The Mathematics Curriculum

At West Point In The Nineteenth Century

V. Frederick Rickey and Amy E. Shell

Department of Mathematical Sciences,

United States Military Academy

 West Point, NY 10996

In the early nineteenth century the United States Military Academy was the only school in the country where one could obtain a technical education. Thus it is not surprising that a significant impact of West Point was through the young officers that were trained here as teachers and then went on to teach at numerous schools across the country after resigning from the Army. This impact was achieved because of the unique mathematics curriculum at West Point. It was unique for the breadth of material taught, for the way it was taught, and for the textbooks used. Experience teaching this curriculum encouraged the faculty to write textbooks to suit their goals, but also impacted the teaching of mathematics across the land.

  The first superintendent of West Point, Jonathan Williams, was aware of the superiority of French textbooks, but because he was unable to procure enough books to supply the cadets, and because they could not read French, the first mathematics textbook used was Charles Hutton’s A Course in Mathematics. When Sylvanus Thayer became superintendent in 1817, he began to wean the cadets and faculty away from Hutton, and French language mathematics texts began to be used, including Lacroix’s Algebra, Legendre’s Geometry and Boucharlet’s Calculus. Soon the entire first year curriculum consisted of Mathematics in the mornings and French in the afternoons (so the cadets could read their mathematics).  All agreed that this was the kind of education that engineers needed, especially military engineers.

A few years after Charles Davies became Professor of Mathematics in 1823, he began publishing mathematics textbooks in English. These began as translations, but then in later editions the names of the original authors disappeared from the books. By mid century, Davies was the most popular textbook writer in the United States and there were colleges where his were the only mathematics textbooks used. Davies was succeeded in 1837 by Professor Albert E. Church, who also wrote a series of textbooks. These textbooks were not so widely used across the country, but they dominated the mathematics curriculum at West Point for the remainder of the century.

  A unique feature of the mathematics curriculum at West Point was the teaching of Descriptive Geometry. It was created by Gaspard Monge, one of the founders of the Ecole Polytechnique and brought to this country by West Point instructor Claudius Crozet. Both Davies and Church wrote textbooks on this topic and it spread throughout the US engineering curriculum in the nineteenth century. Today it has evolved into the subject of engineering drawing.

Two men dominated mathematics at the United States Military Academy in the nineteenth century. They were Charles Davies and Albert E. Church. Between the two of them they headed the mathematics department for over half a century. Moreover, their influence lasted another half a century. They were also influential across the United States because of the textbooks they wrote. The textbooks of Charles Davies and Albert Church were widely used in the teaching of mathematics in American high schools and colleges in the later half of the nineteenth century.

  Davies graduated from West Point in 1815 and then taught mathematics at West Point for over two decades, serving as department head from 1823 to 1837, resigning because of his health to devote the rest of his life to textbook writing. Church graduated first in the class of 1828 and then taught mathematics at West Point for half a century until his death in 1878. Before dealing with them in detail, we should say a few words about mathematics in the first years of the Academy, a period, which is much maligned, and little appreciated.

  Five weeks after Thomas Jefferson became president, his secretary of War, Henry Dearborn, wrote to George Baron, a friend from Maine, to ask if he was interested in a position as teacher of mathematics at an annual salary of about $700. “West Point on the Hudson,” Dearborn wrote, “will probably be the position for the school.” Baron bickered over the salary, but after Dearborn pointed out that it was fixed by congress, and offered several perks including a house, twelve to eighteen cords of firewood, and a place to summer his cow, they came to an agreement.  On June 6, 1801, Dearborn sent Baron his commission as "Teacher of the Arts and Sciences to the Artillerists and Engineers.” Dearborn also requested that Baron purchase "any number of copies not exceeding fifteen or twenty" of Charles Hutton's A Course in Mathematics for use at West Point. It is interesting that the choice of the first textbook at West Point was not left to the “Gentleman well skilled in the mathematics” but was dictated by those in Washington City.[1]

