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What does the natural in the last C in the second line cancel?
This is from Schumann's Album für die Jugend.
Since the natural is immediately followed by a sharp, which is in the key, it does not make sense to me unless it is cancelling the double sharp on the the second C in the same bar or something, but that double sharp should not affect one an octave below, afaik.
It's a courtesy accidental to make extra clear that the double-sharp doesn't apply. There is sometimes confusion, especially with beginners, about whether an accidental in one octave applies to other octaves. You're correct that it's otherwise unnecessary.
The notation is included in Henle's Urtext edition, so we can suppose Schumann wrote it that way originally, particularly since there is no mention of it in the critical commentary (PDF page 6). A quick perusal of editions on IMSLP shows that the German second edition does not include the natural — just the sharp — but Clara Schumann's edition, which is considered authoritative, does.
As Aaron states, it's a courtesy accidental. Completely unnecessary in theory, but put there to remind the reader that although the upper, earlier C is a double sharp, that one is not. However, since it's in a different octave, it would never have been affected by that x anyway - accidentals always only affect the notes in the same octave.
Given that the key signature already contains C♯ (all C notes should be C♯ unless changed), there should be no need to put anything before that lower C(♯). And, if it's a cautionary/courtesy accidental, it would help to be in parentheses - and maybe even not need the natural sign, only the sharp, telling it's reverted to 'normal' for that piece.
