Some Talks given by Fred Rickey


The following list is given in reverse chronological order. It is a representative sampling of the talks that I have given in the past two decades, not the complete list. Duplicates have been consolidated and some talks have been omitted entirely. 

 Talks given in 2011:


"The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof."  
Plenary Speaker, Canadian Undergraduate Mathematics Conference, Laval University, Quebec City, 15-19 June 2011.

History and Pedagogy of Mathematics Meeting, American University, Washington, DC. May 12, 13, 2011.


"Logic in Warsaw, 1915-1939," 
Pohle Colloquium, Aldelphi University, May 4, 2011. Slides.
Stanislaw Lesniewski (1886-1939) and Jan Lukasiewicz (1878-1956) joined the University of Warsaw faculty soon after it was converted from a Russian language university to Polish. They were good teachers and were doing interesting research so a school started to form around them. A few years later they were joined by Alfred Tarski (1901-1983), who was Lesniewski's only Ph.D. student. Since Tarski immigrated to the US, his later work is well known, but the early work of the three is not so well known. After setting the scene, we will describe some of the work of Lukasiewicz on Aristotle's syllogistic and many-valued logics as well as the foundational system of Protothetic, Ontology, and Mereology, that Lesniewski developed from his own analysis of the Russell antinomy.

"Rare Books at West Point," April 27, 2011.


"Calculus at West Point in the Twentieth Century,"  Special Session on the History and Philosophy of Mathematics, AMS Spring Eastern Section Meeting (meeting #1070), College of Holy Cross, Worcester, MA, April 9-10, 2011. This session is organized by Jim Tattersall and myself.  Joint work with LTC Tina Hartley. PowerPoint.
 

"Logic in Warsaw, 1915-1939," 
Regional Meeting of the American Mathematical Society, University of Iowa, 18--20 March  2011.

Stanislaw Lesniewski (1886-1939) and Jan Lukasiewicz (1878-1956) joined the University of Warsaw faculty at the time that it was converted from a Russian language school to Polish. They were good teachers and were doing interesting research so a school started to form around them. A few years later they were joined by Alfred Tarski (1901-1983), who was Lesniewski's only Ph.D. student. Since Tarski immigrated to the US, his later work is well known, but the early work of the three is not so well known. After setting the scene, we will describe some of the work of Lukasiewicz on Aristotle's syllogistic and many-valued logics as well as the foundational system of Protothetic, Ontology, and Mereology, that Lesniewski developed from his own analysis of the Russell antinomy.


 "Logic in Warsaw, 1915-1939," 
Regional Meeting of the American Mathematical Society, University of Iowa, 18--20 March  2011.

Stanislaw Lesniewski (1886-1939) and Jan Lukasiewicz (1878-1956) joined the University of Warsaw faculty at the time that it was converted from a Russian language school to Polish. They were good teachers and were doing interesting research so a school started to form around them. A few years later they were joined by Alfred Tarski (1901-1983), who was Lesniewski's only Ph.D. student. Since Tarski immigrated to the US, his later work is well known, but the early work of the three is not so well known. After setting the scene, we will describe some of the work of Lukasiewicz on Aristotle's syllogistic and many-valued logics as well as the foundational system of Protothetic, Ontology, and Mereology, that Lesniewski developed from his own analysis of the Russell antinomy.


"Why do we use "m" for slope?"  History and Pedagogy of Mathematics Meeting, American University, Washington, DC. March 12, 13, 2011.  Power Point.

We don't know! But we know of no historical justification for the perennial claim that it comes from the French 'monter', or from any other hypothesis that is sometimes given. So what do we know? We have attacked the question of  when the letter "m" and the word "slope" were used and can put temporal bounds on when they were first used. The earliest dictionary use of the mathematical word 'slope' occurs in the Mathematical Dictionary and Cyclopedia of Mathematical Science (New York, 1855) by Charles Davies and William G. Peck, so the word is older than this. However it does not occur in any eighteenth century book that we have examined. Rather than just dealing with the historically uninteresting question of who was first, we shall discuss the historical development of equations of straight lines and the more important question of why we introduce definitions at all.

Special Session on the History and Philosophy of Mathematics, AMS Spring Eastern Section Meeting (meeting #1070), College of Holy Cross, Worcester, MA, April 9-10, 2011. This session is organized by Jim Tattersall and myself. 

Talks given in 2010:

Mathematical Treasures in the West Point Library, for cadets in MA100, 6 November 2010.

From the Mathematician's Study to the AP Classroom: 300 Years of Learning and Teaching Calculus. History and Pedagogy of Mathematics (HPM) meeting, Huntington Library, San Marino, California. October 23-24, 2010. Power point.

The Bernoulli brothers, through dogged self-study in the seventeenth century, were the first to master the calculus of Leibniz. Countless others – famous names and nameless practitioners – followed their path from autodidact (in this arena) to teachers of calculus. But when did calculus start being taught in the classroom? Why did it happen? What textbooks were used? After a quick survey of the past, we concentrate our attention on the twentieth-century. In particular, we address the question of how calculus became a subject taught in the U. S. high school.

The Impact of Ballistics on Mathematics,” Cadet Forum, 18 October 2010. Power Point.

The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof.  SUNY/Oneonta, October 8, 2010, 3:00 PM. Power point.

The Fundamental Theorem of Calculus (FTC) was a theorem with Newton and Leibniz, a triviality with Bernoulli and Euler, and took on the concept of "fundamental" when Cauchy and Riemann defined the integral.  FTC became part of academic mathematics in the 19th century, but waited until the 20th century to take hold in classroom mathematics.  We will discuss the transition from clear intuition to rigorous proof that occurred over three centuries.Rare Books at American University, September 29, 2010. This show-and-tell session will be held from 10 to 12.

Computing Pi Three Ways, for Army Research Lab Scientists, 10 September 2010, 1330-1500 in room 504 of the Jefferson Library. A combination talk and rare book show with power point and a text.

The Fundamental Theorem of Calculus. August 17, 2010.

A Mathematical Tour of Greece and Rome, July 10-25, 2010. Day by day schedule. Things to try to see. Bibliography.

201 Year of Mathematics at West Point, Faculty Development Seminar, 8 July 2010.

"Trigonometry and Calculus," DC FAME, June 23 – July 2 and July 28 – August 6, 2010.

"The Archimedean Palimpsest," June 8, 2010, SUNY Ulster County Community College. Power Point. pdf.

Machin's Formula for Computing Pi, History and Pedagogy of Mathematics (HPM) meeting, Washington, DC, 13 March 2010. PowerPoint, Text, Library Tour.

Around 1610, John Machin found an interesting arctangent formula that he expanded using Gregory's arctangent series and then computed π "True to above a 100 Places." We shall prove Machin's formula, discuss how it has been used to compute π, and discuss its interesting history and ramification. We shall refrain from the "indoor sport" of deriving variants of Machin's formula but will discuss a few of them.

Eric Temple Bell's Men of Mathematics:  From Influential to Infamous. MAA Session of Mathematical Texts: Famous, Infamous, and Influential, AMS/MAA meeting, San Francisco, January 13, 2010.

Talks given in 2009:

The Fundamental Theorem of Calculus, Faculty Development Workshop, USMA, 16 July 2009.

201 years of mathematics at West Point, Faculty Development Workshop, USMA, 8 July 2009.

Choosing Department Heads at West Point, CSHPM, Saint John's, Newfoundland, June 6, 2009.

If you were looking to hire a mathematician to teach at your institution around 1800, who would you hire and why? Even thought this was a Presidential appointment, why would Lagrange or Lacroix move from cosmopolitan Paris to West Point, New York, an isolated outpost on the Hudson river 90 Kilometers north of New York City? There was an aversion in the United States army against hiring anyone French. Gauss was famous for discovering Ceres, but not yet for his mathematics. An English speaker was needed, but it was not considered fair to steal a faculty member from another school even by the President. This problem has arisen and has been solved 21 times since the Military Academy was founded in 1802. Were the methods of solution all the same or has the hiring process changed over time? Our purpose here is to illustrate the dramas involved: sometimes the mathematician had a connection with the U.S. President, sometimes there was family connection, and on occasion the person selected was the most qualified army officer for the job. The most interesting case was when a national search was conducted, a search that included some prominent civilian mathematicians.

Qualitative Graphing Techniques, 4th Annual Spuyten Duyvil Undergraduate Mathematics Conference, SUNY New Paltz, April 25, 2009. 

Graphing calculators are wonderful. Computer algebra systems are even better. They easily produce nice graphs. But do these graphs display the salient features of the function? Not always. Without an initial idea of what the graph looks like, one might miss essential features. We present here a way to easily sketch qualitatively correct graphs of most of the rational functions which occur in a calculus course. Calculus then can be used to refine these graphs. Handout. Graphing Polynomials. Graphing Rational Functions.

Jared Mansfield: Ohio's First Mathematician, Meeting of the Ohio Section of the MAA, April 3, 2009. Bowling Green State University, Bowling Green, OH.

