Title: Amy Ackerberg-Hastings
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Chapter Five
The Two Circles Will Touch Each Other Internally:
Charles Davies at the Art and Business of Teaching Geometry
“[W]ith scientific attainments fit for a head schoolmaster, he [Charles Davies] has exactly the kind of talent fit for making Columbia & its Board the instrument of his own selfish needs.” “[W]ith all his selfishness and mischief making, he is, and always was a fool. I have known him from boyhood, before he became a professor at West Point, and when he was brought into Columbia, I felt assured that his influence would ruin all your hopes, for he is essentially a [small] minded man, and incapable of generous impulses or enlarged views.” The average person who managed to so antagonize prominent men at a time when one would expect them to have been more concerned with the American Civil War would likely have found his career at an end. The situation proved to be no impediment to a born salesman and teacher like Davies, however. He not only survived this episode with his reputation intact to continue to write and revise wildly successful mathematics textbooks, he eventually talked the Columbia Board into granting him the prestige of emeritus status when he retired in 1865, a title Davies used proudly to the end of his life. 

J. B. Barnard to G. Kemble, 26 October 1863, Hamilton Fish Papers, Rare Book and Manuscript Library, Columbia University, New York.

G. Kemble to Hamilton Fish, 31 October 1863, Hamilton Fish Papers (cit. n. 1).

Many of the same characteristics which irritated some acquaintances were the features which distinguished Davies from Jeremiah Day and John Farrar. Most notably, Davies viewed writing textbooks as one of his primary occupations, rather than as something he did out of necessity or for a short time only. He then actively and openly marketed his books, building up his own reputation in a manner that Day and Farrar would have rejected as unseemly even though they lived just as comfortably on the profits from their textbook series. An exceptionally prolific author, Davies also differed from Day and Farrar because he was educated in a military environment and then moved around to several institutions during his career, he revised his textbooks over time, and he wrote separate books for children, secondary students, and college students. Like Day and Farrar, though, Davies was largely self-taught in mathematics, was a young man when he became a professor, and was beloved in the classroom. In his geometry textbook, Elements of Geometry and Trigonometry, Davies mixed together content from French, Scottish, and English textbooks with the goal of training the intellect. While his treatment of geometry could be superficial at times, the influence of “Davies’s Legendre” ran deep as it was overwhelmingly adopted in the United States in the mid-nineteenth century. Yet, overall, what shaped all these aspects of Davies’s career was the fact that he saw himself simultaneously as a professor and a businessman, like two touching circles with one inside the other.
The Shallow Roots of West Point Mathematics
Military education in the United States did not command the inherent prestige Harvard and Yale held through those institutions’ longevity. Rather, after two other short-lived attempts at formal, hierarchical training, a military school was not opened until 1801. At West Point in the Hudson Highlands of New York, a Revolutionary War fort (the plans of which Benedict Arnold had attempted to sell to the British), the school’s twelve cadets were instructed by George Baron, who previously taught at the Royal Military Academy in England with Charles Hutton and who was later known for nitpicking over the computational details in Nathaniel Bowditch’s Navigator. Baron’s contentious personality led to his firing in February 1802. The school was then replaced by the United States Military Academy, as established by the Military Peace Establishment Act passed by Congress on March 16, 1802, to create the Army Corps of Engineers. President Thomas Jefferson used the powers granted him by the Act to choose officers and cadets whom he believed would model the Academy on France’s École Polytechnique.

Albert L. Mills, “Introduction,” in The Centennial of the United States Military Academy: 1802-1902, vol. 1 (Washington, DC: Government Printing Office, 1904), pp. 1-10; Edward R. Hogan, “George Baron and the Mathematical Correspondent,” Historia Mathematica 3 (1976): 403-415.

Histories of the Military Academy include Stephen E. Ambrose, Duty, Honor, Country: A History of West Point (Baltimore: The Johns Hopkins University Press, 1966); and Centennial (cit. n. 3). On the founding of this institution, see also Allan R. Millett and Peter Maslowski, For the Common Defense: A Military History of the United States of America, rev. ed. (New York: The Free Press, 1994), pp. 75, 104.

While Jefferson had an educational connection to Scottish mathematics—William Small, one of Jefferson’s teachers at William and Mary, emphasized Euclid’s Elements as the foundation of mathematics and sound reasoning of any sort and trained Jefferson both in Newtonian science and Common Sense philosophy—his influence upon the United States Military Academy was mainly limited to hiring Jonathan Williams and endorsing the Academy’s scientific society; I. Bernard Cohen, Science and the Founding Fathers: Science in the Political Thought of Jefferson, Franklin, Adams, and Madison (New York: W. W. Norton & Co., 1995), pp. 68-72. See also Florian Cajori, The Teaching and History of Mathematics in the United States (Washington, DC: Government Printing Office, 1890), p. 84; Peter Michael Molloy, “Technical Education and the Young Republic: West Point as America’s École Polytechnique, 1802-1833” (Ph.D. diss., Brown University, 1975), pp. 172, 273-274. On Jefferson’s direct mathematical influence on the University of Virginia, see David Eugene Smith, “Thomas Jefferson and Mathematics,” Scripta Mathematica 1 (1932-1933): 3-14; and “A Note on Thomas Jefferson,” Scripta Mathematica 1 (1932-1933): 87-90.

By July, William A. Barron, Jared Mansfield (1759-1830), and Colonel Jonathan Williams (1750-1815) were settled at West Point and offering courses. Barron was a Harvard graduate and tutor who had worked closely with Samuel Webber, while Williams was perhaps most likely to introduce the French influence, given his background and position as Superintendent of the Academy. He had worked for Benjamin Franklin in Europe in the 1780s and returned to the United States to become an active member of the American Philosophical Society. Williams’s collection of technical books formed the basis of the Academy’s library. A Yale graduate, Mansfield was already known as a surveyor and as the author of what some historians have called the first piece of American mathematical research, the 1801 Essays, Mathematical and Philosophical. The War Department bestowed a military commission as captain of engineering upon him so that he could teach engineering and mathematics at the Academy. In the mathematics part of the course, Mansfield taught algebra and Barron covered geometry, both from Charles Hutton’s A Course of Mathematics. Stephen Ambrose called the entire Academy course of study “superficial,” noting that cadets took the graduation examination whenever they felt they were ready. The course material remained at this same informal level into the 1810s. From 1802 to 1813, there was a Military Philosophical Society organized by Williams, but most members were civilian and held their bi-weekly meetings away from West Point. Only the top cadets attended the discussions of military engineering with the faculty.

Ambrose, Duty, Honor, Country (cit. n. 4), pp. 21-22, 25. An extended biography of Williams is Molloy, “Technical Education” (cit. n. 5), pp. 199-238.

Biographies of Mansfield include “Mansfield, Jared,” in The National Cyclopedia of American Biography (hereinafter cited NCAB), vol. 3 (New York and Clifton, NH: James T. White and Company, 1898-1984), p. 214; and Alois F. Kovarik, “Mansfield, Jared,” in Dictionary of American Biography (hereinafter cited DAB), ed. Allen Johnson, vol. 6, part 2 (New York: Charles Scribner’s Sons, 1927-1936), pp. 256-257.

Ambrose, Duty, Honor, Country (cit. n. 4), pp. 24-37.

Samuel E. Tillman, “The Academic History of the Military Academy, 1802-1902,” in Centennial (cit. n. 3), pp. 223-241, on pp. 223-230.

Molloy, “Technical Education” (cit. n. 5), pp. 271-308; Ambrose, Duty, Honor, Country (cit. n. 4), p. 30. See also Sidney Forman, “The United States Military Philosophical Society, 1802-1813,” William and Mary Quarterly 2 (1945): 273-285.

Academy faculty generally did not remain at the institution for long. Mansfield was relieved in 1803 when he was appointed surveyor-general of the Northwest Territory, while Williams resigned briefly in the same year. Williams departed a second time in 1812, after the Academy had begun to suffer from the actions of a hostile Secretary of War, William Eustice. For instance, in 1810 Eustice took away commissions from Academy graduates, forcing them to enter active service as privates, and transferred several hundred artillerymen into the cadets’ and faculty’s quarters at West Point. Another faculty member who remained at the Academy only briefly was the Swiss-born Ferdinand Hassler, who replaced Barron in 1807 but then himself left to set up the United States Coast Survey in 1810. Thus, there were in fact no instructors—or students—remaining in West Point when professional military leaders were needed for the War of 1812 except for Alden Partridge (1785-1854), who had become assistant professor in mathematics in 1806 after studying for three years at Dartmouth College and one at the Academy. One of the first two graduates of the Academy, Joseph G. Swift, succeeded Williams as Superintendent but stayed away from West Point pursuing his interests in military politics and then the war effort. Swift was also Chief of Engineers for the United States Army.
Reacting in shock and concern at the nation’s military unreadiness, members of Congress began to act to regularize the institution. The number of cadets had already been optimistically limited to 250 on April 29, 1812 (there had been only eighty-eight graduates in the Academy’s first ten years), and an Academic Board was granted the authority to confer degrees. Reading, writing, and arithmetic were established as entrance requirements. Furthermore, formal provisions were made for a permanent and enlarged faculty. Partridge was the first to accept a full professorship, of civil and military engineering, in 1813. He was then named Superintendent of the Academy on January 31, 1815; he had been serving as the acting superintendent in Swift’s absence. In the meantime, Mansfield was called back to West Point in 1812 as Professor of Natural and Experimental Philosophy, taking up his post in 1814, while Andrew Ellicott was hired as Professor of Mathematics to succeed Partridge. Although he also used only Hutton’s A Course of Mathematics as a textbook, Ellicott took his students all the way from algebra and geometry through trigonometry and logarithms to calculus during the three hours of mathematical instruction held each day. Yet, cadets continued to enter the Academy at their convenience throughout the year and nearly always left or were commissioned long before they completed the four-year course.

Ambrose, Duty, Honor, Country (cit. n. 4), pp. 38-50; Silvio A. Bedini, Thinkers and Tinkers: Early American Men of Science (New York: Charles Scribner’s Sons, 1975), p. 364; Tillman, “Academic History” (cit. n. 9), p. 227, 241; Mills, “Introduction” (cit. n. 3), p. 10.

Davies’s Mobile Career
Meanwhile, Charles Davies was born in Washington, Connecticut, on January 22, 1798. While he was still quite young, his parents moved to St. Lawrence County, New York, where his father farmed and became county sheriff. Davies attended public schools near his home and helped his father until the War of 1812. At that time, General Swift became acquainted with the Davies family while he was supervising fortification construction along the St. Lawrence River. Swift was favorably impressed by the young Davies and recommended him for appointment to the Military Academy, which was accomplished on December 27, 1813.

Biographies of Davies include: “Davies, Charles,” in NCAB (cit. n. 7), vol. 3, p. 26; “Charles Davies,” in Biographical Register of the Officers and Graduates of the U. S. Military Academy at West Point, N.Y., George W. Cullum, 3d ed., vol. 1 (Boston and New York: Houghton, Mifflin and Company, 1891), pp. 151-155; E. D. Mansfield, “Charles Davies,” in Eighth Annual Reunion of the Association of the Graduates of the United States Military Academy at West Point, New York, June 14, 1877 (New York: A. S. Barnes & Co., 1877), pp. 23-27; “Presidents of New York State Teachers’ Association,” American Journal of Education 15 (1865): 477-487; and “Charles Davies,” in Davies Memoir: A Genealogical and Biographical Monograph on the Family and Descendants of John Davies of Litchfield, Connecticut, Henry Eugene Davies (Privately Printed, 1895), pp. 67-73. See also “Prof. Charles Davies,” American Journal of Science 112 (1876): 320.

Register of Cadets Admitted, Special Collections and Archives, United States Military Academy, West Point; “Davies,” Davies Memoir (cit. n. 12), pp. 67-68; “Davies,” Eighth Annual Reunion (cit. n. 12), p. 23.

