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Math on the Plain

by Colonel David Arney

There are numerous perspectives one can take to view the beauty of West Point. Some of the obvious things to consider as you walk around the West Point grounds (the cadet area and large flat open area called "the plain") are the rich military history of this fortification, the massive Gothic architecture of the fortress and buildings, the unique geology and geography of the river and the rock, the history of the people (cadets and faculty) who have passed through the gates of the Academy, nature's blessing of flora and fawn in the heart of the Hudson Highlands, and especially the great impact this place has had on our nation and the world. One less obvious perspective is to view and taste some of the beauty in West Point's role and contributions in mathematics and mathematics education. You have to look carefully, but there is plenty of beauty in a mathematical tour of West Point. Please join me for a short walking tour of West Point with this mathematics perspective in mind.

We start at the library corner where we plainly see a 12-foot tall bronze statue of George S. Patton (USMA 1909), dressed for combat with pistols on his belt, holding binoculars, looking as if he is ready to lead an armored attack across the plains of Europe. However, in this location, he is intently watching the entrance to the West Point Library. If Patton could use his binoculars to peer through the wall of the 4th floor of the library, he would see a bust of one of his World War II colleagues, Omar Bradley (USMA 1915). When it comes to appreciating, understanding, and using mathematics, these two colleagues could not be more different.

Bradley was more than just a mathematics educator (he taught mathematics at USMA from 1919 to 1923). He also showed he was a real mathematician by using and developing mathematical ideas in performing his military duties as he rose to the rank of General of the Army (5 stars). It has been reported that Bradley was a superb teacher; so good that he was extended a 4th year in the Mathematics Department to help develop his fellow faculty members as well as teach cadets. Just as impressive were his talents in using mathematics to solve problems. After leading our Allies' efforts in Europe during World War II, he reflected that he often made his operational decisions by thinking of them in terms of constrained optimization problems and utilizing many of today's foundations of operations research. Who knows, in the process of his mathematical thinking, he could have been one of the first to think about the mathematical technique of linear programming.

On the other hand, Patton was a weaker mathematics student. He struggled so much in all his subjects that it took him 5 years to complete the 4-year academy curriculum. However, the mere fact that he was given the extra time meant that Academy officials saw something special in Patton that made the investment in this young leader worth pursuing. If Patton had an academic strength, it was in history. He was quoted as saying, "to be a successful soldier, you must know history."

It should be noted that both Patton and Bradley studied similar mathematics programs at USMA. The mathematics studies were quite intense and supported the required general engineering curriculum in place at the Academy. Bradley and Patton went to mathematics class 6 days per week for 80-85 minutes per day for the first 2 years for a total of about 612 in-class hours. This could be equivalent in today's semester system to about an amazing 45 credit hours of mathematics. Their courses included algebra, trigonometry, geometry, descriptive geometry, differential and integral calculus, differential equations, and linear perspective. So let's give Patton some credit for his mathematics talents. He survived a rigorous program that surely left him with powerful thinking and problem-solving skills to clearly and logically solve tough quantitative problems faced in his military career. Given the contributions of these two diverse personalities, it was fortunate that the USMA education system found room for both of them to succeed.

Today the Academy tries to develop the future Bradleys and Pattons. The formula has not changed much. All cadets take 16 credit hours of core mathematics while engineering and science students take even more. Special cadets with great interest in mathematics, as Bradley had, can major in mathematics and take 15 or more courses in undergraduate mathematics. At West Point mathematics, science, engineering, humanities, and social sciences are for everyone.

As we move west on the street, we approach the clock tower on the north end of Pershing Barracks. This building is named for John J. Pershing (USMA 1886) who as General of the Army (5 stars) commanded the American Armies in World War I. This clock tower plays a big role in a story told about Douglas MacArthur (USMA 1903). It seems that MacArthur was a multi-talented cadet. He was so outstanding in academics that he finished first in his class with an all-time high academic average and was the top cadet in each of his mathematics courses. He must have been an outstanding leader and engineer as well. During his first year at the Academy, he led a group of fellow plebes on a spirit mission. A spirit mission is best described as an activity on the boundary of regulatory acceptance that demonstrates teamwork, creativity, and, thereby, the spirit of the cadets involved and instills spirit in the rest of the cadets. Another opinion that is sometimes held by the Academy leadership is that spirit missions often end up being misguided pranks. At West Point, spirit missions are more frequent during the week of the Army-Navy football game. This may be the setting for MacArthur's caper as well. MacArthur and his team sneaked out of the barracks late at night, carried a heavy and bulky revelry cannon across "the plain", and placed the cannon of the top floor of the clock tower, at least 60-feet high. The story continues that it took the Academy's engineers a week to remove the cannon from the clock tower and return it to its proper place on "the plain". MacArthur must have been a mathematics instructor's dream student. He was bright, creative, studious, and motivated. When he missed answering a question correctly, it must have been a bad question. Of course, MacArthur had another side which was revealed in his fiercely competitive nature. It was said that he and classmate Ulysses S. Grant III (USMA 1903), who was 6th in the class and grandson of the former president, competed in all aspects of the cadet life.

