mounting a rifle scope ?

The height over bore is not a major consideration as far as how the rifle will shoot, but can be a huge consideration as to how the rifle fits you, and also a consideration as to how the bolt handle clears the eye piece (on a bolt action)

What rifles are you talking about and what scope?
 
The height of the scope should be determined by where your eye is, with a proper cheek weld on the stock.
And that's a whole 'nuther topic altogether.
 
Important and there is more !!!

How important is it to mount the scope low and close to the bore ?
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Just taking your post at face value, it is very important. Now then, it doesn't stop there as there are "additional" considerations. There are some excellent videos in YouTube that might help with any additional questions that you will have. I can send you one link, as you wish. ..... ;)

I once had what I thought, was a perfect mount until I tried to install the objective scope cap. I had to cut away a bottom section of the cap. I see that Leopold makes a scope whose front bell, partially straddles the barrel. .... :)

Be Safe !!!
 
A scope can only be as low as the front ocular allows. Too high creates cheek weld and sighting issues. Those see through rings do it instantly.
 
You bet. Those see through rings were one of the dumbest ideas to come along. I don't know how many people I have run into that put them on and then never bothered to see if the iron sights were lined up. I do remember one guy that tried it when scopes were steel tubes with no rubber at the eyepeice. Cut a ring in his forehead.
 
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