Jared Mansfield (1759-1830) was educated at Yale, taught school in New Haven and Philadelphia, and wrote Essays, Mathematical and Physical (1801). These came to the attention of President Thomas Jefferson who appointed him the first faculty member at the military academy at West Point. He stayed but 18 months till Jefferson appointed him Surveyor General of the United States. Thus began his career in Ohio. He was responsible for setting up the survey in Ohio and the Northwest Territory. In 1812 he returned East and became Professor of Natural and Experimental Philosophy at West Point. We will reveal Mansfield's colorful personality and his contributions to Ohio. PowerPoint.

History can make your class sparkle, Ohio NExT. April 3, 2009. Bowling Green State University, Bowling Green, OH

There are many ways to use history in the classroom. If you report what happened on this day in the history of mathematics, your students will love it, but won't let you skip a day. Quotations are always great fun. Both of these allow you to mention a wide range of people and ideas that the students are otherwise unlikely to encounter. The bulk of the presentation will be a discussion of several tested classroom examples at various levels: completing the square, trigonometry from Archimedes to Euler, Bolzano and the intermediate value theorem, and designing a clepsydra with calculus. There will be time for lots of questions, at the dinner, and at the meeting, so bring yours along. PowerPoint.

Minicourse #14: Teaching a course in the history of mathematics, presented with Victor J. Katz, University of the District of Columbia at the annual MAA meeting, Washington DC, January 6 and 8, 2009.

Many schools are introducing courses in the history of mathematics and asking faculty who may never have taken such a course to teach them. This minicourse will assist those teaching history by introducing participants to numerous resources, discussing differing approaches and sample syllabi, providing suggestions for student projects and assessments, and giving those teaching such courses for the first time the confidence to master the subject themselves and to present the material to their students. For additional information see:                                                                                    http://www.dean.usma.edu/departments/math/people/rickey/hm/mini/default.html

Talks given in 2008:

Ferdinand Hassler's Fabulous Library, Canadian Mathematical Society, Ottawa, December 6-8, 2008.

Ferdinand Hassler came to the United States in 1805 with a library of several thousand technical books. Sadly, poverty forced him to sell many of them. In 1825, the United States Military Academy Library purchased 405 books from him. Unfortunately, no list of those books has been found and he did not write his name in his books, so they cannot be definitely identified. By comparing library catalogs and correspondence for other book purchases, I will conjecture about which treasures in the West Point Library came from Hassler.

"Show and Tell: Rare Mathematics Books from the Martin and Michalowicz collections," American University Library, 21 November 2008.

"Recruitment and Retention of Under-Represented Faculty in the Mathematical Sciences," Patriot League STEM Conference, Lafayette College, 8 November 2008. Joint work with Donald A. Outing.

The Department of Mathematical Sciences at the U.S. Military Academy, West Point, NY, has an impressive record of strengthening minority representation.  One-third of our faculty are individuals from under-represented groups, to include four African Americans and ten females with PhDs in Mathematics.  We will discuss our recruitment and retention efforts.

"Historical Notes on Archimedes, Trigonometry, and Algebra," colloquium, United States Military Academy Preparatory School, 29 October 2008.

"The impact of ballistics on mathematics. The work of Robins and Euler in the eighteenth-century," 16th ARL/USMA Technical Symposium, 23 October 2008. Text. Power Point.

"Professor Ferdinand Hassler of West Point and Union College," colloquium, Union College, 30 September 2008.

Ferdinand Hassler (1770-1843) was a remarkable individual. He learned surveying in his native Switzerland before coming to the United States with his remarkable library. He taught mathematics at the United States Military Academy from 1807 to 1809 and while teaching started to write the first analytic trigonometry book published in the U.S. Then he taught at Union College for a year. From 1815 to 1817 and also from 1832 to 1843 he was Superintendent of the U.S. Coast Survey. His scientific work was very important in the development of the United States. In addition to his scientific accomplishments, he was a colorful and idiosyncratic individual. In closing we will identify his great-great-grandson.  PowerPoint.

 "201 years of mathematics at West Point," Faculty Development Workshop, USMA, 15 July 2008.

 

Talks given in 2007:

"Trigonometry and Calculus," DC FAME  July 23 - August 10, 2007

"The Impact of Ballistics on Mathematics," Center for Faculty Development, USMA, 29 April 2007. Joint work with Shawn McMurran.  PowerPoint.

"In Celebration of Euler's 300th Birthday," Colloquium, Furman University, Greenville, SC, 26 April, 2007.

"Why Euler created trigonometric functions," Special Session on Euler, New Jersey Section of the MAA, 31 March 2007. PowerPoint.

In 1735, Euler found the differential equation  k4 d4y/dx4 = y  to be "rather slippery."  In 1739, he "rather unexpectedly" found the full solution, a solution involving trigonometric functions. Previously there were only trigonometric "lines" in a circle. Euler's views on trigonometry matured and in 1748 in his Introductio in analysin infinitorum, he introduced the trigonometric functions on the unit circle just the way we introduced them today. By the time he published his Institutional calculi integrals (three volumes, 1768-1770), he had a full command of the solutions of first-order linear differential equations with constant coefficients.

"Spicing up your Math Classes with History," SUNY and Suffolk Community College, Selden, NY, March 23-24, 2007. PowerPoint.

How might one use history in the classroom? A variety of examples will be provided:  If the topic of the day is the product rule, then we ought to let students know what Liebniz can teach them on this topic. If the Intermediate Value Theorem is the topic, don't fail to mention Bolzano. A good way to start a class is to talk about what happened on this date. How about using quotations to make a point about mathematics. Which famous mathematicians were born on this date? The latest Smithsonian magazine has an article about the Archie, so lets mention that famous palimpsest. Finally since this year is the 300th anniversary of Euler's birth, let's talk about his work on trigonometry.

"In Celebration of Euler's 300th Birthday," Rowan University Colloquium, 12 February 2007.

Leonhard Euler changed mathematics. He made functions the central concept in analysis, gave us trigonometry as we know it today, wrote a series of seminal textbooks that influenced both the teaching and research in mathematics, and did much more. He did a great deal of applied mathematics, including work on gunnery. We will describe his life and some of his numerous contributions to mathematics.

"A punishment tour for "P" Echols," AMS-MAA special session on the history of mathematics at the joint meetings in NOLA, 7-8 January 2007. This is joint work with Shawnee McMurran who is visiting from California State University, San Bernardino.

Charles P. Echols became head of the mathematics department at West Point in June 1904. At the end of the fall semester 40 % of the freshman class failed mathematics. As a result he was sent on a study tour of five U.S.  colleges in the spring term and then to European schools the next academic year. Detailed reports of both of these trips were written. We will report on these reports, comparing the curricula and teaching methods of the schools he visited.

"The Practice of Math History," a panel discussion with Joseph W. Dauben and Karen H. Parshall sponsored by the SIGMAA on the History of Mathematics, Annual joint meetings, New Orleans, 5 January 2007.

"Teaching a History of Mathematics Course," MAA minicourse at the annual meeting,  New Orleans, 5-6 January 2007. Presented with Victor J. Katz, University of the District of Columbia.

Many schools are introducing courses in the history of mathematics and asking faculty who may never have taken such a course to teach them. This minicourse will assist those teaching history by introducing participants to numerous resources, discussing differing approaches and sample syllabi, providing suggestions for student projects and assessments and giving those teaching such courses for the first time the confidence to master the subject themselves and to present the material to their students.

 

Talks given in 2006:

"Some History of the Calculus of the Trigonometric Functions," Canadian Mathematical Society special session on the history of mathematics, Toronto, 11 December 2006. PowerPoint.

Can you evaluate the integral of the sine using Riemann sums?  Do you think Archimedes could? Is it intuitively clear to you that the derivative of the sine is the cosine? If not, why not? What did Newton and Leibniz know about sines and cosines? When did sines become the sine function? Who is the most important individual in the history of trigonometry?  Answers will be provided.

"Who was the most bizarre math P West Point ever had?"  Cadet Math Forum, 5 December 2006. Gabe.

"Rare Mathematics Texts at Columbia University," for Sandra Boer's History of Mathematics Class at BMCC, 28 November 2006.

"The Impact of Ballistics on Mathematics,"  The Frederick V. Pohle Colloquium in the History of Mathematics at Adelphi University, 1 November 2006. This is joint work with Shawnee McMurran, who is visiting West Point from California State University, San Bernardino.

In 1742 Benjamin Robins published New Principles of Gunnery, the first book to deal extensively with external ballistics. Subsequently, Frederick the Great asked Euler for a translation of the best manuscript on gunnery. Euler chose Robins’ book and, being true to form, tripled the length of the work with annotations. The annotated text was translated back into English, which, two and a half centuries later, brings us to the theme of this lecture.

"Some History of the Calculus of the Trigonometric Functions," banquet speaker, joint meeting of the Metropolitan NY and Seaway sections of the MAA, 13 October 2006. Power Point for talk. DRAFT of text.

Can you evaluate the integral of the sine using Riemann sums?  Do you think Archimedes could? Is it intuitively clear to you that the derivative of the sine is the cosine? If not, why not? What did Newton and Leibniz know about sines and cosines? When did sines become the sine function? Who is the most important individual in the history of trigonometry?  Answers will be provided.

