Introduction: Teaching, Research, and Primary Sources

If the number of books, papers, and conference sessions in recent years is any indication, the history of mathematics is growing ever more popular. Mathematicians have long been fascinated by stories of the masters who helped to build our discipline, but in recent years an increasing number of people have been using original sources when teaching and studying the history of mathematics. Original sources here refers to the papers and books in which mathematicians stated conjectures, proved theorems, and generally went about their work. Studying original sources is fun and exciting; seeing the raw first versions of major ideas in mathematics often gives us more insight into the creation of mathematics than does studying re-written, streamlined textbooks.

In general, original sources in the history of mathematics play two important roles. The first of these is fairly classical: collections of original sources are used by scholars in the field to study and understand the work of the original writers. The second role is newer: there has been an increasing effort in recent years to include original sources in the mathematics classroom. In part due to the work of David Pengelley, Janet Barnett, and Reinhard Laubenbacher (see [5], [7], [8]), a growing number of mathematics students are learning their field by reading original sources written by some of the greatest mathematicians of all time. For a particularly exciting example of this work, and for a chance to get involved, see the 16 projects in the Convergence article, Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science or, for even more projects, Teaching Discrete Mathematics via Primary Historical Sources. Both are collaborative efforts by scholars at New Mexico State University, Old Dominion University, and Colorado State University - Pueblo.

Where does one find original sources? Traditionally these have been available only in dusty library stacks, and in source books—single volumes of original sources deemed to be of general interest. More recently, projects such as Google Books are making more of these original sources available online. Google Books, however, is far from being finished with its goal of scanning the world’s books into its database. Additionally, the very size of the database makes it difficult to browse, and the lack of meta-data sometimes makes the works hard to interpret.

If one decides to study original sources in mathematics, one can hardly do better than to read the words of Leonhard Euler, one of the greatest didactic writers in the history of mathematics. More than perhaps any other mathematician, Euler wrote to be understood. His works brim with examples, computations, and even dead-ends in his thinking process—the kind of digression academics are usually taught to avoid in their publications. Unfortunately, most of Euler’s works have never been collected in source books, and only a few are available on sites such as Google Books. It was precisely to correct this large gap in the availability of sources in the history of mathematics that the Euler Archive was created.

This article will introduce the reader to the Euler Archive, give some specific examples of how it has been used in mathematics teaching and research, and suggest some ways that researchers and teachers can take advantage of the site.

Introduction to the Euler Archive

The Euler Archive is dedicated to making the work of Euler available freely and online. At its core are 866 webpages, which we refer to as Eneström pages, or just E-pages, one for each of Euler’s works identified by Gustav Eneström in his catalog of Euler’s works. A sample page is shown here:

Each page gives some basic bibliographic information about the work. Most pages provide a short summary, and most of them (832 of the 866 pages, as of April 2011) provide a link to download a scan of the original work. If applicable, there are also links to translation(s) of the work, and references to modern scholarship that refer to it.

These E-pages can be accessed in a variety of ways. Users can browse the archive by subject, by the date the paper was written (or published), by the place where the paper was published, or by its “Eneström number.” For those interested in the details of Euler’s life, we also provide many historical pages, about the societies for which he worked, the journals in which he published, and the cities in which he lived. For more details about the history of the Euler Archive, see [6].

How the Archive Has Been Used

Over the last few years, the directors of the Euler Archive have been thrilled by the many creative uses to which it’s been put. Most obviously, perhaps, it has been useful to people already interested in Euler’s work – for these researchers, the Euler Archive simply makes easier the work they were already doing. Much more exciting for us is the research that was done because of the existence of the Archive.

Translations, Commentaries, Research Papers

The most obvious examples of research done because the Archive was available are the many translations of Euler that have been made over the last five years. More than 100 of Euler’s papers and books have been translated just since 2002. Many (though certainly not all) were done because the Euler Archive provided both the original sources and a forum in which to publish the translation when it was finished. A related example of scholarship is the growing number of detailed commentaries and summaries that have appeared. At Rowan University, students and faculty have made the preparation of these commentaries a significant part of their undergraduate research program. Readers interested in preparing such commentaries are encouraged to see some of the excellent examples available, listed here with their Eneström numbers:

●      E314: Conjecture into the reasons for some dissonances generally heard in music

●      E352: Remarks on a beautiful relation between direct as well as reciprocal power series

●      E745: On the continued fractions of Wallis

●      E796: Research into the problem of three square numbers such that the sum of any two less the third one provides a square number

We’ve been more surprised by the amount of research mathematics that has come out of the recent renewal in the study of Euler’s works. Eight of the examples we know of are given in our Bibliography (see [2], [3], [9] – [13], [18]), and there are many more. One of the especially exciting aspects of this research is that only a very small portion of Euler’s works have been studied by modern scholars. We have every reason to believe that many more gems—springboards to exciting work in both the history of mathematics and modern mathematical research—are waiting to be discovered.