  The first mathematics lesson at West Point — indeed, the first lesson in any subject — was taught by George Baron on 21 September 1801, six months before the United States Military Academy was legally founded on 16 March 1802.  He used a “standing slate” to teach algebra to a few cadets, this being the first recorded use of the blackboard in the United States.[2] When Joseph G. Swift, who became the first graduate of West Point, arrived on October 14, 1801, Baron furnished him with a copy of Hutton’s Course and “a specimine of his mode of teaching at the blackboard in the academy.” Soon thereafter, Baron and Swift got in a shouting match involving “coarse epithets” and Baron was, for a variety of reasons, court-martialed and fired. “His name was set upon the public building as a disgraced officer.”[3]  Thus ended the short career of the zeroeth professor of mathematics at West Point.

Of Baron we know very little. He was born in England in 1769, and taught school there — but not at the Royal Military Academy at Woolich as is often stated — before immigrating to Hallowell in the District of Maine, a small town in southern Maine some five miles from where Henry Dearborn lived.[4]­  After leaving West Point, Baron moved to New York City where he taught school and founded The Mathematical Correspondent, the first mathematics periodical published in the United States[5]. It survived for only one volume, partly because of the acerbic personality of Baron.

The Hutton text continued to be used at West Point for two decades, until 1823. At the time it was the best text to be had. The first of the two volumes consists of 533 pages.  The main topics are arithmetic, logarithms, algebra, and geometry.  The second volume deals with trigonometry and fluxions, which is Isaac Newton’s name for the calculus, in 622 pages. While this may not sound like very advanced mathematics, its level was far above what was taught at Harvard and other schools.

We have information about how the Hutton book was used from a 39-page notebook in the archives at the West Point library that belonged to Abraham Wendell. He was from the State of New York and attended West Point from September 2, 1813 until March 2, 1815, when he graduated, there then being no set time to completion of the program. The notebook bears the signature “Abra: Wendell, West Point June 17^th 1814.” This copybook deals with algebra and geometry. Specific topics include “Sir Isaac Newton's Rule for raising a Binomial to any power whatsoever,” “Infinite Series,” “Arithmetical Proportion,” “Application of Arithmetical Progression to Military Affairs,”  “Of Computing Shot and Shells in a Finished Pile,” “Quadratic Equations,” and “Resolution of Cubic and Higher Equations.”  

Also in the archives are two folders of materials relating to Henry W. Griswold who also graduated from West Point on March 2, 1815. There is a geometry notebook and two trigonometry notebooks.[6]  The geometry manuscript, which is 12 pages long, contains 52 propositions of Euclidean geometry, but only the enunciations of the theorems; there are no proofs. There are diagrams for the first 31 of these theorems, and spaces for the remaining diagrams, but they were never drawn into this notebook. The first of the trigonometry notebooks consists of 18 pages of carefully written notes. There are two pages of definitions followed by five pages of problems. Then there are three pages devoted to “Heights and Distances,” which include some lovely drawings, and eight pages of spherical trigonometry. The second notebook is similar.

Both the Wendell and the Griswold notebooks can be precisely matched up with Hutton’s text. They contain the same peculiarities of language, bizarre capitalization, and choices of special cases for the diagrams. Thus it seems that the instructor had a copy of A Course in Mathematics in his hands and dictated or wrote out on the board the various propositions, while the cadets copied them down verbatim and then returned to their rooms to prepare a good copy in their notebooks. On Sunday morning the cadets had to show their manuscripts to Professor Ellicott. “No questions were by way of examination asked as to how the results were obtained, but if our manuscripts were neat and presentable we were patted on the head and treated like good little boys. Of course we gave ourselves up to chirography rather than to the mastering of arithmetic.”[7]  

