Hume was deeply concerned to distinguish empirical reasoning from math and logic (matters of fact vs relations of ideas). Induction doesn't work in the sense that it doesn't guarantee its conclusion but of course even Hume thought it works whenever you are ok with something less than a guarantee (which he thought we are most of the time).

Also, the problem of induction is only one part of this marvelous tome. Protip: get the new Fate-Norton Oxford edition and not the Selby-Bigge-Nidditch.