418. If the human race after the flood multiplied from 6 people and if we assume that two hundred years later the number of people increased to a million, the question is: by what fraction should the population have increased annually?

b4

## Malfatti's Problem.
418. This problem is found in "Introductio in Analysin Infinitorum". Euler's solution: "Suppose the population increased annually by $\frac{1}{x}$ of its own part, then after 200 years we get:

$$
\left(\frac{1+x}{x}\right)^{200} \cdot 6=1000000
$$

from which $\frac{1+x}{x}=\left(\frac{1000000}{6}\right)^{\frac{1}{200}}$, and therefore

$$
\lg \frac{1+x}{x}=\frac{1}{200} \lg \frac{1000000}{6}=\text { n, } 0261092
$$

hence, $\frac{1+x}{x}=\frac{1061963}{1000000}$ and $1000000=61963 x$, from which $x$ is approximately 16. This would be the multiplication of people if the annual increase were one sixteenth of the population, which cannot be considered too large. However, if the human race were to grow in such a proportion for 400 years, the number of people would have to reach 166666666666. Such a population could not be sustained by the entire Earth."

Malfatti (1731-1807), the Italian geometer, whose name is associated with the problem because he was the first to solve it.