Frenchman Claude Crozet joined the faculty on September 20, 1816 as Assistant Professor of Engineering, joining Alden Partridge who was the first Professor of Engineering in the United States[8]. How Crozet happened to come to West Point is unclear. It is sometimes said that Thayer is responsible, but we know of no documentation to suggest that they met in France or that Thayer had anything to do with his appointment[9]. Crozet knew little English on his arrival, but by the force of his willpower and drive he started to teach cadets engineering. He quickly learned that they knew nothing of descriptive geometry, the subject created by Gaspard Monge and taught to Crozet at the École Polytechnique in Paris. Descriptive geometry was created to provide geometric solutions to problems such as the defilement of fortresses that had been previously solved by elaborate arithmetical methods. Soon Crozet was teaching descriptive geometry to cadets and writing a text for them. He was the first in America to teach this subject. His A Treatise on Descriptive Geometry was published in 1821, the year he left the faculty. This was the first textbook on the subject in English, and the first textbook written by a West Point faculty member while teaching there.[10]

Crozet’s text was used at West Point for several years, but there was never a second edition. It was replaced in 1832 by Elements of Descriptive Geometry, a text written by Charles Davies in 1826. It is unclear why it took six years for Davies, a member of the Academic Board since 1823, to introduce this text. Davies, in turn, was replaced by a text by Albert Church in 1864, which was used until 1929.

The 1821 Board of Visitors report indicates that “Hutton and Simpson’s Algebra” were being used with the fourth class cadets, although the third and fourth section only used Hutton.[11] This brings up another facet of education at West Point. Not only were the cadets sectioned by their order of merit or class rank, but the upper and lower sections sometimes used different books, and when they used the same book they often covered different portions of the book. The General Regulations and Orders for the Army for 1821 specified that cadets were to study mathematics from sunrise to 7 a.m., attend mathematics class from 8 to 11, study from 11 to noon, and again from one-half hour past sunset to 9:30 p.m.[12]

  Charles Davies, Professor of Mathematics, 1823-1837

Charles Davies was born 22 January 1798 in Washington, Litchfield County, Connecticut. When he was a youth, the family moved to Black Lake, St. Lawrence County, in northern New York State, where he was educated in local schools. During the war of 1812, when the Chief of Engineers, Joseph G. Swift, was preparing the defense of Ogdensburg, he met Davies’ father, the County Sheriff, and took an interest in young Davies and “personally aided in securing his appointment” at West Point.[13] Davies came to West Point in December 1813 and graduated in December 1815. There were no openings in the engineering corps when he graduated so he took a less desirable position in artillery, serving a year in garrison duty before being transferred to the Engineering Corps in August 1816. He resigned from the army on 1 December 1816 to accept a position at West Point teaching mathematics.  He served under department heads Andrew Ellicott and David Douglas, and then taught Natural and Experimental Philosophy for two years, before becoming the mathematics professor in May of 1823 when Douglas became professor in Philosophy.

Davies set to work writing a sequence of textbooks that would be used at West Point and across the country:

1826.                  Elements of Descriptive Geometry, with Their Application to Spherical Trigonometry, Spherical Projections, and Warped Surfaces.

1828.                Elements of Geometry and Trigonometry. Translated from the French of A. M. Legendre, by David Brewster. Revised and Adapted to the Course of Instruction in the United States.

1830.         Elements of Surveying. With the Necessary Tables.

1832.         A Treatise on Shades and Shadows, and Linear Perspective.

1833.                  The Common School Arithmetic, 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.

1835.                  Elements of Algebra: Translated from the French of M. Bourdon. Revised and Adapted to the Course of Mathematical Instruction in the United States.

