Euclid and the Sword
I have written, often, about one of my personal heroes from history, Euclid of Alexandria, who wrote a textbook called Elements which would serve as the foundation for all Western mathematics for 2000 years. You may recall that, outside of his name and a list of his writings, we know almost nothing about Euclid. We know nothing of his birth, or his schooling, or his politics. We don’t know if he traveled extensively or if he was relatively sedentary. We don’t know if he was tall, short, fat, skinny, handsome, or ugly. However, one thing we do know is that Euclid’s work, though purely mathematical, bore a tremendous influence on a wide variety of fields of knowledge.
Euclid’s Elements set out to prove the whole of mathematics deductively from very simple definitions, axioms, and postulates. Deductive logic provided a sound and absolute basis by which mathematics operated for every man, whether rich or poor, high-born or peasant, male or female, famous or obscure. During the 16th and 17th Centuries, this strong foundation became lauded and sought after by philosophers, who began attempting to provide all philosophy with the rigor one found in the Elements. The appeal was obvious: if one could deductively prove his philosophical system, in the manner that Euclid had proved his geometry, then one would be left with incontrovertible conclusions to questions which had previously been highly disputed. Such extremely notable philosophers as Thomas Hobbes and Baruch Spinoza, amongst countless others, attempted to replicate the Definitions-Postulates-Proofs format Euclid had employed in order to settle questions of morality and ethics and governance.
Martial philosophy was no less affected, in that period. The sword and swordplay, especially, underwent a dramatic evolution during that same time. Just as Hobbes and Spinoza attempted to replicate Euclid for ethics, fencing masters similarly moved toward a more rigorous and geometric approach towards understanding combat. And, in my opinion, they were far more successful in that endeavor than the philosophers had been.
; and a few even remember that the c in that equation refers to the hypotenuse of a right triangle, while the a and b refer to the other two legs. However, most of the time, people were just taught to memorize this theorem– they weren’t taught how to prove that it was actually true. Now, the Internet is full of all kinds of really clever visual proofs involving rearranging copies of the triangle in order to form the different squares, but I’m not really a huge fan of these. They make it very easy to see that the Pythagorean Theorem is true, but they don’t really make it easy to see why the Pythagorean Theorem is true. So, today, I wanted to discuss my favorite proof for the Pythagorean Theorem, which comes to us by way of Euclid’s Elements, 
Boxing Pythagoras 