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Great question! According to the United States Naval Observatory, the First Quarter and Last Quarter Moon phases are named as such because of their position in the Moon's orbit. The First Quarter is 1/4 of the way through its orbit from starting position (New Moon), and the Third Quarter is 3/4 of the way along in its orbit.
http://aa.usno.navy.mil/faq/docs/moon_phases.php
The Full Moon is named instead for the percent of surface illuminated from our viewpoint (100 percent, full disc), and occurs when the Moon is 1/2 the way along in its orbit.
New Moon--named so because in this case it is the start of the lunar cycle; we can see no surface illumination from our position on Earth.

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