"A visit to the rare book room," American University, Washington, DC, 9 October 2006.

"Some tested examples for using history in your classroom," MathFest 2006 in Knoxville, TN, August 10-12, 2006.

The clepsydra is an ancient device for measuring time. But what is the proper shape for the bowl of a water clock? This question was not answered until after the mechanical clock was in use, but it does provide an interesting example of the early use of the calculus. The problem was solved by Emde Marriotte in 1686 and by Vincenzo Viviani, in an unpublished manuscript, in 1684. The solution provides a lovely use of solids of revolution, inverse functions, the second fundamental theorem of the calculus, and the chain rule. This sounds terribly complicated but the solution relies on the ideas and not on computation. This is a wonderful example of modeling and shows that a problem can have multiple solutions. This is but one example of how history can be used to motivate and explain mathematical ideas. We shall also deal with the cables on a suspension bridge, a neat picture for the geometric series, and Bolzano's ideas about the intermediate value theorem.

"201 years of mathematics at West Point," Faculty Development Workshop, USMA, 12 August 2006. Power Point for talk.

"Euler's Introductio of 1748," AMS Regional Meeting, San Francisco, April 29-30, 2006. PowerPoint for talk.

Leonhard Euler's Introductio in analysin infinitorum is a seminal book that helped to rework the calculus into the form that we use today. For a century after its publication in 1748 it was widely read by aspiring mathematicians. Today, thanks to John Blanton, it is available in English translation. To encourage aspiring historians to read this famous work, a reader's guide has been prepared. It summarizes the contents of the individual chapters of the Introductio, explains points that the reader might miss, points out antecedents of the work, and details how the work influenced later mathematics. We will discuss a few of the highpoints of the book.

"Researching the History of Mathematics: Yesterday, Today, Tomorrow," a panel discussion at the Research Precession of the annual meeting of the National Council of Teachers of Mathematics, Saint Louis, MO, April 25, 2006, 3:00 to 4:30 PM. Co-panelists are Profesors Alexander P. Karp (Teachers College, Columbia University), Gert Schubring (Bielefeld University, Germany) and Eileen F. Donoghue (College of Staten Island).

"The Bernoulli Brothers and the Calculus," Undergraduate Mathematics Conference for the New York City area, Manhattan College, Saturday, March 25, 2006 and Messiah College, Grantham PA, March 28,  2006.  PowerPoint for talk.

The 1690s were the decade when the calculus was formed. It was a decade of seminal problems, great mathematics, fascinating personalities, and bitter quarrels. The first papers on the calculus were published in the 1680s by Leibniz and the first individuals to truly master them were the Bernoulli brothers, Jakob (1654-1705) and Johann (1667-1748). As soon as they absorbed these groundbreaking papers they did what all good mathematicians do: they posed new questions, answered them, communicated their results, and initiated students into the newest mathematics.Through a sketch of the lives of these fascinating individuals, we shall see how their cooperative work led to intellectual competition and then to brotherly hate. Through a survey of their scientific work, we shall see what great contributions they made to mathematics, including the brachistochrone problem and L'Hospital's rule. Through a detailed look at several of the problems they created, we shall see how their work can be used in our classrooms today, including the origin of the word "integral," and several interesting elementary calculus problems of a geometric nature.

"Teaching a History of Mathematics Course," MAA minicourse at the annual meeting, San Antonio, TX, January 2006. Presented with Victor J. Katz, University of the District of Columbia.

Many schools are introducing courses in the history of mathematics and asking faculty who may never have taken such a course to teach them. This minicourse will assist those teaching history by introducing participants to numerous resources, discussing differing approaches and sample syllabi, providing suggestions for student projects and assessments and giving those teaching such courses for the first time the confidence to master the subject themselves and to present the material to their students.

Talks given in 2005:

"Isaac Newton: Man, Myth, and Mathematics," The 2005 Sehnert Lecture, Northern Kentucky University, October 24, 2005.

"Agnesi vs. Euler: Out with the old, in with the new," The Middle Atlantic Symposium on the History of Mathematics, Villanova University, October 13-15, 2005. The Theme will be Euler and “Introductio in Analysin Infinitorum, Book 1.”

"Dürer's magic square, Cardano's rings, Prince Rupert's cube, and other neat things," presented at the MAA Short Course "Recreational Mathematics: A Short Course in Honor of the 300th Birthday of Benjamin Franklin," Albuquerque, NM, August 2-3, 2005.  Text of talk. PowerPoint for talk.

Recreational mathematics is as old as mathematics itself, so a survey of its history is out of the question. Instead we discuss a few neat things, setting each in its historical context and explaining their significance. As a benchmark for looking forward and back we shall take Charles Hutton's Recreations in Mathematics, which in turn is based on works of Ozanam and Montucla on recreational mathematics.

"Mathematics at West Point in the Mid Twentieth Century," regional AMS meeting, Santa Barbara, CA, April 16, 2005.

Mathematics has been a substantial portion of the curriculum at West Point since its founding in 1802. In the mid twentieth century every cadet took four semesters of mathematics. This provides motivation to look at all aspects of the department of mathematics: Who were the faculty? What was their education and experience? What was the curriculum? Which textbooks were used? How were the classes conducted? How did the department interact with the national mathematical community? How did USMA differ from the national community? How did world events impact the department?

"Isaac Newton: Man, Myth, and Mathematics," Gettysburg College, 17 March 2005.

Isaac Newton (1642-1727) did important work in mathematics and physics, but did you know he also worked in alchemy and church history? He wrote two of the greatest scientific works ever published, Philosophia Naturalis Principia Mathematics (1687) and Optics (1704), neither of which was primarily a mathematical work. We will describe his life, his education, and his work, especially his discovery of the calculus. Persistent rumors that Newton was Irish have not yet been confirmed. Power Point slides.

"West Point and the Hudson River Valley," Hillcrest Elementary School (sixth grade), Peekskill, NY, 16 February 2005. Power Point slides are available.

"Mathematics at West Point in the Early Twentieth Century," Special Session on the History of Mathematics, annual AMS meeting,  Atlanta, January 2005.

In 1902 the United States Military Academy celebrated its centennial, but was it a vibrant intellectual center or a school with a hundred years of tradition unimpeded by progress? Since the study of mathematics occupied a substantial portion of the education of every graduate, we are motivated to look at all aspects of the department of  mathematics: Who were the faculty? What was their education and experience? What was the curriculum? Which textbooks were used? How were the classes conducted? How did the department interact with the national mathematical community? How did USMA differ from the national community? How did world events impact the department?

Talks given in 2004:

"201 Years of Mathematics at West Point," Columbia University, Teacher's College, 13 December 2004.

Since its founding in 1802 every cadet at the United States Military Academy has studied a considerable amount of mathematics, usually for two full years. At times the Academy was a leader of the nation in teaching mathematics; at other times it lagged the nation. We shall explore the reasons. The governance, faculty, cadets, textbooks, and curriculum will be described. Primarily we focus on the nineteenth century, but we also bring the story up to date by describing the current curriculum. A history of the department, written for historians, not mathematicians is available at http://www.dean.usma.edu/math/people/rickey/dms/talks/201YrMathWP.pdf . Here is a copy of the power-point slides I used during the talk.

The Olivier Models at Union College and West Point, Scientific Instrument Collections in the University, Dartmouth College, June, 24-27, 2004.

Descriptive geometry was created by Gaspard Monge (1746-1818) to graphically find the intersections of three dimensional surfaces and his disciple, Theodore Olivier (1793-1853), created dynamic string models to help students visualize these intersections. Olivier's personal collection of his models are now owned by Union College (some 40 models). The United States Military Academy ordered 26 Olivier models from Fabre de Lagrange in Paris in 1857 and 24 of these are extant. A typical model is based on a wooden box (60 cm wide, 27 cm deep, 28 cm high for example) with a bronze structure on top (say 47 cm high). The bronze bars and rings are pierced with small holes through which strings hang into the box and are held tight with lead weights. As the bars and rings are moved to create different surfaces, the strings remain taught. These were designed for classroom use, yet little detail is available about how they were actually used. We will discuss these models and indicate how the set in London influenced the sculpture of Henry Moore (1898-1986).

How Teachers Can Use History to Motivate and Inspire their Students, Preparing Mathematicians to Educate Teachers (PMET) conference, Bowling Green State University, Bowling Green, OH, June 21, 2004. Some thoughts about what to present.  Thoughts about what to discuss.

George Baron, an early American mathematician, Bicentennial Celebration of Mathematics Journals in America, American Philosophical Society, Philadelphia, PA, April 30, 2004.

Marvels of Mathematics History, Center for Lifetime Study, Marist College, Poughkeepsie, NY, Wednesdays, April 7 to May 26,  2004.

Mathematics has a long and fascinating history. We will look at several individuals and events in mathematics that have positively influenced our modern world. We will consider Archimedes the Master, the contributions of the Arab world, Fermat's Theorem, the magnificent Euler, mathematics on the plain of West Point, Lewis Carroll's logic, women in mathematics,  and Alan Turing's provocative question of whether computers can think. This course is designed for the inquisitive lay person who is willing to do a bit of advanced preparation and who wants to actively participate in an intellectual dialogue. 

