# Mathematics Education at West Point: The First Hundred Years

## Introduction

Mathematics is the study which forms the foundation of the course [of study at the United States Military Academy]. This is necessary, both to impart to the mind that combined strength and versatility, that peculiar vigor and rapidity of comparison necessary for military action, and to pave the way for progress in the higher military sciences. All experience shows that the mind, in order that it may act with efficiency, must be accustomed to exertion. It should be taught gradually to develop its own powers, and as it slowly learns their capacity and the manner of employing them, the increasing lights which are thrown upon its course will enable it to go on for an unlimited extent in the path of improvement.
Committee on Military Affairs, US Military Academy, May 17, 1834[1]

As the first engineering school in the United States, the United States Military Academy (USMA) at West Point had a uniquely technical curriculum for its time.  The first two years of the curriculum was dominated by mathematics.  The first superintendent of the Academy, Jonathan Williams, was aware of the superiority of French mathematics, engineering, and military science textbooks.  However, 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 (in part so that the cadets could read their mathematics).  All agreed that this was the kind of education that engineers needed, especially military engineers.

West Point influenced the young nation in many ways.  The infrastructure of the nation (roads, railroads, bridges) were designed and constructed by the Corps of Engineers, trained at West Point.  The Academy also had a profound influence on education.  Many of its faculty went on to instruct at, and even head, many colleges and schools across the land.  A third manner in which the Academy influenced education was through textbooks. A few years after Charles Davies became professor of mathematics in 1823, he began publishing mathematics textbooks in English. These began as translations, but in later editions the names of the original authors disappeared from the books. By mid century, Davies was the most popular author of upper-level mathematics textbooks 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 texts were not so widely used across the country, but they dominated the mathematics curriculum at West Point for the remainder of the century.  These texts, as well as the men who wrote and used them, helped lay the ground-work for technical education in the United States.

[1]Annual Report of the Superintendent of the United States Military Academy, Washington, 1896, p. 47.

## The Founding of the Academy

Although the United States Military Academy was officially established by an act of congress and signed into law by Thomas Jefferson on 16 March 1802, its evolution and birth go back a quarter of a century to 1776 when George Washington, the father of the United States, Henry Knox, his Chief of Artillery, and Alexander Hamilton, then an artillery officer, saw the need for an academy to train military engineers.

On 20 June 1776, Congress passed an act forming a “Corps of Invalids” (disabled veterans) that were to be taught by officers in their spare time. It declared that

When off duty, [the invalids] shall be obliged to attend a mathematical school, appointed for the purpose, to learn geometry, arithmetic, vulgar and decimal fractions, and the extractions of roots.[1]

This training took place at the strategic Revolutionary War fort at West Point, on the Hudson River, fifty miles north of New York City. Located at a tight S-curve in the river, West Point was essential to the revolutionary forces.  If the British occupied West Point, they gained control of the Hudson River, thus effectively cutting the colonies in half.  The Commanding General of West Point became infamous when he tried to deliver the plans for West Point into British hands.  The General in question?—Benedict Arnold. Besides studying mathematics, the officers were also obliged to contribute one day’s pay per month to procure a regimental library. Thus began the oldest government library in the country (the Library of Congress was not founded until 1800). It is unclear what became of this library. Perhaps a few of its volumes survive in the present USMA library, but no evidence for this has yet been found.

Between 1784 and 1794, West Point was occupied by a single company of soldiers whose primary activity was maintaining the decaying fortifications. Then, in 1794, a Corps of Artillerymen and Engineers was created at West Point. In March of 1796, fire claimed the only structure suitable for indoor instruction and classes were suspended. They did not resume on a regular basis until July 2, 1801 with the ordering of cadets to West Point by Secretary of War, Henry Dearborn.

In May of 1802, plans were drawn for the Academy, which was to cost $1,500 for the construction of a mathematics room, a drafting room, quarters for the cadets, two mess rooms, and quarters for the officers, teachers, surgeon and their families. The early curriculum consisted of the rudiments of mathematics and military fortification. Some newly arriving cadets could not read or write, while most had only a basic knowledge of arithmetic and grammar. Unlike the French and English military, Jefferson’s view of the Academy was in keeping with his strong republican (the Republican Party of that time was more akin to the current Democratic Party) leanings, the Academy should be open to young men of all backgrounds, not just the affluent. Thus the admission requirements had to be at a level to allow those from rural backgrounds admission.[2] The admissions requirements were kept at a low level for most of the 19th century. In addition, it was only required that officers of the day have the most basic, practical understanding of mathematics to lay artillery correctly, to construct simple fortifications and to draw rough maps.[3] Five weeks after Thomas Jefferson became president, his Secretary of War, Henry Dearborn, wrote to George Baron, a friend from the District of 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.” For the first decade or so of the Academy’s existence, there was strong lobbying, both in Washington and by some of the early Superintendents, to move the Academy to the Capital, Washington City (later Washington D.C.). 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 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[4] "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.[5]

Dearborn's letter to Baron

Title Page of Hutton's Course of Mathematics

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, a small town in southern Maine some five miles from where Henry Dearborn lived.[6]­

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 Academy was legally founded.  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.[7] 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 specimen 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 buildings as a disgraced officer.”[8]  Thus ended the short career of the zero'th professor of mathematics at West Point. 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.[9] It survived for only one volume, partly because of the acerbic personality of Baron.

[1]Florian Cajori, The Teaching and History of Mathematics in the United States, Washington, Government Printing Office, 1890.

[2]The first written entrance requirements that we have found are from 1810.  See the section on “the New Academy”.

[4] All images courtesy of the USMA Library Archives.

