"Coming Soon to a Library Near You: The NEW Euler-Goldbach Correspondence," Euler 2014. The annual meeting of the Euler Society, St. Edward's University, July 21-23, 2014.

While Euler and Goldbach corresponded
over a period of 35 years, vey few of their letters have been
translated into English. This is about to change with the publication
of volume IVA4 of Euler's Opera Omnia, edited by Franz Lemmermeyer and Martin Mattmüller.

The biggest change from Fuss's 1843 edition of the correspondence is that now the opening and closing portions of the letters are included. They don't deal with mathematics, but they do contain a great deal of interesting information about the two men and their circle. While this information was included in the 1965 edition by Ju\vskevi\vc and Winter, it is not well known. We shall discuss some of it.

This presentation will deal primarily with the rich ideas in the correspondence and attempt to convey the excitement of mathematics being created that is so obvious when one reads through their correspondence.

The biggest change from Fuss's 1843 edition of the correspondence is that now the opening and closing portions of the letters are included. They don't deal with mathematics, but they do contain a great deal of interesting information about the two men and their circle. While this information was included in the 1965 edition by Ju\vskevi\vc and Winter, it is not well known. We shall discuss some of it.

This presentation will deal primarily with the rich ideas in the correspondence and attempt to convey the excitement of mathematics being created that is so obvious when one reads through their correspondence.

"Reassembling Humpty Dumpty Again: Putting George Washington's Cyphering Books Back Together Again," CSHPM, Brock University, May 25, 2014. Joint work with Ted Crackel and Joel Silverstein.

Soon after we began the study of George
Washington's cyphering books we realized that some of the pages were
out of order and that others were missing entirely. We shall describe
some of the detective work that helped us in locating sources for
Washington's mathematics and finding a few missing leaves. Along the
way we shall describe some of the mathematics in the cyphering books,
especially things that are not easily understood by the modern
mathematician.

"Washington and Mathematics,"PASHOM at Villanova, 20 March 2014.

There are many interesting things in
the cyphering books that George Washington compiled as a teenager: His
study of decimal arithmetic is straightforward, but understanding some
of the errors he made can be a fun. He had a technique for
partitioning a plot of land into two equal pieces, but it was wrong ---
and so was his source. Some things are hard to understand for time has
passed them by. For example, his pre-Eulerian trigonometry is a mystery
today, so we shall elucidate it. But some of his work shows that his
sources knew some calculus; a formula for gauging casks relies on
Simpson's rule and a standard surveying technique uses numerical
integration. We shall present one page of his cyphering books which the
Library of Congress neglected to digitize and another that ended up at
Dartmouth.

"The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof," St. Joseph's University, 19 March 2014.

The Fundamental Theorem of Calculus
(FTC) was a theorem with Newton and Leibniz, a triviality with
Bernoulli and Euler, and only took on the meaning of "fundamental" when
Cauchy and Riemann defined the integral. FTC became part of
research 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. Most importantly, we shall explain why both parts
of the FTC are vital.

“Machin’s Formula for Pi,” La Salle University, Philadelphia, March 18, 2014.

This is a story that began in my
Analysis class. One exercise was to prove Machin’s formula. The proof
was not too hard, but it gave no insight into where the formula came
from. So off to the library I went to explore the origins of this
amazing formula. Surprisingly, every reference I wanted was in the West
Point library. What was discovered, was that 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, show how it has been used to
compute π, and discuss its interesting history and ramifications.

"The Notebooks of George Washington on Arithmetic, Geometry, Trigonometry, Logarithms and Surveying," Joint AMS/MAA annual meeting, Baltimore, January 2014. HOM SIGMAA guest speaker.

Just as Grant Wood portrayed Parson
Weems pulling the curtain back on the life of George Washington, we
shall illuminate Washington's mathematical education. We are blessed
with 179 pages of handwritten manuscript in Washington's youthful hand
and while this material has frequently been mentioned by scholars, it
has never been analyzed; we shall present an abundance of detail.
Little is known about Washington's youth, so these papers provide a way
of learning about his education.

The individuals and organizations that have controlled these papers have organized and reorganized them into disorder. Is it reasonable for a thirteen-year-old --- living in plantation Virginia where there were few schools --- to begin his mathematical education with the study of formal geometry? Could he have learned surveying before studying arithmetic and trigonometry? Using physical evidence, handwriting analysis, and mathematical context, we shall present our conjectured order of the pages of the manuscript.

