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23a Accepting a role finally that one may do in the afternoon.
T?a ?a?t? Must be tea party but can't quite parse . A role would be "a part" but can't get the tey.
4d An irrational king executed a mathematician . ??l?r Think it is euler. Could be king lear with the a removed but that leaves eu . Even though 52% of the UK considered the EU irrational , I doubt if that is it.

Thanks

"a role" = "a part" inside the last three letters of "that one may."

(tha)t on(e) (ma)y
e (the base of natural logarithms is an "irrational" (number)) + (r)uler

Disturbing as I find the surface of the clue, I believe "irrational" = e and "king" = "rule" anagrammed (executed.)

ginge's parsing makes more sense; executed = beheaded so take the r off ruler.

Executed = beheaded

Hi mathprofrockstar glad you got there, your last post wasn't there when I began typing.

Incidentally, it was Euler who proved that e was irrational, Bernoulli having discovered the constant some years earlier

22a. Perfect sounding local delicacy (6) -N-A-T
Intact is perfect sounding but what about local delicacy ?

INTACT 'sounds like' INN, local + TACT, delicacy