How was addition and multiplication of natural numbers defined before 1870 (Cantor and modern set theory)?

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).

• You may want to clarify what you mean by "define" (and try your best to avoid imposing today's standards on past mathematicians). Jan 6 at 21:50
• I don't know where to begin searching --- Google-book search words/phrases such as "arithmetic", "algebra", "calculus", "number system", etc. (continued) Jan 7 at 6:27
• date restricted to 19th century (or otherwise; also French and/or German equivalents, if you can read those languages). Look at the prefaces (or introductions, note to the reader, etc.) and table of contents to see if a book might include comments about addition and multiplication of natural numbers. After some experience in doing this, you'll begin to learn about some of the most significant authors and more useful words/phrases to search with (just replace 'arithmetic', 'algebra', etc. in the URL's for those other searches, (continued) Jan 7 at 6:28
• rather than entering into a google-search bar, unless you want to change the date range or something else). For two specific examples that might be useful, see the 3rd result (for me, at least) for the above "number systems" search, which is The Number Concept. Its Origin and Development by Levi Leonard Conant (1896), (continued) Jan 7 at 6:28
• and see [5] in my answer here (where you might also find other books of possible relevance) -- The Spirit of Mathematical Analysis, and Its Relation to a Logical System which is an 1843 English translation of an 1842 German book by Martin [Marcin, Martinus] Ohm. Jan 7 at 6:28