Talks Given Since Retiring

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.

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. 

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 (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. 

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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:

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 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.

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.

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.

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.  

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.

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.

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.