As we stand on the road in front of Pershing Barracks and look south along Thayer Road, we see Bartlett Hall, Mahan Hall, and Grant Hall. These buildings are named for William Bartlett (USMA 1826), Dennis Mahan (USMA 1824), and Ulysses S. Grant (USMA 1843). While Bartlett was a physicist and Mahan was engineer, both wrote mathematics books. Bartlett's book was entitled Analytic Mechanics, which essentially contained the methods of calculus and differential equations used to solve application problems from mechanical engineering. Mahan wrote a book entitled Descriptive Geometry, as Applied to the Drawing of Fortification and Stereotomy, which covers the geometric bridge between mathematics and engineering. This course is basically a mathematical treatment of engineering drawing. The third honoree mentioned, Grant, never wrote a mathematics book, but it may have been one of his dreams. Grant tried several times to join the USMA Mathematics Department, but was turned down. He had finished 21st in his graduating class of 39, doing best in mathematics where he stated "mathematics was easy for me" as he finished 10th out of 53 cadets studying mathematics. His mathematics classes lasted all morning, 6 days per week, for the first two years, covering algebra, geometry, trigonometry, descriptive geometry, surveying, analytic geometry, and calculus. All of Grant's mathematics books were written by Charles Davies (USMA 1815), who, as department head from 1823-1837, was a prolific writer and national leader in mathematics education from primary school through college. Grant's mathematics professor was Albert Church (USMA 1828), who was also a textbook writer and department head for 41 years, 1837-1878. In response to Grant's requests to return to West Point and teach, Church told him to remain with the field Army because he just did not have the academic credentials to teach mathematics. Even Grant's persistent letter writing campaign for the teaching job did not overcome his mediocre undergraduate mathematics performance. However, it was sufficient to be elected President of the United States.

We now enter the cadet area by walking by the north end of Pershing Barracks toward the 1st division of the old cadet barracks. The 1st division is all that remains of the old central barracks. This horseshoe-shaped barracks was built in 18880. Traditionally, 1st division housed the Brigade First Captain (overall cadet in charge). First Captains of some mathematical note are MacArthur, Pershing, John Barnard (USMA 1833), and Hans Pung (USMA 1995). Barnard was an accomplished scientist, prolific author, and fortification engineer, who among other mathematics and scientific endeavors helped establish the National Academy of Science. Pung was a mathematics major and a Marshall Scholarship winner, which provided him a 3-year graduate fellowship to study mathematics at Oxford.

Directly behind First Division is a 15-foot tall bronze statue of a French soldier defending Paris in 1815 with his sword held high, holding a flag, standing next to a cannon. This statue was a gift from the Ecole Polytechnique in Paris after World War I and is a replica of a statue at the Ecole. The relationship between Ecole Polytechnique and West Point began a century earlier, when Sylvanus Thayer (USMA 1807), an assistant professor of mathematics, visited European military schools to learn about the French and English engineering curricula, obtain some of their best undergraduate books, and see how they taught their cadets. On the faculty of the famous French Military school were several mathematicians of note. So impressive was this group that it may have been the most illustrious mathematics faculty ever assembled. Among them were Gaspard Monge, Joseph LaGrange, Augustine Cauchy, Theodore Olivier, Marquis de LaPlace, and many other famous French mathematicians.

