Most runners are fascinated by distance and will go to great lengths to 
find out the precise distances of their running courses. Such finicky 
measurements require equipment: EDMs, bicycle counters, GPS
systems, mapping software, etc. At the opposite end of the precision scale,
is it possible to make very rough, but useful, estimates of distance using
nothing but the naked eye?

Long before precisely measured tracks and certified courses, British harriers
set up race courses by choosing a distant church steeple. The course rules
were simple -- get to the steeple by any route available. (Supposedly, this
practise was the origin of the name "steeplechase" for today's cross-country 
like track race.) Thus, visibility of distant objects ultimately determined
the length of the race.  

At sea, visibility is mainly limited by the curvature of the earth. To a person
standing in a rowboat, the apparent horizon is about 5 Km away. (A rough rule
is to multiply the square root of the height in feet of one's eye above sea
level by 1.5. This gives the approximate distance to the horizon in miles.) 
Taller objects, such as large ships, can be seen from further away, especially
by an observer standing some distance above the waves (perhaps on the deck of
another large ship.) Seafaring observers in ancient times were aware of the
curvature of the earth because masts of approaching ships became visible well
before the hull appeared over the horizon. The practical limit for visibility
of objects at sea appears to be about 10 nautical miles (11.5 statute miles.)

Similar considerations apply to visibility over land in very flat regions such
as deserts or the great plains of the United States. From most locations,
however, tall objects such as hills or mountains can be viewed from much
greater distances. In such cases visibility is limited by the clarity of the
atmosphere. Dust, aerosols, water droplets, and other impurities in the air
scatter light and reduce the contrast between the outline of distant objects
and the background sky. Over sufficiently long distances the contrast degrades
so much that the object cannot be distinguished from the sky and becomes
effectively invisible. (Visibility of lights at night is limited by
absorption.)  

Under typical atmospheric conditions, land visibility of distant mountains
ranges between 20 and 100 miles. Under exceptional conditions peaks up to
250 miles away have been sighted. For example, as I write this article on
June 4, 2001, the web camera located at

	http://www.aqd.nps.gov/ard/parks/grsm/lookRockWeather.htm

shows the outline of distant peaks in the Great Smokey Mountains. The site
reports that visibility today is 41 miles. More information and real-time views
from other locations that track visibility can be found at
http://www.epa.gov/air/vis/. 

Here is a simple procedure, based on elementary trigonometry, for making very
rough distance estimates with the naked eye: ordinary telephone poles are
about 40 feet tall. A person's thumb (about 2 inches long) held at arm length
(2.5 feet) will just cover a telephone pole that is about 500 feet away. To
pick out a point that is about a half mile away, find a telephone pole that
is covered by one's thumbnail (about 1/2 inch) held at arm's length. For a mile
use one half of one's thumbnail, etc.