Mathematical advances during the U.S. Civil War

SUMMARY: The U.S. Civil War saw numerous advances in firearms, cryptography, and strategy.

The U.S. Civil War, also sometimes known as the War Between the States, was a conflict fought from 1861 to 1865 between 11 Southern U.S. states that seceded from the Union to form the Confederate States of America and the remaining United States. Precipitating causes of the war centered on economic issues and states’ rights versus federal power, often symbolized by the central dividing issue of slavery. More than 600,000 men on both sides died, which is greater than the combined U.S. losses in all subsequent wars and military conflicts through the beginning of the twenty-first century, though World War II exceeds this count if the metric is combat deaths versus deaths from all causes. Some also refer to the Civil War as the first “industrialized war” because of the extensive use of the telegraph, railroads, and mass-manufactured goods and weaponry.

Mathematics was instrumental in this war in many ways. The introduction of the Spencer repeating rifle has been cited by many as the turning point toward the eventual Union victory. Ciphers and code-breaking efforts were important in communicating military strategies and plans. The U.S. Army Signal Corps, founded in 1860, used both telegraphy and line-of-sight methods, such as the wig-wag signaling system in which the left, right, or upward positions of a single flag represented the numbers 1–3 and specific number combinations corresponded to letters. The ranks of leaders on both sides were filled with mathematically educated graduates of schools like the United States Military Academy at West Point. Mathematics education and research were also impacted by the war.

Weaponry

Changes in small arms and artillery that were occurring in Europe and the United States at this time had a tremendous impact on the war. Many different types of smoothbore or rifled artillery were used during the Civil War. One way in which they were distinguished was by their bore size, which was the diameter of their barrels, usually expressed in inches. Another differentiating feature was the weight of the projectiles they fired, in pounds. Different classes of weapons also had different trajectories. Cannons known as “guns” had relatively flat trajectories. Mortar rounds followed a steeply arcing path. Howitzers fell between the other two because the possible angles of inclination and powder charges could be varied more than the other two types.

The most common artillery piece was the Napoleon, a howitzer named after Napoleon Bonaparte, an avid student of mathematics who had revolutionized infantry and artillery warfare. Artillery ammunition included solid shot or balls, grape, canister, shell, and chain shot. Canister and grape shot could be particularly devastating to humans, since they disintegrated into smaller, scattering projectiles along a number of trajectories when fired. At the start of the war, both sides relied primarily on muzzle-loading rifled muskets such as the 58 Springfield, though some smoothbore muskets were also in use. Rifled weapons had greater range—nearly half a mile—versus the 100-yard range for smoothbores, which affected infantry tactics. The breech-loading Spencer repeating rifle was a major innovation, and was considered the most advanced weapon of its time. It used metal cartridge ammunition and could hold seven cartridges at a time, which greatly increased rate of fire and accuracy, though the Union was initially concerned about the corresponding increase in the demand for ammunition.

Revolvers also replaced muzzle-loading pistols, with similar effects. Minié ball ammunition, named for French military officer Claude Minié, was used extensively in the Civil War. Previously, rifles had been difficult to load because the bullets fit tightly in the bore of the weapon, which was necessary for them to be propelled effectively by explosive powder charges. Despite being called a “ball,” the Minié projectile was conical. It was smaller in diameter than older ammunition, and also had grooves that allowed it to fall smoothly and quickly into the barrel of the rifle. A hollow indentation extending from the base caused it to expand to the size of the gun’s bore when fired, optimizing the combination of loading time and accuracy.

Cryptography

One problem addressed by mathematical problem-solving approaches was the terrible problem of “hacking.” Increasingly, communications were being relayed by the newest technology—the telegraph and Morse code. It was relatively easy for someone to climb the telegraph pole, connect a telegraph key to the wires, and intercept messages that were being relayed back and forth from the front lines to the base camp. Messages needed to be coded so that interceptors could not interpret them, which was not a new problem. The problem of encoding military messages can be dated to at least Julius Caesar and earlier to the Spartan military.

The governors of Ohio, Indiana, and Illinois were close enough to the Confederate border that they felt the need to have their messages encoded. Governor William Dennison of Ohio asked Anton Stager to prepare a cipher (a code) that could be used by these three governors. Stager adapted a transposition coding system that had been used in Great Britain years earlier. Words were rearranged into a grid. The first word of the message was the “key” to indicate how many columns were to be formed and in what order they were to be read. Instructions for these codes were printed on cards about the size of a standard index card, which were the precursors of codebooks. Included on the cards were the route, the keys, the code words, and words used to check the cipher.

This system underwent a number of modifications, and Stager’s route cipher was eventually adopted as the Union’s official cipher. Increasingly sophisticated ciphers were created during the war and, as a result, the instructions could no longer fit on cards. Some of the resulting codebooks were 48 pages long. The messages were intercepted by the Confederacy and sent to Richmond, Virginia, the capital of the South. By twenty-first-century coding standards, they appear to have been relatively easy codes to break, but evidence suggests that the South did not have either sufficient manpower or mathematical knowledge to decode them.

