Over 5000 years, the world population has grown from 10 million to 8 billion. This has taken place in two distinct phases that begin with steeper-than-exponential growth due to the population–technology interaction but later came to a halt, blocked by limits in Earth’s carrying capacity. These two phases stand out when data are graphed in a novel way, with time as well as population on logarithmic scales. Each phase can be fitted using a novel “tamed quasi-hyperbolic” T-function. The ancient phase reached a population of 220 million by year 1, dropping to 190 million by year 400. The present phase slowly picked up speed, reaching its maximal relative (percent) growth rate around 1970 and absolute (in million) around 2010. It projects to a population maximum of 11 billion. The entire process conforms to a “PCT model”: population–carrying capacity–technology interaction. Population growth comes from population itself, as long as population is smaller than capacity. Capacity growth comes from increase in technology, as long as capacity falls short of an ultimate carrying capacity. Technology growth comes from technology itself but also from population.