Euclid’s Elements present a somewhat confused presentation of the concept of number, proportion and magnitude from the modern perspective. Euclid lacks the sharp clarity of Grassmann, Dedekind, Peano and others who crafted the modern concepts of natural number, rational number and real number.

Euclid’s books 1, 2, 5, 7 and 10 are the most important to read for the concept of natural number, proportion and magnitude.

David Joyce, whose notes on Dedekind are so valuable, also has notes on Euclid that are valuable and give us a quick overview.

http://aleph0.clarku.edu/~djoyce/java/elements/toc.html

The shortcomings of Euclid’s concepts of natural number, proportion and magnitudes are discussed by Morris Kline.

Kline, Morris (b. 1908)

*Mathematical thought from ancient to modern times.* Oxford University PRess, New York, 1972.

According to Kline Euclid does not have the concept of adding two ratios of magnitudes.

http://www.math.hawaii.edu/~adolf/sl09usem.pdf

What concept of 0 and 1 did Euclid have? The following discusses arithmetic texts in Iceland since the 13th century.

http://www.comap.com/historyjournal/pdf/Vol2No2TheNumbersOneandZero.pdf

Didactical Phenomenology of Mathematical Structures

By Hans Freudenthal

Starting on page 75, the above has comments on Euclid.

Didactical Phenomenology of Mathematical Structures

By Hans Freudenthal

Number and its algebra: syllabus of lectures on the theory of number and its …

By Arthur Lefevre

See page 65 on Euclid.

Euclid probably never clearly unified his concepts of ratio and number.

In paragraph 85, he recommends teachers ponder their methods at this point. For Lefevre, in paragraph 84,

following Euclid it may be shown that there is a combination of proportions that satisfy the same laws of addition as primary numbers, or of fractions of concrete magnitudes.

Different authors may have different views of Euclid and some may be projecting our ideas onto a confused presentation.

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Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition.
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