The Geometry of War

Back in the Renaissance, mathematicians used geometry for all kinds of things, including war. People designed forts, entrenchments, and even weapons using geometry. The ability to make a heavy gun out of a single metal casing capable of longer range and higher accuracy (aka the cannon) greatly aided the armies of Europe. Leonardo da Vinci and Francesco di Giorgio are among the people that contributed to this field, whether it was tools, new weapons, new defenses, or new equations. Leonardo is well-known for his paintings, but he was also an avid military technology researcher. He made designs for bombs, explosive cannonballs, rapid-firing guns, giant catapults and crossbows, submarines, flying machines, and scythe-equipped chariots. These were mostly text-book ideas, though, and were never put into practice. Also, the newly found way of rotating the cannon vertically let the gunners greatly increase range. Instead of having one area where they could shoot, there was now a long line where they could aim. Besides the weapons themselves, there were geometric tools for measuring the cannons, angle, shot, and powder. Specialized rulers, calipers, ladles for the powder, gauges, quadrants, sights, and levels were all tools that gunners used to predict the range and angle the cannons needed to hit the enemy. The distance from the enemy posed a problem, since they decided walking over to the enemy position was a bad idea. Then, mathematicians came up with the idea of using triangles to find distances. They had a measured baseline, and by measuring the angles made between the two ends and the enemy, they could find out how far away they were. This was vital to the gunners because they didn’t have guided missiles and radar back then so they had to manually aim. Besides offense, they also used geometry for defensive entrenchments, walls, and other fortified protection. The typical walls of the day were high, offering protection from ground soldiers. But these high, wide, thin walls were vulnerable to artillery attacks. If you used short walls, then the enemy ground soldiers could attack. The resolution was to have short, thick walls with protruding turrets on the corners. These turrets could shoot parallel to the wall and could defend the walls from ground troops. Italy started producing books that gave instruction for defensive fortresses invulnerable to both artillery and ground forces. They said that the best way for defending from both types was to have arrow-shaped protruding turrets that caught the enemy at the walls in a crossfire. The actual shape of the fortress (square, rectangle, etc.) depended mostly on the site it was built on. It could be a regular rectangle, like the one shown below, and it could also be a long, thin rectangle if it is between hills (which would be an awful place to put it because you could be ambushed easily). As you can see, geometry has greatly contributed to the field of war, for both attackers and defenders.

Francesco di Giorgio and Leonardo da Vinci

This short animation I made shows how protruding turrets in the fort work.(the beginning is where it says 'this is the fort')