Why are f-stops not linear?

Why don't we use a linear scale for f-stops, rather than one based on square roots? A linear scale will be easy to remember (in fact, there will be nothing to remember) and to calculate how much brighter one stop is over another. That is, f/6 will have half the brightness as f/3, which will have two-thirds the brightness of f/2.

This will still have the advantage of the current f-stop scale, which is that we can compare them independent of focal length / angle of view.

What am I missing? Or is the only answer to my question that it's an accident of history?

Answer

The f number is the inverse square root of the light collection efficiency. Why such a crazy relationship?

First - let me define "light collection efficiency". You build an image by collecting photons (light particles) with your lens, and focusing them onto the film / sensor. If you double the area of your lens, you collect twice as many photons - so you can increase your shutter speed by 2x and still collect the same number of photons as before.

Now the f number is the ratio of the focal length of a lens to its diameter : that's a nice easy thing to measure. A 25 mm diameter lens with a focal length of 50 mm has a theoretical f number of 2.0. In practice it will be less, because you lose some light on the surfaces of the lens.

If you double the focal length, the image you create is bigger. In fact, the area of the sensor (for the same field of view) would have to be 4x greater to see the same image. Consequently, each "pixel" on your film/sensor sees 1/4 of the photons it saw before.

Similarly, if you double the diameter of the lens, you increase the area of the lens by 4x. If you do both - double the lens diameter, and double the focal length - then you collect 4x as much light and have to distribute over 4x as much area, so light per unit area stays the same: you have the same "light collection efficiency".

If you just want a factor 2x change in light collection efficiency, then you need to change the f stop by the square root of 2. And this is why you have the series that we tend to use: 1.4, 2.0, 2.8, 4.0, 5.6, 8, 11, 16, 22, ...

The answer to your question, then, consists it two components:

1) Historically, the light gathering power of a lens was most easily described by the ratio of focal length to lens diameter - it is something that is easy to measure. That f-number became the standard, and several attempts to dislodge it have been unsuccessful.
2)And the reason that this (generally accepted) scale is not linear is explained above.

So why don't we use the square of the ratio? Sometimes that would make sense - at other times it would not. The math can actually be easier this way. For example if you have a flash with a guide number of 45 at an ISO of 100, you can compute the f stop needed as (guide number / distance). To shoot a subject at 12 meters you would need a f stop of 4. At 4 meters you would need 11. On the other hand if you don't use a flash then there is no advantage on the math: a shutter speed of 1/125 and an f number of 8 gives the same exposure as 1/250 with f 5.6 - in other words shutter speed times f number squared is constant. I agree that is not helpful.

So back to the original answer. "Just because history".