What was the idea behind Logistic growth model?

The Wikipedia page for Verhulst has links to his original writings. In the first paper from 1838, he basically only says (on p. 115) that since the growth rate of the population decreases as the number of inhabitants increases, we can subtract an unknown function $\varphi(p)$ from the right-hand side of the differential equation, $$ \frac{dp}{dt} = mp - \varphi(p) , $$ and that the simplest hypothesis that one can make about the form of this function is that $$ \varphi(p) = n p^2 . $$ (He also mentions some other possible forms, like $np^\alpha$ or $n \log p$.)