The uncooperative New England militia and failed American offensives on the Canadian front certainly left Swift feeling the urgent need for training new officers, but the Davies family was not as anxious to send their teenager downstate to an institution which offered no consistent system of instruction. Swift had written the family in November 1813 to advise them of Davies’s impending admittance. He wrote another letter on February 18, 1814, asking for Davies’s acceptance or nonacceptance of his appointment and requesting Davies to report to West Point by April 1. Davies did then begin his studies, but, despite the end of the war in the winter of 1814-1815, Davies graduated and was commissioned as a Brevet Second Lieutenant of Light Artillery in December 1815 without completing the entire Academy course. Ellicott sent him to service in New England with a testimonial: “As a regular student Mr. Davies had but few equals in this institution, and his progress was such, that it may be considered both unfortunate to himself, and his country, that he was called away before he had completed a full course of scientific education.”

On the course of the War of 1812, see Millett and Maslowski, For the Common Defense (cit. n. 4), pp. 106-119.

Joseph G. Swift to Thomas John Davies, 18 February 1814, Charles Davies, Miscellaneous Papers, Special Collections and Archives, United States Military Academy, West Point. This letter mentioned Swift’s previous missive.

“Davies,” Davies Memoir (cit. n. 12), p. 68. “Davies,” Eighth Annual Reunion (cit. n. 12), pp. 23-24. Davies was one of forty Academy graduates in 1815; there was no class in the following year; Register of Graduates and Former Cadets of the United States Military Academy (West Point, NY: The West Point Alumni Foundation, Inc., 1964).

Andrew Ellicott about Charles Davies, 20 December 1815, Charles Davies, Miscellaneous Papers (cit. n. 15).

Indeed, the professors at the Military Academy held Davies in high enough esteem that, after he was promoted to Second Lieutenant and transferred to the Corps of Engineering on August 31, 1816, he was invited to return to West Point as an assistant professor of mathematics when John Wright resigned in December 1816. Davies, though, found himself in the midst of a storm when he arrived at the Academy. Partridge had proven to be a rather autocratic superintendent, putting the cadets through endless drills and administering stern punishments. He lost the support of Mansfield and Ellicott even further by trying to force his novel educational ideas upon the other men. Meanwhile, the War Department was clarifying the Academy’s regulations, or organizational principles, and published the new version in July 1816. When it appeared by 1817 that Partridge was balking at implementing the new rules, President Monroe appointed Sylvanus Thayer (1785-1872) to replace Partridge upon Thayer’s return from a two-year study of military schools, armies, and fortifications in Europe. Partridge left the Academy on leave after the appointment was announced, but he returned shortly before Thayer’s appointment began on July 28, 1817, convinced the cadets to take his side, incited them to arrest the professors, and seized command.

George Graham to Joseph G. Swift, 8 January 1817, Thayer Papers, The West Point Thayer Papers, 1808-1872, ed. Cindy Adams, et al, vol. 2 (West Point, NY: Association of Graduates, 1965), Special Collections and Archives, United States Military Academy, West Point; “Charles Davies,” Biographical Register (cit. n. 12), p. 151.

Cajori, Teaching and History (cit. n. 5), p. 86; Ambrose, Duty, Honor, Country (cit. n. 4), pp. 38-48. The biography of Partridge in NCAB glosses over this portion of Partridge’s career, while Thomas M. Spaulding in DAB was more critical. “Partridge, Alden,” in NCAB (cit. n. 7), vol. 18, pp. 322-323; and Thomas Marshall Spaulding, “Partridge, Alden,” in DAB (cit. n. 7), vol. 7, part 2, pp. 281-282.

Regulations, 2 July 1816 (West Point: U. S. Military Academy, 1802-1816), Special Collections and Archives, United States Military Academy, West Point.

Like Partridge, Thayer had completed the classical course at Dartmouth and then spent a year at the Academy, graduating in 1808. He assisted with construction of coastal fortifications in New England and New York until Swift sent him to Europe in 1815. See Gustav Joseph Fiebeger, “Thayer, Sylvanus,” in DAB (cit. n. 7), vol. 9, part 2, pp. 410-411; and “Thayer, Sylvanus,” in NCAB (cit. n. 7), vol. 7, p. 87; as well as Register of Graduates (cit. n. 16), a memorial volume devoted to Thayer.

Ambrose, Duty, Honor, Country (cit. n. 4), pp. 50-61; Tillman, “Academic History” (cit. n. 9), p. 241; Sylvanus Thayer to George Graham, 4 August 1817, Thayer Papers (cit. n. 18), vol. 2.

Although Thayer quickly enlisted reinforcements including General Swift, garnered an order to arrest Partridge, and restored order at the Academy, it took some time to sort out Davies’s role in the affair. On Partridge’s recommendation, General Swift had given him the temporary appointment as assistant professor of mathematics, which was rubber-stamped by George Graham, the acting Secretary of War, in January 1817. After his inauguration, President Madison received complaints about the manner in which Davies was given the position and ordered through Graham that Davies’s qualifications be verified through examination. Then, already suspected to be a crony of Partridge, Davies had the misfortune to be away from the Academy when Partridge seized control and arrested the professors at the end of July. The War Department believed that Davies had helped Partridge and consequently ordered the young man’s arrest. Swift and Mansfield intervened on his behalf, reporting that he was not involved, while Swift also asked Thayer to merely tell Davies he was arrested rather than to take him into custody, in order to save him “too much mortification.” Ultimately, Davies was called as a witness at Partridge’s court martial and was to undergo an investigation himself during the October trial. Partridge was convicted, but his November 1817 sentence was remitted by President Madison. Partridge resigned from the Army on April 15, 1818, to go on to found a series of short-lived military preparatory schools. Davies had already been carrying on with his duties for several months by that point and the controversy died away so much that it was not mentioned in biographies of Davies.

Spaulding, “Partridge” (cit. n. 19).

George Graham to Joseph G. Swift, 8 January 1817; Jared Mansfield to John O’Connor, [3] September 1817, both in Thayer Papers (cit. n. 18), vol. 2.

George Graham to Joseph G. Swift, 14 February 1817, Thayer Papers (cit. n. 18), vol. 2.

Sylvanus Thayer to George Graham, 14 August 1817, Thayer Papers (cit. n. 18), vol. 2. George Pappus reported that Davies was with the cadets who welcomed Partridge when he docked at the Academy; George S. Pappus, To the Point: The United States Military Academy, 1802-1902 (Westport, CT: Pareger Publishers, 1993), p. 92. Apparently, though, Davies saw Partridge as a friend and mentor and, lacking teaching experience at that time, did not fully appreciate the differences between Partridge and the professors. He was, after all, only nineteen years old.

Daniel Parker to Joseph G. Swift, 3 September 1817; Jared Mansfield to John O’Connor, [3] September 1817; Joseph G. Swift to Sylvanus Thayer, 8 September 1817, all in Thayer Papers (cit. n. 18), vol. 2. Yet, despite his defense of his protégé, Swift was apparently willing to sacrifice Davies if that would help establish control over the cadets. See Joseph G. Swift to Sylvanus Thayer, 13 September 1817, Thayer Papers (cit. n. 18), vol. 2.

Joseph G. Swift to Sylvanus Thayer, 20 September 1817; George Graham to Sylvanus Thayer, 25 September 1817; George Graham to Joseph G. Swift, 25 September 1817, all in Thayer Papers (cit. n. 18), vol. 2.

Spaulding, “Partridge” (cit. n. 19).

See, for example, a September 28, 1817, report on the progress of the cadets to the War Department signed by Davies as well as the rest of the faculty in Thayer Papers (cit. n. 18), vol. 2.

Historians nearly unanimously agree that a new era began at the Academy when Thayer became Superintendent. Thayer took Williams’s somewhat vague conception of an “American École Polytechnique” and threw himself into replicating the features he had observed in France. As a first step, he implemented the reforms called for by the 1816 Regulations: he established a five-member Board of Visitors to attend the two general examinations to be held each year, set up the chain of command at the Academy, and brought in a course of instruction for leading American cadets through French textbooks in engineering and military science. The entrance examinations were given orally beginning in 1818. Recitation sessions were regularized, divided by ability, and organized around the blackboard. Rather than only passively hearing lectures, cadets additionally would be required to master a portion of the assigned text and recite from it to the instructor. Professors, on the other hand, were freed from overseeing an individual program for each cadet once there was a uniform admission calendar. To keep track of student performance in the recitation sections, Thayer imposed a detailed arithmetical grading system based upon the French model. The students were marked in each class each day on a scale from “perfect” (3.0) to “complete failure” (0.0). These scores were totaled in weekly reports, weighted by subject from engineering, natural philosophy, and mathematics (2.0) to French (0.5), and combined with remarks on the amount of work completed by each student and on individual character. Finally, the aggregate marks, together with scores on the semi-annual examinations, were used to sort the cadets by order of merit. The cadets resisted Thayer’s disciplined system for years, trying tactics from a mutiny in 1818 to a failed attempt to launch a cannonball into Thayer’s house in 1821, but Thayer’s reforms were wholly implemented by the mid-1820s.

Regulations, 2 July 1816 (cit. n. 20). See also Keith Hoskin, “Textbooks and the Mathematisation of American Reality: The Role of Charles Davies and the US Military Academy at West Point,” Paradigm no. 13 (1994): 11-41, on pp. 26-30.

On Thayer’s grading system, see Tillman, “Academic History” (cit. n. 9), pp. 223-241; Hoskin, “Textbooks” (cit. n. 31), pp. 27-28. For examples of the system in use, see Staff Records, No. 1, 1818-1835, Special Collections and Archives, United States Military Academy, West Point.

Ambrose, Duty, Honor, Country (cit. n. 4), pp. 62-86; Hoskin, “Textbooks” (cit. n. 31), p. 27.

Perhaps Thayer’s most sound move in recreating the École Polytechnique was encouraging the former student in the institution who had found himself at West Point. Named Professor of Engineering on March 6, 1817, Claudius Crozet (1790-1864) was a natural to establish the twin pillars of Thayer’s course, the French language and mathematics. Crozet soon discovered that the cadets were not prepared to learn the advanced engineering he wanted to teach. He had to review elementary mathematics with them first, although, as Thayer’s system became normal practice at the Academy and as the cadets adjusted to reading mathematics textbooks in French, at least the best section of students was able to complete the mathematical subjects listed in the 1816 Regulations (logarithms, algebra, geometry, plane and spherical trigonometry and applications to problems, infinite series, conic sections, fluxions, analytical geometry, and mensuration) and learn analytical trigonometry, differential and integral calculus, and mechanical principles from Crozet by the early 1820s. Crozet also introduced the subject of descriptive geometry developed by Gaspard Monge, beginning a tradition at West Point by writing a textbook on the subject in 1821.

Regulations, 2 July 1816 (cit. n. 20); “Crozet, Claudius,” in NCAB (cit. n. 7), vol. 18, pp. 393-394.

Tillman, “Academic History” (cit. n. 9), pp. 223-241; Ambrose, Duty, Honor, Country (cit. n. 4), pp. 87-105; “Crozet” in NCAB (cit. n. 34); Thomas Marshall Spaulding, “Crozet, Claude,” in DAB (cit. n. 7), vol. 2, part 2, pp. 580-581; Claude Crozet, Treatise on Descriptive Geometry (1821).

Davies embraced this renewed curriculum while he carried out his duties in the department of mathematics and, from October 31, 1821, to April 29, 1823, as assistant professor of natural and experimental philosophy under Mansfield. Ellicott, the professor of mathematics, had been succeeded by his son-in-law, David Douglass, in 1820. When Crozet left the Academy in 1823, Douglass became Professor of Engineering, and Davies was promoted to Professor of Mathematics. In this capacity, Davies proved as hardworking as the other mathematics professors, Day and Farrar. Most notably, Davies followed Crozet’s example and produced a series of textbooks in the late 1820s and early 1830s by translating or co-opting others’ translations of French mathematics textbooks. (See Table 5.1 for a list of Davies’s publications.) Among Davies’s daily duties, he supervised four to five assistants, perhaps dropping in on their sections as Ellicott had, standardizing the instruction offered across sections, and apparently also teaching some of the classes himself. He and the Academy faculty were persistent over the years in requesting that the assistants be drawn from graduated officers rather than from the elder cadets, so that they would be better able to keep order and explain the lesson in the recitation room. All his labors still wore on Davies, though, and rumors circulated as early as 1834 that he 
Table 5.1. Publications by Charles Davies.						

