I'm kind of curious as to where you're at. I ask this because the quality of white birch varies by locale. Living in central Alaska, I had read about many folks making good bows out of birch, so I knocked down a 6" diameter tree and split out a bunch of staves. The two bows I made broke out in compression fractures up and down both limbs as soon as I passed 22" on the tillering tree. I later learned that the Athabaskan natives of the area made straight bows with string blocks to achieve brace height, so the bows were under no tension at brace. I wondered how, if birch was such a poor wood, why others seemed to have great luck with it. Upon further research, I discovered that the subspecies in my area was exceptionally weak. The subspecies found in the northeastern US appears to be a crossbreed with yellow birch, and is stronger than average and much longer lived than the rest of the species. I've seen a few in Vermont that were nearly 2' in diameter, putting them well over 100yo. Aside from the extreme northwest, white birch across North America is supposed to be good bow wood. The usual recommendation I've seen is to keep the limbs wide, like most of the other white woods. Hope this helps,
Julian