In the colleges of Cambridge before world war II, what is the meaning
of the degree BA Cantab?

Hello addie09 and thank you for your question.

" 'Cantab' is the abbreviation which, placed after such academic
letters as B.A. or M.A. indicates that the degree in question was
obtained in Cantabrigiensis - the Latin for Cambridge."
http://www.cantabile.co.uk/members.asp

Also see:

"Cantab.
abbreviation 
1. Especially used in abbreviations of academic degrees:
Cantabrigiensis (Latin), belonging to Cambridge.

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THX1138

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short, clear and extremely helpful. Precisely what I needed.