When did humans first learn the size of the Earth? Or, for that matter, when did it become clear that the Earth was a sphere?

To those not attuned to science history, it may come as a surprise to learn that the Earth’s size and shape were settled 2,200 years ago, and by the same person. In what was one of the great intellectual feats of antiquity, the Greek Eratosthenes took the measure of the Earth using only simple tools and mathematics, together with his creative mind. This post recounts how Eratosthenes did it.

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The great mathematician Leonard Euler (1707 – 1783) was comfortable working in any branch of mathematics (and he even invented some new branches, such as topology). This post is about one of his many results in number theory. I will describe a formula that he discovered in 1737 which involves fractions formed from prime numbers. The formula is an odd and surprising result, and its derivation illustrates Euler’s remarkable ingenuity.

For more than one hundred years, this formula was just another curiosity among Euler’s many results. Then, in 1859 Bernard Reimann used it as the starting point in his landmark paper on prime numbers.

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