I know how to define addition and multiplication of natural numbers using set theory, but I think that before Cantor, mathematicians did not try to use set theory as a foundation for mathematics (I could be wrong). How did mathematicians define addition and multiplication of natural numbers before Cantor and Peano?
I chose mathematics before 1870 as an arbitrary cutoff, after Cauchy’s death and before Cantor and “modern” set theory, since rigorous mathematics started with Cauchy (I think). This era should have some somewhat rigorous definitions of addition and multiplication.
I also want to know how addition and multiplication were defined in Newton’s or ancient times (if there exist a definition).