This is a case study in mid-eighteenth century mathematical education in the American Colonies. We shall contrast the surprising depth of his theoretical education --- including logarithms and trigonometry --- with his practical use of mathematics as a field surveyor.

After serving two terms as President, Washington took pains to preserve his papers for he believed that someday they might be "of interest.'' You will be the judge of how interesting these papers are.

The individuals and organizations that have controlled these papers have organized and reorganized them into disorder. Is it reasonable for a thirteen-year-old --- living in plantation Virginia where there were few schools --- to begin his mathematical education with the study of formal geometry? Could he have learned surveying before studying arithmetic and trigonometry? Using physical evidence, handwriting analysis, and mathematical context, we shall present our conjectured order of the pages of the manuscript.

This is a case study in mid-eighteenth century mathematical education in the American Colonies. We shall contrast the surprising depth of his theoretical education --- including logarithms and trigonometry --- with his practical use of mathematics as a field surveyor.

After serving two terms as President, Washington took pains to preserve his papers for he believed that someday they might be "of interest.'' You will be the judge of how interesting these papers are.

"Reading, Writing and Doing the History of Mathematics: Learning the Methods of Historical Research," MAA Short Course, Baltimore, January 2014. Organized by Amy Shell-Gellasch. As one of the six speakers on this two day program I will discuss Historical documents and sources and also pedagogy.

"Pi, Trigonometry, and the Calendar: A Glimpse at Euler's Mathematics," December 6, 2013. SUNY / Ulster.

Leonhard Euler (1707-1783) was the most
prolific mathematician of the eighteenth-century and his work has
influenced the way we do and the way we teach mathematics. Euler
changed mathematics by dealing with functions, rather than curves and
introduced many symbols we use today, including that for pi. Although
trigonometry is an ancient subject, Euler changed it significantly when
he created the trigonometric functions and this led to important new
areas of mathematics such as set theory. Finally, his work on continued
fractions explains why our calendar is the way it is.

"The Cyphering Books of George Washington," Pohle Colloquium, Adelphi University, Wednesday, November 6, 2013.

Between the ages of 13 and 15, George
Washington (1732--1799) compiled two cyphering books consisting of 179
manuscript pages. These give a fascinating insight into the education
of a young man who was preparing to be a surveyor. But what did a
mid-eighteenth century surveyor need to know? Measuring angles and
distances was crucial. Of course, area was what the landholder cared
about. Trigonometry was sufficient for computing area, but was it
essential? We will describe the content of these cyphering books
in detail and argue that Washington had not just the mathematical tools
to be a successful surveyor, but an outstanding technical education for
his day.

"George Washington's Use of Trigonometry and Logarithms," CSHPM Meeting at MathFest, 1-3 August 2013. Hartford, CT. Joint work with Theodore Crackel and Joel Silvergerg.

You will probably remember from your
grade school education that George Washington spent several of his
youthful years as a professional surveyor. But how much mathematics did
he know and how did he use it as a surveyor? Thanks to two "cyphering
books" he compiled as a teenager, we are able to show what he learned
of trigonometry and surveying. The combined use of these subjects is
very perplexing to the modern reader, so we shall illustrate and
explain the methods he used. Finally, in contrast to what one would
expect, we will argue that he did not use trigonometry in surveying.

"The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof," Saint Louis University, 13 February 2013; Wooster College, 21 February 2013; Youngstown State University, 22 February 2013, West Point, 20 August 2013.

Special Session on the History of Mathematics, AMS Regional Meeting, Boston College, April 6-7, 2013. Co-organized with James Tattersall.

"What mathematics did George Washington know before he became a professional surveyor and how did he use it," Special sessions on History of Mathematics at the Joint AMS-MAA Meetings, San Diego, California, 11 January 2013, 1:00-5:50 PM Organized by Patti Hunter, Deborah Kent, and Adrian Rice. Joint work with Theodore J. Crackel, Editor-in-Chief Emeritus, The Papers of George Washington.

Between the ages of 13 and 15, George
Washington compiled two cyphering books consisting of 180 manuscript
pages. Whether he learned mathematics from tutors, from teachers, from
his half-brother Austin, or on his own from various books, we do know
that he mastered a good deal of arithmetic, geometry, surveying and
other material. We will describe in detail what he learned and show how
he used it --- or did not use it --- in the youthful surveys in his
cyphering books.

