Sarah Graham, of Leestream School, asks :-
Why does a calendar refer to the lunar phases being "first quarter" and "last quarter", when in fact it's half of the Moon that appears bright at those times?
Duncan Steel, an astronomer in Canberra, Australia, who wrote the book "Eclipse", responded.
The answer is that we refer to these phases as being "quarters" because they occur when the Moon is at quarter-fractions of a lunar month. That is, measuring a lunar month from one "new moon" to the next (when the crescent moon is just visible in the western sky shortly after sunset), the "first quarter" occurs about seven days later, the western half of the lunar disk being visible in the evening sky. At that time it is closest to overhead at about 6pm, and sets near midnight. The "last quarter", also termed the "third quarter", happens a couple of weeks later, with the eastern half of the lunar disk appearing bright. Then, the Moon rises at about midnight in the east, is closest to overhead near 6am, and sets in the west at midday.
In between the two quarters is "full moon", when our celestial neighbour appears brightest, its whole earthward disk being illuminated. A lunar month, measured between two full moons, or between two new moons, lasts for just over 29-and-a-half days, on average.
This is the origin of the time interval we call a "month" (i.e. a "moon-th" although the calendar month has been divorced from the motion of the Moon for over two thousand years, since the Roman era. It is because of this that the dates of new and full moon slip gradually earlier. Sometimes there are two full moons in one 31-day calendar month, and the second is often called a "blue moon."