Euler's errors?

Solution 1:

Euler apparently had some trouble deriving the Jacobian used in change of variables for double integrals.

He began by considering congruent transformations consisting of (affine) linear functions, and got something like $$\mathrm{d}x\,\mathrm{d}y=m\sqrt{1-m^2}\,\mathrm{d}t^2+(1-2m^2)\,\mathrm{d}t\,\mathrm{d}v-m\sqrt{1-m^2}\,\mathrm{d}v^2$$ which he described as "obviously wrong and even meaningless." He then got

$$\mathrm{d}x\,\mathrm{d}y=\left(\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}\right)\,\mathrm{d}t\,\mathrm{d}v$$ which was not symmetric in the variables, and therefore would not do. Finally, he derived the correct

$$\mathrm{d}x\,\mathrm{d}y=\left|\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}-\frac{\partial y}{\partial t}\frac{\partial x}{\partial v}\right|\,\mathrm{d}t\,\mathrm{d}v$$ and lamented that simply multiplying out $$\mathrm{d}x\,\mathrm{d}y=\left(\frac{\partial x}{\partial t}\,\mathrm{d}t+\frac{\partial x}{\partial v}\,\mathrm{d}v\right)\left(\frac{\partial y}{\partial t}\,\mathrm{d}t+\frac{\partial y}{\partial v}\,\mathrm{d}v\right)=\left|\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}+\frac{\partial y}{\partial t}\frac{\partial x}{\partial v}\right|\,\mathrm{d}t\,\mathrm{d}v$$ and shredding the squared differentials yielded an incorrect but annoyingly close answer.

But let us remember, if Euler committed errors it was only because of the unrivaled breadth of his work. If I could finish with a quote from the article cited below: "As a developer of algorithms to solve problems of various sorts, Euler has never been surpassed."

Source: For an excellent review of the history of the Jacobian, and to learn more about the details of what I have written, I highly recommend reading this article by Prof. Victor J. Katz (Internet Archive, jstor.

Solution 2:

Euler conjectured that for $n=2\pmod 4$ there are no mutually orthogonal Latin squares of size $n\times n$. Bose and Shrikande disproved it by construction and earned the name Euler's Spoilers. See http://en.wikipedia.org/wiki/Graeco-Latin_square