Keith Hoskin argues that Davies was so young—not quite nineteen when he became an assistant professor—that Thayer could work his magic of disciplinarity as well on him as on the cadets. Hoskin, “Textbooks” (cit. n. 31), p. 30.

Tillman, “Academic History” (cit. n. 9), pp. 241-245.

Davies published so many textbooks with such similar titles that compiling a list of them is a challenge. The titles and dates in Table 5.1 are “best guesses” based on The National Union Catalog (hereinafter cited NUC), 753 vols. (London: Mansell, 1976); Henry Barnard, “American Textbooks,” American Journal of Education 13 (1863): 625-640; 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, History of Mathematics Vol. 18 (American Mathematical Society and London Mathematical Society, 2000); and copies of Davies’s textbooks which were available during this study.

Cajori, Teaching and History (cit. n. 5), p. 115; Staff Records, 15 June 1835 (cit. n. 32).

For instance, see Staff Records, 9 September 1833 and 22 September 1835 (cit. n. 32).

“Demonstration of a Problem in Conic Sections.” American Journal of Science 6 (1823): 280-282.

Elements of Descriptive Geometry. Philadelphia, 1826. 28 printings.

Elements of Geometry and Trigonometry.  New York, 1828. More than 33 printings.

Elements of Surveying With the Necessary Tables. New York: J. & J. Harper, 1830. 7 printings. 26 printings as Elements of Surveying and Navigation after 1841. 8 printings as Elements of Surveying, Navigation, and Levelling after 1870.

Treatise on Shades and Shadows, and Linear Perspective. New York: J. & J. Harper, 1832. 18 printings.

Common School Arithmetic. Hartford, 1833. 4 printings.

Elements of Algebra: Translated from M. Bourdon. New York: Wiley and Long, 1835. 41 printings.

Elements of Analytical Geometry. New York: Wiley and Long, 1836. 25 printings.

Elements of the Differential and Integral Calculus. New York: Wiley and Long, 1836. 16 printings. 3 printings as Elements of Analytical Geometry and of Differential and Integral Calculus after 1859.

Mental and Practical Arithmetic. Hartford: A. S. Barnes & Co., 1838. 3 printings.

First Lessons in Algebra. New York: Wiley and Long, 1838. 5 printings.

First Lessons in Geometry. Hartford: A. S. Barnes & Co., 1839. 2 printings.

Practical Geometry. Philadelphia: A. S. Barnes & Co., 1839. 7 printings.

First Lessons in Arithmetic. Hartford: A. S. Barnes & Co., 1840. 13 printings.

Elementary Geometry. Philadelphia: A. S. Barnes & Co., 1841. 9 printings.

Arithmetic, Designed for Academies and Schools. Hartford, 1841. 12 printings.

Elementary Algebra. Philadelphia: A. S. Barnes & Co., 1842. 22 printings.

Table 5.1. (continued)									

Key to Davies’s Elementary Algebra. Philadelphia: A. S. Barnes & Co., 1842. 2 printings.

Key, Containing the Statements and Solutions of Questions in Davies’s Elementary Algebra. Philadelphia: A. S. Barnes & Co., 1844. 14 printings.

Key to Davies’s Arithmetic. Philadelphia: A. S. Barnes & Co., 1844. 6 printings.

The Case of Frederick Emerson versus Charles Davies and Alfred S. Barnes. Boston, 1845.

University Arithmetic. New York: A. S. Barnes & Co., 1846. 21 printings.

Key to University Arithmetic. New York: A. S. Barnes & Co., 1846. 10 printings.

Elements of Drawing and Mensuration. New York: A. S. Barnes & Co., 1846.

The Arithmetic Table Book. New York: A. S. Barnes & Co., 1848. 3 printings.

The Logic and Utility of Mathematics, With the Best Methods of Instruction Explained and Illustrated. New York: A. S. Barnes & Co., 1850. 8 printings. 1 printing as The Nature and Utility of Mathematics in 1873.

Grammar of Arithmetic. New York: A. S. Barnes & Co., 1850. 3 printings.

Lecture on the Duties and Relations of Parents, Teachers and Pupils, as Connected with Education.  Detroit:  Free Press Book and Job Office Print., 1852.

Practical Mathematics with Drawing and Mensuration. New York: A. S. Barnes and Co., 1852. 6 printings.

School Arithmetic, Analytical and Practical. New York: A. S. Barnes and Co., 1852. 14 printings.

Intellectual Arithmetic. New York: A. S. Barnes and Co., 1854. 14 printings.

Primary Arithmetic and Table Book. New York: A. S. Barnes and Co., 1855. 3 printings.

With William Guy Peck. Mathematical Dictionary and Cyclopedia of Mathematical Science. New York: A. S. Barnes & Burr, 1855. 10 printings. 

Table 5.1. (continued)									

Key to Bourdon’s Elements of Algebra. New York, 1856. 7 printings.

Key to Arithmetic for Academies and Schools. New York, 1856.

Key to School Arithmetic, Analytical and Practical. New York: A. S. Barnes and Co., 1856. 7 printings.

New University Arithmetic. New York, 1856. 8 printings.

Key to New School Arithmetic. New York: A. S. Barnes and Co., 1856. 3 printings.

“Inaugural Address, Charles Davies, LL.D.” In Addresses of the Newly-Appointed Professors of Columbia College. Intro. William Betts. New York, 1858. Pp. 117-151.

University Algebra. New York: A. S. Barnes and Co., 1858. 14 printings.

Key to Davies’s University Algebra. New York: A. S. Barnes and Co., 1859. 2 printings.

New Elementary Algebra. New York: A. S. Barnes and Co., 1859. 16 printings.

Differential and Integral Calculus. New York, [1860]. 3 printings.

Primary Arithmetic. New York: A. S. Barnes and Co., 1862. 5 printings.

Practical Arithmetic. New York: A. S. Barnes and Co., 1863. 13 printings.

Elements of Written Arithmetic. New York: A. S. Barnes and Co., 1863. 5 printings.

Metric System Explained. New York: A. S. Barnes and Co., 1867. 1 printing.

Outlines of Mathematical Science. New York: A. S. Barnes and Co., 1867. 1 printing.

Metric System of Weights and Measures. New York: A. S. Barnes and Co., 1870. 3 printings.

Key to the Practical Arithmetic. New York: A. S. Barnes and Co., 1870. 2 printings.

Table 5.1. (continued)									

An Examination of the Demonstrations of Davies’s Legendre. New York: A. S. Barnes and Co., 1873. 1 printing.

Address to  Fifth Annual Reunion of the Association of the Graduates to the United States Military Academy at West Point, New York.  June 11, 1874.  New York:  A. S. Barnes & Co., 1874.

Address to  Sixth Annual Reunion of the Association of the Graduates to the United States Military Academy at West Point, New York.  June 17, 1875.  New York:  A. S. Barnes & Co., 1875.

First Book in Arithmetic. New York: A. S. Barnes and Co., 1877. 1 printing.

The Complete Arithmetic. New York: A. S. Barnes and Co., 1877. 1 printing.	

planned to leave West Point. In late 1836, suffering from a bronchial infection, he embarked on a recuperation tour of historical sites in Europe which lasted at least seven months, while his family stayed with the Mansfields, who had retired to Cincinnati in 1828. When he returned at the end of May 1837, Davies officially resigned from the Academy, and the Davies clan moved to Hartford, Connecticut. He was succeeded at the Academy by Albert E. Church, one of the assistant professors. By 1839, all the mathematical textbooks taught at the Academy had been edited by Davies: Bourdon’s Elements of Algebra (published in 1835), Legendre’s Éléments with plane and spherical geometry (published in 1828), Elements of Descriptive Geometry (published in 1826), A Treatise on Shades and Shadows (published in 1832), Elements of Surveying (published in 1830), Elements of Analytical Geometry (published in 1836), and Elements of the Differential and Integral Calculus (published in 1836). Although he left West Point with the intention of focusing on writing more textbooks, Davies also spent a good portion of each day in Connecticut teaching his three children. One wonders how they responded to the mathematics lessons!

Sylvanus Thayer to Charles Gratiot, 6 July 1834, Thayer Papers (cit. n. 18), vol. 6.

Charles Davies to Mary Ann Davies, 10 December 1836 and 5 May 1837, Charles Davies, Letters, Special Collections and Archives, United States Military Academy, West Point. Although Henry Davies said that Davies was also studying current research in mathematics during the European trip, Davies’s letters to his wife do not mention any scholarly pursuits; “Davies,” Davies Memoir (cit. n. 12), p. 69. See also Ambrose, Duty, Honor, Country (cit. n. 4), pp. 87-105.

Tillman, “Academic History” (cit. n. 9), p. 244.

“Davies” in Biographical Register (cit. n. 12), p. 153; Mary Ann Davies to Elizabeth Mansfield, 28 November 1837, Charles Davies, Letters (cit. n. 42).

The Academy textbooks which appeared under Davies’s name soon were purchased by colleges as well, and Davies additionally began to write arithmetics and at least one guide to geometry aimed at children. He seems to have issued several schoolbooks from a printing firm he established together with a relative of his eventual son-in-law, William Guy Peck. In the meantime, Alfred S. Barnes, who moved from Hartford to New York in the early 1830s to try to establish himself as a publisher, had decided that his niche in the business should be to publish only the best textbooks. When he met Davies in 1838, the pair naturally partnered. Well-versed in mathematics himself, Barnes suggested that Davies’s works be marketed as a national series of standard books. A. S. Barnes & Co. immediately issued its own printings of Davies’s college textbooks: Elements of Algebra and Elements of the Differential and Integral Calculus in 1838, and Elements of Geometry and Trigonometry, Elements of Analytical Geometry, Treatise on Shades and Shadows, and Elements of Surveying in 1839. Barnes then took the textbooks on the road to show them directly to colleges and academies. His salesmanship and teachers’ perception of the superior content of Davies’s works made them unusually popular. Even as Barnes moved his publishing house to Philadelphia in 1840 and then back to New York in 1845, the business continued to grow.

This printing house apparently also sold other mathematics textbooks, for Hezekiah Howe sold the copyright for Day’s An Introduction to Algebra to Davies & Peck and Collins, Keen & Co. in 1838; Hezekiah Howe to Leavitt, Lord & Company, 8 January 1838, Hezekiah Howe & Co. Letter Book, 1833-1838, Beinecke Rare Book & Manuscript Library, Yale University, New Haven.

Hellmut Lehmann-Haupt, Ruth Shepard Granniss, and Lawrence C. Wroth, The Book in America: A History of the Making, the Selling, and the Collecting of Books in the United States (New York: R. R. Bowker Company, 1939), pp. 187-188.

Elements of Descriptive Geometry was not published by A. S. Barnes & Co. until 1844.

John Barnes Pratt, A Century of Book Publishing 1838-1938 (New York: A. S. Barnes and Co., 1938), pp. 1-7; Lehmann-Haupt, Granniss, and Wroth, Book in America (cit. n. 46), pp. 187-188. Keith Hoskin names other “market opportunities” for the textbooks, including the Academy’s pre-eminence in science, the inner virtue of the textbooks, and the link between the Academy and the first modern high school, Central High in Philadelphia; Hoskin, “Textbooks” (cit. n. 31), pp. 32-34.

Unable to stay out of the classroom, Davies became professor of mathematics at Trinity College in Hartford in 1839—ironically, Trinity had been founded several years earlier in a controversial outreach effort by Yale during Day’s presidency. Davies had also become more deeply involved in politics and legislation, an interest which dated back to his days as a professor at the Academy developing contacts as well as enlisting the support of lawmakers for improvements and salary increases. For example, Davies campaigned for William Henry Harrison in the 1840 presidential election and then was one of the few Whigs to urge for conciliation with John Tyler when the former Democrat ascended to the presidency. A. S. Barnes & Co. had also published its first new Davies schoolbook in 1840, First Lessons in Arithmetic. Around the same time in 1841 that the next textbook appeared, Davies had fallen ill again and resigned his professorship at Trinity. He then served on the Board of Visitors at the Military Academy in the summer of 1841 and was recommissioned in the Army as a paymaster in November in order to return to West Point permanently as the Academy’s treasurer. While serving in his new post, Davies authored several more schoolbooks.

