  In this note we will present how Euler's investigations on various different
subjects lead to certain properties of the Legendre polynomials. More
precisely, we will show that the generating function and the difference
equation for the Legendre polynomials was already written down by Euler in at
least two different papers. Furthermore, we will demonstrate that some familiar
expressions for the Legendre polynomials are corollaries of the
before-mentioned works. Finally, we will show that Euler's ideas on continued
fractions lead to an integral representation for the Legendre polynomials that
seems to be less generally known.