SIGMAA Panel discussion: Using mathematical archives and special collections for research and teaching, Joint Mathematics Meetings, San Diego, 11 January 2013, 9:00-10:55 AM. The panel also includes Victor Katz (UDC emeritus), Carol Mead (Archives of American Mathematics), Dominic Klyve (Central Washington University), Peggy Kidwell (Smithsonian) and Shirley Gray (National Curve Bank). Organized by Janet Beery and Amy Shell-Gellasch.

"The History and Impact of IHMT, The Institute in the History of Mathematics and its Use in Teaching," Joint Mathematics Meetings, San Diego, Wednesday January 9, 2013, 8:00 a.m.-11:00 a.m. MAA Session on Writing the History of the MAA. Joint work with Victor J. Katz, Professor Emeritus, University of the District of Columbia.

An NSF proposal for an "Institute in
the History of Mathematics and Its Use in Teaching" was submitted
under the auspices of the MAA, where the co-directors were visiting
mathematicians in 1993-94 and 1994-95 respectively. The
Institute strove to aid college faculty in teaching courses in the
history of mathematics and in using the history of mathematics in the
classroom to motivate students. Forty participants came for three weeks
during the summers of 1995 and 1996; an additional forty attended in
1996 and 1997. A new group of forty attended for two weeks
in the summers of 1998 and 1999. The faculty for this Institute
consisted of distinguished historians of mathematics. While the
institute was designed to improve the teaching of the history of
mathematics and show faculty how to use history in teaching mathematics
classes, there was one result that surprised the co-directors: A
considerable number of the participants became so interested that they
started doing research in the history of mathematics. We will describe
the structure, content, and results of IHMT and discuss its substantial
impact, both on the careers of the individuals involved and on the
teaching and use of the history of mathematics in colleges and
universities throughout the country.

"How George Washington prepared to be a surveyor: A study in mid-eighteenth century mathematics education," History and Pedagogy of Mathematics Meeting, University of California, Berkeley, 26-28 October 2012. Joint work with Theodore J. Crackel, Editor-in-Chief Emeritus, The Papers of George Washington.

What did a mid-eighteenth century
surveyor need to know? Measuring angles and distances were crucial. Of
course, area was what the landholder cared about. Trigonometry
was sufficient for computing area, but was it essential? Based on
the cyphering books that George Washington prepared as a teenager, we
will argue that he had not just the mathematical tools to be a
successful surveyor, but an outstanding technical education for his
day.

“The Education of George Washington,” Associated Colleges of the Chicago Area (ACCA), North Park University, 24 October 2012 (after dinner). Joint work with Theodore J. Crackel, Editor-in-Chief Emeritus, The Papers of George Washington.

It is unclear how much formal education
George Washington received: among the possibilities are an ABC school
operated by one of his father’s tenants, some tutoring by a transported
convict, and few years in a school operated by the Reverend James Marye
in Fredricksburg, but the details in each case are sketchy. In
addition, after his father died in 1843, his mother sent him to live
with his half-brother Austin who had very recently been an assistant
teacher at Appleby Grammar School in the north of England. Fortunately,
from these early teenage years, 13 to 15, two cyphering books survive
and from them we know what he learned of geometry, trigonometry,
surveying, and other subjects. This education prepared him for a good
job as a surveyor.

"The Palimpsest of Archimedes,” Associated Colleges of the Chicago Area (ACCA), North Park University, 24 October 2012 (before dinner).

In 1998 a palimpsested manuscript
containing several works of Archimedes was sold at auction. Soon the
owner deposited it at the Walters Art Museum in Baltimore for
conservation, imaging, and scholarship. The first two phases of this
project are now complete, resulting in an award winning popular book
(The Archimedes Codex, by Reviel Netz and William Noel), a magnificent
exhibition at the Walters (which, sadly, is now over), scores of
scientific papers, and a two volume (with three to come) scholarly set
dealing with the manuscript. This talk will be accessible to all and of
interest to librarians, philosophers, art historians, chemists,
photographers, and, of course, everyone interested in mathematics.

"Did George Washington know more mathematics than anyone else in Colonial America in 1750?" Joint work with Theodore J. Crackel, Editor-in-Chief Emeritus, The Papers of George Washington. Ohio Section of the MAA, Baldwin-Wallace College, 19-20 October 2012. Also, colloquium, Bowling Green State University, 22 October 2012.

