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The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight.

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# Teaching and Research with Original Sources from the Euler Archive

## The Euler Archivists' Top 5 Picks

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 Euler 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.

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Klyve, Dominic, Lee Stemkoski and Erik Tou, "Teaching and Research with Original Sources from the Euler Archive," Loci (April 2011), DOI: 10.4169/loci003672