[5]Dearborn to Baron, April 11, May 11, and June 6, 1801. Miscellaneous Letters Sent, Secretary of War, M370, RG 107, NA. We would like to thank Theodore J. Crackel for providing copies of these letters. Transcriptions available at: http://www.dean.usma.edu/math/people/rickey/dms/DeptHeads/Baron-George.htm

[6]Theodore J. Crackel, West Point: A Bicentennial History, Lawrence: University Press of Kansas, 2002, pp. 308-9. The statement that Baron did not teach at Woolich is based on Crackel’s examination of the list of faculty who taught at Woolich in "The Shop;" The Story of the Royal Military Academy, by F. G. Guggisberg, London and New York: Cassell and Company, limited, 1900. The date of Baron’s birth is known from a portrait that was distributed with the eighth issue of The Mathematical Correspondent, but it exists today only in some copies, e.g., that at American University. The copy on microfilm in the American Periodical Series II, reel 26, does not have the portrait.

[7]Joe Albree, David C. Arney, and V. Frederick Rickey, A Station Favorable to the Pursuits of Science: Primary Materials in the History of Mathematics at the United States Military Academy. Providence: American Mathematical Society and London: London Mathematical Society, 2000, p. 11.

[8]The Memoirs of Gen. Joseph Gardner Swift, LL.D., U.S.A., First Graduate of the United States Military Academy, West Point, Chief Engineer U.S.A. from 1812 to 1818: 1800-1865: to Which is Added a Genealogy of the Family of Thomas Swift of Dorchester, Mass., 1634 by Harrison Ellery. Worcester, MA: F.S. Blanchard & Co., 1890. See pages 27-28, 31.

[9]Edward R. Hogan, “George Baron and the Mathematical Correspondent,” Historia Mathematica, 3 (1976), 403-415.

## Charles Hutton

Charles Hutton (1737 – 1823) was born in Newcastle, England, the youngest son of an overviewer (supervisor) of a coal mine. When he was seven, Hutton was involved in a street-brawl and severely dislocated his left elbow. He hid this injury from his parents and by the time they learned of it, it was too late to treat it properly, so the injury became permanent. Since Hutton was unable to join his older brothers in the mine, he was sent to school to learn to read. After several years the teacher left and Hutton replaced him, thus beginning a habit of teaching by day and learning by night.[1]

One pupil that Hutton attracted was Robert Shafto. He made his private library available to Hutton and then encouraged him to publish. Hutton’s first work, The Schoolmaster’s Guide, or a Complete System of Practical Arithmetic, appeared in 1764 and became the standard school text in England for half a century. During the Christmas holiday of 1666, Hutton advertised that “Any schoolmaster, in town or country, who are desirous of improvement in any branch of the mathematics, by applying to Mr. Hutton, may be instructed.”[2] This in-service training was repeated the next year. That there was ample audience is attested to by the 59 schoolmasters from the Newcastle area who were subscribers to his next book, A Treatise on Mensuration (1767). Besides its mathematical interest, this work is noted for the woodcuts by the young Thomas Bewick, who became one of the great masters of the woodcut. Alas, this just makes the book more expensive for the historian of mathematics to acquire.

In 1760, Hutton opened his own school in Newcastle. This became a success and he became known as an excellent teacher. His patron, Shafto, suggested that he should move to London and apply for a vacancy at the Royal Military Academy in Woolich. The position was to be filled by competitive examination. Bishop Horsley, the editor of Newton’s works, and Nevel Maskelyn, the Astronomer Royal, examined the eleven candidates. Half were judged satisfactory for the post, but Hutton stood out, and he obtained this professorship in 1773.  He remained at Woolich for 34 years.

Howson so nicely tells one event in Hutton’s career that we shall quote the passage in its entirety:

In 1786 Hutton began to suffer from pulmonary disorders. The RMA was situated near the river and dampness began to affect his chest; his predecessor Simpson had in fact died from a chest complaint. Hutton decided then to move, and bought land on the hill south of the river overlooking Woolich. There he built himself a house and also others for letting. No sooner had he done this than it was decided to move the Academy from the damp riverside to the hilltop. A magnificent new building was erected, but, in the eyes of George III, its attractiveness was spoiled by the presence of Hutton’s houses. These were therefor sold to the crown who promptly demolished them, leaving Hutton with a hefty profit from his speculation, sufficient to guarantee his financial future. Thus a physical disability turned him to mathematics and ill-health made him rich.[3]

Hutton’s most important work was his Mathematical and Philosophical Dictionary. This appeared in two volumes in 1795.[4]  This work is a comprehensive survey which includes biographical information on mathematicians and a fair amount of mathematics history for its time.  Margaret E. Baron writes that the Dictionary:

…is probably the best known of Hutton’s works. Although it was criticized as unbalanced in content, unduly cautious in tone, and somewhat lacking judgement, the dictionary has served as a valuable source for historians of mathematics.[5]

Hutton is also famous as editor of The Lady’s Diary from 1773 to 1818, a total of 45 years.

His A Course in Mathematics was lauded before it appeared.  In its ‘Notices of works in hand’ the Monthly Magazine (August 1798) stated:

From Dr H’s talents and long experience in his profession, there is every reason to expect that this will not only be a most useful and valuable work, but will completely supersede every other of the same description.[6]

It proved to be popular, appearing in numerous editions over fifty years. There were several editions that were published in North America and there was even an Arabic edition.  It is not surprising that this text was used in the United States, for the British influence on American education was extremely strong at this time.

[1]A. G. Howson, A History of Mathematics Education in England, Cambridge University Press, 1982, p. 189.

[2]Howson, p. 63

[3]Howson, p. 67

[4]The USMA library copy of the first volume is not the first printing.

[5]Margaret E. Baron, Dictionary of Scientific Biography, VI, p. 577.

[6]Howson, p. 67.

## Hutton and the Notebooks

The Hutton text was 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, 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.  American schools at the time taught a classical curriculum, consisting primarily of the classical languages and authors.  West Point was the first school to teach a highly technical curriculum that prepared its students to be engineers.