Classroom Projects

Teaching from primary sources is a common approach in the humanities and social sciences, where "Great Books" programs and curricula are well-known. Many of the benefits of reading original sources are universal, applying to mathematics as well as other disciplines. One major benefit is that the act of interpreting and distilling results originally presented in a non-textbook style, and phrasing them in modern terms, can lead to better understanding; the extra time and effort required to do this may also help students retain the knowledge longer. Reading primary sources also personalizes the development of ideas, thus inspiring students with the knowledge that they also have the ability to create new results. In addition, Pivkina notes that using original sources may encourage students to be more engaged with the material, as "original sources by their nature invite questions while textbooks usually do not" [15].

When reading original sources in mathematics, the works of Leonhard Euler are particularly suitable, as he

• wrote to be understood (rather than to impress),
• did not assume extensive background,
• used lots of examples, and
• gradually built ideas upon one other.

Almost all of Euler’s works are written at or below the level of the average junior undergraduate mathematics major. A great many are accessible to first-year students. Essentially all are available on the Euler Archive. The papers that are the most accessible and written at the lowest level mathematically are those most likely to be translated already. We strongly urge anyone interested in using these sources in the classroom to peruse the list of translated papers. Something is likely to catch your interest!

For examples of ways to use Euler's original works in the classroom, see [1] and [17]. Even better, write your own, and let us know! The Euler Archive is happy to consider publishing new projects which make use of the resources we make available.

Undergraduate Research – Translations

Working on a translation of a paper from the history of mathematics is a wonderful basis for an undergraduate research project. In doing so, the student is forced to look closely at work done by a world-class mathematician and understand it on a sentence-by-sentence level. In addition to the excitement generated by such an endeavor, the benefits to the student of carefully engaging with mathematics can’t be overstated.

Of course there is one big limitation to such work; the student must have facility in a foreign language. In the case of Euler, translation almost always requires Latin or French, although a few of his German papers also remain untranslated. However, experience has shown that in a student-professor collaboration, it is not necessary for the professor to have any knowledge of the source language.

A very nice model of student-professor translation has been developed by Thomas Osler of Rowan University. Using his model, many other groups have had considerable success. Osler has written a nice summary of his methods, and the interested reader is encouraged to see his Experience Translating Euler’s Papers in the Euler Archive.

Another model worth noting demonstrates the broad range of students for whom translation projects can be appropriate. In 2004, Dartmouth undergraduate student Greta Perl worked with Dominic Klyve to translate into English the German Eneström Index. As was mentioned above, this is the definitive list of all of Euler’s works, giving each a unique number in (roughly) the order in which they were published. As he put together the list, however, Eneström added valuable information about translations, reprintings, and notes from academy records describing when Euler first publicly presented each work. While the Index is not in any way mathematical, it is an invaluable resource for Euler scholars. A number of other non-technical (but useful) documents remain untranslated, and could be used as the foundation for a very interesting interdisciplinary collaboration.

Undergraduate Research – Mathematics

While translation projects are the most obvious avenue for undergraduate research, the mathematical and scientific content can be a fertile ground for more in-depth analysis. Once a paper or book of Euler's has been made available in English (or any modern language), several important questions come immediately to the fore: Is Euler's mathematics correct? Did his work predate anyone else's? How does his terminology and notation compare to his contemporaries? These questions can lead easily into a more comprehensive appraisal of Euler's works, as such a research project marries the mathematics with the historical context in a way that contributes to today's body of knowledge.

An even broader question may be asked here: How did Euler's work fit into the general milieux of the times, and how did that historical moment impact later understandings of the topic? This type of research has been done by several people. One example is Bruce Petrie's research on the emergence of mathematical transcendence—a comparative analysis involving the works of Euler, Lambert, Liouville, and others [14]. A similar research program was undertaken by Sandro Caparrini, in which he examined the works of Euler in his chronicling of the origins of vector calculus [4].

While we are currently unaware of an undergraduate research project of this form, a good model for this type of research is Ed Sandifer’s celebrated MAA column, How Euler Did It [16]. Most columns begin by summarizing a paper of Euler’s, and then expand to include the wider history of the topic. A good example is his February 2009 column on the estimation of \(\pi\). After explaining the methods Euler used in his E705 paper, Sandifer continued the narrative by connecting it to another paper of Euler’s (E706) and to the work of Slovenian mathematician Jurij Vega. This type of project can easily begin with a translation, but this need not be the case. At any rate, the majority of the project’s time would involve historical research—finding contemporary sources, both original and secondary, and comparing their mathematical content. While this type of project diverges somewhat from the pure mathematics that many readers may find most familiar, the experience of historical research provides valuable skills for student and professor alike.

