Geometry of War

Leonardo da Vinci (1452-1519) described war as "una pazzia bestialissima" (a most bestial folly). That folly was vividly depicted in Leonardo's huge fresco of a battle between the Florentines and the Milanese, which took place in 1440 at Anghiari.

"As Leonardo saw it, the painter had to reach the onlooker's heart, move him, force him to think, and edify him," Serge Bramly writes in Leonardo: The Artist and the Man. "He hated war and in his fresco sought to communicate to others the horror it inspired in him."

That fresco no longer exists. Copies made by other artists (http://www.visi.com/~reuteler/vinci/anghiari.jpg) and Leonardo's own preliminary sketches (http://www.visi.com/~reuteler/vinci/battle3.jpg) provide glimpses of the fury, brutality, violence, and rage evident among the men and even the horses strewn across Leonardo's vast battle scene.

At the same time, Leonardo was a military engineer, capable of preparing designs for horribly murderous devices. He spent a great deal of time pondering machines of destruction--bombs, explosive cannonballs, rapid-firing guns, giant catapults and crossbows, and scythe-equipped chariots. He devoted many pages to studies of antique weapons, drawings of sharp blades and cruelly pointed pikes and spears, and depictions of artillery (http://www.visi.com/~reuteler/vinci/artillery.jpg).

Leonardo was not alone in his fascination with military technology. The University of Oxford's Museum of the History of Science is now featuring a fascinating online exhibition called The Geometry of War, 1500-1750 (http://www.mhs.ox.ac.uk/geometry/content.htm). The 81 illustrations, accompanied by an essay and extensive notes, demonstrate how mathematical science was used on the battlefields of early modern Europe.

The exhibition reveals that the same geometric and technical ingenuity that went into the development of perspective painting and the deployment of navigational and surveying devices during the Renaissance also informed the invention of instruments for warfare.

The display "seeks to draw attention to the fact that the advent, not of gunpowder directly, but of its more effective use in improved ordnance, precipitated a new variety of practical mathematics and a new disciplinary discourse that flourished through the sixteenth and seventeenth centuries. . . . This geometrical discipline engaged a great many practitioners in the period and was associated with a range of technological developments applied throughout Europe, both in military campaigns and in large works of military engineering."

According to the exhibition guide, "Heavy guns manufactured in single metal castings were longer, capable of more accurate fire, and were adjustable in elevation. Consequently, gunners needed instruments to measure both the inclination of the barrel and the distance to the target, together with a means of relating these two measurements. Geometers offered a variety of solutions to these problems, as well as designs for fortifications to withstand attack from the new artillery."

The online display focuses mainly on three areas: gunnery, rangefinding and surveying, and fortification. Additional material highlights the mathematics of troop formations and reminds us that the telescope was originally introduced as an instrument of war.

Gunnery, for instance, was a complex and dangerous art. Gunners were often castigated for their inability to fire reliably and accurately. Though mathematicians couldn't remedy variations in powder or flaws in individual guns, they did try to improve the gunner's lot by devising instruments to measure the size of shot, determine the elevation of guns and mortars, and calculate of the range of fire. The bewildering variety of weaponry then available, however, often made that a complicated task for both gunner and mathematician.

One handy device was the gunner's folding rule (http://www.mhs.ox.ac.uk/geometry/cat3.htm). Used to measure lengths, make calculations with dividers, and determine elevation, the rule typically incorporated artillery tables, which provided such data as the weight of shot for different artillery pieces and the amount of powder required for both ordinary use and preliminary testing. An additional table inscribed on the rule's surface related the size of iron cannonballs (ranging from 1 to 8 inches in diameter) to their weight.

In their efforts to market their inventions, mathematical instrument makers often couldn't resist combining several capabilities in one package. That sometimes led to awkward, impractical hybrids. One German device made in 1595 allowed the gunner to read the weight of shot directly from a measurement of its diameter, with separate scales for iron, lead, and stone (http://www.mhs.ox.ac.uk/geometry/cat7.htm). The same device also included a sundial and tables of astronomical and calendrical information.

Curiously, the precision, elegance, and delicacy of many of the instruments contrast sharply with the harsh, unruly environment of a battlefield. It's unclear how usable many of them would have been in practice. Perhaps, like Leonardo's imaginative creations, they were more textbook exercises in ingenuity than devices of true military utility.

References:

Bramly, Serge. 1994. Leonardo: The Artist and the Man. New York: Penguin.

You can visit the Museum of the History of Science's exhibition on The Geometry of War, 1500-1750 at http://www.mhs.ox.ac.uk/geometry/content.htm.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.