Leonhard Euler’s Research on the Multiplication of the Human Race with Models of Population Growth

The renowned Swiss mathematician Leonhard Euler created three variations of a simple population projection model, including one stable model and two non-stable models, that consider a couple with different fertility behaviors and life-spans. While one of the models was published by a German demographer, Johann Peter Süßmilch, in his book “The Divine Order”, the other two are not widely known in contemporary literature. This paper compares and reanalyzes the three variants of Euler’s population projections using matrix algebra, providing diagrams and tables of the population time series and their growth rates, as well as age structures of selected years. It is demonstrated that the non-stable projection models can be explained in the long run by their geometric trend component, which is a special case of strong ergodicity in demography as described by Euler. Additionally, a continuous variant of Euler’s stable model is introduced, allowing for the calculation of the age structure, intrinsic growth rate, and population momentum in a straightforward manner. The effect of immortality on population size and age structure at high growth rates is also examined.