Fact About Equality Proven by Euler in 1748 (Context: Integer Partitions)

Question

I'm currently reading Integer Partitions by Andrews and Eriksson. In the introductory chapter (p.2), there is the following statement:

... The table would have a more efficient design:
\begin{array}{r|r} 1 + 1& 2\\ 1 + 1 +1 & 3\\ 3 & 2+1\\ 1+1+1+1&4 \\ 3+1 & 3+1 \end{array}

The fact that there will always be as many items in the left column as in the right one was first proved by Leonhard Euler in 1748.

Which proof by Euler are they talking about?

Answer 1

The number of partitions (of $n$) into odd parts equals the number of partitions into distinct parts.

Answer 2

The proof they are referring to is that the number of partitions with odd parts is equal to the number of partitions into distinct parts