Kapteyn thought that because he couldn't see any stars farther than 5 kpc away, that there were no stars farther than 5 kpc away. In fact, the scattering of light by interstellar dust keeps us from seeing more than 5 kiloparsecs away in the disk of our galaxy. Shapley noted that the globular clusters that swarm around our galaxy all tend to lie on one side of the sky, centered on the constellation Sagittarius. If the Sun were at the center of the galaxy, we would see roughly the same number of globular clusters in all directions. Shapley was also able to measure the distances to globular clusters.
Globular clusters are too far away to measure their distances by parallax.
However, globular clusters contain RR Lyrae variable stars. Shapley's plot of the distribution of globular clusters is shown below. The yellow blotch where the axes cross is the position of the Sun. The red X off to the right is the center of the galaxy, as defined by the globular clusters. Shapley concluded that the center of the galaxy is 17 kiloparsecs away.