What if your scope was mounted on the side, parallel with bore? How would your example work?

Wow, that's a tricky little brain teaser. I guess up and down scope movements would fit my previous example...left and right movements would probably not have any affect. LOS angle would change, but not in the same direction as gravity.

I could be wrong

The sight height would be zero with the line of sight only crossing the bullet path at one point after leaving the gun,but now your windage would only be correct at one distance.

Bullet will start at LOS and fall away never to intersect. If you could raise your scope 1" exactly with no adustment to scope the bullet would start 1" below line of sight and continue to fall away with exactly the same path as the last shot only 1" lower at every point in its path.

So if I made adjustable scope mounts and could move this scope up and down by one inch icrements, wouldn't every time I moved the scope the arc would shift by that much?

I believe the answer your looking for is that sight height is important in calculating trajectory because you are using the line of sight as your aiming reference. If you disregarded the sight entirely and instead relied upon your actual cartridge performance to aim same way as an old artillery man who could detrmine impact by the distance, height and angle of the cannon we would not need to take sight height into effect. However then you would be required to take into account height of the barrel from ground, altitude, and angle of barrel relative to target, all very difficult things to do without tons of practice and all of which would require spotters and many rounds to determine the actual performance.However since its much easier and FASTERfor us to aim by eye with ascope and then fine tunethe point of impact, sight height is important. Its your line of sight which provides the referrence to calculate trajectory and adjust the barrel and bullet trajectory with the aimed for target.Its only because it seems so intutitve that your friend feels he doesn't need to know sight height. Same reason why its much easier to catch fly balls in the outfield when you hold your glove directly in your line of sight( in front of your face)andwaist high basket catches are considered risky show off maneuvers. Think of catching a ball as the inverse of hitting a target with a rifle. you always gotta know where your gloved hand is relative to your face( and why you are always told to get in front of the ball so its path is directly into your line of sight and thus lowering the sight height and minimizing any calculations your brain would have to do and thus making you react faster.) Imagine seeing a ball hit and the simply running to a spot where you calculated it would land and sticking out your glove without looking and expecting the ball to land. like an old fashioned cannon man it would take a lot of balls until you finally put yourself into the right spot for that to work.As with a scoped gun its much easier and faster to simply get into the general area where the ball was hitand then fine tune once your there so the ball is proceeding directly into your line of sight and put your mitt in front of your face and pray. Hope it helps you visualize the difference. Help me if I'm wrong guys, just trying to put it ionto a way that hecan feel and understand the difference. take care.

The scope might be parallel,but you still have internal adjustments that would allow you to have the bullet path cross the line of sight.

Correct about the scope adjustments, but my point was what would happen if you raised and lowered the scope theoreticly(could not be done) without adjustingthe scope.

Or you can shoot your rifle sideways.