We have information about how the Hutton book was used from student notebooks of the time.  One such notebook in the USMA Archives is a 39-page notebook that belonged to Abraham Wendell.[1] He was from the State of New York and attended West Point from September 2, 1813 until March 2, 1815, when he graduated. At the time there was no set program or graduation requirements.  Cadets were graduated when the professors felt they had obtained the requisite knowledge.  Thus students that had prior college experience were often graduated within a year or two of coming to the Academy.

The notebook bears the signature “Abra: Wendell, West Point June 17th 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.[2]  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.  These theorems correspond exactly to the propositions in the 1812 edition of the first volume of Hutton.

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 watercolor drawings, and eight pages of spherical trigonometry. There is a trigonometry problem in one of the notebooks that asks the cadet to calculate the height of Fort Clinton at West Point given information about the level of the Hudson river below.  This is the only problem not taken directly from the text.

The second notebook is similar, however the writing is different, and though this notebook is catalogued with the Wendell papers, his name does not appear in it.  The material covered is similar, but not the same, though still taken directly from Hutton.  We believe that this may be the notebook of a cadet a year or so ahead of Wendell, and given to him to study from when the owner no longer needed it.[3]  Since so much emphasis was placed on accurate, clean drawings, it would not be surprising if cadets who made good drawings loaned their work to other cadets.  The following comment was made by Professor Albert Church (USMA class of 1828) just days before his death at a meeting of the United States Service Institute, 1864.

Of course, the real teachers in these subjects were those cadets who made careful notes, finished their drawings early in the day, made the demonstrations to their classmates, and lent their drawings for copying. Great skill was acquired in making these copies. A clear and large pane of glass was placed on the top of the washstand, a lighted candle underneath the finished drawing on the glass, and the paper for the copy on top, and every point quickly marked with a pencil.[4]

Both the Wendell and the Griswold notebooks can be precisely matched up with Hutton’s text.

Page from Hutton

Griswold Notebook

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 mornings 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.”[5] The trigonometry notebooks do not follow Hutton as closely as the geometry notebooks do, only covering portions of the text.  For the most part, few proofs are provided.  The notebooks contain largely propositions and examples. (Other Cadets pages: Griswold Notebook 2, McLean Notebook)

Griswold and McLean notebooks

[1] Wendell and Griswald notebooks, USMA Special Collections.

[2]We would like to thank Christine E. Coalwell, Research Associate at Monticello, The Thomas Jefferson Memorial Foundation, for calling these three notebooks to our attention.

[3]  This tradition is still maintained at the Academy.  After completing the core mathematics courses, each cadet passes their math portfolio on to the Plebe they are in charge of.

[4]Annual Report of the Superintendent of the United States Military Academy, Washington, 1896, p. 61.

[5]George D. Ramsay, “Recollections of the U. S. Military Academy at West Point, New York, 1814-1820,” typescript in West Point archives. This is paraphrased in George W. Cullum, “The Early History of the United States Military Academy,” pp. 465-672 of volume 3 of his Biographical Register of the Officers and Graduates of the U. S. Military Academy, 1891, esp. p. 621.

## Andrew Ellicott

Andrew Ellicott (1754-1820) was professor of mathematics from 1813 until his death in 1820 and used Hutton.  At this time each department had only one professor and several instructors, thus to be the “Professor of Mathematics” meant to be the department head. He had the reputation as a good teacher, and it appears that under his leadership, the teaching of mathematics was much less tied to the text.  Ellicott was a kindly and friendly man who was well liked by the cadets for he was full of interesting stories. They nicknamed him “Old Infinite Series,” revealing that the topic was indeed taught to the Corps. He was famous for the perfect geometrical constructions that he made at the blackboard with cord and straightedge. He even had a small slate and sponge attached to his buttonhole.

Andrew Ellicott

Here is how Ellicott was described by Cadet E. D. Mansfield, the son of Jared Mansfield, professor of mathematics at the Academy from 1817 to 1828:

There are some who will recollect Professor Ellicott sitting at his desk at the end of a long room, in the second story of what was called the Mess Hall, teaching geometry and algebra, looking and acting precisely like the old-fashioned school-master, of whom it was written,

“And still they gazed, and still the wonder grew

That one small head could carry all he knew.”

In the other end of the room, or in the next room, was his acting assistant, Stephen H. Long.  *   *   *  The text-book used was Hutton’s Mathematics, and at that time the best to be had.   *   *   *   It was a good text book then, for there were no cadets trained to pursue deeper or more analytic works.[1]

In a letter to Joseph G. Swift, the first USMA graduate (1802) and Superintendent from 1812 to 1814, dated 10 February 1815, Ellicott writes that they have kept 80 cadets together for vacation. “[T]hey have made great progress, --- these classes are in Conic sections, one of which will be in fluxions before the end of the next month.”[2]  This is the earliest reference to the teaching of calculus at USMA that we have found. Whether it was actually taught as a class, or as a seminar, and precisely what was taught is unknown.

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.[3] 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.[4]  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.  Cajori comments that:

At West Point, during the first few years of its existence, neither fluxions nor calculus received much attention. As late as 1816 it is stated in the West Point curriculum that fluxions were “to be taught at the option of professor and student.” In 1817, Claude Crozet, trained at the Polytechnic School in Paris, became teacher of engineering. A few times, at least, he used in print the Newtonian notation, as for instance, in the solution, written in French, of a problem which he published in the Portico, of Baltimore, in 1817.[5]

By 1825 all cadets were learning some calculus.

[1]Quoted in Cajori, 1890, p. 115; the stars, which indicate an ellipsis, are in Cajori.

[2]Thayer Papers, vol. 2, 1808-1817, p. 79

[3]Peter Michael Molloy, Technical Education and the Young Republic: West Point as America’s École Polytechnique, 1802-1833. Ph.D. dissertation, Brown University, 1975, p. 377. UMI number 7615673.