Thayer never saw the faculty of the Ecole in action, since Napoleon's defeat at Waterloo resulted in a temporary closing of the school. Napoleon had helped establish the Ecole and several other military technical schools and recognized the benefit of a rigorous program in mathematics and engineering. Napoleon was an accomplished mathematician and always kept plenty of mathematics talent, such as some of those just mentioned from the faculty of the Ecole, on his military staff. This closing of the Ecole gave Thayer plenty of time to meet with the faculty to discuss the engineering, science, and mathematics curricula and pedagogy. He discovered how to incorporate the Ecole's philosophy that rigorous mathematics was the key foundation to a successful military engineering and science program. He recognized the need for a comprehensive mathematics program to bridge the gap between the United States secondary school mathematics and the study of engineering. There in Paris, the seeds were planted for the new mathematics and science program at USMA and the principles developed for the Thayer method of teaching. While Thayer was in Europe he bought approximately 1000 volumes of the best books he could find for an undergraduate library. Thayer's books are now beautifully bound and displayed in the West Point Room of the library, around the corner from Bradley's bust. Possibly because he was a mathematician and saw the need for a demanding mathematics program, Thayer's original collection and the additional volumes he bought after he became Superintendent of the Academy were rich in mathematical works in terms of quantity and quality. Today the West Point Library holds one of the finest collections in the United States of pre-20th century mathematics textbooks, references and treatises. However, equally impressive is the intensive mathematics program that remains today with roots firmly planted in the rigorous French school guided by the best mathematicians of the 18th and 19th centuries and Thayer's brilliant insight in adapting the program to America.

We move north about 200 feet through a sallyport in the new cadet barracks that takes us to the edge of the plain. At the corner of this wing of the barracks stands an impressive statue of Dwight Eisenhower (USMA 1915), Bradley's classmate and Bradley's and Patton's commander during World War II. Given the diverse mathematics talents of Bradley and Patton, it is not surprising that Eisenhower's mathematics record lies between the two. Eisenhower had developed a very analytic mind from in depth study in mathematics in high school and at West Point. MacArthur used Eisenhower as an analyst on the Army staff, working on quantitative issues like resource allocation, mobilization plans, and the impact of technology on air power, mechanization, and the industrial base of the military. Eisenhower had become a successful operations research analyst before such a profession even existed. Another of Eisenhower's great talents was to find the right person to get a specific job accomplished--for example: Bradley to plan operations, Patton to execute them. When faced with finding officers to perform war planning, design the postwar Army composition, run our nation's and Army's scientific intelligence program, and set up and organize the Central Intelligence Agency, he called on USMA mathematics professors William Bessell (USMA 1920) and Charles Nicholas (USMA 1925). Both men used their analytic skills to help General Eisenhower (another 5 star) and then returned to mathematics teaching at West Point. Bessell headed the mathematics department 1947-1959 and served as Dean from 1959-1965. Nicholas succeeded Bessell as department head and is famous for authoring a comprehensive, integrated mathematics program (15(?) volumes), affectionately referred to by cadets as "the green death". Like Grant, Eisenhower's mathematics talents did not earn him a teaching position on the West Point faculty, were sufficient for election to the Office of the President of the United States.

We move down the cement apron to the center section of the fortress called Washington Hall. This building holds the cadet mess hall where all 4000 cadets can sit and simultaneously eat their meals. In front of the building stands an impressive 30-feet tall statue of George Washington riding a horse. No; Washington was not a graduate of the Academy, but he was no stranger to West Point. His Revolutionary War headquarters was located here so that he could defend the key terrain linking the northern and southern colonies and prevent the British forces from using the Hudson River by blocking it with a great chain that extended across the river at West Point. After the war, President Washington proposed a military academy for our new nation, conveniently located at West Point where some basic military training had continued after the war and troops were garrisoned. Washington felt the country's fledging Army needed professional, competent leaders educated at a military academy. The chief antagonist of the Washington plan was Thomas Jefferson. Jefferson, much like Napoleon, was an accomplished mathematician who appreciated its value. Jefferson recognized the need for more schools of higher education in America, but was afraid a national military academy would produce and benefit an elite class of Army officers. Jefferson's plan was for a national technical school to help educate the nation's common people, who after graduation would build our country's infrastructure. Neither plan was implemented until Jefferson, as President of the United States, agreed that one school could accomplish both goals--educate both professional officers for the Army and competent engineers for our growing nation. In 1802, Congress passed and Jefferson signed the law establishing the United States Military Academy at West Point. The Academy has been producing graduates to meet those needs envisioned by Washington and Jefferson ever since.

