Archimedes famously used inscribed and circumscribed polygons to approximate the circumference of a circle. He then repeatedly doubled the numbers of sides to get an approximation for π.

In 1667, James Gregory did the same, but he used areas: He discovered the following beautiful double-recurrence relation that can be used to compute the areas of inscribed and circumscribed n-gons:

Gregory’s Theorem. Let I k and C k denote the areas of regular k-gons inscribed in and circumscribed around a given circle. Then for all n, I 2 n is the geometric mean of I n and C n , and C 2n is the harmonic mean of I 2n and C n ; that is,

and

We can use these formulas to approximate π. For instance, a square inscribed in a unit circle has area I4=2 and a square circumscribed about the unit circle has area C 4 =4. Applying the recurrence relations, we obtain the following sequence of bounds:

n I n C n 4 2 4 8 2.828427125 3.313708499 16 3.061467459 3.182597878 32 3.121445152 3.151724907 64 3.136548491 3.144118385 128 3.140331157 3.14222363 256 3.141277251 3.141750369 512 3.141513801 3.141632081 1024 3.14157294 3.14160251 2048 3.141587725 3.141595118 This summer I tweeted this theorem: Let I_n and C_n be the areas of the inscribed and circumscribed regular n-gons for the unit circle. In 1667James Gr… twitter.com/i/web/status/1… —

Dave Richeson (@divbyzero) June 20, 2018 My friend Tom Edgar—a mathematician at Pacific Lutheran University and a master at finding “proofs without words”—emailed me to see if I wanted to try finding a proof without words of Gregory’s theorem. This is what we came up with. The two parts of Gregory’s theorem follow from the two parts of the following lemma. We give the proof… without words. Lemma. and Proof.