The historic total global human population dataset is available on Wikipedia and provides an opportunity for modelling with simple models such as the exponential and logistic differential equations for population. Using the per-capita population growth rate (PPGR) predicted by these two models and estimated PPGR from the data, we are able to estimate parameters for the exponential model and discover that we cannot estimate parameters for the logistic model. However, we are able to create a new differential equation that models this data using ideas from the derivation of the logistic model. This is the superexponential model that goes to infinity in finite time. While the initial exponential and superexponential models are good for approximating the population data, neither really matches the PPGRs estimated from the data. We switch to a hybrid model that better matches the estimated PPGRs. It is superexponential for the first portion of the data and logistic for the rest. This hybrid model greatly reduces the modelling error and can extrapolate from the data. We note that the model switches between superexponential and logistic around 1960, the year the FDA-approved oral contraceptives, a fascinating historical tie-in with our modelling.