Examples of letters between Davies and the Academy faculty and members of Congress include: Faculty to John C. Calhoun, 20 January 1818, Thayer Papers (cit. n. 18), vol. 3; and Henry Clay to Charles Davies, C. F. Smith, and N. Tillinghash, 6 February 1834, Charles Davies, Letters (cit. n. 42).

Joseph Trumbull to Charles Davies, 28 February 1840 and 19 April 1840; John Tyler to Charles Davies, 1 October 1841, all in Charles Davies, Letters (cit. n. 42).

Keith Hoskin drops a tantalizing hint by observing that Davies was also one of three members of an 1841 inspection board which recommended that Daniel Tyler’s efficiency measures at the Springfield Armory be adopted; Hoskin, “Textbooks” (cit. n. 31), pp. 37-38.

Davies retired from the Academy a second time in September 1845. Even though he purchased a country home at Fishkill-on-the-Hudson in New York and wrote more schoolbooks, Davies allowed himself to be called back to work again in September 1848 for a one year appointment as professor of mathematics at the University of the City of New York. In 1850, he traveled to Europe for another six-month tour of points of interest, this time accompanied by his wife and one of his daughters. Davies continued to produce textbooks in the 1850s, and he also found time to finish correcting a calculus textbook left behind by his former student and colleague at the Academy, Edward Courtenay (1803-1853). In addition, he served as president of the New York Teachers’ Association from 1853 to 1854.

See L. Bruwish, William Noys, and Robert Kelly (of the University of the City of New York) to Charles Davies, 19 September 1848, Charles Davies, Letters (cit. n. 42); and “Davies” in Biographical Register (cit. n. 12), p. 154. Davies sometimes referred to the location of his home as “Fishkill-on-the-Hudson” and sometimes as “Fishkill Landing.” The site of his house is now lost.

“Davies,” Davies Memoir (cit. n. 12), p. 70.

Sylvanus Thayer to Charles Davies, 18 March 1855, Charles Davies, Letters (cit. n. 42). Courtenay also taught mathematics at the University of Virginia for eleven years; the textbook was the 1855 Treatise on Differential and Integral Calculus and Calculus of Variations. “Courteney [sic], Edward Henry,” in NCAB (cit. n. 7), vol. 5, pp. 519-520.

“Presidents of New York” (cit. n. 12), p. 480. Apparently, Davies remembered this date incorrectly as 1843 to 1844 when he submitted his biography to George Cullum for the first edition of Biographical Register (cit. n. 12) (Charles Davies to George W. Cullum, 12 June 1867, Charles Davies, Letters (cit. n. 42)), and the mistake is repeated through all existing biographies of Davies. He could not have been president in the 1840s, for the Teachers’ Association had not yet been founded.

The strength of Davies’s reputation led to a professorship at Columbia College in 1857. The same course of study was in place there for most of the nineteenth century, although a new effort was made in the 1830s to ensure that there was truly content rather than just lip service to the subjects. In mathematics, first- and second-year students studied algebra and geometry, while third-year students turned to spherical trigonometry, conic sections, analytical geometry, and fluxions. Students could continue to study fluxions in the fourth year if they wished. Robert Adrain, the American editor of Hutton’s A Course of Mathematics, was Professor of Mathematics and Natural Philosophy there early in the nineteenth century, but the professorship of mathematics and astronomy was divided into two positions shortly after Davies was hired as the “Professor of Higher Mathematics.” The 156 students attending Columbia in 1858 followed the required course for the first three years in order that their minds might be disciplined and invigorated. Seniors were then to apply their skills toward acquiring a permanent body of knowledge in either the School of Letters, the School of Science, or the School of Jurisprudence. When he began to teach in February 1858, Davies delivered an address on the nature, language, and uses of mathematical science.

J. Howard Van Amringe, “History of Columbia University,” in Universities and Their Sons, ed. Joshua L. Chamberlain, vol. 1 (Boston: R. Herndon Company, 1898), pp. 569-731, on pp. 569-656.

Statutes of Columbia College, Revised and Passed by the Board of Trustees, March, 1821 (New York, 1821).

Van Amringe, “History of Columbia” (cit. n. 56), pp. 657-666.

[Charles Davies], “Inaugural Address, Charles Davies, LL.D.,” in Addresses of the Newly-Appointed Professors of Columbia College, intro. William Betts (New York, 1858), pp. 117-151.

Davies took his son-in-law, William Guy Peck, with him to Columbia as an assistant professor. When he suggested in October 1863 that the Trustees reorganize the Department of Higher and Pure Mathematics so that Davies would fill merely a supervisory role at a reduced salary, a full professorship for another of his assistants, J. Howard Van Amringe (1835-1915), would be created, and Peck would be promoted into a vacant professorship in the Department of Mechanics and Physics, Davies helped set off a chain of events which ultimately led to his own resignation. When he learned of the proposed plan, General J. B. Barnard, the brother of another candidate for the physics position, angrily argued to members of the Board of Trustees that Peck had few scientific qualifications, if any, and complained that Davies was successfully stalling the Board’s decision. Apparently, most people involved believed that Charles King, president of Columbia, was planning to retire soon and that the Professor of Physics would have the inside track to succeed him. Frederick A. P. Barnard was not elected to the physics professorship, but he was hired for the presidency when King did retire in the spring of 1864. Davies threatened to leave Columbia before the whole affair was settled, and even after college politics quieted down, he became increasingly reluctant to leave his country home and an ailing daughter who would pass away in late 1865. When Columbia’s Committee on Faculty decided that Davies’s $2,000 annual salary was not worth the two hours of teaching per week he was providing, he submitted his resignation. It was accepted in September 1865, and an emeritus professorship accompanied by no salary or duties was arranged for him.

Van Amringe, “History of Columbia” (cit. n. 56), p. 666.

Charles Davies to Hamilton Fish, 13 October 1863, Hamilton Fish Papers (cit. n. 1). Fish (1808-1893) was a prominent lawyer, former governor of New York, and nearly permanent member of the Columbia Board of Trustees. See Joseph V. Fuller, “Fish, Hamilton,” in DAB (cit. n. 7), vol. 3, part 2, pp. 397-400. On Van Amringe, who was later an interim president of Columbia, see Milton Halsey Thomas, “Van Amringe, John Howard,” in DAB (cit. n. 7), vol. 10, part 1, pp. 148-149; and “Van Amringe, J[ohn] Howard,” in NCAB (cit. n. 7), vol. 29, pp. 137-138.

J. B. Barnard to G. Kemble, 24 October 1863 and 26 October 1863, Hamilton Fish Papers (cit. n. 1).

G. Kemble to Hamilton Fish, 23 January 1864 and 20 May 1864, Hamilton Fish Papers (cit. n. 1); Van Amringe, “History of Columbia” (cit. n. 56), pp. 657-670. Barnard had to survive a controversy of his own when his previous position at a college in Mississippi called his loyalty to the Union into question. See G. M. Ogden to Fish, 30 August 1864, 8 September 1864, and 20 September 1864; Charles King to Hamilton Fish, 31 August 1864; Frederick A. P. Barnard to Hamilton Fish, 29 September 1864, all in Hamilton Fish Papers (cit. n. 1).

Charles Davies to Hamilton Fish, 8 January 1864, Hamilton Fish Papers (cit. n. 1); Eunice Davies to E. D. Mansfield, 29 September 1864, Charles Davies, Letters (cit. n. 42).

Charles Davies to Hamilton Fish, 6 June 1865; and Beverly Haight to Hamilton Fish, 9 June 1865, Hamilton Fish Papers (cit. n. 1). The daughter was Louisa H. Scudder, the third of Davies’s five children.

William A. C. Bartlett to Hamilton Fish, 20 September 1865 and 27 September 1865, Hamilton Fish Papers (cit. n. 1).

Although Davies wrote few textbooks after his resignation and as well tapered off in making revisions for new printings of his works, he continued to be a busy man during his retirement. Due to his contacts from the Military Academy, fame as a textbook author, and prominent brothers—one a judge in the New York Court of Appeals and one a military engineer—Davies was called upon to advise legislators on mathematical and scientific issues. He also found himself in demand as a speaker. For example, he addressed the cadets at the Virginia Military Institute in 1875. Davies additionally instituted a yearly tradition of lectures at the annual reunion of the Military Academy, an event begun in 1869 in part to reunite Army officers who had found themselves on opposite sides in the Civil War. The most famous of these speeches was Davies’s address in 1875 on the one hundredth anniversary of the Battle of Bunker Hill. Davies worked a plea for reconciliation of the Union into his speech. Fittingly, Generals Sherman and Johnston both reviewed and approved a draft of the lecture. Davies died at his country home on September 17, 1876.

Davies reported on his impressions of the nation’s capital during one of his lengthy consulting trips in Charles Davies to [Eunice Davies or Alice Davies], 15 January 1866 and 29 March 1866, Charles Davies, Letters (cit. n. 42).

Charles Davies to Mary Ann Davies, 23 June 1875, Charles Davies, Miscellaneous Papers (cit. n. 15).

Charles Davies, “Address,” in Sixth Annual Reunion of the Association of the Graduates to the United States Military Academy at West Point, New York, June 17, 1875 (New York: A. S. Barnes & Co., 1875), pp. 9-18.

See William Tecumseh Sherman to Charles Davies, 26 February 1875; and Joseph E. Johnston to Charles Davies, 1 May 1875 and 22 May 1875, all in Charles Davies, Letters (cit. n. 42).

Development of Elements of Geometry and Trigonometry
Despite Thayer’s and Crozet’s desires to rely on French mathematics textbooks, the Academy curriculum lacked a firm textual foundation for several years. The mathematics professors may have read the works taught at the École Polytechnique in preparing the material they taught, but cadets did not purchase their own copies of the textbooks. Certainly it was inherently difficult to guide first-year cadets simultaneously through the French language and mathematics. Furthermore, there are no records of a separate geometry textbook in the Academy curriculum. Thus, while Crozet’s Treatise of Descriptive Geometry may have been the only mathematics textbook written in English officially used at the Academy when Davies ascended to the professorship of mathematics in 1823, the other assistant professors of mathematics for the sections containing the less able cadets had continued to teach from Hutton’s A Course of Mathematics in 1818 and 1819 and Samuel Tillman, in a history of the department of mathematics, noted that Hutton’s A Course of Mathematics was not entirely discarded until the years 1823 to 1825. It was not until 1823 that the Academy implemented the 1821 advice of Harvard professors Andrews Norton and George Ticknor and adopted Farrar’s series of translations: Elements of Algebra, based upon Lacroix’s textbook; Legendre’s Éléments; and An Elementary Treatise on Plane and Spherical Trigonometry, from writings by Lacroix and Bézout, for the more advanced of the first-year cadets. Students in the second year turned to Biot’s Géométrie analytique and Lacroix’s Traité élémentaire du calcul, in French, along with Crozet’s Treatise of Descriptive Geometry, and the cadets studied the principles of perspective, shades, and shadows.

Edward S. Holden and W. L. Ostrander, “A Tentative List of Text-Books Used in the United States Military Academy at West Point From 1802-1902,” in Centennial (cit. n. 3), pp. 439-466; listed no such book before Farrar’s translation of Legendre’s Éléments was adopted in 1823, including no edition of Playfair’s Elements or Legendre’s Éléments. The Military Academy Library does contain the 1813 tenth edition of Legendre’s Éléments, brought from Paris by Thayer, which may have served as a reference for Academy instructors. With respect to the Library’s holdings, the author was greatly aided by an unpublished catalogue of mathematics books at the United States Military Academy which was compiled by Joe Albree, David C. Arney, and V. Frederick Rickey. This catalogue is now generally available as Albree, Arney, and Rickey, Station Favorable (cit. n. 38).

