I was in one of those glows; you know, the one you get 
when someone compliments one of your photographs.
  "Yes, very nice," the area professional repeated.  At 
that point I felt very confident in my work and knowledge 
of photography.
  "By the way," he added, "what was your exposure for this 
photograph?"
  "1/250 of a second at f4," I explained - and then it hit 
me.  I knew what 1/250 of a second was; it was a short 
fraction of time.  I understood what film speed was and 
it's exposure requirements compared to other film speeds. I 
had memorized my f-stops and understood what would happen 
if I used larger or smaller aperatures.  But my balloon 
popped when I pondered that age-old questions that plagues 
all photographers sometime in their life, "Just what the 
heck DOES f4 mean?  F is for what?  Just what are there 4 
of?"
  I didn't feel as smart as I had just a few moments 
before.  I waved and grumbled my thanks as I left the 
photography studio; the photographer now perplexed as to my 
sudden change in mood.  It was at that moment that I 
realized that "F" stood for "Fool."
  I went directly to the library; it can be a valuable tool 
for photographers, especially if you look around the 770's 
in the Dewey Decimal System.  I borrowed three books and 
headed home to begin what was to be a week of study, and 
finally, comprehension.
  "F," I learned, could actually mean "fraction".  The "4" 
in "f4" actually stood for 1/4.  The diameter of the 
aperature at f4 would be 1/4th the focal length of the 
lens.  A 50mm lens set to f4 would have an aperature 
diameter of 12.5mm (50mm X 1/4).  A 1000 lens set at F22 
would have an aperature diameter of 45mm (1000mm X 1/22). 
So with this information, you could derive the formula: 
F-Stop = A/FL; where A is the diameter of the aperature, FL 
is the focal length of the lens, and F-stop is expressed in 
it's fraction form.
  So we now know what the numbers mean, but why do we use 
THOSE particular numbers?  Before we can answer that, we 
have to go back in time and get a little history.  
  
  When the first lenses where used in cameras, a 50mm lens 
was actally 50mm long, a 1000mm lens was actually 1000mm 
long.  Lenses where basically nothing more than a convex 
lens at the end of a tube.  A 50 mm lens would be a convex 
lens at the end of a 50mm tube, and so on....  The above 
formulas would be directly applicable to these types of 
lenses.  But as photography became more sophisticated, 
photographers tired of lugging around huge long lenses. 
  Techniques were developed to use multiple elements in a 
lens to make the effective focal length of a lens much 
longer than its real lengh (example: a 1000mm lens could 
now be made 250mm long).  The above formulas cannot be 
directly used on modern lenses to determine exact aperature 
diameter, but they can be used to express the ratios 
between different aperatures.
 
  Ok, with that out of the way, let's get on to the 
"Inversed Squared Law" (ISL).  Part of the ISL states that 
the area of a circle is directly proportional to the change 
of the diameter squared.  Therefore, if we multiply or 
divide the diameter of an aperature by 1.4, the area of the 
aperature would be twice as big or half as big as it was 
before (1.4 squared equals 2).
  From here it is easy to see that if the area of the 
aperature is twice or half of what it was before, it would 
let in twice or half as much light in; which would equal 1 
stop of exposure either way.
  
  *So to put it in a nutshell: if you multiply or divide 
you aperature diameter by 1.4, you get 1 stop more or less 
in exposure.*
  
  Now, why do we use the f-stops we do?  For ease of 
computation, let's start out with a non-descript lens with 
a focal length of FL.  Let's give the largest aperature 
that lens could have a diameter of FL.  We could use our 
formula: F-stop = Aperture Diameter/Focal Length to get: 
F-stop = FL/FL, which would equal 1 (or 1/1).  So the 
maximum aperature for this lens would be f1.
  To find the aperature that would give us one stop less 
exposure than f1, we would divide the aperature by 1.4; 
this would be 1 divided by 1.4 which is 1/1.4; so the next 
aperature for this lens would be f1.4.
  To find the aperature that would give us one stop less 
exposure than f1.4, we would divide the aperature by 1.4; 
this would be 1/1.4 divided by 1.4, which would equal 1/2. 
So the next aperature for this lens would be f2.
  So far our f-stops on this lens are 1, 1.4, and 2.  If 
you continued this process, you would find that the 
following stops would be 2.8, 4, 5.6, 8, 11, 16, 22.  The 
reason that lenses don't have a maximum aperature of f1 is 
that it is nearly impossible to have a maximum aperature 
diameter that is equal to the focal length of the lens.