As we continue north around the cement apron, we find another statue, nearly symmetric with the building to the Eisenhower statue we just visited. By the way, such symmetry produces beauty in the eye of a mathematician. We see that this monument commemorates Douglas MacArthur. Since we already discussed MacArthur as a cadet, let's now focus on his exploits as an officer. MacArthur influenced the mathematics program at two different times in his career. As Superintendent of the Academy from 1919 to 1922, he instituted a reform of the entire curriculum that integrated courses, utilized technology, and demanded more work by cadets outside class. The statistics of the reform show 10% fewer mathematics classes and class time reduced from 85 to 75 minutes per class. However, the quality of the mathematics program increased under guidance of department head Charles Echols (USMA 1891) and implementation of the new courses by Omar Bradley and his fellow instructors. Key changes in the modernization were emphasis of the use of the slide rule, new textbooks, increased time for and demands on cadet study, and integration of the topics and applications of mathematics and engineering science. Under MacArthur's reform the mathematics program had been modernized while maintaining its original philosophy of rigor and magnitude.

MacArthur left the academy and eventually became the Chief of Staff of the Army from 1930 to 1935. During his tenure in that position, the academy faced a challenge to the quality of its academic program from the President of Harvard. MacArthur, Superintendent William Connor (USMA 1897), department head Harris Jones (USMA 1917), and a mathematics team of 10 yearling "mathletes" saw to it that the challenge was met. In May 1933 Army defeated Harvard in a mathematics competition which was widely reported like an athletic event. MacArthur personally awarded medals, gifts, and privileges to the victorious team for successfully defending his alma mater and reformed curriculum.

Across from MacArthur, we see a beautiful fenced-in garden. At the corner on the wall of the fence is a spot called Constitution Corner. This messages given here are the role of the Army soldier and the purpose of the Academy. West Point teaches cadets how to "support and defend the constitution" through its propose of "producing leaders of character who serve in the common defense." Providing shade for this corner are two trees, a cherry and a peach tree. These trees are dedicated to the Army football teams that won the 1985 Peach Bowl and 1984 Cherry Bowl. It's not totally by coincidence that many of the names that I have mentioned already were Army football players: Patton, Bradley, Eisenhower, and Pung. Over the years Army football players have achieved success in academics, including mathematics, and military undertakings. In 1995, Eric Oliver, a mathematics major and football player, was awarded both a Rhodes Scholarship and NCAA postgraduate scholarship. Recently, we have had many other graduating cadets study mathematics in graduate program under prestigious fellowships. Hertz Fellowship winners include Richard Staats (USMA 1984, PhD MIT 1995), Andrew Fedorchek (USMA 1988, PhD student at Stanford), Thomas Tracyk (USMA 1991, Georgia Tech student), and Marcia Geiger (USMA 1992). Exchange cadet Anton Pineda (USMA 1990, MS RPI) received a similar national fellowship from his native country, the Philippines, and Ray Eason (USMA 1994) is studying mathematics at Oxford under a Marshall Scholarship. Overall USMA has had Rhodes Fellows, Marshall Fellows, and Hertz Fellows.

Overlooking the beautiful gardens is the stately home of the Superintendent, Quarters 100. In deference to the power of mathematics, all buildings on post are assigned a number. Thayer lived here, although there have been same major expansions and renovations since then. The current occupant is General Howard Graves (USMA 1961), who won a Rhodes Scholarship and was an outstanding mathematics student as a cadet. Graves' course of study included about 30 credit hours of mathematics in algebra, trigonometry, analytic geometry, calculus, differential equations, and statistics. He finished at the top of his class of 534 in this mathematics coursework and was awarded the Lee Saber for his efforts. The Lee Saber, named for Robert E. Lee (USMA 1829), has been given to the top cadet in core mathematics since 1930(?). Speaking of Lee, not only was he Superintendent during 1852-1855, but he was also a mathematics instructor as a cadet. When the Academy ran short of faculty in the early 1800, upper-class cadets were used to teach elementary courses. Lee was an excellent mathematics student (4th out of 60) and a natural selection to teach the program he had just completed. Another Superintendent with special talents in mathematics was Richard Delafield (USMA 1818), who served three tours as Superintendent during the period 1838-1861. Delafield was an outstanding geometrer and produced some of the finest work possible in his descriptive geometry class. His drawing portfolio is stored next to Bradley's bust and Thayer's book collection on the 4th floor of the library. Delafield eventually used his talents to design plans for fortifications and other civil engineering projects. Alden Partridge (USMA 1806) was acting Superintendent (1814-1817) and a professor of mathematics (1806-1813). After being replaced as Superintendent by Thayer, Partridge left the Army and established numerous military and science schools, among them Norwich University. Other Superintendents of mathematical note were MacArthur, whom we have discussed, and George Cullum, who will learn about shortly.

