Divergence time estimation is crucial to provide temporal signals for dating biologically important events, from species divergence to viral transmissions in space and time. With the advent of high-throughput sequencing, recent Bayesian phylogenetic studies have analyzed hundreds to thousands of sequences. Such large-scale analyses challenge divergence time reconstruction by requiring inference on highly-correlated internal node heights that often become computationally infeasible. To overcome this limitation, we explore a ratio transformation that maps the original N - 1 internal node heights into a space of one height parameter and N - 2 ratio parameters. To make the analyses scalable, we develop a collection of linear-time algorithms to compute the gradient and Jacobian-associated terms of the log-likelihood with respect to these ratios. We then apply Hamiltonian Monte Carlo sampling with the ratio transform in a Bayesian framework to learn the divergence times in four pathogenic viruses (West Nile virus, rabies virus, Lassa virus and Ebola virus) and the coralline red algae.  Our method both resolves a mixing issue in the West Nile virus example and improves  inference efficiency by at least 5-fold for the Lassa and rabies virus examples as well  as for the algae example. Our method now also makes it computationally feasible to  incorporate mixed-effects molecular clock models for the Ebola virus example, confirms  the findings from the original study and reveals clearer multimodal distributions of the  divergence times of some clades of interest.