How did Elrond figure out the ruin letters were written on Durin's Day in The Hobbit?

In The Hobbit written by J.R.R.
Tolkien, Elrond was an elf lord who "knew about runes of every kind." Bilbo, Gandhi, and
their companion take the map to Elrond to seek his help in reading it. He knows how to
decipher the writing, but the dwarf Thorin explains what Durin's Day
is.

By the light of the moon, Eldrond holds up the map and
proclaims the writing is "moon-letters" that only can be seen when the moon shines
behind them. In some cases, the sophisticated silver lettering invented by the dwarves
can be read only during a "moon of the same shape and season as the day when they were
written." (53)

Based on the letters, Elrond determines the
letter were written underneath a crescent moon in mid-summer a long time ago. Elrond
translates:

readability="8">

"Stand by the grey stone when the thrush knocks,"
reads Elrond, "and the setting sun with the last light of Durin's Day will shine upon
the key-hole."

Elrond asks
the dwarf, Thorin, who is a descendent of Durin, about Durin's Day.Thorin
explains:

"The
first day of the dwarves' New Year," said Thorin, "is as all should know the first day
of the last moon of Autumn on the threshold of winter. We still call it Durin's Day when
the last moon of Autumn and the sun are in the sky
together."