[4]Partridge Papers, Norwich University.

[5]Cajori, A History of Mathematical Notation, vol. 2, 1929, pp. 253-354.

## Sylvanus Thayer and the 'New' Academy

In 1815, Sylvanus Thayer (1785-1872) and William McRee were sent to Europe to study the continental military system and schools.  Most of their time was spent at the École Polytechnique.  Their other mission was to obtain texts.  Thayer and Ray brought back over a thousand books, as well as maps and pamphlets from France.  Most of these now reside in the Thayer Collection of the Academy Library.  Upon returning to the States in 1817, Thayer was made the third Superintendent of the Academy.  He went on to be known as the Father of the Academy for the work he did to revise and revitalize the Academy and its curriculum.  However, it must be said that many of his changes were started or advocated by his predecessors. He modeled many of his changes, both academic and military, on the École.  Through the remainder of the nineteenth century, the Academy had strong ties to the École Polytechnique and St. Cyr, which was founded just a few months after the USMA.

At this time, the entrance age was 14-18 years, and the only entrance requirements were that the cadets be “well versed in Reading, Writing, English Grammar and common Arithmetick, including Vulgar and Decimal Fractions and the extraction of square and Cube Roots.”[1] Though the entrance requirements would change only slowly, when Thayer became Superintendent, the methods of instruction changed quickly. Unlike his predecessors, he did not hold the title of Chief of Engineers, so could devote all of his time and creative energies to getting the fledgling Academy on a solid academic track. He established the Board of Visitors and the Academic Board, which consisted of the Superintendent, the Commandant of Cadets, and the heads of the academic departments.  He also developed strict teaching and living procedures for the Academy.

By 1818, each West Point cadet was taking the equivalent of one hundred credit hours of  mathematics.[2] The General Regulations and Orders for the Army [3] 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., six days a week.[4]

Thayer imposed a precise system for grading cadets’ daily recitations in the classroom, requiring each instructor to submit a weekly compilation of these marks to the superintendent. This not only kept Thayer abreast of the progress of every cadet, it also enabled him to implement another reform: assigning students to sections according to demonstrated competence.[5]

The 1821 Board of Visitors report indicates that “Hutton and Simpson’s Algebra” were being used with the fourth class cadets or freshmen, although the third and fourth section, the lower sections, only used Hutton.[6] This brings up another facet of education at West Point. Cadets were sectioned by their order of merit or class rank.  This idea of ranking the students Thayer brought back from the École.  Cadets were graded in every subject every day.  Periodically, weekly to monthly, they were resectioned so that the first section had the top ten or so students, the next section had the next highest students, and so on.  This system placed students in sections suitable to their ability.  Upper and lower sections covered different portions of the book, and at times even used different books.  For example, in 1825 there were three sections of mathematics for the second year students.  The first or upper section studied calculus out of Lecroix, the second section studied out of Bourharle, and the third or lowest section still used Hutton.

Thayer 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.” 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 to on the material before they were asked to recite.[7]    By the end of the century, mathematics was still allocated three hours per day, however, small sections of up to 15 students met with an instructor form 8:00 to 9:30 or 9:30 to 11:00 for lecture and recitation. The remainder of the time was for individual study.  The professor was in charge of one section, and observed the other sections while instructors taught two sections.[8]

Photo of Recitation (c. 1900)

[1] 1819 Annual Reports of the Board of Visitors to the Untied States Military Academy, Superintendent’s Annual Report, 1896.

[2]Chris Arney, 200 Years of West Point Mathematics: Profiles of Mathematical Soldiers and Trailblazers, AMS Eastern Section Meeting, Boston MA, October 5, 2002.

[3]Quoted on p. 44 of the Annual Report of the Superintendent of the United States Military Academy for 1896.

[4]Over the years, as the curriculum has expanded, these numbers have been reduced. Not until 2003 did Saturday classes end entirely.  Currently all Cadets take a minimum of four semesters of mathematics totaling 17 credits.

[5]James L. Morrison, Jr., The Best School in the World: West Point, 1833-1866, The Kent States University Press, 1998, p. 4.

[6]Annual Report of the Board of Visitors to the United States Military Academy made to Congress and the Secretary of War for the Year 1821, p. 58.

[7]Albert E. Church, Personal Reminiscences of the Military Academy, from 1824 to 1878, West Point, 1879, p. 45.  Available on the United States Military Academy Library web: http://usmalibrary.usma.edu/. This description of teaching methods essentially agrees with one given by Thayer in a letter to President Monroe, October 10, 1828. The West Point Thayer Papers 1808-1872, edited by Cindy Adams, 1965. Available on the USMA Library web.

[8]Annual Report of the Superintendent, 1896. p. 58.

An 1869 article in The New Englander by Robert Keep describes the teaching at West Point during the nineteenth century so well that we will quote it at length.

Let us now look in upon the daily recitations, and select for our examination the fourth or incoming class, which numbers from seventy-five to one hundred. At the commencement of the year it is arranged in alphabetical order, from A to Z, and divided into six or seven divisions or sections, each containing twelve to fourteen men. Each section has two recitations per diem, except on Saturday, when the second recitation is omitted. These two recitations, during the first half of the year, are in Davies's Bourdon’s Algebra, and French. The Mathematical hours are from 8 to 11; French from 2 to 4; and this general division of time holds good through all the classes. Mathematics have the heart of the day—the three best hours—and six recitations per week; English studies, including Law, Ethics, Tactics, and Modern Languages, usually the two hours from 2 to 4 P.M., and but five recitations per week.

