I gotta agree that the OP makes it needlessly complicated.
It's like this...
The F stop refers to the size of the hole that the light goes through. Being a hole, it has two dimensions - height and width.
If you take any shape and enlarge it so it becomes 1.4 times as wide and 1.4 times as high, then it's area will roughly double. So, for example, you can have a square that is 1 foot wide by 1 foot high, and it has an area of 1 square foot. If you want to have a square with an area of two square feet, you can't just double the width and height - then you get four square feet. Instead of multiplying the dimensions by 2, you multiply them by 1.4. A square that is 1.4 feet by 1.4 feet is about two square feet in area.
And if you have a hole that is letting light through, and you increase its size by 1.4 like this, then it will let about twice as much light through. In other words, it lets in an extra stop of light.
And that's why we use that progression of numbers in f-stops.
Easy.