This paper is about infinitival clauses and their subjects in Ancient Greek. AncientGreek has a tripartite paradigm that involves: (a) infinitives with overt accusative subjects(distinct from main-clause subjects and also coreferential but emphatic), (b) infinitives with nullcontrolled subjects that copy the case of their controller (resulting in the operation of caseagreement across copula [CAAC]), and (c) infinitives with null accusative subjects that arereferential and arbitrary. To account for this, I first argue that all infinitival clauses are CPs.Arguments for this include the modal distinctions among Ancient Greek infinitival clauses, thecoordination of infinitival clauses with finite embedded clauses, the existence of infinitivalclauses with overt complementizers, evidence from binding of infinitival subjects as well asenclitic focus particles in Ancient Greek infinitives. Although all infinitives are CPs, I arguethat there is a further distinction between strong- and weak-phase CPs, with phasehood beingrelated to features in the left periphery of the clause. Infinitives with overt and null accusativesubjects are strong phases, C*Ps, whereas control infinitives are weak phases—CPs that aretransparent domains and can therefore allow case agreement to operate across a clauseboundary. I also compare Ancient Greek to Latin and argue that the distinction between strongandweak-phase CPs is also found in the finite domain. The main implication of this proposal isthat the availability of a subject is only a property of a clause, defined as a strong-phase CP.