To teach the Mathematics to the fourth class, there are three of four instructors besides the Professor, each of these instructors hearing two recitations in the three hours—i.e. an hour  and a half is allotted to thirteen men…. Five or six men are at once sent to the black-board, and having taken their places there and written their names, are given each a proposition to demonstrate, while another is called up to recite on the reading matter of the lesson in answer to questions. He is catechised until one of those first sent to the board signifies his readiness, which he does by facing the instructor and assuming the attitude of attention…

He begins by describing in general terms his subject, then he enunciates the theorem or scientific statement, and  lastly follows out the work he has put upon the board, indicating each point of progress by the pointer…

His demonstration concluded, he, having so far proceeded without interruption, is keenly questioned by the instructor. What has been imperfectly understood is elucidated, misconceptions are corrected, and he is at last allowed to take his seat…. Nor do the questions asked cover only the point of demonstration, but run over the review and back-review, including everything in any way hinted at by the recitation, so that one man may often recite half the lesson. The work is now erased, another takes the place with perhaps the same topic, and the second of those first stationed  at the board is called upon for his demonstration, and so on through the section…. It is not meant, it must be understood, that the instructor at West Point does not question and communicate, only that the recitation and instruction are separated…

The instructor marks daily upon a scale as follow: 3, thorough, 2.5, good, 2, fair, 1.5, imperfect, 1, bad, 0, failure, and each Saturday transfers his marks to a printed blank which shows the daily and aggregate rank of each cadet of the section for the week. These blanks are exposed every Monday, at noon, to view… The marks are thus shown every week, in every department, for the four years. Dependent upon this is the system of transfers, which has been in operation some forty years. The sections do not remain the same from week to week. [1]

These weekly grades were also used to resection the cadets. This idea of grading the cadets every day, is at the core of Thayer’s philosophy.  And this method of instruction, which quickly became known as the ‘Thayer method’, was in use in all departments for the entire century. In the technical subjects such as mathematics, science and engineering, cadet boards were graded on a daily basis through the 1980s.

Given a very limited faculty, this class-intensive form of instruction would have taken its toll on the professors.  Thayer outlined his solution to this problem in a letter to President James Monroe dated October 10th, 1828.

It must be very desirable, for the reasons mentioned in your letters, to relieve the professors from these duties. This may be done  1t. [first] by employing a number of young graduates who would not only act as assistant professors but also under the Instruction of the President would perform the more active executive duties. This class of persons would as teachers be eminently useful even now but will be found indispensable whenever the number of students shall amount to several hundreds. A professor can deliver lectures to many more than he can thoroughly teach.  I will illustrate the idea I would convey by supposing a case. A class of 80 students is to be taught Mathematics or Natural Philosophy devoting three hours of each day to the study of the subject at their rooms and three other hours with the Professor. One hour is to be taken up in the Lecture but this alone is not sufficient. Each student should demonstrate a proposition or explain an investigation at the Black-board and also be interrogated to see that he thoroughly understands the principles. This will require, as experience proves, not less that 15 minutes on an average for each student. Now it is evident that only eight students can be examined in the remaining few hours so that each can be examined only about once a fortnight which in effect is merely equivalent to no examination at all. What is to be done? Let the class be divided into at least four parts or sections and let each section attend 3 hours daily with an assistant professor to be examined upon the subject of the Lecture or lessons given on the preceding day. The Professor besides lecturing may either have the recitations of one Section himself or what would be the better practice, he might without taking the immediate charge, be present at the recitations, visiting each section in turn and only occasionally putting questions and giving explanations. You know that this is the system of instruction which has been practiced at West Point during the last ten years with what success I leave it for others to say.[2]

As Thayer indicates, these assistant professor were recent graduates, and in some instances, cadets from the upper classes who had shown a strength in mathematics.  As we will mention latter, some of these cadet instructors went on to other academic posts and helped to spread the USMA curriculum across the country.

In order to facilitate this style of instruction, the classrooms at the Academy were, and are to this day, fairly unique.  The 1896 Annual Report of the Superintendent describes them as follows:

Upon the walls in oak frames, their surfaces flush with the face of the frames, are twelve or fourteen slates, usually 4 feet by 3 feet 6 inches... They are all known by the generic name of blackboards.  From the lower part of each frame projects a shallow chalk tray, having at its bottom still shallower drawers, and above each drawer a galvanized wire grating. The chalk crayons and erasers, when not in use, are kept on the grating in tray, while the dust which there implements always generate falls into racks to support rulers and pointers.[3]

This means that there was a blackboard for each student (photo c. 1900) and several at the front for the instructor.  Several sources claim that Claudius Crozet introduced the blackboard to West Point in 1817, and that West Point was the first place the blackboard was used in the country.[4]  However, as noted earlier, the blackboard was in use at USMA in 1801.  Whether this was the first use of the blackboard in the United States is unclear. It was in use in various areas of the country by the second decade of the century: Philadelphia 1809, Boston 1812, Salem 1820, Bowdoin College 1824.  By the 1830s they were in common use in most schools.[5]

Blackboards (c. 1900)

Another of Thayer’s additions to the Academy was the institution of bi-annual exams, in the style of the École. 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 an 1877 copy of Davies’s Algebra in the West Point Library that was owned by Cadet Acton. On the endpapers is written

Examined Jany 2nd 1879.

Subject – “Logarithms”   “Fessed cold”

The cadet slang “fessed cold” means that Cadet Acton failed the exam. This is confirmed by his Cullum[6] 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!

Along with setting firm guidelines for academic instruction, Thayer set to work immediately to bring the curriculum to a higher level.  The topics taught during this period were: analytic geometry, plane and spherical trigonometry, mensuration, logarithms, conics, surveying, and fluxions.  In particular, the precursor to technical drawing, descriptive geometry became a mainstay of the curriculum for many years.  The Committee on Military Affairs of the Academy in 1834 describe descriptive geometry as: “a science peculiarly necessary in civil and military engineering, and which has been nowhere else cultivated with advantage or assiduity, save in France.”[7]  The first professor of descriptive geometry was Claudius Crozet.

