The Foundations of Arithmetic in Euclid
Call it an essay, a monograph, or a book, it is 142 pages, size A5, 11-point type, text occupying (9/12)2 of each page, dated January 12, 2016:
The Introduction includes a summary of each of section of the book. Briefly, the four chapters concern:
- The philosophy of history, as developed by R. G. Collingwood in several books, as it pertains to the reading of Euclid.
- Some general findings about Euclid, mostly obtained in the process of leading a course in which students read Book I of the Elements.
- More specific findings about Euclid's number theory, as for example that he does prove rigorously the commutativity of multiplication in any ordered ring whose positive elements are well ordered.
- These findings, expressed in more modern terms, along with (in the first section) an argument that the last proposition of Book VII of the Elements is a later addition, precisely because it is too “modern.”
This version is the same as the above, but with less text on each page, because the box of text includes a header: 153 pages, size A5, 11-point type, text-with-header occupying 9/16 of each page, dated July 13, 2015:
Originally the first section of the fourth chapter was in the third chapter, and there was no fourth chapter: this draft was “On the Foundations of Arithmetic in Euclid” (April 17, 2015; 98 pages of size A5):
After preparing that draft, I wrote out the details of the proof of the commutativity of multiplication in another (draft) paper, “Commutativity of Multiplication in Euclid's Arithmetic” (May 5, 2015; 19 pages of size A5):
This shorter paper works out an argument that Euclid proves commutativity of ordinal multiplication in any well-ordered set that is closed under ordinal addition and ordinal multiplication, provided this addition is commutative.
Here is an earlier draft, of January 12, 2015; 97 pages of size A5: