Don't think 1st/2nd/etc radius curves, think curve radius in inches. Much easier to figure what will fit.
You need at least an inch clearance from the centre line of straight and wide radius track. On tight curves, you need 1-1/4" to 1-1/2". So, on a
19" (1' 7") wide board you can run a 16" to 16-1/2" diameter circle, ie, an 8" to 8-1/4" radius curve, which is a very tight radius. (Radius is measured to the centre of track.)
If you plan to run the larger locos and scale length passenger cars, that's much too tight. You should be using 11" radius or larger. (I'd use at least 15" radius for such a layout.) If you are planning to run
0-4-0 tank engines (and the smaller 0-6-0Ts), with 4 wheel wagons and 4 or 6 wheel coaches, 8" radius will do. But any rolling stock longer than about 30 scale feet (2-1/2" or so) will give you serious troubles on such a tight curve.
The smallest commercially available radius was around 7-1/2" (Arnold, IIRC). AFAIK, it's no longer available. The most common smallest radius is 9-3/4", which won't fit on your board. All the major manufacturers make track of this radius and larger. Peco's smallest radius is 9", still too large for your board. So if you must limit yourself to a 1' 7" wide board you'll have to use flextrack.
Very tight. Might just get away with it if you select stock which works well together over tight curves (eg. all vehicles about the same lengths and wheelbases, so the couplings move compatible distances).
If you use flexi-track for such a curve you need to lay it very carefully so that its not got a short bit of even tighter radius.
Going out a bit will help no-end. Is there anyway you can squeeze an extra few inches just where the circle would be widest ?
I think Wolf's sums are wrong. If Peco Settrack comes in 9inch radius, that's 1ft6 diameter, so just in the space available (assuming sufficient over-hang space. However, I would agree about preferring 11 or 15inch minimums if at all possible.
Any larger loco will struggle round very tight curves.
Nail a short piece of 2"x1" timber with tapered ends on edge where the curve extends to the edge of the baseboard. Paint them to match the baseboard quickly before anyone notices. Centre the track so your trains miss the back edge and blindly claim you haven't gone over the 17" limit. No one will notice!