A Multicultural History of Algebra, at a festival entitled "Crossing Borders: Globalization in the Arts, Sciences, and Society," SUNY Potsdam, March 31 to April 3, 2004.

Mathematics always crosses borders. In fact it seems to recognize no international borders. While some specific bits of mathematics are special to a particular culture, most of mathematics is independent of culture, common to all cultures. To argue this theme we consider the example of the solution of polynomial equations. This topic has a long and interesting history that meanders across many borders. We shall begin in Mesopotamia, move to the Aral Sea, then to Persia, Italy, Germany, Norway, France, and finally to the United States. We will mention a host of interesting characters and describe, for a general audience, the mathematics they did.

The History of Mathematics and Its Use in Teaching, Oklahoma-Arkansas Section of the MAA, Conway, Arkansas, March 26, 2004, 8:30-11:00 AM.

What is the shape of the cables on a suspension bridge? This is a fun question to ask because so many people get it wrong. Even those who get it right can't always justify their answer. So now we are ready to do mathematics. We shall present both a geometric and an analytic solution to the problem and place it in a historical setting so that its importance is clear. There are many questions that grab the interest of the student and our theme today will be to present a selection of such problems. Of course every problem has a history and telling that history makes the problems all the more interesting. Did you know that Daniel Bernoulli used differential equations to explain the spread of smallpox? What were Bernhard Bolzano's views about proof and how can they be used to present the intermediate value theorem in a calculus class? Why did Newton create his law of cooling? Why is the Mercator map projection so useful and what did it contribute to calculus? What shape should a clepsydra be? Why does the sine function have a simple derivative? These and other historical questions will be answered. All of the examples are proven classroom winners.

Carl Friedrich Gauss: Classic Hard Figurer, Rose-Hulman Undergraduate Mathematics Conference, Rose-Hulman Institute of Technology, Terre Haute, Indiana, March 19-20, 2004.

Gauss made important contributions to astronomy, celestial mechanics, surveying, geodesy, geomagnetism, electromagnetism, mechanics, optics, the design of scientific instruments, and actuarial science. And yes, he was also a mathematician, making significant advances in number theory, algebra, geometry, analysis and probability. It is hard to believe one person could do all this and it is certainly too much to describe carefully in one hour so we shall concentrate on some of his mathematical work and try to see how you the student can benefit from the study of the biography of Gauss.

The Archimedes Palimpsest, Rose-Hulman Undergraduate Mathematics Conference, Rose-Hulman Institute of Technology, Terre Haute, Indiana, March 19-20, 2004.

Archimedes is undoubtedly the greatest mathematician of antiquity, but his writings were difficult and so survive in very few copies, some in unique copies. The most fascinating of these contains his wonderful work that we call "the Method." This manuscript was created in the tenth century, palimpsested in the twelfth, discovered and published in the early twentieth, and sold at auction in 1998. Currently it is undergoing restoration and extensive scholarly study at the Walters Art Museum in Baltimore. Scholars have great hope for what the manuscript will reveal. All of this makes for a fascinating story.

History of Mathematics and the Teaching of Calculus, Bowling Green State University, Bowling Green, OH, March 17, 2004.

Mathematics at West Point in the Early Twentieth Century (a very preliminary report), Philadelphia Area Seminar on the History of Mathematics, Villanova University, February 19, 2004.

The United States Military Academy celebrated its centennial in 1902 but was it a vibrant intellectual center or a school with a hundred years of tradition unimpeded by progress? Since the study of mathematics occupied a substantial portion of the education of every graduate, this motivates us to look at all aspects of the department of mathematics: Who were the faculty? What was their education and experience? What was the curriculum? Which textbooks were used? How were the classes conducted? How did the department interact with the national mathematical community? How did world events impact the department?

Contributions of Ferdinand Hassler to Early American Science, Special Session on the History of Mathematics, annual AMS meeting, Phoenix, AZ, Saturday, January 10, 2004, 1:00 to 1:20 pm.

Ferdinand Hassler (1770-1843) was a remarkable individual. He learned surveying in his native Switzerland before coming to the United States. He taught mathematics at the United States Military Academy from 1807 to 1809 and while teaching started to write the first analytic trigonometry book published in the U.S. From 1815 to 1817 and also from 1832 to 1843 he was Superintendent of the U.S. Coast Survey. His scientific work was very important in the development of the country. In closing we will identify his great-great-grandson. [Abstract number 993-01-1007]

Tell the Truth, Tell Nothing But the Truth, But Whatever You Do, Don't Tell the Whole Truth, MAA Contributed Paper Session on Truth in Using the History of Mathematics in Teaching Mathematics, Annual MAA meeting, Phoenix, AZ, Thursday, January 8, 2004, 9:00 to 9:15 am.

This former juror and current teacher knows that standards are different in different venues. The mathematics teacher would never lie about mathematical facts (well, hardly ever); similarly no lies should be told about historical facts. When using the history of mathematics to motivate students care need to be taken not to provide too much detail. Tell what is relevant, important, and inspiring. Then stop. A variety of examples, both good and bad, will be given to support this position.

Talks given in 2003:

The Mathematics Curriculum At West Point: The First Hundred Years, joint work with Amy Shell-Gellasch, Special Session on the History of Mathematics, Spring Eastern Sectional Meeting of the AMS. The meeting will be held in New York, NY, at the Courant Institute on April 12-13, 2003.

In the early nineteenth century the United States Military Academy was the only school in the country where one could obtain a technical education. Thus it is not surprising that West Point influenced education throughout the nation. This impact was achieved because of the unique mathematics curriculum at West Point. It was unique for the breadth of material taught, for the way it was taught, and for the textbooks used. Experience teaching this curriculum encouraged the faculty to write textbooks to suit their goals. These teachers and their texts impacted the teaching of mathematics across the land.  [Abstract number 986-01-38]

The Olivier String Models at Union College and West Point, joint work with Amy Shell-Gellasch, Hudson River Undergraduate Mathematics Conference, Union College, Schenectady, NY, April 12, 2003.

Descriptive geometry was created by Gaspard Monge (1746-1818) to graphically find the intersections of three dimensional surfaces and his disciple, Theodore Olivier (1793-1853), created dynamic string models to help students visualize these intersections. Some of these models are on display here at Union College, others are at West Point. We will discuss these models and indicate how another set influenced the sculpture of Henry Moore (1898-1986).

The Archimedes Palimpsest, a presentation for the USMA librarians, March 5, 2003.

This manuscript, which contains a unique work of Archimedes, was written in the tenth century and palimpsested in the twelfth with a religious text. Currently it is under conservation, imaging, and study at the Walters Art Museum in Baltimore. The speaker will relate the sale of the manuscript, describe its earlier history, indicate what has been done to it recently, and explain what historians of mathematics hope to learn from it.

Cantor’s leap from calculus to set theory, Moravian College Student Mathematics Conference, Saturday, February 15, 2003. 

After Georg Cantor (1845-1918) received his PhD in number theory and took a position at Halle, his senior colleague Eduard Heine (1821-1881) got him interested in the question of representing a function in a unique way as a Fourier series. He proved several theorems of the form “If the Fourier series converges except on such and such exceptional points, then the representation is unique.” Soon however, the sets of exceptional points became of interest in themselves, and this led to such concepts as the derived set, and thus to set theory. We will discuss Cantor’s first (and almost unknown) proof of the uncountability of the reals, and also his proof that there are as many points on a line as in the plane. Some glimpses of later work will be discussed.

"Mathematics in the Ancient World," the Short Course before the annual meeting of the Mathematical Association of America, Baltimore, MD, 13-14 January 2003. I am the organizer. 

Nearly everyone who has taken an interest in the history of mathematics becomes fascinated with some facet of ancient mathematics. But only a few have the mathematical preparation, historical sensitivities, and linguistic skills to do original work. The speakers at this short course will give an expository survey of their special area of ancient mathematics. They will discuss some areas of current research, point out open questions, and provide guidelines to help you delve into the expository and research literature. Those of you who have taught history of mathematics, will undoubtedly learn that some of what you read in older literature has been superseded by modern scholarship. Thus you will have much to carry back to your classroom.

Speakers:

Talks given in 2002:

"A Reader's Guide to Euler's Introductio," Euler 2K + 2 Conference. Countdown to the Tercentennary, Rumford Maine, 4-7 August 2002. The Euler Newsletter has a  report on this talk.

Leonhard Euler's Introductio in analysin infinitorum is surely one of his most famous works. For a century after its publication in 1748 it was widely read by aspiring mathematicians. Today, thanks to John Blanton, it is available in English translation. To encourage aspiring historians to read this famous work, a reader's guide will be distributed. It will summarize the contents of the individual chapters of the Introductio, explain points that the reader might miss, point out antecedents of the work, and detail how the work influenced later mathematics.

"George Baron, One of America's First Mathematicians," Invited Paper Session on the History of Mathematics organized by Paul Wolfson (West Chester University) and Roger Cooke (University of Vermont), at the Burlington VT, MathFest, Friday, August 2, 2002, 3:15 pm - 6:15 pm.

