Two striking differences between humans and our closest living relatives, chimpanzees and gorillas, are the size of our brains (larger by a factor of three or four) and our life span (longer by a factor of about two). Our thesis is that these two distinctive features of humans are products of coevolutionary selection. The large human brain is an investment with initial costs and later rewards, which coevolved with increased energy allocations to survival. Not only does this theory help explain life history variation among primates and its extreme evolution in humans; it also provides new insight into the evolution of longevity in other biological systems. We introduce and apply a general formal demographic model for constrained growth and evolutionary tradeoffs in the presence of life-cycle transfers between age groups in a population.