Tillman, “Academic History” (cit. n. 9), pp. 242-244; Cajori, Teaching and History (cit. n. 5), pp. 115-116. In one of his last actions as assistant professor of natural and experimental philosophy, Davies submitted his only mathematical or scientific article to American Journal of Science—a correction of one of Hutton’s proofs in the conic sections chapter of A Course of Mathematics to make it general; Charles Davies, “Demonstration of a Problem in Conic Sections,” American Journal of Science 6 (1823): 280-282.

Helena M. Pycior, “British Synthetic Vs. French Analytic Styles of Algebra in the Early American Republic,” in The History of Modern Mathematics, ed. David E. Rowe and John McCleary, vol. 1 (San Diego: Academic Press, 1989), p. 138.

Although Thayer purchased the Cambridge Analytical Society’s English translation of Lacroix’s calculus textbook during his sojourn in Europe, Albree, Arney, and Rickey believe that the cadets studied from an 1820 Paris edition of the book. The materials used for perspective, shades, and shadows were probably manuscript notes by Crozet which no longer exist. Albree, Arney, and Rickey, Station Favorable (cit. n.38), pp. 15, 153; An Elementary Treatise on the Differential and Integral Calculus (Cambridge: J. Deighton and Sons, 1816); Silvestre-François Lacroix, Traité élémentaire de calcul différential et de calcul intégral (Paris, 1802).

Although these textbooks worked out well enough and the Academy’s reputation for offering an excellent scientific and practical education increased dramatically in the 1820s as all the cadets were required to study the more advanced textbooks, Davies gradually became interested in putting his own name on textbooks. For one thing, he married Mansfield’s daughter, Mary Ann (1807-1897), in 1825, and she gave birth to two of their five children by the end of 1828. Therefore, Davies may have desired an additional route to financial security, especially if one considers that he had earned only ten dollars per month as an assistant professor. Davies’s first publication, Elements of Descriptive Geometry, was based upon Crozet’s Treatise and appeared in 1826. In addition, he and his own assistant, Edward C. Ross, became disenchanted with some of Farrar’s translations. To remedy the translation errors in Elements of Algebra which had been pointed out by Jasper Adams and because the material in Lacroix’s textbook had become dated, Ross prepared a translation of Louis Pierre Marie Bourdon’s textbook on algebra which was published in 1831. Davies’s response was slightly different and reflected his preparation of Elements of Descriptive Geometry. He chose to seek out textbooks and existing translations which he could adapt and republish, as he even did with Ross’s own book in 1835 (after Ross had left the Academy to enter active service in 1833).

Mary Ann studied at Emma Willard’s Troy Female Seminary, and she was actually scolded by her mother for choosing to study Paley’s Natural Theology over Euclid’s Elements; Elizabeth Mansfield to Mary Ann Davies, 9 March 1823, Charles Davies, Letters (cit. n. 42). Like Eliza Farrar, Mary Ann fretted to her mother that Davies preferred another young woman. Elizabeth responded, “I have heard the same report respecting Mr. Davies’ attentions to Mary Piston that you have, but do not believe that there is any other foundation for it than his having rode out with her several times.” She went on to describe Davies: “[H]is standing in society is highly respectable for one of his years and if he is not an adonis in his person or a Chesterfield in his manners, I have no doubt that he will make an excellent husband. He is besides all this a man of business. . . .” Elizabeth Mansfield to Mary Ann Davies, May 1823, Charles Davies, Letters (cit. n. 42); emphasis in source.

Regulations, 2 July 1816 (cit. n. 20). Davies had supplemented that income at a woolen manufacturer in Glenham, New York, a few miles north of West Point and across the Hudson River. See Charles Davies to Mary Ann Davies, 8 September 1823, Charles Davies, Letters (cit. n. 42). He had even been admitted to the New York bar in 1828, but he apparently only ever argued one case. Davies also spent the 1820s developing contacts by recommending friends and former students for jobs and by traveling to Albany and Washington, D.C. See Charles Davies to Samuel Southard, 18 January 1825, Charles Davies, Miscellaneous Papers (cit. n. 15); Charles Davies to Sylvanus Thayer, 26 July 1825 and 22 June 1827; Edward H. Courtenay to Sylvanus Thayer, 17 June 1827, all in Thayer Papers (cit. n. 18), vol. 4; Charles Davies to Elizabeth Mansfield, 18 October 1825; Marian Foot to Mary Ann Davies, 15 January 1828, both in Charles Davies, Letters (cit. n. 42).

Charles Davies, Elements of Descriptive Geometry (Philadelphia: Carey and Lea, 1826).

Elements of Algebra, Translated from the French of M. Bourdon, for the Use of the Cadets of the U. S. Military Academy, trans. Lt. Edward C. Ross (New York: E. B. Clayton, 1831). See Pycior, “British Synthetic” (cit. n. 73), p. 138. Farrar published a shorter translation of the same textbook in the same year: Elements of Algebra (Boston: Hilliard, Gray, Little and Wilkins, 1831). For a biography of Ross, see Robert Kelly, Life and Character of the Late Prof. Edward C. Ross, LL.D. (New York, 1851).

Charles Davies, Elements of Algebra: Translated from the French of M. Bourdon, New York: Wiley and Long, 1835.

For example, Davies’s second textbook was a republication of Thomas Carlyle’s translation, Elements of Geometry and Trigonometry, which was published in New York in 1828 by James Ryan and sold by a variety of booksellers. While Helena Pycior argued that Davies freely modified the content of Ross’s translation, Elements of Algebra to, as Davies put it, “unite . . . the scientific discussions of the French, with the practical methods of the English school,” Davies was more conservative in the first American printing of Elements of Geometry and Trigonometry. Davies wrote a new preface, in which he admitted the responsibility of altering such a celebrated work. He claimed that, unlike Legendre, Brewster, and Farrar, he avoided reference to particular diagrams so that beginners could improve their faculty of abstraction. In other words, Davies removed diagram labels from the statement of theorems so the theorems would be general (he would say “every straight line” rather than “every straight line CD”). Indeed, in general, “Geometry is not studied merely for the facts which it teaches . . . but, because it disciplines the untrained intellect, and conducts the untaught mind to the temple of truth.” Yet, otherwise, the only changes Davies made to Carlyle’s translation were cosmetic and made his Elements of Geometry and Trigonometry resemble Farrar’s translation: in addition to several minor alterations in wording, Davies renumbered all the propositions continuously and moved the three theorems and appendix at the end of Book VII to the appendix in Book VIII, just as Farrar had. The diagrams were re-engraved, with the shading that had been in Carlyle’s translation removed. Davies lastly added four corollaries or scholia and gave the division of the circle in sexagesimal instead of decimal notation in an abridged treatise on trigonometry.

Davies, Elements of Algebra (cit. n. 79), pp. iii-iv; quoted in Pycior, “British Synthetic” (cit. n. 73), p. 139. For a less positive view of Davies with respect to Ross’s efforts, see Lao Genevra Simons, Bibliography of Early American Textbooks on Algebra, Scripta Mathematica Studies No. 1 (New York: Scripta Mathematica, 1936). See “Davies and Mathematical Styles” in this chapter for a discussion of whether Davies truly meant “English” or “British” in the contrast he drew.

[Charles Davies, ed.], Elements of Geometry and Trigonometry; With Notes. Translated From the French of A. M. Legendre, Revised and Altered for the Use of the Military Academy at West Point, [trans.Thomas Carlyle], ed. David Brewster (New York: James Ryan, 1828), pp. i-v.

[Davies], Elements of Geometry and Trigonometry, 1828 (cit. n. 81); [John Farrar, trans.], Elements of Geometry, by Adrien-Marie Legendre (Cambridge, MA: Cummings & Hilliard, 1819).

[Davies], Elements of Geometry and Trigonometry, 1828 (cit. n. 81), pp. 121-142.

Davies’s slightly-edited Elements of Geometry and Trigonometry was received more warmly in the United States than the Brewster and Carlyle version was in Great Britain. Even though James Ryan assembled several mathematics textbooks at his press, a different New York firm, White, Gallagher, and White, reissued Elements of Geometry and Trigonometry in 1830. The names of N. & J. White; Collins & Hanney; Collins & Co.; and James Ryan appeared on the 1832 third printing. Then, in 1834, Davies sold the right to publish to another firm, Harper & Brothers, which had purchased Davies’s next work, Elements of Surveying, in 1830. James and John Harper also issued Davies’s Treatise on Shades and Shadows in 1832. Although Davies signed over exclusive rights for Elements of Geometry and Trigonometry to the Harpers for ten years, his next three college textbooks were sold to Wiley & Long as Davies began to produce volumes in a rush of activity—Elements of Algebra in 1835 (the revision/co-option of Ross’s translation of Bourdon’s textbook), Elements of Analytical Geometry in 1836, and Elements of the Differential and Integral Calculus in 1836—making Davies Wiley’s second mathematical author, after Day. Indeed, Harper & Brothers’ contract with Davies apparently soon fell apart even though the firm was otherwise quite profitable before the Civil War. Wiley & Long also printed and sold the 1834 Elements of Geometry and Trigonometry, as did young bookseller Alfred S. Barnes, who was soon to be Davies’s business partner.

Although Ryan’s name appeared on several nineteenth-century textbooks, and he was also the person behind the short-lived mathematics journal, Mathematical Diary (c. 1832), no known biographies of him exist.

Contract for Elements of Surveying, 1830, Harper & Brothers Archives, Rare Book and Manuscript Library, Columbia University, New York. The book sold as Charles Davies, Elements of Surveying with the Necessary Tables, New York: J. & J. Harper, 1830.

Charles Davies, Treatise on Shades and Shadows, and Linear Perspective, New York: J. & J. Harper, 1832.

Contract for Legendre’s Geometry and Trigonometry, 1 April 1834, Harper & Brothers Archives (cit. n. 85); H. B. Phillips, Guilford L. Spencer, and Dick Wick Hall, “Numbers: Books on Mathematics,” in First One Hundred Fifty Years: A History of John Wiley and Sons, Incorporated 1807-1957 (New York: John Wiley & Sons, Inc., 1957), pp. 69-74, on p. 70. Davies is mentioned repeatedly as one of the first Wiley textbook authors in various technical fields. See also, for example, Richard S. Kirby, “Graphic Communication: Books on Drawing and Descriptive Geometry,” pp. 83-89, on p. 84; and James Kip Finch, “The World of Construction: Books on Civil Engineering,” pp. 110-122, on p. 120. Davies also published Common School Arithmetic on his own in Hartford in 1834 as his first school textbook.

Lehmann-Haupt, Granniss, and Wroth, Book in America (cit. n. 46), pp. 171-172.

According to the catalogue by Albree, Arney, and Rickey (cit. n. 38), p. 65; the Harper & Brothers version was thirty pages shorter than the copies of Elements of Geometry and Trigonometry sold by other printers and booksellers; the chief difference was that it did not contain the two sections Davies added on mensuration, (cit. n. 38), p. 65. The plates for Elements of Geometry and Trigonometry remained Davies’s property, which may have aided him in simultaneously selling the textbook to other printers; Contract for Legendre’s Geometry and Trigonometry, 1 April 1834, Harper & Brothers Archives (cit. n. 85).

Davies became a more significant editor with the 1834 printing of Elements of Geometry and Trigonometry. First of all, Davies restored the traditional numbering system by book for the propositions. He substituted in material from Edinburgh Encyclopaedia in Book V and from Encyclopaedia Metropolitana in Book II, also adding sixty-two pages of logarithmic and trigonometric tables and a table similar to those in Carlyle’s translation comparing the organization of propositions in Euclid’s Elements with that of Legendre’s Éléments. Davies added eight axioms, several from Playfair’s Elements or Simson’s The Elements of Euclid: six statements on the relationships between equals and unequals, “Through the same point, only one straight line can be drawn which shall be parallel to a given line,” and “All right angles are equal to each other,” eliminating the first proposition from Carlyle’s translation, which proved right angles are equal. Repeatedly, Davies elaborated on the reasoning process used to arrive at conclusions within proofs or in additional scholia. He also rearranged a very few definitions and propositions in the earlier books. 

