Malthus observes that humans tend to like having sex, which means that inevitably (especially before modern contraceptives) humans would likely continue to make children at a constant rate. But since two people can have more than two children, and each of those children can have even more children, population growth is not arithmetical, but rather geometric.
What Malthus means by 'arithmetical' and 'geometric' is simply that some systems produce at the level of addition and subtraction, and other systems work differently. If it works by process of addition, it is arithmetical, and if it works by process of multiplication, such as population growth, it is geometric.
So Malthus concludes from that basic study of the systems of population growth that we can expect the population to double every 25 years. By the way, Malthus's mathematical analysis is understood by most people to be incorrect, but his idea is still powerful. Could humans populate at such a rate that eventually, we exhaust our resources, and what would happen if that were the case?
He then explains that he is not predicting a doomsday, or an apocalypse, but rather, that given our understanding of mathematics, humans should begin to consider their effect on the environment as an exponentially powerful animal, since our biological existence means that we must sustain ourselves by using natural resources.
Malthus continues by explaining that many of the variables in the sustainability question are fluid, such as the potential development of new technologies to advance agriculture and infrastructure. Instead of offering a creative solution, he leaves the question open for audience participation, because after all, we're all on the same planet, so overpopulation is a risk that could potentially effect the entire race if not treated with scientific attention.
