What does Mercutio say about Tybalt's style of fighting?

Mercutio says about Tybalt's style of fighting that his "fiddlestick" will "make [Tybalt] dance," and that Tybalt is "a dog, a rat, a mouse, a cat, to scratch a man to death! a braggart, a rogue, a villain, that fights by the book of arithmetic!" (III.1).

Most "modern" translations say that "fights by the book of arithmetic" simply means that Tybalt "fights without inspiration, but merely by the book." Scholars of swordplay, however, have a more interesting...

Mercutio says about Tybalt's style of fighting that his "fiddlestick" will "make [Tybalt] dance," and that Tybalt is "a dog, a rat, a mouse, a cat, to
scratch a man to death! a braggart, a rogue, a villain, that fights by the book of arithmetic!" (III.1).

Most "modern" translations say that "fights by the book of arithmetic" simply means that Tybalt "fights without inspiration, but merely by the book." Scholars of swordplay, however, have a more interesting explanation.

There are two main schools of fencing: Italian (adapted to English and known as the classical school), and Spanish. English gentlemen (i.e., Shakespeare's audience) thought the Spanish style barbarous and ungentlemanly. (Of course, the characters in the play are Italian, and would thus have the same assumptions about fencing schools as Shakespeare's audience.) Almost all of the characters in the play (except for old Capulet, who asks for someone to bring him his "long sword," an antiquated weapon) fight with rapiers and they use the Italian style.

The Italian style consists of low stances with the rapier held low and withdrawn a bit. They are primarily linear in their movements--darting in and out--and try to deliver a solid stab then withdraw out of range. This is how Mercutio is fighting.

The Spanish style had a far more creative approach, using short stances and moving a lot to different angles. Some of their swordplay books actually map out the preferred angles of attack and how to get there most effectively (hence, "by the book of arithmetic"). Compared to their Italian-fencing opponents, they looked like they were "dancing," which Mercutio uses as an insult. Another important aspect of this style is that the arm is held almost at full length, keeping the point of the rapier in the opponent's face.

The difference explains how Tybalt kills Mercutio, too. As Mercutio had just delivered his stab attempt and Romeo came between the two, he would have to withdraw his rapier for another stab attempt. But Tybalt, having his sword at face level, has trained to flip his wrist to quickly recover his blade (to that position) when it is parried, which means his jab was probably an automatic (trained) movement. It also explains why Mercutio says that the cut that kills him is "a scratch," that it isn't a very deep cut, but "'twill serve."

Understand also that the difference in fencing styles was a clue to Shakespeare's audience that Tybalt was a barbarian, an ungentlemanly cur. Also, such a swordfight would look really cool on stage.