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1a,What a courier brings over (8) ---i---i-s

deliveries

Thanks Xwordfan

4d. An irrational king executed a mathematician (5)
E-L-R Must be EULER but can’t parse.

‘e’ is an irrational number. King executed (headless) is (r)ULER.

Interestingly it was Euler who proved ‘e’ to be irrational

Excellent and thank you jono. Incidentally I have an M.Sc in Maths but wasn’t expecting to see irrational numbers crop up in this crossword ! but there you go.
Indeed Euler considered among the greatest mathematicians ever.