How can one possibly answer this
question? But I shall try. The cyphering books that George Washington
compiled between 1745 and 1748 when he was between ages 13 and 15
provide detailed information about what mathematics he had learned:
arithmetic through square roots, geometry, trigonometry, logarithms,
and surveying. But what mathematics did others know at the time,
including college graduates, their professors, and others educated in
Europe. We shall shed light on these questions.

"Historical Notes for the Calculus Teacher," Calculus Seminar, Bowling Green State University, Monday 24 September 2012. The "sun day" is 25 October (If it is possible to go sailing on the 24th, the seminar won't be that day). slides.pdf.

After some introductory material (11
slides), we will present examples of historical topics that have been
used successfully in the classroom: The AM-GM Inequality (2 slides),
The Intermediate Value Theorem (12 slides), The Clepsydra (8 slides),
The Tractrix (5 slides), The Calculus of the trigonometric functions
(23 slides), The product rule (7 slides), and L’Hospital’s rule (5
slides).

"The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof,” Bucknell University, 6 September 2012. slides.pdf.

"Rare Mathematics Books at Columbia University," A show-and-tell for a history of mathematics class at New York University conducted by Florence Fasanelli, May 24, 2012.

“The Education of George Washington,” AMS Special Session on Relations between History and Pedagogy of Mathematics, organized by David L. Roberts and Kathleen M. Clark, George Washington University, Washington, DC, March 17-18, 2012 (Saturday - Sunday).

At a meeting at George Washington
University, what could be more fitting than talking about George
Washington. It is unclear how much education he received: among the
possibilities are an ABC school operated by one of his father’s
tenants, some tutoring by a transported convict, and few years in a
school operated by the Reverend James Marye in Fredricksburg, but the
details in each case are sketchy. In addition, after his father
died in 1843, his mother sent him to live with his half-brother Austin
who had very recently been an assistant teacher at Appleby Grammar
School in the north of England. Fortunately, from these early teenage
years, 13 to 15, two copybooks survive and from them we know what he
learned of algebra, trigonometry, surveying, and other subjects. The
copybooks will be discussed in detail.

"Machin's Formula for Computing Pi," Randolph-Macon College, Ashland, VA, 22 February 2001.

The Calculus Network, Richmond, Virginia, 21 February 2012. Here are the topics for the day:

February 21 in Mathematical History.

(1). "The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof.”

The Fundamental Theorem of Calculus
(FTC) was a theorem with Newton and Leibniz, a triviality with
Bernoulli and Euler, and only took on the meaning of "fundamental" when
Cauchy and Riemann defined the integral. FTC became part of
research 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. Most importantly, we shall explain why both parts
of the FTC are vital. Slides.

(2). "L’Hospital’s Rule.”The plan here is to read the short
passage from the first calculus book Analyse des Infiniment Petits Pour
l’intelligence des lignes courbes (1696) where L’Hospitals rule is
introduced. We will take turns reading the text, in French or English
as you wish, and discussing what we have read. By “reading the masters”
you will see that there is a simple intuition underlying L’Hospital’s
rule, something that is lost in modern textbook presentations. We will
also discuss the history of the rule.

(3). "Using History in Teaching Calculus: Some Examples.”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
present several tested classroom such as: completing the square,
trigonometry from Archimedes to Euler, Bolzano and the intermediate
value theorem, and the brachistochrone. Hopefully, there will be time
for lots of questions during the day, so bring yours along. Here are
some files:

History of the brachistochrone. How to build a brachistochrone. "The Palimpsest of Archimedes,” a presentation to the new inductees of Mu Alpha Theta, the mathematics honorary society at Maggie L. Walker Governor’s School, Richmond, VA, 21 February 2012.

In 1998 a palimpsested manuscript
containing several works of Archimedes was sold at auction. Soon the
owner deposited it at the Walters Art Museum in Baltimore for
conservation, imaging, and scholarship. The first two phases of this
project are now complete, resulting in a popular book (The Archimedes
Codex, by Reviel Netz and William Noel), a magnificent exhibition at
the Walters (which, sadly, is now over), scores of scientific papers,
and a two volume (with three to come) scholarly set dealing with the
manuscript. This talk will be accessible to all and of interest to
librarians, philosophers, art historians, chemists, photographers, and,
of course, everyone interested in mathematics.

"The Clepsydra," Virginia Commonwealth University, Richmond, VA, 20 February 2012.

The clepsydra is an ancient device for
measuring time. But what is the proper shape for the container 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. Slides in .pdf format.

