## 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.

Throughout most of the 15th and 16th Centuries, the treatises written on fencing and combat were either untitled, or else bore a general description as a name. These have come to be known by their German description, “Fechtbuch,” which translates simply as “fencing book,” but some bore other simple titles, like *Trattato d’uno Schermo* (“Treatise on Defense”). However, during the 16th Century, we begin to see treatises taking on more philosophical titles. A first and very notable example is Agrippa’s *Trattato di Scientia d’Arme, con vn Dialogo di Filosofia* (“Treatise on the Science of Arms, with a Dialogue in Philosophy”) from 1553. In the 1580’s, we see the appearance of books like *Sobre el arte de la escrima* (“On the Art of Fencing”) and *Nobilissimo discorso intorno il scherma* (“Most Noble Discourse on Defense”), and in 1599 we find George Silver’s *Paradoxes of Defense*. Indeed, the content of the works begin to shift, as well. Accompanying the usual discussion and illustration of particular techniques is an analysis of the philosophy and intention behind those movements. Strategy and tactics, purpose and planning, began to be the focus of these works.

Then, in the 17th Century, an Italian fencing master named Salvator Fabris was in the employ of the King of Denmark, who implored Fabris to set his ideas to page. The fencing master set to work, with the King’s court artist on retainer to provide illustration. On the 25th of September, 1606, Salvator Fabris published *Lo Schermo, overo Scienza d’Arme* (“On Fencing, or the Science of Arms”), a massive work which– to this day– remains one of the most important and influential treatises ever written on the use of the Italian rapier. Though I know of nothing which directly links Fabris to Euclid’s *Elements*, it is known that he had been educated at the University of Padua– the very same institution which produced such noted mathematicians as Copernicus and Galileo. Given that, if Fabris had any training in mathematics, at all, he would likely have been familiar with the *Elements. *Certainly, the format of *Lo Schermo* is somewhat reminiscent of the Definitions-Postulates-Proofs layout which other notables had been trying to borrow. The treatise begins by laying out definitions of the different positions for guard and footwork. It carries on by postulating the proper timing and measure of the defined techniques. Finally, the book sets out illustrations of numerous encounters, accompanied by step-by-step descriptions of the actions which led to the engagement. Each section is very clearly and rigorously described, in stark contrast to the terse and bare descriptions often found in earlier works. In further evidence of Euclid’s influence, Fabris focused greatly on the geometry of fencing, describing the lines of attack and defense for blade, guard, and footwork. Even the very illustrations show evidence of geometric influence, being far more precise than had been found in earlier works, and adding a gridded floor pattern and shadow to allow the reader a sense of perspective.

Just a few years later, another Italian fencing master would publish his own treatise, which would make the parallels to Euclid’s *Elements* undeniable. Ridolfo Capo Ferro‘s *Gran Simulacro dell’Arte e dell’Uso della Scherma *(“Grand Representation of the Art and Use of Fencing”) was certainly not as long or comprehensive as Fabris’ massive *Lo Schermo* had been, but it follows a similar Definitions-Postulates-Proofs format. However, in even greater similarity to Euclid than had been shown in Fabris, Capo Ferro enumerates each individual thought and topic under discussion. For example, we see by #24 that Capo Ferro defines “fencing” as “the art of defending oneself well with the sword,” while #47 tells us “the *misura stretta *in *pie fermo* is when I can strike the adversary by only pushing the body and legs forward.” Furthermore, the *Gran Simulacro*‘s illustrations, composed by the magnificent Rafael Schiamirossi, bear even greater precision than those of Fabris, and are often labeled alphabetically in the manner of Euclid’s diagrams. The geometry is not only a focus, but is absolutely integral to Capo Ferro’s descriptions of technique. The image from Figure 2, above, is taken from the 5th Plate of the *Gran Simulacro*, and Euclid’s influence is wholly obvious in its design.

In 300 BCE, a relatively obscure scholar wrote a comprehensive treatise on mathematics. That work was so incredibly influential and so thoroughly amazing that it left its imprint even upon fields which, to the modern eye, seem wildly disassociated with geometry and number. While philosophers and theologians attempted to emulate the rigor of the *Elements* in their works, they were largely unsuccessful. Most of the questions they had sought to answer deductively and absolutely remain open questions in those fields even today. However, the fencing masters did a far better job of adapting Euclid’s formula to their own work. The foundations set down by Fabris, Capo Ferro, and others in the 17th Century revolutionized the sword. The rapier became the weapon of the civilized man, soon to be followed by the smallsword. The smallsword then evolved into the fencing foil and the epee, both of which continue to be used even to this day in modern fencing. The principles of blade, guard, and footwork which were rigorously laid out by these masters four centuries ago remain true even for today’s Olympian athletes. Were it not for the efforts of a relatively unknown mathematician over two millennia ago, we may never have seen the full beauty of the sword, today.

Reblogged this on Saving school math and commented:

If you want to know more about Euclid and his influence, especially outside mathematics, read this. It is fascinating.

Fascinating. I have taken the liberty of reblogging it on my site. Thanks.