# Teaching and Research with Original Sources from the Euler Archive - The Euler Archivists' Top 5 Picks

Author(s):
Dominic Klyve (Central Washington University), Lee Stemkoski (Adelphi University), and Erik Tou (Carthage College)

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.

Dominic Klyve (Central Washington University), Lee Stemkoski (Adelphi University), and Erik Tou (Carthage College), "Teaching and Research with Original Sources from the Euler Archive - The Euler Archivists' Top 5 Picks," Loci (August 2013), DOI:10.4169/loci003672