I agree with your idea. It's all about ratios and relationships. Very simple. Forget about all the crazy math, it's a friggin simple arch. Measure the width and heighth of the big arch, divide the width of the bookcase into the width of the big arch. Reduce the heighth by the same ratio. Basically scale it down.
The problem with just copying two feet of the big arch is you won't get much of an arch at just the two feet. It will appear closer to being flat.
The advantage of the math is that it produces a radius for the arch, which you will not get by simply duplicating the arch onto a piece of plywood. This becomes an issue when you have to cut the arch into your bookcase. If you plan to cut it freehand, no problem, you trace the arch onto your wood and cut. However, if you want a better than freehand cut, you need to decide how you are going to actually make the cut.
Assuming you do the "crazy math" and calculate the radius, you can cut the arc using a router and a radius cutting jig (see a recent episode of this old house where Norm Abrams cuts a radius on an outdoor support using a router and a templating bit with a radius jig). This will give you a perfect cut, which you may want since you are building a bookcase, and a freehand cut will be very hard to get to look right.
Alternatively, if you don't care to do the math, you can trace the outline onto your bookcase, draw a perpendicular bisector line, and set the desired radius of cut by eye. If you use a router with a circle cutting jig, you will get a perfect cut.