# Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it’s just so simple that it’s surprising no one thought of it sooner.

The example that makes me ask is the 2011 paper John Baez mentioned called “Two semicircles fill half a circle”, which proves a fairly simple geometrical fact similar to those that have been pondered for thousands of years.

This proof of the irrationality of $$√2\sqrt 2$$ appears to have been discovered in 1892 by A.P. Kiselev: If $$√2\sqrt2$$ is rational, let $$△ABO\triangle ABO$$ be the smallest possible isosceles right triangle whose sides are integers.

Construct $$CDCD$$ perpendicular to $$AOAO$$ with $$AC=ABAC = AB$$. $$△OCD\triangle OCD$$ is another isosceles right triangle.

$$AC=ABAC=AB$$, therefore $$ACAC$$ is an integer, therefore $$OC=OA−ACOC = OA - AC$$ is an integer. $$△OCD\triangle OCD$$ is isosceles, so $$OC=CDOC=CD$$ and CD is an integer. $$CDCD$$ and $$BDBD$$ are equal because they are tangent to the same circle, so $$BDBD$$ and $$OD=BO−BDOD = BO-BD$$ are integers. But then $$△OCD\triangle OCD$$ is an isosceles right triangle with integer sides, contradicting the assumption that $$△ABO\triangle ABO$$ was the smallest such.

I am amazed that this wasn’t found by the Greeks, because it is so much more in their style than the proof that they did find.