## Euclid Propotions in Math foundations Videos N. J. Wildberger

Euclid Proportions in Math foundations Videos N. J. Wildberger

MathFoundations22: Difficulties with Euclid

http://www-history.mcs.st-and.ac.uk/Extras/Russell_Euclid.html

Ratio and Proportion in Euclid
Louisiana State University

http://www.lamath.org/journal/vol5no2/Ratio_Proportion.pdf

Further reading. Useful commentaries on Euclid but to address issues that are
associated with teaching include Alexander John Ellis, “Euclid’s conception of ratio
and proportion,” in Algebra Identified with Geometry, London: C. F. Hodgson & Sons
1874, and Augustus De Morgan, The Connection of Number and Magnitude: An
Attempt to Explain the Fifth Book of Euclid. London: Taylor and Walton 1836.
Reprints of these books are available on line.

Alexander John Ellis, “Euclid’s conception of ratio
and proportion,” in Algebra Identified with Geometry, London: C. F. Hodgson & Sons
1874,

http://quod.lib.umich.edu/u/umhistmath/ACV0529.0001.001?view=toc

Augustus De  Morgan

http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=De%20Morgan%2C%20Augustus%2C%201806-1871

Note that the prior blog articles on Euclid have been edited to take out some overly dogmatic statements about what is or is not in Euclid.   It is better for me to quote others who are more knowledgeable and stick closer to their statements.  The overall argument though I still believe is sound.  In particular, that

1) Euclid is overly complex and difficult to follow on ratios, fractions and proportions however we interpret him.

2) Mathematicians developed a new approach to number in the 19th century for natural numbers, rationals and reads.

3) School teaching still has elements of Euclid’s way of thinking.

4) School teaching does not properly show these as precursors leading to better 19th century ideas of number especially of rational number that overcame prior problems or limitations.

5) Current school teaching avoiding 19th century number notions leaves students with a muddled concept of fractions that has something in common with the confusions in Euclid.

6) This confused concept of fractions can only be fixed by explicitly teaching the 19th century concepts of natural number and then rationals as pairs of naturals with rules and procedures crafted to give desired properties, overcome limitations, and still provide for the utility of fractions in everyday uses that extend back even before Euclid.