George Baron taught at West Point in 1801, but was fired before the United States Military Academy was founded on March 16, 1802. He used a Black Board in his teaching and may have been the first in the country to do so. After leaving West Point he founded the first mathematical periodical in the United States, the short lived Mathematical Correspondent. The contents of this journal and what little is known of Baron's life will be described.

"We Must Use History in Teaching Mathematics: Examples for the Calculus Classroom," Study the Masters meeting, Kristiansand Norway, 12-15 June 2002. 

Advocating the use of the history of mathematics as motivational and learning device for teaching mathematics has become a commonplace in the mathematical community, albeit it has not been supported (indeed there has been little attempt) by careful research done by mathematics educators. But the feedback from teachers has been overwhelming.

The present audience does not need to be convinced of the value of history as a motivational device in the classroom, so I shant talk about that, but I shall argue the point in writing, so that the participants can criticize and augment my argument. Rather, I shall give examples of how history can be used to motivate mathematics so that our students will learn more mathematics, and, more importantly, learn how mathematics is done.

The clepsydra is an ancient device for measuring time. But what is the proper shape for the bowl of a water clock? Alas this mathematical question was not answered until after the mechanical clock was in use, but it does provide an interesting example of the early use of the calculus. The problem was solved by Edme Marriotte in 1686 and by Vincenzo Viviani, in an unpublished manuscript, in 1684. The solution provides a lovely use of solids of revolution, inverse functions, the second fundamental theorem of the calculus, and the chain rule. This sounds terribly complicated but the solution relies on the ideas and not on computation. This is a wonderful example of modeling and shows that a problem can have multiple solutions.

Daniel Bernoulli's 1670 work on the spread of smallpox is a fine classroom example. This was the first attempt to explain the propagation of an infectious disease and involves the logistic equation a century before Velhost did his work in 1854. An example that has been very successful in the classroom is Bolzano's work on the intermediate value theorem, so I shall describe how I present this topic. Finally, I shall return to a question left open in 1988: Who showed that the cables on a suspension bridge form a parabola? The elegant and elementary geometric solution of Pardies (1673) will be given.

"Characters of Mathematics," Wisconsin Mathematics Council Green Lake Conference, May 2, 2002. 

The history of mathematics is salted with a variety of interesting characters: Pythagoras didn't discover his theorem. No one named Fibonacci ever existed, but he had an interesting question about rabbits. Pascal was engrossed in theology, but a toothache changed that. Cardano did bizarre things, and so did Kepler. Hardy did some rational things that border on the irrational. Erdös had a strange collection of idiosyncrasies. Graham was president of the International Juggling Association. But one thing all of these people shared was a passionate interest in mathematics. I will tell some stories about these folks, partly to amuse you, but mostly to encourage you to take an interest in their mathematics, for that is what their lives were all about.

"The Creation of the Calculus: Who? What? When? Where? Why?," Mathematics Retreat, University of Wisconsin, Eau Claire, April 8, 2002. 

The creation of the calculus is credited to Isaac Newton and Gottfried Wilhelm Leibniz in the latter part of the seventeenth-century, but it hardly began -- or ended -- with them. Their seminal work will be discussed so that we can see precisely why history has placed such great value on their work. This will necessitate that we show what their predecessors did and how their successors developed and extended their ideas. Then we will be in a position to frame a definition of the calculus of Newton and Leibniz and to contrast it with the calculus we do today.

"The Palimpsest of Archimedes," Mathematics Retreat, University of Wisconsin, Eau Claire, April 8, 2002.

Archimedes is undoubtedly the greatest mathematician of antiquity, but his writings were difficult and so survive in very few copies, some in unique copies. The most fascinating of these, which contains his wonderful work that we call "the Method," was sold at auction by Christie's in 1998. This manuscript was created in the tenth century, palimpsested in the twelfth, discovered and published in the early twentieth, and sold at auction in 1998. It has been exhibited at the Walters Art Museum in Baltimore and the Field Museum in Chicago and is presently under restoration and extensive scholarly study. Scholars have great hope for what the manuscript will reveal. All of this makes for a fascinating story.

"Antiderivatives, Bridges, Clepsydra, and other Historical Examples for the Calculus Classroom," University of Toledo, March 29, 2001.

The history of mathematics is a wonderful classroom tool for it motivates students to remember the mathematics they learn, encourages them to want to know more mathematics, and helps them understand how mathematics is done. We will begin by arguing that history is a necessary classroom tool but will concentrate on examples of how you can use history in your calculus classroom.

"The Mathematics Curriculum at West Point in the Nineteenth Century," Making History: West Point at 200 Years, A Department of History Symposium, 7-9 March 2002. Joint work with Dr. Amy E. Shell. Full text of the talk in word. HTML with overheads.

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

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

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

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

"The Palimpsest of Archimedes," colloquium, Vassar College, 29 January 2002. 

Archimedes is undoubtedly the greatest mathematician of antiquity, but his writings were difficult and so survive in very few copies, some in unique copies. The most fascinating of these, which contains his wonderful work that we call "the Method," was sold at auction by Christie's in 1998. This manuscript was created in the tenth century, palimpsested in the twelfth, discovered and published in the early twentieth, and sold at auction in 1998. It has been exhibited at the Walters Art Museum in Baltimore and the Field Museum in Chicago and is presently under restoration and extensive scholarly study. Scholars have great hope for what the manuscript will reveal. All of this makes for a fascinating story.

Talks given in 2001:

"Clever historical ideas which will motivate your students," MAA New Jersey Section meeting, to be held at Middlesex County College, Edison,  New Jersey, October 27, 2001. 

 

Did you know that the Mercator map projection led to the discovery of the integral of the secant? Telling this story to your students will make this difficult to motivate integral quite natural. And what about the integral of the sine function? Can you do it with "Riemann sums"? Pascal did. Perhaps you have seen the Infinite Acres film and so know of an object that you can't paint the outside of, but the inside can be filled with paint. But do you know how to make "a drinking glass, that had a small weight, but that even the hardiest drinker could not empty." Examples like these will make your calculus students enjoy, learn, and remember your class. Along the way they will learn some interesting history and something about how mathematics is created by some of the best mathematicians who ever lived.

"Calculus at West Point in the Nineteenth Century," Special Session of History of Mathematics, AMS 2001 Fall Eastern Section Meeting, Williamstown, MA, October 13-14, 2001. AMS Meeting #971.

Only six textbooks were used at West Point in teaching calculus in the nineteenth century. They were by Charles Hutton, Silvestre François Lacroix, Jean Louis Boucharlat, Charles Davies, Albert E. Church, and Edgar W. Bass. The first of these authors was British, the next two French, and the final three Americans, in fact, West Point faculty members. A careful analysis and comparison of these texts will be presented.  We will examine what is unique about each of these texts and what they have in common by examining particular definitions, results, and problems.

"Characters of Mathematics," The Math Forum, USMA, October 2, 2001, 1930-2030.

The history of mathematics is salted with a variety of interesting characters. Cardano was one of the craziest, Kepler did some weird things, Hardy did some rational things that border on the irrational, and Erdös had a strange collection of idiosyncrasies. But one thing all of these people shared was a passionate interest in mathematics. I will tell some stories about these folks and a variety of others, partly to amuse you, but mostly to encourage you to take an interest in their mathematics, for that is what their lives were all about. 

"The British Influence on Mathematics at West Point," John Fauvel Memorial Conference at Colorado College, September 21, 2001. 

Unfortunately the England-West Point connection is not terribly strong, but because this meeting is in honor of my dear friend John Fauvel (1947-2001) I want to speak about that connection. The first mathematics teacher here was from Woolich, but he was fired before West Point actually began. However Hutton's Course of Mathematics was used for about 20 years here at West Point and I thought it would be interesting to talk about what it contains. There are some fluxions and also some material on descriptive geometry. The later later (pronounce those two differently and this sentence will work) became an important subject in the curriculum here at West Point. 

"History of mathematics, especially history of calculus and using history in the classroom," Summer Seminar 2001, North Central Section of the MAA, Bemidji State University, Bemidji, MN, July 24-27, 2001.

Details of the program are available. 

"The First Century of Mathematics at West Point," conference on the History of Undergraduate Mathematics in America (HUMA), West Point, NY, June 21-24, 2001. 

Mathematics has been taught at West Point since before the United States Military Academy was founded in 1802 and has been a central part of the curriculum ever since. The textbooks used at West Point were first British, then French, and finally American; in fact, many of the American texts were written by West Point authors. We intend to discuss the textbooks used, the design of the curriculum, the teaching methods used, and the influence of the program on other schools. Especial attention will be paid to two subjects, calculus and descriptive geometry. 

"Calculus Classroom Chronicles: Catenaries, Clepsydrae, and Cycloids," Northeastern Section of the Mathematical Association of America, Norwich Vermont, June 9, 2001

History is a marvelous tool in the calculus classroom for it shows students that the ideas which they are learning have been useful in the past and thus may be useful in the future. The incorporation of history allows students to learn about the great masters of the field, to get glimpses of the works they wrote, to see mathematical modeling in action, and to really see how mathematics is done. This will be a talk of examples of how you can incorporate history in your classroom.

