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That “a population subject to invariant age-specific mortality and fertility will asymptotically approach an exponential rate of balanced growth or decay with a stable relative age distribution” was first established in 1911 by Lotka and Sharpe. (Euler had clearly glimpsed this truth in a private communication of around 1760.) Bortkiewicz proved in 1911 the quite different and more obvious proposition: ‘Any population subject to invariant age-specific mortality, must be growing exponentially in all its parts.” Kuczynski, an eminent demographer in his own right, understandably irritated Lotka by his repeated attributions of the Lotka result to Bort­kiewicz, thus leading Lotka to deny incorrectly the truth of valid third theorems like the following: “A population with invariant age-specific mortality and with exponential growth in its total numbers will asymp­totically approach a stable age distribution. The present analysis surveys the misunderstandings, sorts out the issues and errors in the debates, and supplies overdue generalizations and extensions.