the number of days to the first quarter moon, then the number of days to full moon. The Latin which means "to call”, is the word "calend”, from which we get the word "calendar”.
The word "week" comes from the word "wik.” Wik can be defined to mean "to change, a bend.” We not only get the word week from wik, but also wicker. Watch the moon from one night to the next and observe how the terminator, that part of the moon where it's light side meets the unilluminated side, will change from a steep curve, the crescent moon, to seven evenings later it will look like a straight line, the first quarter moon.
Our calendar is made of months which correspond to the time it takes the moon to go around the earth once. Four weeks corresponds to the four distinct phases of the moon. New moon to first quarter is seven days, first quarter to full moon takes seven days, full moon to last quarter another seven days and finally last quarter to new moon our final seven days.
Then why aren't all the months 28 days (7x4) long? If we count the number of days from one phase of the moon to the corresponding phase a month later, we would count 29.3 days. This measurement is called the synodic period of the moon. If we use 29.3 days divided by the number of days in the year, we arrive at 12.3 months in the year. When we try to couple the moon month to a solar year, there is a third of a month too many days. Days must be added to the moon month to make a lunar and solar calendar balance. This juggling of the two calendars into one calendar is quite a complicated affair.
If we note when the moon appears close to a star, then count the number of days for the moon to return to that same star we would have counted 27.3 days. The measurement of the moon with respect to background stars is called the siderial period of the moon, 27.3 days. Divide the siderial period into the number of days in the year we find not 12 months in the year but 13. Note the siderial period of the moon is almost 28 days.
It is the menstrual cycle. Words associated with the menstrual cycle reflect the connection. Menos is one of the many Egyptian and Sythian goddesses of the moon, pause means to stop. Menopause means quite literally "to stop the moon." A Latin word for "moon" is "mensis”, Old Church Slavonic is "meseci," Greek is "mene" and Lithuanian is "menesis."
Can there be any wonder a connection between the moon and female is strong in cultures around the world? Goddess centered celebrants would go into the forests and woodlands to honor what the full moon represents. Often patriarchal societies would frown on such lunar activities, interpreting the full moon as a symbol of evil and fright.