[1]Keep, R., The System of Instruction at West Point: Can it be Employed in our Colleges, The New Englander, XXVIII (CVI), 1869, pp. 4-18. Available online at Cornell University Library “Making of America” website.

[2]The West Point Thayer Papers 1808-1872, ed. C. Adams, 1965.  Available online the USMA Library and at: http://www.dean.usma.edu/math/people/rickey/dms/doc/1828-10-10-Thayer-Monroe.htm

[3]Annual Report of the Superintendent of the United States Military Academy, 1896, p. 75.

[4]Edwin L. Dooley, Jr.,  Claudius Crozet: Disseminator of French Technical Education to the United States, Proceedings of the Consortium on Revolutionary Europe, 1750-1850, 1986, p. 452.

[5]Anderson, C., Technology in American Education: 1650-1900, 1962.

[7]Annual Report of the Superintendent of the United States Military Academy, Washington, 1896, p. 47.

## Claudius Crozet

Frenchman Claudius (Claude) Crozet (1790-1864) joined the faculty of the Academy on September 20, 1816 as Assistant Professor of Engineering, joining Alden Partridge who was the first Professor of Engineering in the United States.[1]  Crozet biographers Edwin Dooley and Robert Hunter indicate that it is unclear why Crozet left France, but that he met General Simon Bernard on the trans-Atlantic voyage and Bernard had met Thayer in Paris and so encouraged Crozet to come to West Point.[2]Beyond this, it is not known how Crozet happened to come to West Point.[3]

Claudius Crozet

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.  As Crozet attempted to teach descriptive geometry, he found that the cadets did not have the requisite knowledge to use the existing descriptive geometry texts such as Monge.  So he found himself teaching some algebra and geometry before he could teach descriptive geometry.  This led him to write an introductory descriptive geometry text at a level more suitable for the cadets.  Crozet was the first in America to teach this subject, and his A Treatise on Descriptive Geometry was published in 1821, the year he left the faculty.[4] This was the first textbook on the subject in English, and the first textbook written by a West Point faculty member while teaching there.[5] Jared Mansfield, the second professor at USMA, appointed May 1802, wrote a rather nice book, Essays, Mathematical and Physical, that was published in 1802 just before he joined the faculty. It was the first mathematics book published in the United States that contained original work.

Crozet’s text was used at the Academy for several years but 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. The Davies’ text, in turn, was replaced by a text by Albert Church in 1864, which was used until 1929.

[1]American State Papers. Military Affairs, vol. 1, p. 386.

[2]Dooley, E. and Hunter, R, Claudius Crozet: French Engineer in American 1790-1864, University of Virginia Press, 1989, p 15.

[3]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. The Crozet-Thayer connection is reported, without documentation, in Stephen E. Ambrose, Duty, Honor, Country; A History of West Point, Baltimore: The Johns Hopkins University Press, 1966. Reprinted in paperback, 1999, p. 97.

[4]Crozet spent the majority of the rest if his life in Virginia as an engineer and one of the founders of the Virginia Military Institute.

[5]For additional information about Mansfield and his book, see Joe Albree, “Jared Mansfield (1759-1830): Janus figure in American mathematics,” pp. 73-94 in History of Undergraduate Mathematics in America: Proceedings of a Conference Held at the United States Military Academy, West Point, New York, June 21-24, 2001, edited by Amy Shell-Gellasch.

## Charles Davies, Mathematics Professor, 1823-1837

Charles Davies was born 22 January 1798 in Washington, Connecticut. When he was a youth, the family moved to Black Lake, 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.[1]  Davies came to West Point in December 1813 and graduated in December 1815. There were no openings in the Corps of Engineers when he graduated so he took a less desirable position in artillery, serving a year in garrison duty before being transferred to the Corps of Engineers in August 1816. He resigned from the Army on 1 December 1816 to accept a position at West Point teaching mathematics.[2]  He served under department heads Andrew Ellicott and David Douglas, and then taught natural and experimental philosophy for two years, before becoming professor of mathematics in May of 1823 when Douglas became professor in philosophy.

While at West Point, Davies started a long and lucrative career as the author of text books, first for use at the Academy, and then across the country. The ones used at West Point were:

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.

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.[3]

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.”[4]

All of the works listed above, except the arithmetic, were used as textbooks at West Point, as well as many other schools. This is due in part to the dozens of West Point graduates who taught mathematics at schools across the country.[5]  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.”[6] 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.”[7]

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.[8]  In his lifetime Davies published 49 different titles.[9] 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.[10], (a firm that flourished by publishing the Davies texts) had sold about 7,000,000 texts by Davies, and were selling about 350,000 every year.[11] Is it any wonder then that he dominated mathematics textbook writing in the nineteenth century?

In 1831 two other translations of Bourdon’s Algebra appeared, one by Farrar at Harvard and one by Edward C. Ross.  Ross (1801-1851) taught at the Academy his last year as a cadet and graduated in 1821.  He remained at West Point as a mathematics instructor until 1833. After leaving the Academy, Ross went on to teach mathematics at Kenyon College in Ohio and then the Free Academy in New York City. [12]

The Ross translation, Elements of Algebra Translated from the French of M. Bourdon, for the use of cadets of the U.S. Military Academy (1831), was significant in that it freed the cadets from the necessity of studying French in order to read their mathematics texts. The Ross translation was used until Charles Davies produced his own in 1835 — based partly on that of Ross. Davies’ text, 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 made few substantial changes. (See the appendix for a listing of texts used at West Point and the date they were introduced into the curriculum.)

[1]Henry Eugene Davies, Davies Memoir. A Genealogical and Biographical Monograph on the Family and Descendants of John Davies of Litchfield, Connecticut. Privately Printed, 1895.

[2]This is curious, for in 1809 Ferdinand Hassler was forced to resign the professorship because he was not in the army. Probably the law changed in the meantime.

[3]For information about editions of these books, see Albree et al.

