# Where did Euler prove 'his' theorem on homogeneous functions?

Where in Eulers writings can I find a proof of his homogeneous function theorem: $$y$$ is a homogeneous function of degree $$k$$ in $$x_1,\ldots,x_n$$ iff $$ky = \sum_{i=1}^n x_i\frac{\partial y}{\partial x_i}$$? None of the sources I've seen cite him.

I'm curious because in his Introduction to the analysis of the infinite he defines a homogeneous function as one "in which each term has the same degree" and goes on to discuss several examples (not just polynomials). But I don't see him writing down the "modern" definition $$f(\lambda{\bf x})=\lambda^k f({\bf x})$$ and wonder how he would prove the result without this. (I know he used function notation, but very rarely.)

• Thanks. So it starts in par. 220 and indeed he does not use $f(x)$ notation! – Michael Bächtold Jan 22 at 8:56
• Though as far as I can tell from that book he only proves the direction $\implies$. – Michael Bächtold Jan 22 at 10:10