I must mention a couple cadets who processed great mathematics talent, but were not so successful as cadets and did not graduate from the Academy. Edgar Allan Poe entered the Academy in 1830, but left in 1831. What is interesting for this tour is not his great poetry (he did use some mathematical concepts and names in his poetry), but his tremendous talents as a cryptographer. Poe spent several years designing and breaking codes for a cryptology section of a couple mathematics journals. He was king of the hill when it came to code breaking. While core mathematics at West Point can not take all the credit for his success, Poe studied and did well in the rigorous mathematics program designed by Thayer and Davies and was taught personally by Davies. James Whistler's story is similar. He lasted longer, leaving West Point in his third year of study in 1854 after disciplinary problems and failing Chemistry. Whistler's mathematics program was designed and taught by Church. He used his mathematics, especially descriptive geometry, in his first job as a draftsman for the United States Geodetic and Coastal Survey. An agency filled with the best scientists and mathematicians in the country, many of whom were West Point graduates. I am sure he was a good draftsman and produced beautiful maps, but soon after he moved on to Paris and then England where he hung around with artists like Claude Monet and produced magnificent artwork.

Let me cover one other side note. Davies and Church were the mathematics department heads during the formative and glory years of the department and Academy (1823-1878). Both men were prolific authors, together authoring over 40 different mathematics textbooks, and dedicated educators who made tremendous contributions to the Academy and the mathematics education community. It was said that every schoolboy in America knew about West Point because they studied from one of Davies' most popular books, The Common School Arithmetic, which had on its title page "Prepared for the Use of Academies and Common Schools in the United States, and also for the Use of the Young Gentlemen who may be Preparing to enter the Military Academy at West Point." Davies and Church were the first to establish entrance exams in mathematics, first verbal and then written, which were the predecessors of the modern SATs. Davies went on to teach at Trinity College, New York University, and Columbia.

The next stop on the tour is real special. At the corner of Jefferson Road and Washington Road stands a statue of Sylvanus Thayer, "Father of the Military Academy." Since we already discussed his visit to Europe, let's learn about his days as a cadet, faculty member, and Superintendent. Thayer had already graduated from Dartmouth as valedictorian of his class when he arrived to become a cadet. It took him just 14? months of study to complete the unstructured program of the new academy and become West Point's 33rd graduate. When he was appointed Superintendent in 1817, he immediately went to work improving the curriculum he had taken and taught as an assistant mathematics professor in 1811. Some of his pedagogical reforms included sectioning cadets with homogenous ability, daily tests, competitive class rank system, interactive classrooms, and great use of the blackboards by cadets to practice and demonstrate their skills. It was said that the "most singular characteristic" of the Thayer system was its emphasis on mathematics using the Ecole Polytechnique as its model. Only a crazy mathematics professor would ever think that USMA could stand for United States Mathematics Academy, but many cadets in the middle of the 19th century, when West Point built its reputation as "the best school in the world", probably spent more time studying mathematics than performing military training. By the way, The Best School in the World is a book by James Morrison about West Point in the pre-civil war period (1833-1866), but the quotation is taken from President Andrew Jackson.

During graduation week each year a special ceremony is held at this statue. West Point alumnae gather and march across the plain to assemble in front of Thayer. The oldest living graduate present at the service lays a wreath at the base of the statue while cadets and alumnae salute together and sing the alma mater. Thayer's insights and leadership not only gave West Point a great foundation, but also helped establish the foundation in the United States for mathematics, science, technology, and engineering education. Some refer to him as the "Father of American Technology."

If we look to the West, we can see the Dean's house. West Point has had 11 Deans of the Academic Board since the first one was appointed in 1945. Three were former mathematics department heads: Harris Jones (USMA 1917), William Bessell (USMA 1920), and John Dick (USMA 1935).

As we walk east along Washington Road, we can view the cannons (they are war trophies) that were the weapons of war in many of our country's battles. The cannons are grouped chronologically on Trophy Point by which war they were used and are arranged around the other memorabilia on the point, like the tall shaft of Battle Monument, Sheridan's statue, and links of the great chain. The big, cumbersome, inaccurate cannons of the Revolutionary War are fun to climb on today, but must have been painful to move or shoot in their days of use. Military engineers improved the cannon over the years, and the Civil War cannon are a bit smaller, but had more range and accuracy. The good work of military engineers, scientists, and mathematicians is in clear evidence as one walks from era to era seeing first hand the advances in weaponry. Possibly the ultimate and sublime in this development were the Paris guns designed, built, and used by the Germans to fire rounds 60 miles from the front lines of World War II into Paris. Guns that could be called America's equivalent of the Germany's "Big Bertha" were the shore batteries that were built to protect our country's shorelines. West Point had its own shore batteries that are still buried under an edge of "the plain" just across from Battle Monument.

