The author considers a certain class of algebraic solutions of the sixth Painlevé equation

${P}_{VI}$ (in Hamiltonian form), for which he presents a determinant formula. The entries of the determinant are essentially the Jacobi polynomials. The well known fact that each of the Painlevé equations can be obtained from

${P}_{VI}$ by a coalescence procedure is then used to obtain, from this family of algebraic solutions of

${P}_{VI}$, rational solutions of

${P}_{V}$,

${P}_{III}$ and

${P}_{II}$. Finally, the author considers the connection with the Umemura polynomials for

${P}_{VI}$.