"The Brachistochrone Problem," Pi Mu Epsilon initiation and reunion, Saint Joseph's University, Philadelphia, PA, 20 April, 2001. 

In 1696, Johann Bernoulli posed the following problem: Find the curve connecting two given points in space such that a frictionless body sliding along the curve will get from one point to the other in the minimum time. Several mathematicians solved this easily; others were stumped. We shall discuss the history of this problem and give a solution. This talk is accessible to all who are willing to exercise their integral calculus.

"History Tour: Mathematics on the Plain," Eleventh Annual Service Academy Student Mathematics Conference, West Point, 13 April 2001.

This tour was originated by former department head Chris Arney. This was the first time I did it by myself. When we were in front of Quarters 100, where the Superintendent lives, Mrs. Christman, his wife, came out and invited the group in to see the house. This was a rare thrill for all concerned. The text of the tour along with some pictures is now on the web. 

"History of Mathematics as a Pedagogical Tool, Part II,"  Ohio Section of the MAA, Bowling Green State University, Bowling Green, OH, March 24, 2001, 9:00-10:00 AM. 

Part I of this lecture was presented in 1979 when I was first invited to speak to this section. There was some philosophy about why you should use historical ideas while teaching calculus and then there were examples such as the intermediate value theorem, the history of trigonometry, and the integral of the secant. Perhaps I can be pardoned for revisiting this topic, for I have learned a bit more in the past two decades. Now I can show you how easy it is to differentiate and integrate the trigonometric functions, why the cables on suspension bridges are parabolic, what shape a clepsydra is, and how the Schwartz paradox changed the way we define surface area. It is my hope that you will take these examples into your classrooms and show your students.

"The Palimpsest of Archimedes," Ohio Section of the MAA, Bowling Green State University, Bowling Green, OH, March 23, 2001, Evening Address, 8:00 PM. I am very much looking forward to this meeting as I am Distinguished Teaching Professor Emeritus at BG.

Archimedes is undoubtedly the greatest mathematician of antiquity, but his writings were difficult and so survive in very few copies, some in unique copies. The most fascinating of these, which contains his wonderful work that we call "the Method," was sold at auction by Christie's in 1998. This manuscript was created in the tenth century, palimpsested in the twelfth, discovered and published in the early twentieth, and sold at auction in 1998. It has been exhibited at the Walters Art Museum in Baltimore and the Field Museum in Chicago and is presently under restoration and extensive scholarly study. Scholars have great hope for what the manuscript will reveal. All of this makes for a fascinating story.

"The Creation of the Calculus: Who? What? When? Where? Why?," Ramapo College, March 7, 2001. 

The creation of the calculus is credited to Isaac Newton and Gottfried Wilhelm Leibniz in the latter part of the seventeenth-century, but it hardly began -- or ended -- with them. Their seminal work will be discussed so that we can see precisely why history has placed such great value on their work. This will necessitate that we show what their predecessors did and how their successors developed and extended their ideas. Then we will be in a position to frame a definition of the calculus of Newton and Leibniz and to contrast it with the calculus we do today.

Talks given in 2000:

"The big math attack," Keynote Address, 50th Annual Conference of the Association of Mathematics Teachers of New York State (AMTNYS), Nevele Grand Resort, Ellenville, NY, October 26, 2000, 8:00 PM.

Mathematicians are people too. They eat, drink, and do mathematics. Sometimes all at once. This romp through history, which concentrates on food and mathematics, is designed to convince you that we can find mathematics wherever we choose to look. 

"What your students can learn about the history of mathematics on the web," invited speaker, 10th Annual Kansas City Regional Mathematics Technology EXPO, Kansas City, October 5-7, 2000.

The world wide web is a fantastic tool to enrich the resources available to teachers. There are many good sites for the history of mathematics  --- and we shall point out some of them --- but one must exercise care about the accuracy and quality of what you find. Advice to help you and your students make critical judgments will be given.
"Teaching with technology at West Point," 10th Annual Kansas City Regional Mathematics Technology EXPO, Kansas City, October 5-7, 2000.

This presentation will be a discussion about how we use the calculator (HP48G up to now; TI89 in the fall) and computer (Word, Excel, MathCad, Minitab) in the classroom. This will include a discussion of projects that are designed to push the cadets knowledge of mathematics and technology. 

"Fun, Interesting, and Historical Examples of Infinite Series and Improper Integrals," MAA Student Workshop, Mathfest 2000, UCLA, Saturday, August 5, 2000, 1:00 pm - 2:50 pm. Also given as a seminar at Teacher's College, Columbia University, September 25, 2000, 7:30-9:00 PM.

Examples are the heart of mathematics. They lead to understanding and to theory. So we shall look at early uses of the geometric series, at examples where the divergence of the harmonic series is important, and the example of "a drinking glass, that had a small weight, but that even the hardiest drinker could not empty." These examples will give us a close look at some interesting history and let us explore how mathematics is done.
"Minicourse on Using History to Aid the Learning of Mathematics," Illinois Section of the MAA,  North Central College, Naperville, IL, March 31, April 1, 2000. Similar topics presented at the University of Connecticut.
After a philosophical discussion of why you should use history to motivate and inform your students, a multitude of examples will be given of how you can use history in your classrooms. History can explain notation (why do we write f'), illuminate mathematical vocabulary (why do we say 'complete the square'?), explain where examples come from (why was the Folium of Descartes created?), provide wonderful stories (why do we call it L'Hospital's rule?), show that mathematicians make mistakes (why is the product rule so complicated?), and explain where definitions come from (why did Cauchy get the proof wrong?). These examples -- and others such as Koch's snowflake, Brounckner's series, and the Clepsydra -- are designed to convince you that your students will benefit if you interject historical examples into your teaching.
Talks given in 1999:

"The History of Infinite Series and Examples for Classroom Use," joint Seaway Section/NYSMATYC meeting, Adirondack Community College in Queensbury, NY,  Nov 5-6, 1999.

The use of infinite series in mathematics is as older than the Greeks and newer than Ramanujan. Some of the historical high points of this important, exciting, and convoluted topic will be discussed, but always with one eye on the classroom. We will discuss the geometric series, a variety of way of showing the harmonic series diverges, how to sum the Swineshead series, some of the nutty ideas of Grandi, the sum of the reciprocals of the triangular numbers, series for pi, and how Fermat used series to integrate x-to-the-n using Riemann Sums. Every student of the calculus, from sophomore to seasoned faculty, should find something new in this historical presentation.
"Mathematics at West Point: The first fifty years,"   USMA Colloquium,
Mathematics has been a central part of the curriculum at the United States Military Academy since its founding in 1902. When Sylvanus Thayer became Superintendent in 1817, he reorganized the Academy using the Ecole Polytechnique as a model. He changed the way the cadets were taught as well as the textbooks they used. Now there was an emphasis on French textbooks. Gradually these were replaced by American texts, most notably those of West Point faculty members Charles Davies and Albert Church. Our purpose today will be to explore the changes that took place at West Point in its first half century.
"Early Textbooks at West Point,"  AMS regional meeting, Providence College, Providence, RI, October 2, 1999.
Mathematics instruction at West Point began in 1801 when George Baron taught a few cadets some practical applications of algebra. After the United States Military Academy was founded in 1802, mathematics became a central part of the curriculum and the textbooks were of British origin. When Sylvanus Thayer became Superintendent in 1817, he reorganized the Academy using the Ecole Polytechnique as a model. There were many academic changes, not least of which was an emphasis on the use of French textbooks. These textbooks were gradually replaced by American texts, many of which were written by USMA faculty, most notably Charles Davies. The changes that took place at West Point during the first half of the eighteenth-century in pedagogy, textbooks, and the mathematical content of the curriculum will be discussed. Abstract  947-01-139.

"The Creation of the Calculus: Who, What, When, Where, Why,"

The creation of the calculus is credited to Isaac Newton and Gottfried Wilhelm Leibniz in the latter part of the seventeenth-century, but it hardly began -- or ended -- with them. Their seminal work will be discussed so that we can see precisely why history has placed such great value on their work. This will necessitate that we show what their predecessors did and how their successors developed and extended their ideas. Then we will be in a position to frame a definition of the calculus of Newton and Leibniz and to contrast it with the calculus we do today.
"Teaching a Course in the History of Mathematics," a minicourse presented with Victor Katz at the MAA national meeting in San Antonio, January 1999.
Many colleges and universities are introducing courses in the history of mathematics and asking mathematicians without a strong background in history to teach them. This minicourse will assist those teaching history by introducing participants to numerous resouces, discussing differing approaches and sample syllabi, providing suggestions for student projects, and, in general, giving those teaching such courses for the first time the confidence to master the subject themselves and to present the material to their students.

"Why do we use 'm' for slope?" Presentation, with my colleague Rickey A. Kolb, at the AMS Special Session on the History of Mathematics arranged by Karen Parshall and Victor Katz at the AMS meeting in San Antonio, TX, Friday and Saturday, 15-16 January 1999.

