Our one-day conference investigates the central role played, in the life and work of Leonhard Euler (1707-1783), by the description, calculation and analysis of sites or places (topoi). Active for many years in Saint Petersburg and Berlin and fluent in at least three languages, Euler was no stranger to changes in location and to the negotiation of different cultural and scientific contexts and their respective rhetorical, cultural, and scientific conventions. However, Euler's more specifically scientific activity in mathematics, astronomy, geometry, engineering and philosophy is also linked, in more ways than one, to the problem of topoi and their notation. Not only is his solution to the problem of the "Seven Bridges of Königsberg" considered one of the early foundations of mathematical topology, he also devoted much energy to ship building and navigation.

CONFERENCE TOPICS:

Brian Hopkins (St. Peter’s College, NJ), “The Beginning of Graph Theory”

Wladimir Velminski ( Humboldt University, Berlin), “ Leonhard Euler’s Strategies of Visualization”

Matt Wickman ( Brigham Young University, UT), “ Burns's Nature, Euler's Path”

Julian Havil (Winchester/UK), “Euler: the Basel Problem, the Polyhedral Formula and 'Proof' ”

John Glaus ( Rumford, Maine), “How Leonhard Euler Balanced the Three Fundamentals of People, Places and the Sciences”

Stacy Langton ( University of San Diego, CA), “ Euler as Popular Science Writer: The Shape of the Earth”

Brian Hopkins

Graph theory is a comparably young area of mathematics where relations of adjacency are arbitrary and standard measures from geometry are irrelevant. Euler discussed two problems now considered to be within the purview of graph theory: the bridges of Königsberg and knight's tours. The primary topic in this talk will be the puzzle about seven bridges that spanned the Pregel River. In addition to exploring Euler's solution to the problem (different from today's treatment), we will consider his correspondence preceding the article: Euler called the problem "banal," so why did he bother solving it? Euler's contribution to the recreational mathematics study of chessboard problems seemed unrelated to the bridges in both setting and solution method, but we will sketch the development of graph theory to sufficient generality that encompasses both topics.

Matt Wickman

I will explore the relationship between Euler’s thought and the Scottish Enlightenment, particularly as it involves the theory of attraction. Specifically, I will discuss an often-overlooked aspect of Euler’s work - his metaphysics (e.g. of ether as a quasi-material, half-spiritual fluid filling the universe), relative to the Scottish Enlightenment theory of sympathy. Some might argue that scholars ever since Daniel Bernoulli have disregarded Euler’s metaphysics with good reason. My aim here is less to rehabilitate Euler’s “ethereal” thought than to indicate its provocative correspondence with Scottish moral philosophy and popular poetry. Ultimately, I will suggest that “attraction” in these contexts distills an atmospheric quality which resembles and partly foreshadows a modern concept of environment.

Julian Havil

I propose to point out the difficulties inherent in the concept of rigorous mathematical proof and indicate what 'proof' meant in the 18th century and most particularly, to Euler. I will show a couple of simple examples of divergent series with which he struggled and move to his famous Basel Problem result, again indicating the difference between the modern view of acceptability of proof and that of Euler et al. I will then move to Polyhedral Formula and reinforce the point; also, I should like to use the formula to establish two great results (each in a couple of lines) as an indication of its power: the first, that there at most 5 Platonic solids and the second a result that is central to any proof of
the Four Colour map theorem. So, a bit of easy but significant mathematics as well as historical and anecdotal stuff.

John Glaus

The names of those Euler wrote to and of those who wrote to him are most of the significant mathematicians and physicists, philosophers and metaphysicians, theologians and apologists, aristocrats and monarchs. His letters place Euler at the scientific epicenter for the half-century spanning 1727-1783. Euler was aware that, irrespective of where the correspondence was going and to whom it was addressed, the content of the letters constituted a focused direction. These letters were not papers but rather documents that provided the initial scope of a range of subjects and interests, and they are to be seen as the continuous integration of scientific, philosophical, pedagogical and religious knowledge in discrete contributions that reveal the continuum of knowledge.