The main limiting factor in digitizing Euler's works is the sheer volume of material. Eneström's index lists 866 distinct works, a number which includes 31 books and over 30,000 pages of material. What's more, this count does not include the more than one thousand letters written by Euler or the nearly two thousand received! Naturally, any attempt at archiving and organizing this material will require the user to make use of many different search features.

There are several different facets of any work of Euler's: the form in which it appeared (book, article, pamphlet, or private letter), the places it was published (Berlin or St. Petersburg, academic or private journal), the date it was produced or published (often very different dates), or the subject matter it contains (anything ranging from mathematics to music theory to philosophy). Taken together, these data comprise what might rightly be called a "fingerprint" for any of Euler's works. To clear through this thicket, the Euler Archive begins with a discrete list of categories that any user may employ in their search. These are:

Subject: Are you most interested in Euler as a mathematician? His mathematical papers and books are subdivided by discipline, including topics such as number theory, differential equations, and infinite series. Are you interested in Euler's works in physics and astronomy? These works are also subdivided, into topics ranging from planetary motion to optics to mechanics. A sample of this list is shown below:

Source type: Are you looking for a paper? If so, you may search first for the journal in which it appeared. Or, you can find a list of Euler's books. Was it a letter? We have a list of Euler's correspondence. In the case of journal articles, the reader will find information about the various incarnations of these journals, which changed names along with the political turbulence of the times:

Date of publication: When did the work appear in print? What other works of Euler appeared near the same time? Where was Euler working and living at the time?

Date of presentation: If you're looking for a paper: when did Euler present it to the Academy for peer review? Often, the publication date is many years after the date of presentation (in some cases as many as 50 years!), so this is an important datum to have.

Index Number: Eneström's index of Euler's works assigns a number to 866 of Euler's books and articles. For those who are well-versed in Euler scholarship, this can provide a shortcut to the work in question.

Translation: Which of Euler's works have been translated into English? Are there commentaries for these works? The Euler Archive maintains a list of known and pending translations of Euler’s works. Those interested in attempting their own translation can “sign up” on the Archive, to minimize unnecessary duplications. Those interested in analyzing a work that doesn’t require a translation may explore this listing of over 150 papers and books.

Clearly, the wide range of available searches offers any user the ability to quickly find a particular work, and also identify that work's fingerprint. However, the Archive can be used for casual browsing as well. Any user interested in finding Euler's original work can simply choose the subject material that she finds appealing, or the type of publication that is most amenable to her needs (letters, books, or articles).

For those who want more context for Euler’s works, the Euler Archive also provides a series of short, historical vignettes on topics ranging from Euler’s life to his contemporaries to histories of the Berlin and St. Petersburg Academies. Each of these articles provides a list of sources for further reading.

As with any large collection, the collectors are bound to have some favorites. So too with the staff of the Euler Archive. Having spent several years crafting and growing the Archive, there are a handful of sources that we feel are particularly important and/or fascinating. We present five of these as our “top picks.”

#1  On the sums of series of reciprocals [E41]

In this paper, Euler solved the “Basel Problem”: finding the sum of the series \[{\sum_{n=1}^{\infty}}{\frac{1}{n^2}},\] which he showed to be \(\frac{{\pi}^2}{6}\). He went on to calculate the exact value of this series with the exponent \(2\) replaced by each of \(4,6,8,10,\) and \(12\).

#2  The solution of a problem relating to the geometry of position [E53]

This paper contains Euler’s solution to the Königsberg Bridge problem, in which he showed that no route exists that crosses each of the Königsberg bridges exactly once.

#3  Elements of the doctrine of solids [E230, E231]

In these two papers, Euler established his famous theorem for polyhedra, in which the number of vertices plus the number of faces exceeds the number of edges by two: \(V+F=E+2.\)

#4  Investigations on a new type of magic square [E530]

In this 100-page paper, Euler first described then systematically studied many aspects of Graeco-Latin squares. In the end, he tried and failed to prove that a Graeco-Latin square of order \(6\) (or any order of the form \(4k+2\)) cannot exist, and he settled for a plausibility argument.

#5  Letters to a German Princess [E343]

Over the course of 234 letters, Euler laid out for his student (the German princess) the fundamentals of motion, mechanics, sound, electricity, music, philosophy, and the nature of evil. Along the way, he gave explanations for everything from why the moon looks larger near the horizon to the cause of tides. A pleasure to read from cover to cover, or to dip into at random.

