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Malthus is often quoted and rarely read. His central argument is a piece of school maths: a geometric progression against an arithmetic one. This post works through the numbers in his 1798 essay, shows the mistakes students make with them, and explains why a book about human population ended up shaping biology.
Who was Thomas Malthus?
Thomas Robert Malthus lived from 1766 to 1834, according to the Library of Economics and Liberty biography. He is remembered above all for An Essay on the Principle of Population, first published in 1798. Its full title says it treats population "as it affects the future improvement of society", with remarks on the speculations of William Godwin, the Marquis de Condorcet and other writers.
What did Malthus argue?
His argument rests on two statements, which he gives in exactly these words:
"Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio."
For the rate of growth he used the United States, where he reported that the population had been found to double itself in 25 years. For food, he took Britain and allowed, generously he thought, that its produce might grow every 25 years by an amount equal to what it already produced. Doubling is geometric. Adding the same amount each time is arithmetic.
A worked example: Malthus's own numbers
Malthus started with Britain at about seven million people, with food enough for exactly that number. Each step below is 25 years.
Population doubles each step, so it is 7 × 2^n million after n steps: 7, 14, 28, 56, 112.
Food rises by 7 million people's worth each step, so it feeds 7 × (1 + n) million: 7, 14, 21, 28, 35.
After 25 years (n = 1): 14 people and food for 14. Balanced.
After 50 years (n = 2): 28 against 21.
After 100 years (n = 4): 112 against 35. The essay says this "would leave a population of seventy-seven millions totally unprovided for". Check: 112 − 35 = 77.
He then did the same for the whole world, writing the two sequences as 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 and 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. After two and a quarter centuries, he wrote, population would be to food "as 512 to 10", and after three centuries "as 4096 to 13".
Check the second claim: 225 years ÷ 25 = 9 steps, so population is 2^9 = 512 and food is 1 + 9 = 10. Then 300 years ÷ 25 = 12 steps, so 2^12 = 4096 and 1 + 12 = 13. His arithmetic holds.
Common mistakes:
Counting terms instead of steps. After 9 steps the population is 2^9, not 2^10. The first term, 1, is the starting point at step zero.
Mixing up the formulas. For an arithmetic progression the nth term is a + (n − 1)d; for a geometric one it is ar^(n − 1). Our guide to sequences and series for JEE drills both.
Reading the gap as a forecast. Malthus wrote that the effects of the two powers "must be kept equal". The 77 million show the pressure that checks on population have to remove, not a crowd he expected to see.
Where did Malthus go wrong?
The Econlib biography argues that Malthus, writing before industrial farming, did not fully appreciate how technology would raise food production. It also argues that he is often misread: his actual conclusion was that because humans had not all starved, choices such as later marriage must be checking population growth, and those choices were worth studying.
Biology made the same correction in its own terms. NCERT's Class 12 chapter on organisms and populations describes unchecked "exponential or geometric" growth, then explains that no population has unlimited resources, so growth levels off at a carrying capacity. That logistic model came from Pierre Verhulst, who started from Malthus; see our post on Pierre Verhulst and the logistic curve.
Why did Darwin care?
In his autobiography, Charles Darwin wrote that in October 1838 he "happened to read for amusement 'Malthus on Population'". Prepared by years of observing the struggle for existence, "it at once struck me that under these circumstances favourable variations would tend to be preserved, and unfavourable ones to be destroyed." He added: "Here then I had at last got a theory by which to work". Our post on Darwin, Wallace and natural selection covers what came next.
What can a student take from his way of working?
Make your argument checkable. Malthus gave numbers anyone could test. That is why we can check them, and why critics could find the weak assumption.
Question your assumptions, not just your arithmetic. His sums were right; the assumption that food could only grow arithmetically was the weak point.
Read outside your subject. Darwin found his key idea in a book on population.
If geometric and arithmetic progressions still blur together, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning behind each formula.
Try it now: rerun Malthus's table with food rising by 10 million every 25 years and find the step when population first overtakes food.