We don't know! But we know of no historical justification for the often made claim that it comes from the French 'monter.' We have much better information on when the letter 'm' and the word 'slope' were first used. The earliest use of 'm' for slope that we have located is in an 1844 British text by M. O'Brien entitled A Treatise on Plane Co-Ordinate Geometry. The mathematical word 'slope' occurs in the Mathematical Dictionary and Cyclopedia of Mathematical Science (New York, 1855) by Charles Davies and William G. Peck, but we have been unable to locate an earlier usage. Rather than simply present the historically uninteresting question of who was first, we shall discuss the historical development of equations of straight lines.

Talks given in 1997:

"The Beginnings of the Leibnizian Calculus," Tenth Anniversary Midwest Conference on the History of Mathematics, University of Akron, April 4-5, 1997. This conference has been rescheduled from October 18-19, 1996.
In 1684 Gottfried Wilhelm Leibniz (1646--1716) presented his ``Nova methodus,'' his new method of maxima, minima, and tangents to the scientific community. We shall explore the life of the author, the content of this famous paper, and some of his other work. Our intent is to show that the work of Leibniz was a watershed; mathematics in succeeding generations was considerably different than mathematics before him. This will show why Leibniz is justly cited as one of the creators of the calculus.
"Euler and the calculus," Florida Section of the MAA meeting at Florida State University, Talahassey, Florida, 28 February to 1 March, 1997
Leonard Euler (1707-1783) was the most prolific mathematician of all time. Besides his hundreds of research papers, he wrote seminal books that reworked the calculus into a form that we can recognize today. After sketching his life, we shall concentrate on the three works which he wrote on the calculus.
"Johann Bernoulli's Calculus Texts," AMS Special Session on the History of Mathematics organized by Karen Parshall and Jim Tattersall at the Joint Winter meetings in San Diego, January 10 and 11, 1997.
When Johann Bernoulli (1667-1748) was in Paris in 1691-92, he wrote manuscripts, for the use of L'Hospital (1661-1704), on the differential and integral calculus: Lectiones de calculo differentialium and Lectiones mathematicae, de methodo integralium, aliisque. The first of these was the model for L'Hospital's Analyse des infiniment petits, the first calculus textbook (1696), while the second was not published until 1742 in the third volume of Bernoulli's Opera omnia. [These are most readily available, in German translation, as Ostwald's Klassiker, #211 (1924) and #194 (1914) respectively.] While these works have been much discussed for their impact on L'Hospital, their contents are little known. This presentation will sketch the contents of these works and then present a few of the many wonderful problems they contain.

Talks given in 1996:

"Happy 300th Birthday to the First Calculus Text," Seaway Section of the MAA, Geneseo College, Geneseo, NY, November 8, 1996 and at Western Kentucky University, Bowling Green, KY, November 1, 1996.
This year we celebrate the three-hundredth anniversary of the first calculus text, the Analyse des Infiniment Petits. It was pulished by Guillaume-Francois-Antoine de l'Hospital 1661-1704) who is best known today for the rule which bears his name. The authorship of this famous rule came into question a few months after L'Hospital's death when Johann Bernoulli wrote a paper generalizing "my rule." These two words did not go unnoticed by his contemporaries, but Bernoulli had such a reputation as a scoundrel that his claim was not believed. Today we know the rule is indeed the work of Bernoulli. After we trace the history of L'Hospital's rule, we shall examine the contents of L'Hospital's text as well as the text of Bernoulli's Lectiones de calculo differentialium, all with an eye for classroom examples.
"Using History in Teaching Calculus," Western Kentucky University, Bowling Green, KY, November 2, 1996.
The calculus has a fascinating history that can be used to motivate and instruct our students. After a discussion of why you should use history in teaching calculus several specific examples will be given: Why the sine has a simple derivative, the Schwarz area paradox, Bolzano and the intermediate value theorem, Torricelli's solid and improper integrals, and Bruckner's series.
"Perils and pleasures of the Internet," The abstract is too long to have on this page.

"The History of Improper Integrals," Kansas Section of the MAA and the Kansas Mathematics Association of Two Year Colleges, McPherson College, McPherson, Kansas, 19-20 April 1996.

As with so many fundamental concepts in analysis, it was Cauchy who gave precise definitions of improper integrals. The concept however, has its roots in the seventeenth-century. We will discuss Torricelli's example of an infinite solid with finite volume and Sluse's glass "that had a small weight, but that even the hardiest drinker could not empty." The talk will be understandable to all who have studied improper integrals in calculus.
"Ways of Using History to Motivate Calculus," Kansas Section of the MAA and the Kansas Mathematics Association of Two Year Colleges, McPherson College, McPherson, Kansas, 19-20 April 1996.
The calculus has an interesting history that should be used by teachers to improve student motivation and understanding. Rather than argue this point in the abstract, examples will be given of how the speaker has effectively used history in the classroom. Topics will include "What to say on the first day of a calculus class," "Leibniz got the product rule wrong," "Gregory of Saint Vincent and the logarithm" and "How Cotes discovered the derivative of the sine."
"Happy 300th Birthday L'Hospital," HIMED-96, sponsored by the British Society for the History of Mathematics, Lancaster, England, April 12, 1996.
Guillaume-Francois-Antoine de l'Hospital 1661-1704) is best known today for the rule which bears his name, but in his day he was renown for the first calculus text Analyse des Infiniment Petits (1696). The authorship of this famous rule came into question a few months after L'Hospital's death when Johann Bernoulli wrote a paper generalizing "my rule." These two words did not go unnoticed by his contemporaries, but Bernoulli had such a reputation as a scoundrel that his claim was not believed. Today we know the rule is indeed the work of Bernoulli. In this talk we shall examine the contents of L'Hospital's text as well as the text of Bernoulli's Lectiones de calculo differentialium, all with an eye for classroom examples. Contrary to the received opinion, we shall show, by careful analysis of the original texts, that L'Hospital is deserving of much more credit for the Analyse than he usually receives.
"From our calculus it follows," a series of three lectures, at Allegheny College, Meadville, PA, March 25, 26, 27. The individual lectures are: "Fermat's Last Theorem," Meadville Area Senior High, Meadville, PA, March 25, 1996.

"Mathematics and Humor," presented to the Kappa Mu Epsilon student honorary at BGSU, March 19, 1996.

Over the centuries there have been many humorous things that have occured in mathematics and concerning mathematician. A few of these stories will be told. In addition, some references will be supplied where you can find more such stories. There are also web sites dealing with humor in mathematics.
"Rare Books at the University of Michigan", a presentation to Dan Chazan's History of Mathematics Class from Michingan State University, 11 March 1996.

Talks given in 1995:

"The Calculus Texts of Johann Bernoulli," at Bruce Chandler's Seminar on the History of Mathematics, Graduate Center, City University of New York, New York City, 8 December 1995.

"The Development of the Calculus," New York Academy of Science, New York City, 7 December 1995.

It is a commonplace that Isaac Newton and Gottfried Leibniz invented the calculus in the latter part of the seventeenth-century. But to give all the credit to them is unjust. Many individuals before them and still more after them are responsible for the development of the calculus. To make the argument that Newton and Leibniz really did something new and interesting we shall look at one topic as presented by their predecessors and successors, namely improper integrals. We will contrast Torricelli's clever example of an infinite solid with finite volume with the ease with which Euler could handle similar problems. The talk will be understandable to all who have studied calculus.
"The importance of using history in teaching mathematics," at a meeting on "New Trends in the Teaching and Learning of Mathematics," Mathematisches Forshungsinstitut Oberwolfach, Germany, 26 November to 2 December 1995.
Mathematics has a rich and interesting history that teachers must utilize to improve student motivation and understanding. This position could be argued in the abstract, but examples of what has worked for me will be more interesting than philosophy. Here are some examples that could be presented: What to say on the first day of calculus class. What to say on November 27. Gregory of Saint Vincent and the logarithm. Euler and the trigonometric functions. Perrault and the tractrix. Participants and the meeting will be able to select examples for presentation from a longer list.
Institute in the History of Mathematics and Its Use in Teaching, American University, Washington DC, 5-23 June 1995.

"Mathematics on the internet," Michigan Section of the MAA, Grand Valley State University, Allendale MI, 6 May 1995.

