“Three-part” means “three-quarter”:
three-part, adj. b. = three parts, n.
three parts, n. Three out of four equal parts, three quarters.
Oxford English Dictionary.
and “three-quarter moon” means “gibbous moon”, that is, a moon that is more than half but less than fully illuminated. A couple of citations for this sense:
the greater part of the bright side is seen, and it is called “gibbous” and appears as a “three-quarter moon”
George Dalziel & Edward Dalziel (1862). A Handy Book to the Sky, Air, Earth, and Waters, p.14. London: Ward and Lock.
at 8, it shows about three fourths of its size, at 9 it becomes a full moon, at 10 a three-quarter moon […] The moon is sometimes said to be crescent at 7 and 5, and is called gibbous at 8 and 10.
Edwin D. Babbitt (1878). The Principles of Light and Color, p. 202. New York: Babbitt & Co.
In Salamandastron the moon is described as “burgeoning” (growing) which corresponds to the more usual term “waxing” for describing a gibbous moon.