Charles Davies, ed., Elements of Geometry and Trigonometry Translated From the French of A. M. Legendre by David Brewster, Revised and Adapted to the Course of Mathematical Instruction in the United States (Philadelphia: A. S. Barnes and Co., [1834]).

Davies, Elements of Geometry and Trigonometry, 1834 (cit. n. 90), p. iv.

Davies, Elements of Geometry and Trigonometry, 1834 (cit. n. 90), p. 13.

The significant change was his inclusion of a new book between Books I and II to contain expanded and rewritten material on proportion. In other words, this and all subsequent editions of Elements of Geometry and Trigonometry were published with nine instead of eight books. Davies also removed the appendix on isoperimetry which previously followed the book on regular polygons. He omitted four propositions on planes and solid angles and revised the book on polyedrons. Davies switched the positions of the last two books, so that Book VIII in 1828, on the three round bodies, remained Book VIII in 1834. Although most of the propositions remained, Davies rewrote many of the proofs. Davies recast what became the final book, from a focus on the sphere to an examination of spherical triangles and polygons. He reduced the number of definitions from fifteen to seven, kept half of the original twenty-four propositions, and added nine theorems. Davies had also either completely rewritten or borrowed the section on plane and spherical trigonometry. Finally, Davies added five pages on the application of algebra to geometry from Hutton’s work on the same topic and a section with the rules for measuring surfaces and solids. This version was printed at least eleven times and was taught in colleges including Dartmouth, the University of Alabama, and the University of Michigan.

NUC (cit. n. 38); Cajori, Teaching and History (cit. n. 5).

Davies next revised Elements of Geometry and Trigonometry in 1851. The title page for this book contained one of the first times Davies added “LL.D.” after his name to reflect an honorary doctorate awarded by Geneva College in 1849. He also explained in the preface to this edition that he used Legendre’s Éléments as translated by Carlyle as a model and guide but not as a standard, changing the language and arrangement of arguments. He put an increased emphasis on problems for the student to work out and on uniting pure geometry with mensuration, stating, “Practical examples cannot fail to point out the generality and utility of abstract science.” Davies had removed Carlyle’s essay on proportion from the beginning of the textbook in the previous edition, and he now inserted his own introduction on extension. He then revised the list of definitions in Book I and made an effort to simplify his writing, reflecting the younger audience Elements of Geometry and Trigonometry had been gaining in the academies and recently-invented high schools. Davies retained essentially the same body of theorems and proofs, though. After making similarly minor changes in Book II, on proportion, Davies changed a theorem into a postulate in Book III, on the circle and measurement of angles, and rewrote several corollaries and scholia. In Book IV, the proportion of figures, Davies invented a symbol for equivalence of proportions, but it was never adopted by other mathematicians. Davies revised Book V by adding six theorems and removing three to demonstrate the value of π in a more traditional fashion. He then converted two more theorems to postulates in Book VI, the principles that a perpendicular to a plane could be drawn from either a point within or a point without the plane. This book was renamed “Planes and Polyedral Angles” from “Planes and Solid Angles.” After making minor changes to the final three books, Davies wrote a brief essay on direct and indirect proofs for the notes. Davies redivided the trigonometry section into plane, analytical plane, and spherical trigonometry, but he made no changes to the appendix on mensuration. This version went through seven printings, with the University of Mississippi among the colleges which adopted it.

Charles Davies, ed., Elements of Geometry and Trigonometry From the Works of A. M. Legendre, Revised and Adopted to the Course of Mathematical Instruction in the United States (New York: A. S. Barnes & Co., 1851).

“Davies” in NCAB (cit. n. 12).

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), p. iv.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), pp. 1-12.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), pp. 13-46.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), pp. 57-86.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), pp. 87-134.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), pp. 135-173.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), pp. 245-370.

Davies revised Elements of Geometry and Trigonometry once more shortly before accepting the professorship at Columbia in 1858, and he made only a few additional modifications in 1862, the last revision of the textbook published during his lifetime. In this edition, he called Legendre’s Éléments the “preeminent treatise of elementary geometry of the past 100 years” and revised his comments about the reference to particular diagrams to explain that he stated each proposition in general terms and then referred to the figure. Davies also replaced the term “solidity” with “volume.” For the first time, Davies was aided by an assistant editor, William Guy Peck. Davies rewrote the textbook’s introduction to focus on the four types of quantity in geometry: lines, surfaces, volumes, and angles. He also said that operations in geometry were done with signs, as they were in analysis. He reorganized the definitions in Book I and replaced two axioms while removing one. He added an eighth postulate, that it was possible to draw a line parallel to another line. He rearranged some propositions and revised the scholia and corollaries. He abridged Book II, allowed multiplication by a ratio, and removed four propositions while adding two. In contrast, Davies put several symbols back into words in a reorganized Book III and removed his equivalence symbol from Book IV. He also removed nearly half of the problems which had survived since Davies had first reprinted Carlyle’s translation. Although Books V through IX covered the same material as before, Davies significantly revised the structure of each book, substituting in new proofs and/or new theorems. He no longer included any notes on the propositions, but he wrote new instructions for the trigonometry tables and combined the mensuration section into one unit. “Davies’s Legendre,” as the book had come to be called, had sold three hundred thousand copies by 1862, when it was priced at $2.25. After 1862, at least ten additional printings of Elements of Geometry and Trigonometry appeared. Georgetown and Transylvania University were two institutions which turned to the textbook in the 1860s. Furthermore, A. S. Barnes & Co. no longer marketed the textbook merely as part of a series but as a component of “Davies’ national course of mathematics.” The publishing house found seven justifications for adding “national” to a set of books “now rounded to their perfect fruition”: the use of the system in the national military and naval academies; a “quasi” endorsement by Congress; the system’s use in the schools in Washington, D.C.; its utility as a reference in resolving governmental questions about mathematics; its role as educator of the nation’s great soldiers, sailors, and scientists; its larger role as educator of the greatest number of citizens; and the fact that the system was used in every state.

Charles Davies, Elements of Geometry and Trigonometry, From the Works of A. M. Legendre (New York: A. S. Barnes & Co., 1857); Charles Davies, Elements of Geometry and Trigonometry, From the Works ofA. M. Legendre, Adapted to the Course of Instruction in the United States (New York: A. S. Barnes & Co., 1862). Davies wrote a new preface in 1875, but this was not printed until 1882; [J. Howard Van Amringe, ed.], Elements of Geometry From Davies’ Legendre (New York andChicago: A. S. Barnes & Co., [1882]).

Preface to Davies, Elements of Geometry and Trigonometry, 1862 (cit. n. 103).

Davies, Elements of Geometry and Trigonometry, 1862 (cit. n. 103), pp. 9-12.

Davies, Elements of Geometry and Trigonometry, 1862 (cit. n. 103), pp. 13-49.

Davies, Elements of Geometry and Trigonometry, 1862 (cit. n. 103), pp. 50-58.

Davies, Elements of Geometry and Trigonometry, 1862 (cit. n. 103), pp. 59-135.

Davies, Elements of Geometry and Trigonometry, 1862 (cit. n. 103), pp. 136-259.

Advertising supplement to Charles Davies, The Logic and Utility of Mathematics, With the Best Methods of Instruction Explained and Illustrated (New York: A. S. Barnes & Co., 1850), p. 11. This list was organized topically, while the series was termed “the West Point course” and divided into three parts of “common school,” “academic,” and “collegiate” (including “Davies’ Legendre’s Geometry”) textbooks in Davies, Elements of Geometry and Trigonometry, 1862 (cit. n. 103), p. ii.

Advertising supplement to Davies, Logic and Utility (cit. n. 110), p. 11.

Davies and Mathematical Styles
“Davies’s Legendre” ultimately dominated the market so much that it became its own brand name, and yet it can be difficult at first glance to discern any philosophical intent behind the textbook. The series was never reviewed in a journal nor mentioned in the reviews of other mathematical textbooks, Davies never mentioned “analysis” or “synthesis” in his few surviving letters, and Davies did not provide readers with a direct glimpse into his philosophy of mathematics education until he was well into the prolific production of textbooks; this work was the 1850 Logic and Utility of Mathematics. Once he published Logic and Utility, Davies did use “analysis” in print in several ways. Most often, he distinguished analysis as associated with algebra and the differential and integral calculus. He saw geometry as separate from algebra, analytical geometry, and differential and integral calculus, which were all part of analysis, since these subjects dealt with quantities represented by letters. Davies also mentioned that the analytical type of reasoning in logic was better for introducing science, but synthesis was better for memorization, and he raised analysis and synthesis again when he spent the third part of Logic and Utility arguing that the usefulness of mathematics lay in its role in training the intellect.

Davies, Logic and Utility (cit. n. 110). A shortened version of Logic and Utility was published in Great Britain as: Charles Davies, Mathematical Science: Its Logic and Utility, London: W. Kent and Co., n.d. Although there were no reviews of Davies’s series in any of the American review journals, there were two announcements of “new books” by Davies in the short-lived mathematics journal conducted by Charles Gill, Mathematical Miscellany. Gill noted in the first number that he had received Elements of Geometry and Trigonometry and Elements of Algebra from Davies, and he “trust[ed] they will become as popular as their merits deserve” in a phrase indicating the unease with which other mathematics writers seem to have viewed Davies’s production of textbooks. In the second number, Gill announced the appearance of Elements of Analytical Geometry and Elements of Differential and Integral Calculus without further comment; Mathematical Miscellany 1 (1836-1838): 52ff.

See, for example, Davies, “Inaugural Address” (cit. n. 59), p. 124; Davies, Logic and Utility (cit. n. 110), pp. 117-221; Charles Davies and William Guy Peck, Mathematical Dictionary and Cyclopedia of Mathematical Science (New York: A. S. Barnes & Burr, 1855), pp. 22-23. The second edition of Dictionary appeared in 1859 and was used for this study.

Davies, Logic and Utility (cit. n. 110), pp. 261-292.

Davies, Logic and Utility (cit. n. 110), pp. 41-97, 293-340.

Still, it cannot be said that Davies did not consider the three understandings of analysis and synthesis in earlier years; one such example was his desire to propagate the so-called “West Point system of mathematical instruction.” As his textbook series became the backbone of Academy mathematics instruction throughout the nineteenth century, Davies claimed public credit for the success of Academy graduates, ultimately titling his series “The West Point Course” in the 1862 edition of Elements of Geometry and Trigonometry. By this phrase, which turned up frequently in the advertisements and introductions to Davies’s textbooks, Davies meant generally the foundation of French and mathematics taught in highly structured recitation sessions which was established at the Military Academy under Thayer. In case the contributions of this educational system over the decades were not obvious, Davies reminded readers that Academy graduates were desired “wherever science of the highest grade has been needed” and touted the Academy for “scattering science and knowledge over the nation.” Davies’s description of Academy instruction highlighted the feature of generality which was still most often associated with the French treatment of mathematics in the middle of the nineteenth century:

Davies, Elements of Geometry and Trigonometry, 1862 (cit. n. 103), p. ii.

Charles Davies, Elements of Analytical Geometry (New York: Wiley & Long, 1836).

Davies, Logic and Utility (cit. n. 110), pp. 3-4; Davies, “Address,” Sixth Annual Reunion (cit. n. 69), p. 9.

It is of the essence of that system that a principle be taught before it is applied to practice; that general principles and general laws be taught, for their contemplation is far more improving to the mind that the examination of isolated propositions; and that when such principles and such laws are fully comprehended, their applications be then taught as consequences or practical results. . . . In that system Mathematics is the basis—Science precedes Art—Theory goes before Practice—the general formula embraces all the particulars.

Davies, Logic and Utility (cit. n. 110), pp. 3-4.

