It is cross-linguistically common for languages to undergo a diachronic increase in the range of adverbial notions that can be expressed by means of infinitival constructions, and the Romance languages are a good example of this process. Examining the development of adverbial "prepositional infinitive" constructions in Spanish, Portuguese and Romanian, this paper draws on diachronic corpus data to provide detailed evidence for the increase of such constructions, both in terms of types and tokens, over the centuries. It is shown that, in all three languages, different prepositions begin to appear in the infinitival construction individually at different times, before experiencing a gradual rise in token frequency. It is of particular interest that the order in which different prepositions, expressing different adverbial notions, begin to participate in the construction is very similar in all three languages, despite the fact that we are clearly dealing with separate, unconnected developments. A hierarchy [purposive greater than abessive greater than temporal greater than concessive] can be identified, in which the purposive notion is the earliest and most frequent to be expressed by means of an infinitival construction. The second part of the paper is concerned with identifying what causes this development to proceed in such a parallel fashion in different languages, showing that subject coreference between main and adverbial clause is a crucial factor. A diachronic survey reveals that subject coreference is statistically far more likely with some adverbial notions than with others, and that this likelihood is remarkably stable through time and across languages. Those adverbial notions with a greater likelihood of coreference are the earliest and most frequent to be expressed by means of an infinitival construction. It is therefore concluded that the frequency and statistical likelihood of real-world usage patterns is responsible for the hierarchical sequence in which specific adverbial notions come to be expressed infinitivally, and for the overall spread of these structures. Furthermore, it is argued that the cross-linguistically common association of certain adverbial clause types (notably purpose clauses) with infinitival structures is not primarily due to the infinitive's lack of overt tense morphology, but rather to its lack of morphological subject agreement. (Contains 4 tables and 14 figures.)