The internet is a fabulous resource for mathematicians. We can now find out things in minutes that used to take days. This hands-on tour will begin with the MAA gopher and then branch out to other gophers, especially those that contain information of mathematical interest. From there we will proceed to libraries, data bases, and software archives. Finally, we will wander out on the World Wide Web.
"Experiences of a Visiting Mathematician at the MAA," Luncheon address at the Michigan Section of the MAA, Grand Valley State University, Allendale MI, 5 May 1995.
What does a mathematician visiting the MAA do? The short answer is: Whatever needs to be done! Daily tasks were extremely varied, from taking phone calls that required some mathematical expertise to writing back cover copy for MAA books. The large projects that I was involved in were the birthing of Math Horizons and the design and construction of the MAA gopher server. The riches of Washington D.C. allowed me to pursue my interest in the history of mathematics. By describing my experiences at the MAA headquarters (and in D.C), I hope to give you insight into how your organization functions.
"The History of Improper Integrals," colloquium, New Mexico State University, Las Cruses, NM, 7 April 1995 and at the Indiana Section of the MAA, Tri-State University, Angola, IN, 31 March 1995.
As with so many fundamental concepts in analysis, it was Cauchy who gave precise definitions of improper integrals. The concept however, has its roots in the seventeenth-century. We will discuss Torricelli's example of an infinite solid with finite volume and Sluse's glass "that had a small weight, but that even the hardiest drinker could not empty." The talk will be understandable to all who have studied improper integrals in calculus.
"History can make your classroom exciting," colloquium, Wright State University, 3 February 1995.
The history of mathematics is a wonderful tool for motivating students and developing mathematical ideas. Examples will be provided that can be used in the classroom at several levels from high school algebra and geometry through calculus.
"A Theorem of Barrow using the "new method" of Leibniz," National meeting of the American Mathematical Society, San Francisco, 7 January 1995.
The Lectiones geometricae (1670) of Isaac Barrow contains the following theorem: The sum of the intervals between the ordinates and perpendiculars to a curve taken on the axis and measured in it is equal to half of the squares of the final ordinate. In his Methodus figurarum (1685), John Craig tried to simplify Barrow's complicated geometrical proof of this result using the differential notation of Leibniz, but he made a mess of it. This prompted Leibniz to publish his Nova methodus (1686). By reviewing the proofs of this result of Barrow we shall see the tremendous advances that Leibniz made with his new calculus.

Talks given in 1994:

"Leadership," given to the new initiates of Omicron Delta Kappa, the national leadership society, BGSU, 9 December 1994.

"How to trisect angles, and why you can't," presented at The Ohio State University at a National Institute for the Humanities sponsored seminar on "Great Theorems of Mathematics" conducted by William Dunham for high school teachers, Ohio State, 5 July 1994.

"Differentiation and Integration Through History," Freudenthal Institute, Utrecht, The Netherlands, 9 June 1994.

"The Bernoulli Boys and the Calculus," Stafcolloquium, Rijksuniversiteit Groningen, The Netherlands, 7 June 1994, Georgetown University, 25 March 1994, Trinity University, San Antonio, TX, 11 November 1993, Loyola College, Baltimore MD, 17 November 1993, and Miami University, OH, October 2, 1992.

While we today are anticipating calculus reform in the 1990s, the 1690s were the decade when the calculus was formed. It was a decade of seminal problems, great mathematics, fascinating personalities, and bitter quarrels. The first papers on the calculus were published in the 1680s and the first individuals to truly master them were the Bernoulli brothers, Jakob (1654- 1705) and Johann (1667-1748). As soon as they absorbed these seminal papers they did what all good mathematicians do: they posed new questions, answered some of them, communicated their results, and initiated students into the newest mathematics. Through a sketch of the lives of these fascinating individuals, we shall see how their cooperative work led to intellectual competition and then to brotherly hate. Through a survey of their scientific work, we shall see what great contributions they made to mathematics, including the brachistochrone problem and L'Hospital's rule. Through a detailed look at several of the problems they created, we shall see how their work can be used in our classrooms today, including the origin of the word "integral," and several interesting elementary calculus problems of a geometric nature.

"History: A Vital Tool for the Calculus Teacher," at a meeting on Oude wiskunde in modern onderwijs (Old mathematics in the modern classroom), organized by the Landelijk Werkcontact Geschiendes en Maatachappelijke Functie van de Wiskunde, Utrecht, The Netherlands, 4 June 1994.
The calculus has an interesting history that should be used by teachers to improve student motivation and understanding. Rather than argue this point in the abstract, examples will be given of how the speaker has effectively used history in the classroom. Topics will include "What to say on the first day of a calculus class", "What to say on June 4", "The mistake with the product rule, older than you might think", "Gregory of Saint Vincent and the logarithm" and "Euler and the trigonometric functions."
"Math on the Mall," a tour in Washington DC for the MAA staff to celebrate Math Awareness Week, 28 April 1994, for the Secondary Recipients of the Presidential Awards for Excellence in Science and Mathematics, 29 April 1994 , and for the Elementary Recipients of the Presidential Awards for Excellence in Science and Mathematics, 11 March 1994,

"Welcome and Congratulations from the MAA," to Presidential Awardees, 27 April 1994.

"Some History of Fermat's Last Theorem," annual meeting of the Maryland-District of Columbia- Virginia Section of the MAA, St. Mary's College of Maryland, April 15-16, 1994, Bullitt Lecture (he was a noted collector of rare mathematics books), University of Louisville, 7 April 1994, and Frostburg State University , MD, 3 February 1994

Pierre de Fermat (1601-1665) attained fame in his day for his creation of analytic geometry, early work on the calculus and on probability, and for pioneering work in number theory --- all without publishing a single paper. The most famous of his annotations in his copy of the Arithmetica of Diophantus and the work to which it led will be discussed in this general lecture. This will provide an opportunity to mention many famous individuals, including Euler, Cauchy, Wiles, and vos Savant.
"Rare Books in the Classroom," Washington Rare Book Group, American University Rare Book Room, 3 March 1994.

"Leibniz and the Creation of his Calculus, " Pennsylvania State University 24 February 1994, Radford University, Radford VA, 6 October 1993, American University, Washington D.C., 28 September 1993, University of Toledo, 11 May 1993, University of Cincinnati, 4 February 1993, and Michigan Section of the MAA, Saganaw MI, May 8, 1992.

Gottfried Wilhelm Leibniz (1646--1716) is well known to all mathematicians as one of the inventors of the calculus. We shall bring his personality to life by discussing his biography, including the individuals, books, and ideas that influenced him, and then sketch his many contributions to various areas of learning. Naturally, we shall concentrate our attention on his mathematics, especially his contributions to the calculus. We shall see how his study of difference sequences led to the idea behind the fundamental theorem of calculus, and how that idea can enrich our classrooms. After discussing the trouble Leibniz had with the product rule, and how that can encourage our students, we shall take a careful look at his ``Nova methodus,'' his new method of maxima, minima, and tangents. By placing his work in a historical setting, we shall see why he truly deserves credit as one of the heroes of mathematics.

Talks given in 1993:

"Applying to graduate school," Coppin State College, Baltimore MD, 4 November 1993.

"Benjamin Franklin Finkel (1865-1947) and the founding of the Monthly," after dinner talk at the fall meeting of the Ohio Section of the MAA, Ohio Northern University, 22 October 1993.

The MAA was founded in 1915, but for many years before The American Mathematical Monthly had been appearing. The first issue was dated 1894 and it was edited by Benjamin Franklin Finkel? But who was Finkel? Happily for the Ohio Section, he was born in Ohio, educated in country schools, and then attended Ohio Normal University, a school which is now called---after his suggestion---Ohio Northern University. Finkel was a problem solver. He published his solutions and problems in numerous periodicals with long forgotten titles, but was always unhappy that they appeared so irregularly. This is the reason he founded the Monthly. Tonight we will take a close look at Finkel and his problems.
"The secret of my teaching success," Vancouver, British Columbia, at the first joint meeting of the Canadian Mathematical Society with the AMS and MAA, 15 August 1993. Presentation made in conjunction with the reception of the first national MAA awards for "Distinguished College or University Teaching of Mathematics."

"Saint Vincent and the Logarithm," AMS Special Session in the History of Mathematics, Washington, DC, 18 April 1993.

The Opus geometricum quadraturae circuli et sectionum coni (1647) of Gregory of Saint Vincent (1584--1667) contains the most magnificent frontispiece in the whole history of mathematics. The main result of the work, that the circle can be squared, was shown to be incorrect by Leibniz. Yet the book contains much of interest, especially a proof that the area under a hyperbola is given by a logarithm. Some of the contents of the work will be discussed. [Abstract published in the AMS Notices.]
"The Fibonacci Sequence and its history" Lafayette College, Easton, PA, 15 April 1993.

"Rare Books in the Classroom," Annual HPM Meeting, Seattle, Washington, 2 April 1993.

"The Cyclotomic Equation and Regular Polygons," Colloquium, United States Military Academy, West Point, NY, 24 March 1993.

Carl Friedrich Gauss (1777-1855) is often credited with proving that we can construct a regular n-gon with straightedge and compass iff n is a product of distinct Fermat primes times a power of two. In fact he only proved the constructive "if" part. The converse is due to Pierre Laurent Wantzel (1814-1848). We shall discuss the history of construction of regular polygons, provide some of the mathematical details, and indicate how this can be used in the classroom.

Talks given in 1992:

"It takes all the running you can do to keep in the same place," Graduation Talk, BGSU, December 19, 1992.

"How Columbus Discovered America," Math Day, University of Findlay, Findlay, OH, March 19, 1992 and to fifth graders at Mount Blanchard Elementary School, Mount Blanchard, OH, 23 October 1992.

"Lesniewski and Tarski on Truth," Lesniewski ajuhdi, Grenoble, France.


 


A full list of talks given is available.


Posted February 2001.          Send comments to Fred-Rickey@usma.edu

Updated March 2004.