Conclusion

We conclude this survey with a call to action. As evidenced above, Euler’s papers are numerous, varied, important, and largely untranslated. We continue to believe strongly that the act of translating Euler’s works is beneficial for both the translator and for the mathematical community as a whole. An announcement of a new goal was recently made at a meeting of the Euler Society:

the complete translation of Euler’s works into English by 2033, the 250th anniversary of his death.

To accomplish this objective, we need to produce about 32 new translations per year for the next 22 years. Clearly this goal cannot be achieved without the efforts of many people working together, and the Euler Archive stands ready to serve as a guide, a resource, a publication venue, and a source of encouragement for those willing to join us in this great work. Share this goal with your colleagues and students, choose an article, and join the effort!

About the Authors

Dominic Klyve and Lee Stemkoski founded the Euler Archive in 2002 while they were graduate students at Dartmouth College. Klyve earned his Ph.D. in 2007 and is now an assistant professor of mathematics at Central Washington University in Ellensburg, Washington. Stemkoski earned his doctorate in 2006 and is assistant professor of mathematics at Adelphi University in Garden City, New York. Erik Tou, assistant professor of mathematics at Carthage College in Kenosha, Wisconsin, assisted with the Archive while a graduate student at Dartmouth and is now Chief Historian for the Euler Archive. He earned the Ph.D. from Dartmouth College in 2007.

Bibliography

[1] Barnett, Janet. "Euler Circuits and the Königsberg Bridge Problem." Available online as Project #14 at http://www.math.nmsu.edu/hist_projects/#Quick

[2] Blecksmith, Richard, John Brillhart, and Michael Decaro. “The Completion of Euler’s Factoring Formula.” To appear in the Rocky Mountain Journal of Mathematics.

[3] Brillhart, John. “A Note on Euler's Factoring Problem.” American Mathematical Monthly 116:10 (2009), pp. 928-931.

[4] Caparrini, Sandro. “Euler’s Influence on the Birth of Vector Mechanics.” In Leonhard Euler: Life, Work and Legacy. Robert E. Bradley and Ed Sandifer (eds.), Elsevier, 2007, pp. 459-477.

[5] Hopkins, Brian. Resources for Teaching Discrete Mathematics. Mathematical Association of America, 2008.

[6] Klyve, Dominic and Lee Stemkoski. "The Euler Archive: Giving Euler to the World." In Euler at 300: An Appreciation. Robert Bradley, Lawrence D’Antonio, and Edward Sandifer (eds.), Mathematical Association of America, 2007, pp. 33-41.

[7] Knoebel, Art, Reinhard Laubenbacher, Jerry Lodder, and David Pengelley. Mathematical Masterpieces: Further Chronicles by the Explorers. Springer, 2007.

[8] Laubenbacher, Reinhard and David Pengelley. Mathematical Expeditions: Chronicles by the Explorers. Springer, 1998.

[9] Osler, Thomas. “Another look at Euler’s parallel oblique angled diameters.” To appear in The Mathematical Gazette.

[10] Osler, Thomas. “Euler and the functional equation for the zeta function.” The Mathematical Scientist, 34 (2009), pp. 62-73.

[11] Osler, Thomas. “Euler's little summation formula and special values of the zeta function.” The Mathematical Gazette, 92 (2008), pp. 295-299.

[12] Osler, Thomas and Steve Donahue. “Euler’s method of integration by parts.” To appear in The Mathematical Gazette.

[13] Osler, Thomas and Andrew Robertson. “Euler's little summation formula and sums of powers,” Mathematical Spectrum, 40 (2006/2007), pp. 73-76.

[14] Petrie, Bruce J. "Euler, Lambert, and the Irrationality of \(e\) and \(\pi\)." Proceedings of the Canadian Society for History and Philosophy of Mathematics, 22 (2009), pp. 104-19.

[15] Pivkina, Inna. "Original historical sources in data structures and algorithms courses." Journal of Computing Sciences in Colleges, 26:4, April 2011.

[16] Sandifer, C. Edward. How Euler Did It (online column). Mathematical Association of America, 2007. Past columns available online at: http://eulerarchive.maa.org/hedi/index.html

[17] Stemkoski, Lee. "Investigating Euler's Polyhedral Formula Using Original Sources." Loci: Convergence, 6 (April 2009). DOI: 10.4169/loci003297. Available online at:
http://www.maa.org/publications/periodicals/convergence/investigating-eulers-polyhedral-formula-using-original-sources-original-sources-who-what-where-how

[18] Walter, Jacob and Thomas Osler. "A modern look at a neglected summation formula by Euler." The Mathematical Gazette, 93 (2009), pp. 237-243.