[4]Amy K. Ackerberg-Hastings, Mathematics is a Gentlemans Art: Analysis and Synthesis in American College Geometry Teaching, 1790-1840, Ph.D. dissertation, Iowa State University, 2000. University Microform 9977308.

[5]A list of USMA graduates who taught mathematics at other schools and the schools where they taught is being compiled. The information is available at  http://www.dean.usma.edu/math/people/rickey/dms/OldestSchools.html.

[6]Florian Cajori, The Teaching and History of Mathematics in the United States, Washington, Government Printing Office, 1890, p. 120.

[7]Amy K. Ackerberg-Hastings, p. 216. Chapter five of this work, “The two circles will touch each other internally: Charles Davies and the art and business of teaching geometry,” History of Undergraduate Mathematics in America: Proceedings of a Conference Held at the United States Military Academy, West Point, New York, June 21-24, 2001, edited by Amy Shell-Gellasch. See pp. 215-276, for the best available biography of Davies.

[8]Amy K. Ackerberg-Hastings, "Charles Davies, mathematical businessman", pp. 119-132 in Shell-Gellasch. Almost all of the books of Davies appear in multiple “editions” but many are so alike that they should be called “printings.”

[9]Davies’ earliest texts were translations of French authors.  These then evolved into text attributed to Davies alone, and eventually he wrote his own text from scratch.

[10]Note that this is not the Barnes of Barnes and Nobles fame.

[11]First Century of National Existence; the United States as They Were and Are . . .  by an Eminent Corps of Scientific and Literary Men. Illustrated with Over Two Hundred and Twenty-Five Engravings. San Francisco: L. Stebbins. 1875, p. 268. Available on the web through the Humanities Text Initiative: http://www.hti.umich.edu.

[12]Arney, Chris, West Point's Scientific 200: Celebration of the Bicentennial,  Palmetto Bookworks, 2002, pp. 57-58.

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

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.[3] Also, they used Legendre’s geometry,[4] Lacroix’s trigonometry,[5] both in translation, and Crozet’s book on descriptive geometry.[6]  These were all fine books and provided an excellent curriculum.

During the second year, Church and his classmates in the higher sections used Jean Baptiste Biot’s Essai de géométrie analytique, appliquée aux courbes et aux surrfaces du second ordre (second edition 1805) for analytic geometry and Silvestre François Lacroix’s Traité élémentaire de calcul différentiel et de calcul intégral (first edition 1802) for calculus.[7]The lower sections used Jean-Louis Boucharlat’s Elémens de calcul différentiel et de calcul intégral (first edition 1812) for calculus. (Again, the French the cadets studied every afternoon of their plebe or freshman year was so that they could learn their mathematics during their first and second years as well as their later engineering studies.) 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 except for the two years, 1832-34, when he joined his artillery unit. In 1837, he became professor of mathematics, succeeding Charles Davies. Church served as professor until his death in 1878, a total of fifty years.

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.[8]

These textbooks were also used at many other schools, but were not as widely used as those of Davies. Note that these titles were meant to be improvements on what was already being taught. There was 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.  Notable additions to the mathematics curriculum included determinants and the method of least squares in the 1880s.[9]

The texts of Church, and even more so Davies, dominated the curriculum at the Academy for the majority of the nineteenth century.  Within a decade of their introduction at West Point, the Davies and Church texts were being used across the country at schools such as Union College and The University of Michigan. In the closing years of the century, texts from yet another author connected to the Academy were introduced in the curriculum.

Edgar Bass (1843-1918) graduated from the Academy in 1868.  He returned after a year to teach natural and experimental philosophy from 1869 to 1874, and again from 1876 to 1898, succeeding Church as head of the Department of Mathematics in 1878.  The two intervening years in the 1870s he spent on the US expedition to New Zealand to observe the transit of Venus.[10]

Bass marks the beginning of a transition away from the traditional modes of instruction started in Thayer’s time, and away from the traditional texts of Davies and Church.  In order to clarify the older texts, and incorporate more of the calculus, he wrote a series of pamphlets for the use of the cadets. These later became his Elements of Differential Calculus, which replaced Church in 1896. He also wrote Elements of Trigonometry in 1888 for use at the Academy.

[1]The 1821 Annual Report of the Superintendent of the United States Military Academy, as quoted in the 1896 Report, p. 44, indicates “The superintendent was authorized to detail cadets to act as assistant professors, each to receive \$10 per month for extra services.” However, Lester A. Webb, Captain Alden Partridge and the United States Military Academy, 1806-1833 indicates on p. 172 that cadets were already being used as instructors in 1816. There is no evidence that either Davies or Church was a cadet instructor.

[2]Albert E. Church, Personal Reminiscences of the Military Academy, from 1824 to 1878, West Point, 1879. See pp. 39-41.  Available on the United States Military Academy Library web: http://usmalibrary.usma.edu/.

[3]Silvestre François Lacroix, Elements of Algebra, Translated from the French for the Use of Students of the University of Cambridge, New England (first edition 1818). In 1821, neither mathematics professor David Douglas nor Superintendent Thayer were aware that John Farrar had published this English translation of Lacroix [The West Point Thayer Papers 1808-1872, edited by Cindy Adams, 1965, Norton to Thayer, August 13, 1821]. The 1823 Board of Visitors report indicates that an English translation was used, so this confirms Church’s recollection. The 1825 report lists Lacroix’s Algebra, but whether it was in French or English is unclear. The “Tentative List of Text-Books” in the first Centennial volume indicates that a French edition of the work was used, and an 1825 French copy in the library bears the stamp “Textbook West Point 1823 to       ,” but we have come to distrust these stamps, which were probably inserted when Edward Holden was preparing the Centennial volumes. Professor Davies would have to have been very unhappy with the Farrar translation to have the cadets use the original French.

