15. Assume that the center of the circle is at the origin $O(0,0)$, and that the points $A_{1}, A_{2}, \ldots, A_{1975}$ are arranged on the upper half-circle so that $\angle A_{i} O A_{1}=\alpha_{i}\left(\alpha_{1}=0\right)$. The distance $A_{i} A_{j}$ equals $2 \sin \frac{\alpha_{j}-\alpha_{i}}{2} = 2 \sin \frac{\alpha_{j}}{2} \cos \frac{\alpha_{i}}{2} - \cos \frac{\alpha_{j}}{2} \sin \frac{\alpha_{i}}{2}$, and it will be rational if all $\sin \frac{\alpha_{k}}{2}, \cos \frac{\alpha_{k}}{2}$ are rational. Finally, observe that there exist infinitely many angles $\alpha$ such that both $\sin \alpha, \cos \alpha$ are rational, and that such $\alpha$ can be arbitrarily small. For example, take $\alpha$ so that $\sin \alpha = \frac{2 t}{t^{2}+1}$ and $\cos \alpha = \frac{t^{2}-1}{t^{2}+1}$ for any $t \in \mathbb{Q}$.