Linguistic diffusion is commonly equated with contact, and contrasted with genealogy. This article takes a new perspective, by showing how diffusion lies in fact at the heart of language genealogy itself. Indeed, the Comparative method has taught us to identify genetic subgroups based on sets of shared innovations; but each of these innovations necessarily had to diffuse from speaker to speaker across a network of then mutually intelligible idiolects. Such a diffusionist approach to language genealogy allows us to model language change as it really took place in the social and geographical space of past societies. Crucially, the entangled isoglosses typical of dialect continuums and linkages (Ross 1988) cannot be handled by the Tree model, which is solely based on divergence; but they are easily captured by a diffusionist approach such as the Wave model, where the key process is convergence. After comparing the theoretical underpinnings of these two models, I introduce Historical Glottometry, a new quantitative approach aiming to free the Comparative Method from any cladistic assumption, and to reconcile it with a wave-based analysis. Finally, data from a group of Oceanic languages from Vanuatu illustrate the powerful potential of Glottometry as a new method for linguistic subgrouping.