The typically “French” aspects of mathematics teaching at the Academy encompassed what subjects were included in the curriculum as well as the mode in which they were discussed. For example, Davies noted that French authors had devoted much labor and talent to elementary textbooks on analytical geometry, while descriptive geometry was taught in most public schools in France as a discipline essential for architects and engineers. Similarly, these disciplines were a part of the Academy curriculum. In contrast, Harvard and Yale were typical of American liberal arts colleges in not providing instruction in analytical geometry until more than ten years after Thayer enforced the 1816 Regulations at the Academy and in never including descriptive geometry in the college course. In addition, Academy professors and former professors were often described by nineteenth-century observers as the first Americans capable of introducing the modern, usually algebraic techniques gradually perfected during the eighteenth century by Continental mathematicians. For instance, an anonymous reviewer praised Ferdinand Hassler—who was popularly associated with the Academy long after he left West Point for the Coast Survey in 1810—for making analytical trigonometry understandable for students through an algebraic approach because the geometrical method was obsolete.

Davies, Analytical Geometry (cit. n. 117); Davies, Descriptive Geometry (cit. n. 77).

Review of Elements of Analytic Trigonometry, Plane and Spherical; by Ferdinand R. Hassler, American Quarterly Review 1 (1827): 38-54, on pp. 51-52.

However, thanks largely to Davies, the “West Point system” was not based solely on “French” characteristics. Davies regularly described the Academy’s method of mathematical instruction as “the union of the French and English systems of mathematics.” By this, as Helena Pycior argued, Davies meant that the French tendency toward speculation was combined with an English bent for practical results. Note that Davies thus saw English contributions to mathematics in a rather different and more positive light than did someone like Day, who rejected both voluminous and concise English books as unsuited for American students and who appreciated Legendre for including applications in Éléments. Pycior correctly attributed Davies’s perception to his experience with Hutton’s A Course of Mathematics as the archetype of English mathematics. It may be recalled that the second volume was devoted heavily to the “useful” subjects more commonly found at the Royal Military Academy than at Oxford or Cambridge: mensuration of superficies and the circle, gauging, heights and distances, surveying, navigation, dialling, and spherical astronomy.

Davies, Logic and Utility (cit. n. 110), p. 3. See also, for instance, Davies, Elements of Algebra (cit. n. 79), p. iv.

Pycior, “British Synthetic” (cit. n. 73), p. 137.

Pycior, “British Synthetic” (cit. n. 73), p. 152.

In other words, as someone responsible for training soldiers and engineers, Davies sympathized with those filling similar roles in England. Further, as a whole, English higher education represented greater devotion to mixed mathematics than French institutions did—by the time Davies became active, the École Polytechnique would have appeared to be dominated by pure mathematicians who almost excluded applications in their zeal for the fruits of abstract theory. Although instructors such as Cauchy and Poisson created students able to invent new mathematics, the intellectual independence they fostered was not necessarily a good thing for the success of Academy cadets who would be more concerned with putting their knowledge to use in building the growing American infrastructure than with following developments in higher mathematics. Davies thus felt justified in combining his understandings of the advantages of the French and English styles. Davies added features that he considered “English,” or practical, to his series of translated French mathematics textbooks because he believed that, “Practical examples cannot fail to point out the generality and utility of abstract science.” Note lastly that Davies modeled his books on features of mathematics generally found only in England. But although Davies did not acknowledge the influence of Scottish mathematics, it was present through the sources on which he relied.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), p. iv.

While there is no evidence that Davies was even capable of mastering the original mathematical research of his day, he could recognize trends that were currently in favor. Even though Legendre and Playfair both included sections on isoperimetry, Davies removed this topic, noting that problems of maxima and minima properly belonged to the calculus of variations. Davies could also evoke the priorities of eighteenth-century mathematicians such as Euler. In one instance, he experimented with basing Elements of Geometry and Trigonometry on the property of extension. Yet, Davies made this connection between geometry and mechanics in the same era that Benjamin Peirce used “geometer” in the French sense of a mathematician of highest order in his textbook of mechanics.

See Davies and Peck, Mathematical Dictionary (cit. n. 113), p. 279, where Davies grouped isoperimetry with analytical geometry, calculus, and the calculus of variations as part of “higher geometry.”

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), pp. 9-12.

Benjamin Peirce, Physical and Celestial Mechanics: A System of Analytic Mechanics (Boston: Little, Brown, and Co., 1855).

One case study of Davies’s efforts to combine “French” and “English” aspects of mathematics is to observe some of the modifications he introduced through the various editions of Elements of Geometry and Trigonometry. For example, Davies went from admitting he had made some alterations to Legendre’s Éléments in 1834 to firmly pointing out that he had departed from the original in 1851: “In the preparation of the present edition of the Geometry of A. M. Legendre, the original has been consulted as a model and guide, but not implicitly followed as a standard.” In 1834, Davies replaced Carlyle’s essay on proportion with a more substantial treatment from Encyclopaedia Metropolitana. He substituted the key proposition in the book on quadrature, on making a circle and polygon coincide, with a theorem from William Wallace’s Book V in the Edinburgh Encyclopaedia, which was modeled on Euclid’s Elements rather than on Legendre’s textbook. In other areas including regular polygons and polyedrons, though, Davies rearranged the chain of reasoning so much that the influences upon him are no longer recognizable. He also tried to establish himself as an influence upon others by renaming the book on spheres “spherical triangles and polygons” and, later, “spherical geometry,” but this material never really took hold with other professors as a part of elementary geometry despite the market dominance of Elements of Geometry and Trigonometry. Ironically, Davies backed off from noting other influences upon him and the modifications he had made to Legendre’s Éléments in the 1862 edition of the textbook, by which time the departure from Legendre’s work was at its greatest level.

See also Helena Pycior’s discussion of Davies’s presentation of negative numbers in Elements of Algebra, where she concluded that Davies gutted the analytical method of Bourdon in the act of incorporating his own practical concerns; Pycior, “British Synthetic” (cit. n. 73), p. 145.

Davies, Elements of Geometry and Trigonometry, 1834 (cit. n. 90), p. iii; Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), p. iii; emphasis in source.

Davies and Educational Technique
Although Davies emphasized the practical benefits of a mathematical education in the style of that offered at the Military Academy, the reason for teaching geometry and the rest of elementary mathematics which he always valued at least as much as applications was mental discipline. As he explained in Elements of Descriptive Geometry, mathematics was worthy of attention whether it was studied as an introduction to mechanics or in order to exercise the mind. Davies devoted the most space to describing the benefits of mental discipline in Logic and Utility. There, he explained that mathematical science was the preeminent subject to “best develop and steady the intellect of the young” while “at the same time lay the foundations of all that is truly great in the Practical.” Assuming that the importance of practical mathematical results was self-evident to his readers, Davies prepared Logic and Utility “to point out and note the mental faculties which [mathematical science] calls into exercise; to show why and how it develops those faculties; and in what respect it gives to the whole mental machinery greater power and certainty of action than can be attained by other studies.”

Charles Davies, Elements of Descriptive Geometry With Their Application to Spherical Trigonometry, Spherical Projections, and Warped Surfaces, rev. ed. (New York: A. S. Barnes & Co., 1866), pp. ii-viii.

Davies, Logic and Utility (cit. n. 110), p. 13.

Davies, Logic and Utility (cit. n. 110), p. 17.

Further, Davies exhorted in favor of mental discipline regardless of the setting. For instance, he focused most of his inaugural address at Columbia on explaining how the study of mathematics accomplished mental training, by filling the mind with clear ideas expressed in a certain language. Davies returned to these themes in a broader sense when he delivered a lecture at the Normal School in Ypsilanti, Michigan, in 1852. In all of education, he reminded his audience, the faculties of the young mind must be strengthened and developed along with the physical and moral natures. And, as he had said of mathematics in Logic and Utility, “Teach one thing at a time—teach that thing thoroughly—and as far as possible, teach all its connections with other things!” Throughout the speech, Davies advocated the thorough education of children in public schools, urging their future teachers to be faithful to their calling. This was one of the few recorded times that Davies spoke more generally of education rather than concentrating on mathematics. His emphasis on mental discipline, though, was remembered even well after his death. For instance, one of Davies’s assistant professors at Columbia, J. Howard Van Amringe, edited a posthumous edition of Elements of Geometry and Trigonometry in which he described “Davies’s Legendre” as superior to any work for training the logical powers of pupils and instruction in geometrical truth.

Davies, “Inaugural Address” (cit. n. 59), pp. 126-129.

Charles Davies, Lecture on the Duties and Relations of Parents, Teachers and Pupils, as Connected With Education (Detroit: Free Press Book and Job Office Print., 1852).

Davies, Lecture on the Duties (cit. n. 135), p. 7.

Davies, Lecture on the Duties (cit. n. 135), p. 14. See also Davies, Logic and Utility (cit. n. 110), p. 13.

[Van Amringe], Elements of Geometry From Davies’ Legendre (cit. n. 103), p. v.

What about mathematics made it the vehicle to mental discipline? Davies explained that no one could master systematic knowledge of any subject without disciplining the mind. This mental training ought to fill the mind with “clear and distinct ideas” expressed in unambiguous language, and the only candidate for a common language free from error was mathematics. He described mathematical knowledge as founded on the concepts of number and quantity, and he argued that the study of mathematics forced students to develop habits of “close attention, nice discrimination, and certain judgment.” These habits enabled students to grasp abstract concepts, so Davies defined the purpose of Elements of Geometry and Trigonometry as strengthening the “faculty of abstraction” through “the intellectual labor” of learning propositions without reference to particular diagrams. In other writings, Davies also appealed to two different definitions of “analysis” to emphasize the relationship between mathematics and mental discipline. First, he used “analysis” to denote the process of dividing a problem, mathematical or otherwise, into its constituent parts. The process of classifying these elements sharpened the mind. Second, Davies contrasted the “Analytical method [which] is best adapted to investigation, and the presentation of subjects in their general outlines” with “the Synthetical method [which] is best adapted to instruction, because it exhibits all the parts of a subject separately, and in their proper order and connection.” Davies believed that the analytical method was currently employed to develop all branches of mathematics besides arithmetic and geometry, but it could potentially help students understand symbols as based upon the notion of quantity in the entirety of mathematics. 

Davies, Logic and Utility (cit. n. 110), p. 15.

Davies, Logic and Utility (cit. n. 110), p. 106; Davies 1858, pp. 126-127.

Davies, Logic and Utility (cit. n. 110), pp. 14, 342.

Davies, Elements of Geometry and Trigonometry, 1834 (cit. n. 90), p. iii.

Davies, Lecture on the Duties (cit. n. 135), p. 18.

Davies, Logic and Utility (cit. n. 110), p. 106.

But to maximize the value of mathematics in mental training, Davies urged that students be asked to deal with only one subject at a time. First came arithmetic, “the most useful and simple branch of mathematical science.” Then, Davies taught algebra, the universal arithmetic, up to quadratic equations before turning to geometry. Most of the next subjects were “applications”: trigonometry, the application of arithmetic, algebra, and geometry to the measurement of triangles; surveying and levelling, applications of trigonometry; descriptive geometry, which projected lines, surfaces, and solids onto paper; and shades, shadows, and perspective, the application of descriptive geometry. Lastly, students ought to learn analytical geometry, which revealed the full power of algebra and geometry to the mind, and the differential and integral calculus, which gave a new and even greater view of the power of mathematics. Once the pupil had mastered each method of mathematics, he was equipped “to compare different methods with each other.” There was also a proper order to follow in presenting each subject. As Davies explained in his essay on geometry, students had to be exposed to geometrical objects first, and then definitions and axioms, before they were ready to comprehend the architecture of a proof. Unlike Day’s reviewer and probably because many of his textbooks were for a very youthful audience, Davies argued that the content of mathematics could be simplified without sacrificing the development of mental discipline, as will be seen in his approach to the foundations of geometry.

Davies, Logic and Utility (cit. n. 110), p. 345.

Davies, Logic and Utility (cit. n. 110), pp. 346-350.

Davies, Logic and Utility (cit. n. 110), pp. 350-352.

Davies, Logic and Utility (cit. n. 110), p. 345.

Davies, Logic and Utility (cit. n. 110), p. 257.