By contrast, the Union forces had a team to work in breaking the Confederate codes, including many times President Abraham Lincoln. The codes used by the South at the beginning of the war were not standardized, resulting in many messages that were unreadable. The Confederacy eventually settled on a code from 1587, the Vigenère cipher, named for Blaise de Vigenère—although others before him, such as Giovan Bellaso, are also noted as having invented it. This code consisted of a tableau of staggered alphabets. The ease of this code was that the code did not have to change if a coded message was captured, only the code phrase. The problem with the Vigenère code came in errors in transmission over the telegraph. Even though the code was harder to break than the Union cipher, it was more difficult to implement because a missed letter would result in an incoherent message. For instance, General Edmund Smith reportedly spent 12 hours trying to decode a message from General Joseph Johnston during the Vicksburg campaign. The message requested reinforcements, but Smith was unable to read it. He eventually sent a courier, but it was too late for reinforcements; Johnston’s army was already cut off. Revisions to the code to avoid this problem in the future made deciphering easier as well.

Mathematically Educated Leaders

Many of the military leaders for both the North and the South were graduates of the U.S. Military Academy at West Point, which was the United States’ only engineering school for an extended period of time. The Civil War was fought with 359 generals who graduated from West Point. They served on both sides, 217 for the Union and 142 for the Confederacy. This list of elite officers and leaders includes many well-known officers. Ulysses S. Grant had plans to return to West Point to teach mathematics, and these plans were changed by the outbreak of the Mexican War. Robert E. Lee graduated second in his class in 1829 and served as an assistant professor for mathematics for his first two years at West Point. Edmund Kirby Smith (1845) taught mathematics at the University of the South after the war, where he joined another West Point graduate, Josiah Gorgas. Other well-known graduates who served both sides during the Civil War included Confederate President Jefferson Davis, Braxton Bragg (1837), John Bell Hood (1853), Thomas Jonathan “Stonewall” Jackson (1846), Albert Sidney Johnston (1826), James Longstreet (1842), George E. Pickett (1846), J. E. B. Stuart (1854), William Tecumseh Sherman (1840), George G. Meade (1835), George McClellan (1846), Joseph Hooker (1837), Abner Doubleday (1842), George Armstrong Custer (1861), and Don Carlos Buell (1841).

All these men were mathematically educated, which was unique for that point in U.S. history and likely played a role in many aspects of the war. For example, the maps and messages of the military in the Civil War show the influence of what is now referred to as “descriptive geometry,” which was created by Gaspard Monge and was incorporated into the curriculum after engineer Claudius Crozet brought it to the Military Academy. Mathematics textbooks used by most of the leaders on both sides of the Civil War include those written by mathematicians Charles Davies and Albert Church, some of which were adaptations of earlier French works. This education of the leaders of both sides of the conflict may have had a great deal to do with the long length of the conflict and allows historians the opportunity to study other differences in the two sides.

It was not just the military leaders during the Civil War who made use of mathematics. Lincoln purportedly had a great reverence for Euclid of Alexandria and geometry. Some historians assert that he kept a copy of Euclid’s Elements in his saddlebag and studied it by lamplight to develop his logic and reasoning skills. Phrasing in Lincoln’s well-known 1863 address at the Gettysburg battlefield has sometimes been compared to Euclid.

Education

While mathematics undoubtedly influenced the course and outcome of the war, the Civil War also affected mathematics education and research. Antebellum college curriculum in schools such as The Citadel or Harvard consisted of classes in mathematics that were filled with “practical applications,” such as mercantile transactions, navigation, surveying, civil engineering, mechanics, architecture, fortifications, gunnery, optics, astronomy, geography, history, and “the concerns of Government.” These topics were all expanded in one of the common textbooks of the day, An Introduction to Algebra, by Jeremiah Day. Geometry and trigonometry were also commonly taught, and analytic geometry, conic sections, and calculus were often optional classes.

The problems discussed and worked in these classes, both in surveying and in navigation, were carefully chosen and adapted to make them easily done but not extremely realistic. Thus, the navigators and the surveyors being prepared for the Army were ultimately ill prepared to handle the realistic situations of making measurements under fire or in harsh seas. According to the work of Andrew Fiss, the Union army regulations required that surveyors plot the best course for the army to take. The topographical engineers worked so slowly that many of the generals took to asking local citizens for the best directions. By two years into the war, the topographical engineers were incorporated into the Army Corps of Engineers. Likewise, the Navy found that the U.S. Naval Academy, founded near the middle of the nineteenth century, could prepare navigators better than a mathematics department. However, the academy was negatively impacted by its temporary relocation from Maryland to Rhode Island during the war.

After the war, many universities started offering higher level mathematics courses, and some increasingly focused on research. Harvard and Johns Hopkins University graduated doctorates in mathematics within the next decade.

Bibliography

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Fiss, Andrew. “The Effects of the Civil War on College-Level Math Education.” Talk given at the meeting of the History and Pedagogy of Mathematics—Americas Section, 13 Mar. 2010, Washington, DC.

Hochfelder, David. The Telegraph in America, 1832–1920. Johns Hopkins UP, 2012.

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