### 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.