Certainly, the military engineers were the key technologists in the military battles of the 16th through 19th centuries. In the early years, the abilities of the civil engineers (like Delafield and Kosciuszko) to build impenetrable fortresses was of foremost importance. Later, the mechanical engineers had the key role in building cannons and armaments that were accurate and mobile. In World War I, the scientists made their impact, especially the chemists as the new world of gas warfare began. The physicists impact came with the atomic bomb during World War II. The mathematicians developed the control algorithms that had such a great impact on the "smart weapons" in the "Desert Storm War" in the Persian Gulf. What technology will play the key role in the next war? Many think information science will be crucial. Only time will tell. In any case, West Point scientists and mathematicians will be ready to contribute. In addition, we know that the soldiers, real people performing military duties, will ultimately decide the outcome of the next war, just as they have all preceding ones. That's why the focus of West Point is developing leaders of character.

Before we leave Trophy Point, we can glance up the scenic river and scope out links from the great chain, but we can not linger, we need to move on to more mathematics. We do this by walking down Cullum Road to Kosciuszko's monument. High up on the 30-foot monument is a statue of Thaddeus Kosciuszko, the famous Polish-American military engineer who helped Washington design the fortifications of West Point. Kosciuszko overlooks the river at its critical point, where ships have to negotiate a sharp turn to pass up river. Kosciuszko's background of strong geometric mathematics skills and military training at the Warsaw Military Academy gave him the tools to become an expert at designing and building fortifications. That's exactly what Washington had him do for the headquarters at West Point. The fortification was so strong it could never be breached without complete knowledge of its structure. Giving that structure to the enemy British forces was what Benedict Arnold did to earn the label of "traitor". Kosciuszko went on to lead an unsuccessful rebellion against the Russians in his native land of Poland. He also became great friends with Thomas Jefferson, which just might explain why Jefferson finally agreed to the establishment of a military academy. If our academy could produce graduates like Kosciuszko, then both Washington and Jefferson would have their desires for America's first public school of undergraduate education come true.

Our last stop is Cullum Hall, which is just a bit further down Cullum Road. This is the one building we will enter, only because it would be a tragedy to miss its contents. Before we enter, look across the street at Doubleday Field, named after Abner Doubleday (USMA 1842) and believed by some to be the inventor of the game of baseball while a cadet in 1839. Although that theory is very controversial, for purposes of this tour we will assume it to be true. In any case, the inventor of baseball had great intuition of geometry, trigonometry, and possibly calculus. Just think, how the game would be adversely affected if some of the distances or angles on a ball diamond were changed. A closer pitcher's mound would result in batters never hitting the ball. A longer distance between bases would make infield hits and stolen bases obsolete. The inventor of baseball was certainly a fine applied mathematician, something Abner Doubleday and his fellow graduates of USMA were educated to be.

Now we let's investigate the treasures of Cullum Hall. This building is named after George Cullum (USMA 1833) and thanks to him we know he was the 709th graduate of USMA. Cullum was a prolific writer and one of his largest projects was to catalog the careers of all West Point graduates and to write and assemble their biographies. The Cullum index numbers all graduates and his biographies are a great source of historical information about graduates. Thanks to Cullum we know things like: Joseph Swift was the first graduate in 1802; Thayer was the 33rd graduate; in the first 100 years, 4121 graduated from West Point; there are over 52,000 graduates today and all have taken a rigorous, comprehensive program in mathematics. In his spare time, Cullum wrote a mathematics book entitled Problems of Descriptive Geometry.

As we enter the Cullum Hall through the main door, we see hundreds of plaques honoring graduates, mostly for their contributions during war-time and for sacrificing their lives for defense of our country. The main plaque on the right lists the Academy graduates killed in action during the first 100 years of existence (1802-1902). Other plaques display West Point generals who served during various wars, losses of graduates during 20th century wars, superintendents, and professors, including all those mathematics professors we have discussed and many others. The magnificent ballroom on the second floor of Cullum is adorned with numerous plaques and paintings commemorating many of the West Pointers who served with the Union Army during the Civil War.

Well, this concludes the tour. We have seen plenty while walking only a half mile or so. I hope you agree that there is plenty of beauty in West Point mathematics. So much that to commemorate all the geometry that was studied here at West Point, you will support my effort to change the name of "the plain" to its mathematically correct name "the plane".