A precisely additive scheme for describing proton sponge basicity is presented as the sum of the proton affinity of an appropriate reference monoamine, the strain released on protonation, and the energy of the intramolecular hydrogen bond formed on protonation. This approach is then tested at the B3-LYP/631+G**//HF/6-31G** level on six diamine proton sponges (including two novel compounds) that are polycyclic aromatic hydrocarbon derivatives. A key result is that the loss of destabilizing strain energy on protonation is seldom an important contribution to enhanced basicity, and in some cases an increase in strain energy can actually take place which acts to lower the basicity. The scheme is further tested and discussed in the context of other types of proton sponge, including a bridgehead (bicyclic) diamine, a tricyclic tetraamine, and a "resonance-stabilized" vinamidine proton sponge. Linear relationships found between basicity, hydrogen-bond energy, and structural parameters of the free bases and protonated cations offer the possibility of estimating basicity purely from structure.