Davies and Method of Proof
While American professors rarely alluded to the ancient meanings of “analysis” and “synthesis” as the two directions of proof, Davies did do so in the mathematical dictionary he published with his son-in-law, William Guy Peck, in 1855. Although Davies first described analysis as he generally did, as embracing “all of that portion of the science of mathematics in which the quantities considered are denoted by letters, and the operations to be performed upon them are indicated by signs,” he then distinguished between ancient analysis, which was the process of reasoning discussed by Pappus, and modern analysis, which denoted the algebraic means used in contemporary mathematical investigation and invention. Davies quoted Pappus’ definition of “analysis” and gave an example of the analytical and synthetical steps involved in determining a geometrical construction, also taken from the works of Pappus.

On Peck, see “Peck, William Guy,” in NCAB (cit. n. 7), vol. 5, p. 520.

Davies and Peck, Mathematical Dictionary (cit. n. 113), p. 22.

Davies and Peck, Mathematical Dictionary (cit. n. 113), pp. 22-23.

Davies was more normally concerned, however, with the types of proof appealed to in geometry, such as direct and indirect demonstrations. He gave standard definitions for these in the dictionary: 
In the direct method the premises are definitions, axioms, and previous propositions, and by a process of logical argumentation, the magnitudes of which something is to be proved, are shown to bear the mark by which that something may always be inferred; or, in other words, they are shown to fall under some definition, axiom, or proposition previously laid down. . . . In the indirect demonstration, therefore, the conclusion is compared with the truths known antecedently to the proposition in question. 

Davies and Peck, Mathematical Dictionary (cit. n. 113), p. 281; emphasis in source.

Davies illustrated the indirect method, or reductio ad absurdum, with a proposition from Legendre’s Éléments and explained that this process of reasoning was just as conclusive as a direct argument. He had written these definitions for Logic and Utility and reported them in a note which appeared only in the 1851 edition of Elements of Geometry and Trigonometry; his acceptance of indirect proof represented a departure from the viewpoint that reductio ad absurdum should be appealed to as rarely as possible, which was held by Carlyle in the essay on proportion that Davies removed from Elements of Geometry and Trigonometry in 1834.

Davies, Logic and Utility (cit. n. 110), pp. 241-246; Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), pp. 245-246; [Thomas Carlyle, trans.], Elements of Geometry and Trigonometry; With Notes. Translated From the French of A. M. Legendre, ed. David Brewster (Edinbrugh: Oliver & Boyd, 1824), p. ix.

More significantly, Davies tried to avoid reference to particular diagrams throughout the editions of Elements of Geometry and Trigonometry. After noting the responsibility accompanying the alteration of a work as celebrated as Legendre’s Éléments, Davies explained that the most regrettable departure from the method of Euclid in versions of Éléments, including “the original work, as well as in the translations of Dr. Brewster and Professor Farrar, [was that] the propositions are not enunciated in general terms, but with reference to, and by the aid of, the particular diagrams used for the demonstrations.” Davies believed that this practice prevented beginners from exerting enough intellectual labor to develop their faculties of abstraction. In other words, reference to particular diagrams prevented the development of mental discipline, which was the main reason for teaching geometry. Thus, Davies removed any reference to diagram labels from the statements of propositions. For instance, while Carlyle gave, “If, from a point C assumed within the triangle ABC, straight lines OB, OC, be drawn to the extremities of a side BC, the sum of these straight lines will be less than that of the two other sides AB, AC,” for the ninth proposition of Book I, Davies wrote, “If, from any point within a triangle, two straight lines be drawn to the extremities of either side, their sum will be less than the sum of the two other sides of the triangle.” This was the style of the propositions in versions of Euclid’s Elements, including Simson’s The Elements of Euclid and Playfair’s Elements.

Davies, Elements of Geometry and Trigonometry, 1834 (cit. n. 90), p. iii.

[Carlyle], Elements of Geometry and Trigonometry (cit. n. 154), p. 9; Davies, Elements of Geometry and Trigonometry, 1834 (cit. n. 90), p. 18. This was proposition 8 in Davies’s work because he converted the first proposition into an axiom in this revision of Elements of Geometry and Trigonometry (“All right angles are equal to each other”).

It was a style apparently not received as well in the United States, though. Recall that Timothy Walker, who prepared an abridged version of Legendre’s Éléments, had argued in 1828 that one reason Legendre’s work was superior to Simson’s The Elements of Euclid or Playfair’s Elements was that Legendre rendered each proposition “specific and definite by the introduction of letters, referring each part immediately to the diagram; whereas in Euclid the enunciations are all general and without letters.” From Walker’s point of view, Legendre’s method prevented learners from having to engage in the difficult activities of generalization and abstraction too soon. Farrar, the subject of Walker’s review, followed Legendre’s method of referring to the diagram in the statement of the proposition, but one reason Walker raised this point was that he had heard that Davies was preparing Elements of Geometry and Trigonometry, “in which one of the chief alterations will be the omission of the letters,” and Walker wanted to register his preference for retaining the letters. Davies himself allowed reference to particular diagrams when he felt the situation warranted it. For example, in Practical Geometry, Davies decided that general readers could not understand geometrical truths and their exactness unless he “omit[ted] the demonstrations altogether, and rel[ied] on the accuracy of the enunciation and the illustrations of the diagram.” By 1862, Davies was even defending reference to the diagram within the proofs of the propositions in Elements of Geometry and Trigonometry. Instead of rejecting reference to particular diagrams outright, he allowed each proposition to be “first enunciated in general terms, and afterwards, with reference to a particular figure, that figure being taken to represent any one of the class to which it belongs.” Davies felt he had to reiterate that the truths communicated in his propositions and proofs were indeed general, implying that a study of them promoted mental discipline.

[Timothy Walker], review of Elements of Geometry; by Adrien-Marie Legendre, trans. John Farrar, North American Review 27 (1828): 191-214, on p. 198.

[Walker], review of Elements of Geometry (cit. n. 157), p. 198.

Charles Davies, Practical Geometry: With Selected Applications in Mensuration, in Artificers’ Work and Mechanics (Philadelphia: A. S. Barnes & Co., 1839), p. vi.

Davies, Elements of Geometry and Trigonometry, 1862 (cit. n. 103), p. iii.

Finally, in addition to raising the directions of analysis and synthesis and proof techniques, Davies put his stamp on the organization of geometry as a system by adding to the subject’s first principles throughout the editions of Elements of Geometry and Trigonometry. Instead of reducing his dependence on assumptions, Davies asked students to accept more and more without proof. He began by adding axioms from Euclid’s Elements in 1834. (See Figure 5.1 for a comparison of Playfair’s, Legendre’s, and Davies’s axioms.) The examples given by Davies in the chapter on geometry in Logic and Utility indicate that Davies probably believed that equality of lines or angles could not be asserted without referring to these axioms. Davies also more often than not defined additional terms in later printings of the textbook. Davies not only founded all geometrical reasoning on the definitions and axioms from which the propositions were deduced but concluded that this basis for reasoning had to be enlarged to establish sufficient grounds for determining the truth of propositions. By setting elementary geometry upon a maximized foundation, Davies hoped he could accommodate Elements of Geometry and Trigonometry to an audience 

younger than the college students who normally studied the textbook and thus continue to sell textbooks even though American colleges gradually ceased offering instruction in geometry and required prospective students to have already mastered the basic tenets of elementary geometry in a secondary school after the Civil War.

See, for example, Davies, Logic and Utility (cit. n. 110), pp. 237-239.

Davies, Logic and Utility (cit. n. 110), p. 257. To Davies, the principles of arithmetic and algebra were also deduced from definitions and axioms. See Davies, Logic and Utility (cit. n. 110), p. 276.

In 1851, Davies furthermore introduced the three postulates from Simson’s The Elements of Euclid: “Let it be granted, that a straight line may be drawn from one point to another point,” “That a terminated straight line may be prolonged, in a straight line, to any length,” and “Let it be granted that the circumference of a circle may be described from any centre, and with any radius” (the third postulate was placed in Book III, on circles). More importantly, Davies foreshadowed several constructions with postulates, asserting that it was possible to accomplish these necessary tasks:

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), pp. 20, 58.

3. That if two straight lines are unequal, the length of the less may always be laid off on the greater.
4. That a given straight line may be bisected: that is, divided into two equal parts.
5. That a straight line may bisect a given angle.
6. That a perpendicular may be drawn to a given straight line, either from a point without the line, or at a point of a line.
7. That a straight line may be drawn, making with a given straight line, an angle equal to a given angle.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), p. 20.

1. [in Book VI] Let it be granted, that from a given point of a plane, a line may be drawn perpendicular to that plane.
2. Let it be granted, that from a given point without a plane, a perpendicular may be let fall on the plane.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), p. 157.

In other words, Davies was not content merely to do something such as, say, bisect a straight line. First, he had to convince his readers that it was permissible to bisect any line. Additionally, these postulates were not essential presuppositions or hypotheses as mathematicians usually understand them. Rather, Davies’s definition stated that a “postulate grants the solution of a self-evident problem.” Postulates were constructions which were not solved; students were not exposed to the steps to follow in order to accomplish any of the above tasks. Readers did, however, eventually reach the constructions of these objects, in their original location in the text following Book III. If anything, Davies may actually have confused the younger students he hoped to reach by requiring them to accept so much before beginning to master the propositions and the deductive connections between them.

See, for example, Thomas L. Heath, trans. and intro., The Thirteen Books of the Elements, 2d ed., vol. 1 (New York: Dover Publications, Inc., 1956), pp. 119-120.

Davies, Elements of Geometry and Trigonometry, 1851 (cit. n. 94), p. 18; emphasis in source.

Conclusion
Davies was the dominant American mathematics textbook author of the nineteenth century, with his name especially synonymous with Legendre’s Éléments. The popularity of these textbooks even outlived him. For example, Van Amringe published a revision of Elements of Geometry and Trigonometry in 1882, in which he made only cosmetic changes, except for student exercises he inserted at the end of each book and in an appendix containing ninety propositions for the student to demonstrate. The three editions of this version of the geometry textbook were published by the American Book Company, which was formed from A. S. Barnes and Co. and the textbook divisions of a number of other nineteenth-century publishers. Indeed, from his early years enforcing Thayer’s regimented system and introducing translations of French textbooks considered modern at that time through his stops at Trinity College, the University of the City of New York, and Columbia, Davies conducted his career as a teacher. His political connections, which were largely created as he taught the military figures of the nineteenth century at the United States Military Academy, enabled him to serve as a scientific consultant to the federal government, while his professional activity took place in organizations such as the New York Teachers’ Association and not the American Academy of Arts and Sciences or similar societies. Davies was certainly never a candidate for the American Association for the Advancement of Science founded by Benjamin Peirce and the Lazzaroni and devoted to professional science and university research.

[Van Amringe], Elements of Geometry From Davies’ Legendre (cit. n. 103).

Pratt, Century of Book Publishing (cit. n. 48), pp. 12-15.

Although scientific elites tended to look upon Davies as a hack writer, Davies shaped the geometry education of thousands of Americans and imparted his understandings of analysis and synthesis through Elements of Geometry and Trigonometry. He combined French, English, and, implicitly, Scottish styles of presenting geometry into his own version, which he considered the “West Point system of mathematical instruction.” Even though Davies taught mathematics as preparation for engineering at the Military Academy, he placed greater emphasis on teaching mathematics and geometry to develop mental discipline. To him, the analytical process of dividing a problem into parts helped strengthen the young mind, which should only be exposed to one subject at a time and in proper order. Davies knew of analysis and synthesis as directions of proof. He accepted reductio ad absurdum as a method of proof but persistently rejected reference to particular diagrams. Finally, Davies eventually based geometry upon a large number of postulates and axioms because he believed that accepting the obvious without proof made geometry more readily understood by beginning students.
Davies’s choices in geometry reflected the fact that teaching and business were like two internally touching circles to him. He genuinely cared for the students he was remembered as gently correcting in recitation, but at the same time he never lost sight of the need to actively market his textbooks. His skill at maximizing his reputation as a textbook author obscured the more pleasant aspects of his personality in public life. Nevertheless, Davies’s “national course” carried on elements common to Day’s and Farrar’s series and helped ensure that the traditional goals for college liberal education were a part of American life long past 1840.