This version of post #20 is probably easier to read. I removed a few typos and did some revision.
The common rule is a simplification that works fine for small formats, but becomes increasingly inaccurate with increasing format size. More correctly, the rule should depend on magnification, which is where the focal length comes into play. It is, of course, the angular sweep of the lens axis during the time in which the shutter is open that must be limited to some practical value that is the same for all formats and lenses.
The general magnification in terms of focal length and format dimensions is easily reckoned. The format dimension chosen can be: minor dimension, major dimension, or the diagonal, as you choose. In the following, I’ll use the format diagonal as it is most naturally compared to the focal length of the lens.
This will yield the denominator n of the reciprocal representation of the shutter speed 1/n seconds as handheld speeds are most often used.
The magnification is m = f/d where d = format diagonal and f = focal length of the lens.
In general, the required shutter time is
(1) t = k*(d/f)
where k is some constant empirically determined long ago that gave rise to the generic “rule.” The value of k should remain constant for all formats and lens choices to limit the lens to the same angle swept by the lens axis during the shutter-open period. Note that k can be altered so that the results are satisfactory for a particular user.
Using the common situation of a 50mm lens on the 35mm format (24mm x 36mm), the diagonal of the format is 43.3mm. The rule has the minimum safe handheld shutter speed as 1/60 second (actually 1/64 second).
Solving equation (1) for k we get
(2) k = t*f/d
Using the values in the above situation
k = (1/64 second)*50mm/43.3mm = 0.01804
Of course, if we’d used a different format dimension we’d obtain a different constant. That would work fine, so long as we did so consistently for each format considered.
Substituting this value for k in equation (1), we get
(3) t = 0.01804*d/f
Does this work?
For the 50mm lens on the 35mm format (d = 43.3mm) we get
t = 0.01804*43.3mm/50mm = 0.015625 seconds
But we want the denominator n in order to easily select an appropriate shutter speed, so, using the reciprocal key on a calculator,
n = 1/0.015626 = 64
i.e. we want 1/64 second shutter speed which is conventionally marked “1/60” second on the shutter-speed dial..
We can simplify the calculation by reciprocating equation (3) to get the final form of the equation for n.
(4) n = 1/t = 55.424*f/d (expressed as the denominator n)
On the 35mm format we get the following combinations of focal length and minimum handheld shutter speeds:
35mm, 44.8 (1/44.8 seconds, so use the next faster speed, 1/60 second)
50mm , 64 (1/64 second which is the true value of the speed conventionally marked “1/60” second).
100mm, 128 (1/128 second, conventionally marked “1/125” second.)
200mm, 256 (1/256 second is conventionally marked "1/250" second.)
For the 6 x 7cm format (56mm x 69.5mm, diagonal about 90mm) we get:
50mm lens, 30.79 (indicates about 1/31 second. 1/30 second is probably close enough).
90mm, 55.42 (use 1/60 second)
180mm, 110.8 (use 1/125 second).
For the 4” x 5” format (95mm x 120mm, diagonal = 153mm):
150mm lens, 55.4 (use 1/60 second)
300mm lens, 108.7 (use 1/125 second).
Note: The common version of the “Reciprocal of Focal Length Rule” fails badly with increasing focal length. This is obvious when you examine the 4" x 5" results.
So long as you’re satisfied with the results of using a 50mm lens on a handheld 35mm camera at 1/60 marked shutter speed, this formula translates accurately to any format. You can always tighten or relax the standard by altering the constant k as the results dictate. Increasing k gives faster times, while reducing k recommends slower speeds. As has been pointed out, some folks can get good results handheld at slower shutter speeds than others. Too, some cameras or camera-and-lens combinations are much easier to hold steady than others, which can be exploited as safely using a slower speed than otherwise indicated.
The main idea is to limit the angular sweep of the lens axis during the shutter-open period to the same value for all formats and any lens of known focal length f.