The aspect ratio information you seek can be found in the Aspect Ratio Charts in the back of the book. Page 139 explains how to use them to determine aspect ratio without the need for a calculator (yay, no math!).
I'd like to point out that even though jump ring sizes for each weave are suggested in both AWG and SWG, here in the US, SWG jump rings that are commercially available come in a limited range of inner diameter sizes. Therefore, I suggest using the AWG sizes to determine the aspect ratio. You'll get a closer estimate this way.
Here's an example of how you can use the charts to determine what inner diameter (mandrel size) will work for a specific weave, using a specific gauge of wire. Suppose you want to make a Byzantine chain in 20g SWG. The book suggests sizes for 16g and 18g wire only. The recommended AWG sizes for Byzantine are 16g/4.5mm or 18g/3.5mm. Turn to the 16g AWG AR chart in the back of the book, and look in the ID column for 4.5mm. Follow that row across to find the rounded AR is 3.5. Do the same with 18g/3.5mm and you will see that the rounded AR is 3.4. Therefore, the suggested aspect ratio for Byzantine is 3.4-3.5. Since AR is a ratio and not a measurement, you can simply turn to the 20g SWG AR chart and look in the AR column for the number that is closest to 3.4-3.5. Follow that row across to find the inner diameter (mandrel size) that will work for the weave with that gauge of wire - 3.25mm is the closest id/mandrel size. Just by flipping a few pages (no calculator, no math) you can get a recommendation to a different size and convert to a different gauge scale.
If you want to use wire sizes not covered in the charts, the mathematical formulas you need to make the necessary calculations are also included in the book on page 139.
Be aware that slight variations can exist in tools, wire and jump rings from manufacturer to manufacturer and from vendor to vendor, so use the recommendations in the charts as a starting point for testing the materials that you intend to use. From there you can make the necessary adjustments to fit your unique project. I hope you find this information helpful.