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Wikipedia on American wire gauge reads:

Any two successive gauges (e.g. A & B ) have diameters in the ratio (dia. B ÷ dia. A) of \$\sqrt [39]{92}\$ (approximately 1.12293), while for gauges two steps apart (e.g. A, B & C), the ratio of the C to A is about 1.12293² = 1.26098.

My question is: why exactly this ratio? What's the background behind it?

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    \$\begingroup\$ Since the gage number follows a geometrical progression of successive diameters (1.123) and the ratio of every six gage it is also standard (nearly 2), it is relatively easy to calculate resistance, mass or cross section deducting from the next. So every three gage the resistance, mass per length and cross section doubled or halved. \$\endgroup\$ – GR Tech Jan 16 '15 at 19:25
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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.

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