My reference for Middle-earth's moon phases came from here: http://shire-reckoning.com/moon.html.
It was a good thing the paper-stock was hardy, or it would have never have survived all the long and weary way from Khazad-dûm to Dunharrow. Camped under a bright full moon, the Grey Company were slowly retiring, preparing to take the Paths of the Dead in the morrow. Gimli alone had wandered furthest from the group, ostensibly to take his own rest. Instead, he passed to the edge of the camp, where almost no-one was to be found.
He had kept the little sheet of paper well ever since he'd found it, wedged unobtrusively under a stone in the dark of Khazad-dûm, close by Balin's tomb. Nobody had seen him slip the little page safely beneath his clothing. It was not that he desired to keep it a secret, but a part of him wanted this vestige of his kinsmen's halls all for his own.
Perhaps it was foolish of him - but something in his heart told him the blank page kept a closely-guarded secret. His fingers were rough, but they were Dwarven fingers, and they had picked up the difference in several places of the parchment, though he knew not what was written. Every day hence that he was able, he'd quietly removed the paper from where it was secured on his person, wrapped in rough cloth, and carefully unfurled it below the light of the moon. It was a fool's hope, maybe, but trial and error was now his only recourse.
"Have you had any luck tonight?"
Gimli didn't acknowledge the elf, who'd wandered up noiselessly behind him. Unlike Aragorn, he, of course, hardly ever slept, and so had observed Gimli's little ritual several times before. It was not that Legolas' presence upset him. Rather, Gimli disliked having to veil his distress as, time and again, the light of any moon revealed nothing.
The round moon beamed down bright on the land. He drew forth the parchment, now terribly battered and a little torn, and held it out in front of him, ready to fold it again one disappointed moment later.
To his utter amazement, the centre of the parchment began to glow. So there really were moon-letters - and tonight was the very night to read them!
"By my beard...!" he exclaimed. "Legolas, come here!"
He didn't even have to finish his summons before the elf was beside him, bending a little to peer at the writing that was revealed.
"Can you read it?" he asked, almost eagerly.
"I permit you a fair amount of teasing, Master Elf, but do not insult the great effort my father put in to educate me."
Legolas sighed and cast a glance skyward, muttering something in Sindarin under his breath about perpetual obstinacy, and Great effort indeed!, and being unsurprised, before returning his gaze to the parchment. "Go on then. Demonstrate this fabled education."
Gimli cleared his throat, stood up straight and sure, and took a moment to properly scan the words. He almost dropped the page in disbelief.
"What?" his elf-friend urged, as Gimli began to let out a string of mumbled curses in both Khuzdûl and Westron.
"All the way from Khazad-dûm," he grunted, shaking his head. "Just to find this! Nonsense. A little useful, maybe, if such things could be found in these parts, but nonsense..."
"If you do not say what you've read," Legolas said, "I shall not allow you to share my horse in the morrow, and you will be obliged to walk all the way."
Gimli squinted at him, his face red. Who his fury was directed at - the elf, or the sheet of parchment trembling in his hand - it was hard to say.
"I am Balin, son of Fundin," he quoted through gritted teeth. "Let it be noted here the directions required to produce an inimitable salted pork, first devised by my late mother, as precious as any treasure of this Realm, if not greater in value, and consigned to the realm of secrecy, lest it be thieved..."