Stacy Langton

I will review the history of the theories concerning the shape of the Earth, beginning with a glance at Aristotle (De Caelo, II.13, 14), who gave various arguments to support the view that the Earth was
spherical. But is it exactly spherical, or only approximately? The modern discussion begins with Newton (Principia, Book III, Prop. 18), who argued that because of the Earth's daily rotation about its axis, it should bulge along the Equator. Meanwhile Jean-Dominique Cassini, later assisted by his son Jacques, had begun to make measurements along the meridian of Paris in order to settle this question. The Cassinis found that the length of one degree of latitude was shorter in the north of France than in the south, a result which suggested that the Earth bulged at the poles, rather than at the Equator. In 1735 the French Academy of Sciences sent an expedition to Peru to measure the length of a degree near the Equator, and in 1736 another expedition, led by Maupertuis, was sent to Lapland to measure the length of a degree near the North Pole. The expedition of Maupertuis, which returned first, reported that the length of a degree in Lapland was greater than the length of a degree in France, thus decisively confirming Newton's prediction.

In 1727 the young Euler arrived in St. Petersburg to take up his position as adjunct in the Petersburg Academy. (He was promoted to fill Bulfinger's position as Professor of Physics in 1731 and took Daniel Bernoulli's position as Professor of Mathematics in 1733.) Between 1728 and 1742 the Academy published a "supplement" to the St. Petersburg Gazette, called "Notes to the Gazette", which contained scientific articles written at a popular level by members of the Academy. It was published simultaneously in Russian and in German. Until 1738 the articles appeared anonymously; thereafter the author was indicated by an initial. The Russian scholar Judith Kopelevich has suggested that some of the early, anonymous articles were probably by Euler, in whole or in part. In 1738, the year following Maupertuis' return from the Lapland expedition, Euler published in this journal an article "On the Shape of the Earth". This was the Euler's only contribution to the "Notes" which was known to Gustav Enestrom, who listed it as item 32 in his catalogue of Euler's works. (Enestrom in fact knew only the Russian version of the article.) J. J. Burckhardt, who edited the German version of this article for Euler's Opera Omnia, describes it as "a small masterpiece of Euler's expository art, [one that] will afford pleasure to every reader".

In this article, Euler explains the scientific issues involved, and reviews the various means by which the true shape of the Earth can be determined. Curiously, Euler adduces as one reason for trusting the results of the Lapland expedition the circumstance that the members of the expedition had previously been strong opponents of the Newtonian view concerning the shape of the Earth. Actually, Maupertuis had become a convinced Newtonian after a visit to London in 1728, and in 1735 had published a memoir "Sur la figure de la terre" supporting the Newtonian view. (In a historical irony, Maupertuis would a few years later become Euler's boss at the Berlin Academy.)

In my talk, I will not try to describe everything Euler does in this article. One thing that I want to look at particularly, however, is Euler's analysis of the experimental errors in the measurements made by the expedition, in order to decide whether the results were sufficiently accurate to settle the question of the shape of the Earth. Euler certainly had a good opinion of the capabilities of his St. Petersburg readers!

Wladimir Velminsky

Based on Euler's journals and his autobiography, I examine Euler's constructions of visualization with a special focus on topological ideas. From the very beginning of his life, Euler was chained to mathematics. Thus, the written contemplation of the past approaches calculatory graphs. While Euler devoted himself to calculus professionally, he occasionally made discoveries. In one case this took the form of "Rösselsprungproblem" - the problem of the chess board with only one knight left.

Contact Sven Spieker at spieker@gss.ucsb.edu.

Presented by the Department of Germanic, Slavic and Semitic Studies
6206 Phelps Hall
University of California
Santa Barbara, CA 93106-4130
805-893-2131 / Fax: 805-893-2374
www.gss.ucsb.edu

Sponsoring departments: College of Letters and Science; Program of Comparative Literature; Department of English; Department of Film Studies; Department of French and Italian; Department of Germanic, Slavic and Semitic Studies; Department of History; Division of Mathematical, Life and Physical Science; Division of Humanities and Fine Arts; Interdisciplinary Humanities Center; Early Modern Center; Office of Research.

Financial assistance for the conference and for the subtitling and translation of Wladimir Velminsky's film was provided by the ThinkSwiss Program. "Think Swiss ­ Brainstorm the future" is a US wide program on Education, Research and Innovation for 2007 and 2008. It focuses on the exchange of expertise and know­how in academia and business community in Switzerland and the US. The program is under the auspices of Presence Switzerland (PRS), the Swiss State Secretariat for Education and Research (SER) and the Swiss Federal Department of Foreign Affairs. We thank the Consulate General of Switzerland, Los Angeles (www.thinkswiss.org).

University of California, Santa Barbara