“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.}

Jacques’ use of “burgeoning” (growing) instead of the more usual term “waxing” for describing a gibbous moon suggests that he might have been going for a sense of estrangement through unusual word choices, or maybe it was a case of elegant variation.

“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.}

“Three-quarter moon” means “gibbous moon”, that is, a moon that is more than half but less than fully illuminated. A citation 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.}

Jacques’ use of “burgeoning” instead of the more usual term “waxing” for describing a growing moon suggests that he might have been going for a sense of estrangement through unusual word choices, or maybe it was a case of elegant variation.

Concern was expressed in comments that this sense of “three-part” is ambiguous: how do you know that it means three parts out of four, rather than three out of five or however many? Well, language is often ambiguous, and people have to use context to disambiguate. The examples given by the OED include

1843   G. Borrow Bible in Spain I. vii. 129   He was half-intoxicated, and soon became three parts so.
1871   M. Collins Marquis & Merchant x   He rides a three-parts thorough-bred.
1878   R. Browning La Saisiaz 72   There's the stoppage at the inn Three-parts up the mountain.
1887   R. L. Stevenson Memories & Portraits xv. 250   Conduct is three parts of life, they say; but I think they put it high.

In “three-parts thorough-bred” the listener has to deduce that it is three of the horse’s four grand-parents that are referenced.

“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.

“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.}

Jacques’ use of “burgeoning” (growing) instead of the more usual term “waxing” for describing a gibbous moon suggests that he might have been going for a sense of estrangement through unusual word choices, or maybe it was a case of elegant variation.