[4]Adrien-Marie Legendre, Elements of Geometry, Translated by John Farrar, For the Use of the Students of the University of Cambridge, New England (first edition 1818) is the edition that Church used. This is a translation of Éléments de géométrie avec des notes (first edition 1794). The West Point library has the tenth (1813) edition in French in a Thayer binding, indicating that it was purchased by Thayer while in France. For information on which editions are in the West Point library, see Albree et al., cited in note 2.

[5]Silvestre François Lacroix, An Elementary Treatise on Plane and Spherical Trigonometry, and on the Application of Algebra to Geometry; from the Mathematics of Lacroix and Bezout. Translated from the French for the Use of the Students of the University of Cambridge, New England (first edition 1820) by John Farrar.

[6]That Church used these books is indicated in Church, pp. 46-47.

[7]The use of textbooks in the original French, and especially which editions, is difficult to document due to the paucity of records. There is a copy of Silvestre Francois Lacroix’s Traite elementaire de trigonometrie rectiligne et spherique (1813 edition) in the West Point library that was owned by Lt Samuel Stanhope Smith. He graduated in 1818, but the fact that he included his rank may indicated that he procured this book later while teaching mathematics at West Point from 1818 to 1823. After that he taught Natural and Experimental Philosophy until his death in 1828.

[8]For information about editions of these books, see Albree et al.

[9]Annual Report of the Superintendent, 1896, pp. 61-63.

## Conclusion

As the century wore on, emphasis was placed more and more on teaching the cadets the analytic method as opposed to the strict rote learning of traditional instruction.  The new texts introduced by the faculty incorporated this new focus.[1] The 1860 Report of the Board of Visitors states that: “The introduction of the analytic method into the course of Natural Philosophy and into the preparatory course of Mathematics in consequence will probably form an era in the public education of the United States.”

Though topics were expanded, no new subject were added to the mathematics curriculum for most of the century. After the introduction of descriptive geometry and the calculus in the nineteen-teens, the next major topics to be added were determinants and the method of least squares in 1880.

[1]Keep, Robert P., The System of Instruction at West Point: Can it be incorporated in our Colleges, The New Englander, CVI, Jan 1869, pp. 1-18.  Available from “Cornell University Making of America” online.

## Appendix

Mathematical Text Books In Use at USMA in the Nineteenth Century

Dates represent earliest documented use.

Algebra:

 1802 Hutton (Charles), Course of Mathematics, 1800 [S] 1818 Simson (Robert), Elements of Conic Sections, 1792, Note: this is the first mention of Simson in that Hutton replaces Simson for conic sections, [S] 1823 Lacroix (S. F.), Elements D'Algebra, 1820, (Hutton discontinued) [C, S] 1825 Lacroix, Elements of Algebra, 1825, (English translation by J. Farrar) [BV-25, C] 1831 Bourdon (M.), Elements of Algebra, 1831, (English translation by Lient. E. Ross) [C, S] 1837 Davies (Charles), Elements of Algebra..., 1836 (based on Ross' translation of Bourdon) [S] 1881 Davies' Bourdon, 1877 [S] 1900 Smith (Charles), A Treatise on Algebra, 1893 [C]

Geometry and Analytic Geometry:

 1802 Hutton, Course of Mathematics, 1800 [S] 1821 Biot ( J. B.), Essai de Geometrie Analytique, 1826, [S] 1823 Legendre (A. M.), Elements of Geometry and Trigonometry, English translation by Farrar, 1825 [S] 1837 Davies, Elements of Descriptive Geometry, translation of Legendre, 1826, [S] 1852 Church, Elements of Analytic Geometry, 1851, [S] 1864 Church, Elements of Descriptive Geometry, 1870, [S] 1896 Davies, translation of Van Amringe, 1885, note-both Davies and Church texts are used. [S]

Trigonometry (plane & spherical, including conic sections):

 1802 Hutton, Course of Mathematics, 1800 [S] 1821 Simson, [BV-21] Gregory, [BV-21] 1824 Lacroix, (Farrar translation), 1822 [BV-24, S] 1825 Biot [S] 1832 Legendre, Elements of Geometry and Trigonometry, (Farrar 1825 or Brewster 1828 translation) [S] 1839 Davies, (translation of Legendre), 1822 [S] 1881 Church, Plane and Spherical Trigonometry, 1869 [S] 1896 Ludlow (Henry), Elements of Trigonometry, 1888 [S]

Fluxions / Calculus:

 1821 Lacroix, Traite Elementaire de Calcul Differentiel et de Calcul Integral, 1820 [BV-25] 1824 Hutton, [BV-24, S] 1825 Boucharlat (Jean L.), Elemens de calcul differentiel it de calcul integral, 1826, (in French) [BV-25] 1839 Davies, Elements of the Differential and Integral Calculus, 1839, [C, S] 1843 Church (Albert E.), Elements of Differential and Integral Calculus, 1842, (replaces Davies), [C, S] 1878 Boucharlat still used in lower sections [S] 1887 Bass (Edgar), Introduction to Calculus, 1887, (replaces first part of Church), [S] 1889 Bass, Differential Calculus I, (pamphlet), (replaces second part of Church) [S] 1893 Bass, Differential Calculus II, (pamphlet), (replaces last part of Church) [S] 1896 Bass, Differential Calculus, [S]

 1821 Crozet (Claudius), A Treatise on Descriptive Geometry, 1821, [S] 1832 Davies, Elements of Descriptive Geometry, 1826, [C, S] 1852 Davies, A Treatise on Shades and Shadows and Linear Perspecitve, 1851, [S] 1864 Church, Elements of Descriptive Geometry, (replaces Davies in 1865), [S]

Surveying:

 1832 Davies, Elements of Surveying, 1830, [S]

Sources:    S     Report of the Superintendent, 1896

BV-##    Board of Visitors report for the year 18##

C    Centennial Volume[1]

[1] Centennial Volume, USMA Board of Visitors, USMAA, 1900.