I haven't taken a math course since 1972, when I nearly failed the New York State Regents Examination for trigonometry and algebra.
But to figure out the stem length and angle for the new bike project, I needed to use trig. The stem can be imagined as the hypotenuse of a right triangle, with the level extension from the steering tube as one side (the "adjacent", if you remember your trigonometry), and the rise (the difference of the stem height at the steering tube and at the stem's far end) as the other (in trig terms, the "opposite").
I know I want a slightly more relaxed position on this bike than I have on the Yellow Maserati, my titanium bike. I don't want any more extension (the distance my hands are from my hips), and I'd like a little more rise (the distance my hands are above the ground).
I know that the top of the head tube on the project bike is about 19mm lower than the top of the head tube on the Yellow Maserati (the head tube is shorter, but the fork is longer). I can get a stem with a 17º rise (instead of the 6º rise of the stem on the Yellow Maserati), but how will that change the rise and extension?
Well, with a little help from DuckDuckGo (I'm reducing my use of Google in the interests of preserving some privacy), I found this site, where I was able to get enough of a refresher on trig to figure the missing numbers in my imaginary triangle. It turns out I'll want about 40-45mm in spacers above the headset, and the 110mm stem length will bring my hands up about 2cm and back about 4mm (which I can nearly ignore). A 120mm stem would bring my hands higher, but the extension would actually be longer than the Yellow Maserati's, and I purposely bought a shorter stem for that bike because I felt too "drawn out". So the 110 stem it will be.
The name of the linked site where I got the math refresher is Math is Fun. I'm not sure I believe that, but it is useful, if you can get your brains around it. I'm making this a "stuff that works"page... because it did.