1912- Lilly's improved spiral Calculator
Walter Elsworthy Lilly obtained a 1912 patent in the UK (GB191228603A) for a Spiral model that was substantially the same as the earlier GM Andrews model. See this patent on the International Slide rule page. Just like the GM Andrews model, it has a ten-turn spiral, a log mantissa scale on the outer circumference, and the cursors are engraved with a turn count.
What makes it interesting is that this inventor did away with those hard-to-make counting devices and just stuck with the index scales on the cursors.
I started looking at this patent after making my somewhat modified version of the Andrews model. I have to admit that my "Andrews" reproduction looks more like Lilly's patent than Andrew's.
If you want to make this one, you may use the Andrews PDF and omit that pesky turn counter.
The 10 turn spiral happens to be a popular design for spiral calculators. The earliest I know of was by Nicholson in 1797 (ref: Florian Cajori)
One more point: I was recently re-reading some of Ed. Chamberlain's writings on long scale slide rules (proceedings of the 2011 meeting of slide rule collectors), and noticed that he mentioned this particular instrument. There are apparently two known examples of the calculator in existence so it is possible that this instrument made it to commercial production. My guess is that photo Ed Chamberlain had might be the actual patent model.
Walter Lilly was a professor at Trinity College Dublin. He developed the Mechanical and Electrical engineering programs at a time when the Engineering school at Trinity College only had a civil engineering program. More at Mechanical and Manufacturing Engineering in Trinity College.
I have found one more patent from Lilly, a cylindrical calculator: GB191423968A. He held about a half dozen other patents as well.