1836.                    Elements of the Differential and Integral Calculus.

1837.                    Elements of Analytical Geometry: Embracing the Equations of the Point and Straight Line, the Conic Sections, and Surfaces of the First and Second Order.[14]

Not surprisingly, the effort of producing eight books in eleven years left Davies exhausted and with a severe bronchial infection. Thus he resigned in May of 1837 so that he could tour Europe, restore his health and then “continue to write and revise wildly successful mathematics textbooks.”[15] 

The 1828 text, Elements of Geometry and Trigonometry, was his most popular book. In the period 1828-1895 it appeared in 33 editions/printings and some 300,000 copies.[16]  In his lifetime Davies published 49 different titles. If we include his editing of Edward Courtenay’s posthumous Treatise on the Differential and Integral Calculus, and on the Calculus of Variations, then Davies published an even 50 books. These appeared in at least 492 editions/printings.  They covered the ground from elementary arithmetic through college mathematics (though none was higher than calculus). By 1875, A. S. Barnes & Co., a firm that Davies helped found, had sold about 7,000,000 texts by Davies, and were selling about 350,000 of every year.[17] Is it any wonder then that he dominated mathematics textbook writing in the nineteenth century?

In 1831 two translations of Bourdon’s algebra appeared, one by Farrar at Harvard and one by Edward C. Ross who graduated in 1818 and then taught at West Point until 1833. The Ross translation was used until Charles Davies produced his own in 1835 — based partly on that of Ross. That translation, in various editions, was used for the rest of the century. In 1893 it was supplemented by A Treatise on Algebra by Charles Smith that was used until 1905. The use of the algebra text of Davies for 65 years reveals something about education at West Point. The Academic Board had absolute control over the curriculum and once they got a curriculum in place during Thayer’s years as Superintendent, they were very reluctant to change it. They tinkered a bit, but there were few substantial changes. 

All of the works listed above, except the arithmetic, were used as textbooks at West Point. They were also used at schools across the country. Partly this is due to the dozens of West Point graduates who taught mathematics at schools across the country.[18]  More importantly, these books were vastly superior to other texts that were available at the time. Historian Florian Cajori wrote that the books of Davies “were, as a rule, perspicuous, clear, and logically arranged. They were not too difficult for the ordinary student, and contained elements of great popularity.”[19] Aggressive marketing techniques also led to the widespread use of the books. Davies “saw himself simultaneously as a professor and a businessman, like two touching circles with one inside the other.”[20]

  Albert E. Church, Professor of Mathematics, 1837-1878

When Church arrived at West Point in June of 1824 at age sixteen, the entrance requirements were simply arithmetic, reading and writing. During the summer the cadets “received daily and very thorough instruction” in arithmetic by cadet Dallas Bache, and recited daily to him. Bache had served as Acting Assistant Professor of Mathematics the previous year, and was about to enter his firstie year.  It was then the practice to have a few outstanding cadets serve as instructors — and Bache was outstanding, for he became Superintendent of the United States Coast Survey and President of the National Academy of Sciences.[21] Bache knew who the “duly qualified” were before the oral entrance examination in August and so their examination in arithmetic was “hurried and slight” with not more than one or two questions asked. But the proficiency of the weaker cadets was “fully tested.” Every cadet was “required to read and write in the presence of the Academic Board.”[22]  

During his first year Church and his classmates studied algebra, but “the best text book that could be obtained, in the English language, was a poor translation of Lacroix” so they used it.[23] Also, they used Legendre’s geometry,[24] Lacroix’s trigonometry,[25] both in translation, and Crozet’s book on descriptive geometry.[26]  These were all fine books and provided an excellent curriculum.

During the second year, Church and his classmates in the higher sections, use the analytic geometry of Biot and the calculus of Lacroix, both in French. The lower sections used the calculus book of Boucharlat, also in French. Thus the French the cadets studied every afternoon of their plebe year was put to work in learning mathematics during their second year.[27]

It is noteworthy that in 1825 all cadets were learning some calculus. The earliest record of teaching calculus at West Point dates from 1810. During the winter vacation Alden Partridge tutored Alexander Williams, the son of the first Superintendent, Jonathan Williams, in calculus.[28] The first record of a class being taught in calculus at West Point is in the fall of 1815 when Professor Andrew Ellicott examined seven cadets in the subject.[29]  Curiously, although Charles Davies graduated with this group in December 1815, his name is not on the list. Davies likely learned calculus earlier from Ellicott or Partridge.

