American wire gauge - why exactly this ratio?
It appears that they arbitrarily picked a very small size (they thought) of an even 0.005" (AWG 36) and a very large size of 0.46" (0000) and decided 40 sizes would be enough for practical purposes. That means 40-1 steps, and the result follows. That's why exactly that ratio. 

Why approximately that ratio was picked probably had to do with the trade-offs between having more sizes of wire available vs. what was optimum. As it is, we often don't use 'odd' wire sizes at least in sizes AWG 4 and smaller-- the even ones are enough for most purposes (I have a roll of AWG 39 wire, so they are made). By picking a finer step they avoided the ugliness of half-sizes.