So explain why this doesn't work even at closer distances, say 20 miles of atmosphere (vs. 22.7-mile-thick vertically).
It does work. If you've ever seen a body more than ten or so miles away you would know that it's suddenly becomes a shade of blue, shaded by the thickness of the atmosphere.
Just look at a few distant mountains sometime:
http://www.freenaturepictures.com/assets/images/lores/distantmountain1.jpg
http://www.kauaivacationrentals.com/propimages/hbv19_lan2.jpg
http://image56.webshots.com/156/8/7/26/2694807260058778447elkKlO_fs.jpg
Notice how all of the mountains in the background are shaded blue. The farther the body the more obscured by the atmosphere it becomes.
So why can the surface of other planets be seen clearly through a backyard reflecting telescope with a light filter?
Big enough to be seen at 750x magnification.
Proof?
You can't prove it, because the curvature of the earth does not allow you to see an object 90 miles away, unless the total elevation of you and the object equal almost 10,000 feet (elevation of the object being at the highest point on the object). If you are going to ask for proof again, I really can't help you unless I can convince my friend to send me pics from Florida. I'm sure that a test will show that my statements hold true. If I knew enough optics, I could technically figure out the size of something at a magnification and distance, I'm sure.
The vertical vs. horizontal atmospheric density argument doesn't work either. 50% of atmospheric mass is contained in the lower 5.6 km (3.48 miles) of the troposphere (what we see through horizontally with a telescope in this argument). That means that since a telescope can see fine through the 22.7 miles of vertical atmosphere, it should at LEAST be able to see through 6.96 miles of the troposphere. Yet, you cannot see any point on the ocean past (1.17√a)+(1.17√b) nautical miles [where a is your height b is the height of the object you are looking at, in feet] with or without a telescope. If the earth was flat, height should have no effect on your ability to see something within those 6.96 miles. A hypothetical experiment: assuming you were 5 feet 6 inches tall looking at an object 5 feet high (a Rowbotham experiment number): (1.17√5.5)+(1.17√5)=5.36 nautical miles=6.17 miles. That means that on a clear day, at LEAST 0.79 miles beyond the object should be clearly visible with a telescope, which you will find does not hold true in the real world. Before you attack my 'hypothetical experiment' with 'no proof', here's a real example: http://theflatearthsociety.org/forum/index.php?topic=24914.msg552253#msg552253 Rowbotham also fails to explain why the bottom of objects disappear first as shown in the link (no 'similar' colors there, it's pretty obvious that the lower half of the platform is obscured by the water).
Edit: And he's gone!
It's possible to see bodies through the horizontal atmosphere which are at least 40 miles away. I'm not sure where you got your figure of 22.7 miles from.
Only if the total elevation of you and the object are over 1000 feet (elevation of the object being at the highest point on the object). Once again, I can't prove this without a telescope of course, unless I can get someone with a telescope to send me pics. I'm sure it will hold true if you try.
Also, the vertical atmosphere is a gradient which is thinnest at the top and thickest at the bottom. It's not a constant medium as you are supposing. Even the lower 5.6 km of the atmosphere is a gradient of thickness. There's quite a difference when looking up at the sky and across the surface of the earth.
Furthermore, it's possible for a sufficiently bright light to shine through the density of the atmosphere, which is why on a foggy day it's possible to see the light of a lamp in the distance, but not its pole, the buildings, trees, or scenery around the streetlamp. The celestial bodies above our heads are sufficiently bright to shine through the vertical atmosphere above our heads. Otherwise they would not be seen.
No. The 'thickness' excuse still does not hold, because looking through the entire atmosphere vertically does not explain why you cannot see past 3 miles to a point on the horizon if you are 5 feet 6 inches (notice I said point, a solid object will increase this distance at a decreasing rate based on its height). The gradient you are talking about is nowhere steep enough for this to occur:
http://en.wikipedia.org/wiki/Image:Atmosphere_model.pngThe total vertical mass mathematically must be at least slightly greater than the horizontal distance you can see to the horizon. The mass percentage of the atmosphere at ground level must be 50% or greater for your theory to hold, which it isn't, because the 50% must be spread throughout the first 5.6km. Your light explanation also does not hold, because I'm talking about a clear day. Fog is water vapor, the atmosphere is not always foggy in all places. The surface of Mars and other planets can be seen clearly with a light filter through backyard reflecting telescopes, but the light filter will not remove fog, so this does not explain why objects disappear at less than 6.96 miles on the horizon, through a telescope, on a clear day. Here's pictures of Mars through a simple backyard telescope:
http://img152.imageshack.us/img152/9800/owens1xr6.jpghttp://img211.imageshack.us/img211/1111/grafton1im4.jpgWhy hasn't it vanished or become even partially obscured through the atmosphere?