Albert Church graduated first in the class of 1828 and, like Davies, was commissioned in the artillery, there being no vacancies in the corps of engineers. Thayer requested that Church stay at West Point to teach mathematics, and there he remained for the rest of his life except for the two years, 1832-34, when he joined his artillery unit. In 1837 he became professor of mathematics, succeeding Charles Davies who resigned to devote his time to writing. Church served as professor until his death in 1878, a total of fifty years.

The reports on Church as a teacher are not good. Morris Schaff, who graduated in 1862, called him “an old mathematical cinder, bereft of all natural feeling.”[30]  Arthur Hardy, an 1869 graduate, who taught at Dartmouth in the 1890s, says:

The mathematical recitation was a drill room. In my opinion the result was a soldier who knew the maneuvers, but it did not give an independent, self-reliant grasp of the methods of research. In descriptive geometry the academy had a magnificent collection of models, but they were shown to us after the study was finished — in other words, mental discipline was the object — practical helps and aims were secondary.[31]

The reference is to the string models designed by Theodore Olivier, a devotee of Gaspard Monge who created descriptive geometry. This recollection was probably written nearly twenty years after Hardy’s graduation, so his memory of how these models were used could well be faulty, but there seems to be no reason to doubt their existence. The Board of Visitors Report from 1860 requests the purchase of models, so it seems likely that the 24 models that the department still has were acquired in the 1860s.[32]

Professor Church wrote four textbooks, all of which were used by West Point cadets:

1842.         Elements of the Differential and Integral Calculus.

1851.                  Elements of Analytical Geometry.

1864.                  Elements of Descriptive Geometry, with its Applications to Spherical Projections, Shades and Shadows, Perspective and Isometric Projections.

1869.                  Plane and Spherical Trigonometry.[33]

These textbooks were used at many other schools, but were not as widely used as those of Davies. The thing to note about these titles is that they are meant to be improvements on what was already being taught. There is no broadening or deepening of the curriculum. Church himself admits that once the mathematics curriculum was set in place, it did not change substantially for the rest of the century.

Superintendent Sylvanus Thayer, after his arrival in 1817, had organized a curriculum “which he deemed most fitted for an American military education,” a curriculum whose foundation rested on mathematics. “His guiding principle was thoroughness in everything — thorough teaching — thorough learning.” How this was accomplished is far different than what we think of as “The Thayer Method” today. The professors lectured to the whole class and then the cadets were split into “sections of over twenty cadets each” where they asked questions and recited for the rest of the morning. Three hours a day, six days a week, 43 weeks per year was devoted to mathematics in the classroom. Of course the cadets were expected to study the book — they practically memorized it — but they were lectured on the material before they were asked to recite.[34]

Cadets were examined twice each year, in December and June. This took place in front of the Academic Board and, in June, the Board of Visitors also. One section of cadets was examined at a time. In advance, the instructor wrote the main topics of the course on slips of paper and then drew them randomly to decide which question was asked of each cadet. For example, there is a copy of Davies’s Algebra in the West Point Library that was owned by Cadet Acton. On the endpapers is written

Examined Jan^y 2^nd 1879.
Subject – “Logarithms”   “Fessed cold”

The cadet slang “fessed cold” means that Cadet Acton failed the exam. This is confirmed by his Cullum listing as x1882, a non-graduating member of the class of 1882. The question asked of him was to explain the topic of logarithms. Curiously, this section of the text is heavily annotated in a way that indicates the writer understood the nuances of the subject. But perhaps this is not Cadet Acton’s handwriting. The failure does not seem uppermost in his mind, for he continues the annotation:

This study was ordained in hell
To torment those who on earth dwell
And it suits its purpose well.
            Glory hallelujah!!
                    Amen!
                    Amen!

Page prepared 12 March 2002.