Several weeks ago, links to a survey article by Jeffrey Lagarias about Euler’s work and its modern developments and a blog post by Richard J. Lipton that discusses Lagarias’ paper were circulated on Google+. I’d like to thank Luiz Guzman and Joerg Fliege for first bringing these items to my attention.

Lagarias’ paper is full of lots of yummy goodies, but my favorite part is his summary of Euler’s approach to research (see Section 2.6 of the paper or the end of Lipton’s post).

Taken directly from the paper, here is Lagarias’ summary of Euler’s research rules.

- Always attack a special problem. If possible solve the special problem in a way that leads to a general method.
- Read and digest every earlier attempts at a theory of the phenomenon in question.
- Let a key problem solved be a father to a key problem posed. The new problem finds its place on the structure provided by the solution of the old; its solution in turn will provide further structure.
- If two special problems solved seem cognate to each other, try to unite them in a general scheme. To do so, set aside the differences, and try to build a structure on the common features.
- Never rest content with an imperfect or incomplete argument. If you cannot complete and perfect it yourself, lay bare its flaws for others to see.
- Never abandon a problem you have solved. There are always better ways. Keep searching for them, for they lead to a fuller understanding. While broadening, deepen and simplify.

Lipton’s blog post also lists “Euler’s Research Rules.” My main motivation for reposting them here is to remind myself to follow them!

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MAT 220: Math Reasoning

MAT 411: Abstract Algebra

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