Here's the fourth light rule again.
Light intensity falls off:
Gradually, if the light source is far from the subject.
Quickly, if the light source is close to the subject.
• It's a sunny day.
• You're doing a portrait.
• You're at the Empire State Building.
The exposure at street level, in the sun, is f/8 at 1/1000th of a second.
You go up to the observation deck.
The exposure is the same.
When the light source is far away, like the sun, the intensity of the light source is about the same everywhere.
• It's Thanksgiving.
• You're photographing people seated around a twenty-two-foot-long table.
• You're using the pop-up flash on your camera.
Uncle Fud is sitting near you.
Smelly Aunt Dotty is sitting about twenty feet away at the other end of the table.
There's a huge difference in the exposure on Uncle Fud's bald head and Aunt Dotty.
That's because the light source is close to the subjects.
Because your flash is close to Uncle Fud, and far from Aunt Dotty, there's a huge difference in exposure.
The best exposure for Uncle Fud would be f/16.
By the time light from your flash gets to Aunt Dotty, the best exposure is f/1.4.
So, if you expose for Uncle Fud, Aunt Dotty will be too dark.
If you expose for Aunt Dotty, Uncle Fud's bald head will be blaringly overexposed.
Light intensity decreases according to the Inverse Square Law:
Light intensity falls off as the inverse square of the distance.
1.41 is the square root of 2.
Multiply the distance by 1.41 to double the amount of light, or by .707 (1/1.41) to halve the amount.
Here's a chart of the above Thanksgiving table example.
Note how the slope, the rate of change, is steep when the subject is close to the light source.
So, if you're able, place the light source as far away from the subject as possible.
Or, light the space more evenly by bouncing the light off of the ceiling.
For example, I aimed my Nikon SB800 flash off of the ceiling at this meeting of my men's group.
Here's an explanation of this light rule from 1849, written by Henry Snelling, in the History and Practice of the Art of Photography.
Light also moves with great velocity, but becomes fainter as it recedes from the source from which it emanates; in other words, diverging rays of light diminish in intensity as the square of the distance increases.
For instance let a in Fig. 1, represent the luminous body from which light proceeds, and suppose three square boards, b, c, d, severally one, four and sixteen square inches in size be placed; b one foot, c two feet, and d four feet from a, it will be perceived that the smallest board b will throw c into shadow; that is, obstruct all rays of light that would otherwise fall on c, and if b were removed c would in like manner hide the light from d — Now, if b receives as much light as would fall on c whose surface is four times as large, the light must be four times as powerful and sixteen times as powerful as that which would fall on the second and third boards, because the same quantity of light is diffused over a space four and sixteen times greater.