Hawk and Handsaw

There's a line in Hamlet (Yes, the Shakespeare play) where Hamlet himself states

I am but mad north-northeast; when the wind is southerly I know a hawk from a handsaw.

I've heard all sorts of explanations for this metaphore, but I figured I'd throw out one of my own.

If Shakespeare really didn't mean to compare a hawk to a heronsaw (a bird, which seems to be the going theory) - later corrupted to handsaw - I propose instead that perhaps he really did intend Hamlet to say "a hawk from a handsaw". (Crazy, no?) Although things may have changed in the last 600 years or so, last time I checked, Hawks have always been known for their sharp eyes, while handsaws are known for their sharp teeth. Thus, I propose that Hamlet can tell those who are his friends (watching out for him), from those who are posed to do him (and his family) harm, with a sharp bite.

Proof? Actually, I have none, but I think it's an interesting alternative interpretation. Too bad we can't knock on William's door and ask what he was really thinking. There must be equally good, or likely better, explanations. Anyone care to propose any others?