“Using History in Teaching Algebra, Geometry, and Trigonometry,” Professional Development Day, Maggie L. Walker Governor’s School, Richmond, VA, 20 February 2012. Sadly, this presentation never took place as school was canceled due to weather.

We shall begin with a discussion of the
advantages of using the history of mathematics not only to motivate
students but also to help them learn mathematics. Then we shall do
examples to illustrate how history can be used. These include using
calculi to understand factoring and primes, geometric algebra, why we
use slope for m, and some history of solving equations and also
trigonometry.

February 20 in Mathematical History.

“The Archimedes Palimpsest,” Southern Connecticut State University, New Haven, CT, 10 February 2012.

"Polish Logic from Warsaw to Dublin: The Life and Work of Jan Lukasiewicz,” Widener University, Chester, PA, 20 January 2012. PowerPoint. Text.

"The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof,” Northeastern Section of the MAA, Connecticut College, New London, CT, November 18, 2011.

“Machin’s Formula for Pi,” Jefferson Library, USMA, 17 November 2011. Here is: a list of books that we will see in the rare book room, a full text dealing with this material, and the slides I will show you.

This is a story that began in my
Analysis class. One exercise was to prove Machin’s formula. The proof
was not too hard, but it gave insight into where the formula came from.
So off to the library I went to explore the origins of this amazing
formula. Surprisingly, every reference I wanted was right here at West
Point. What was discovered, and the books where it was found, will be
revealed.

"The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof,” Calculus Seminar, Bowling Green State University, 24 October 2011. Slides.ppt. Slides.pdf.

"Why do we use “slope” for"m"?" Meeting of the Ohio Section of the MAA, University of Findlay, 21-22 October 2011. Slides.pdf.

Wait,Wait . . . Don’t Tell Me! Before I
do, I will explain why your answer to this perennial question about “m”
and “slope’ is wrong. Searching for “firsts” is a waste of a historians
time (more than 20 years in my case), but we have persisted. We have
not found any instances of this usage in the eighteenth century but
“slope” is defined in the Mathematical Dictionary and Cyclopedia of
Mathematical Science (1855) by Charles Davies, one of my precursors at
West Point, and his son-in-law William G. Peck. 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.

"The impact of ballistics on mathematics. The work of Robins and Euler in the eighteenth-century," Colloquium, BGSU, 20 October 2011.

In the first half of the 18th century,
Benjamin Robins, a British mathematician and military engineer,
invented the ballistic pendulum. This device allowed for fairly
accurate estimates of the muzzle velocities of muskets and other
artillery. Through this experimental work he discovered that air
resistance should not be neglected. In 1742 he published these results
in New Principles of Gunnery, the first book to deal extensively with
external ballistics. This book motivated a deeper analysis of
projectile motion — a topic tackled by Leonhard Euler and Daniel
Bernoulli. Subsequently, Frederick the Great encouraged Euler to
translate this work of Robins. Euler, true to form, tripled the length
of the work with his annotations and published them in 1745. The
annotated text was translated back into English in 1777, which, two and
a half centuries later, brings us to our theme here.

"Polish Logic from Warsaw to Dublin: The Life and Work of Jan Lukasiewicz,” PASHOM, 15 September 2011. 11-09-15-Dublin-Lukasiewicz-PASHOM.key. 11-09-15-Dublin-Lukasiewicz-PASHOM.pptt.

A few years after earning his Ph.D. in
Lwow under Twardowski, Jan Lukasiewicz (1878-1956) joined the faculty
of the newly reopened University of Warsaw where he became, along with
Lesniewski and his student Tarski, one of the founders of the Warsaw
School of Logic. He did seminal research in many-valued logics,
propositional calculi, modal logic, and the history of logic,
especially concerning Aristotle's syllogistic. He left Warsaw toward
the end of World War II and found a new home at the Royal Irish Academy
in Dublin where he continued his creative work.

"The Impact of Ballistics on Mathematics," annual meeting of The Euler Society, Carthage College, Kenosha, WI, 27 July 2011. This is joint work with Shawn McMurran 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.

"The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof,” seminar for new faculty, USMA, 20 July 2011.

"Polish Logic from Warsaw to Dublin: The Life and Work of Jan Lukasiewicz," Plenary speaker at the joint meeting of the Canadian Society for the History and Philosophy of Mathematics and the British Society for the History of Mathematics, Dublin, Ireland, 15-17 July 2011.