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Of all the puzzles I do, I don't often get problems with this one, but need to confirm these:-
7 Down - "One line of mathematicians with popular appeal". (6)
I have I-O-I- (can only think of IRONIC or ICONIC)
8 Down - "Character in middle of manic state?" (6) N-T-E-
(Maybe NUTTER or NETHER?)
24 Across - "State of excitement that's less half-hearted! (5) -E-E-
(Surely FEVER?)

8 Nutter - character in middle of manic' gives N, maNIc, then add Utter, state, say.

24 Yes, Fever, state of excitement. The rest is that Fewer, less, is half-hearted - ie the W in that becomes a V (half of W!), so fever, state...

7 I think mathematicians refer to Conic things, lines, and I = One, so Iconic.

It'll be ICONIC. one = I; conic = mathematical "line" (is elipse etc)

...and Iconic, with popular appeal.

Many thanks, Malone - a big help, as always! Also thanks to Jolan. The
clues were a bit sneakier than normal! Please stand by for my posting about the Skeleton, the most wily of the lot!

Rocketman, glad we could help. I never see the Skeleton, but like attempting clues from new puzzles - note the 'attempting'.!