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.

Answer

This proof of the irrationality of 2 appears to have been discovered in 1892 by A.P. Kiselev:

diagram

If 2 is rational, let ABO be the smallest possible isosceles right triangle whose sides are integers.

Construct CD perpendicular to AO with AC=AB. OCD is another isosceles right triangle.

AC=AB, therefore AC is an integer, therefore OC=OAAC is an integer. OCD is isosceles, so OC=CD and CD is an integer. CD and BD are equal because they are tangent to the same circle, so BD and OD=BOBD are integers. But then OCD is an isosceles right triangle with integer sides, contradicting the assumption that 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.

(Source: http://www.cut-the-knot.org/proofs/sq_root.shtml#proof7)

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