here is my review of euclid from amazon:
This work of Euclid, made highly accessible in this edition is indeed incredible. How many people know that Euclid had a thorough grasp of the concept of a tangent line to a circle comparable to that of Newton? Certainly not me. I am currently teaching from this book and reading it in detail, to my great pleasure. In Proposition III.16, Euclid essentially shows the following 4 properties are equivalent for a line L meeting a circle C at a point P:
1) L meets C nowhere else, 2) L does not "cut" C, i.e. L contains no points interior to the circle, 3) L is perpendicular to the radius of C through P, 4) L makes angle zero with the circle C at P, i.e. the greatest lower bound of the angles between L and all secants of C through P is zero,
and his proof also shows: 5) the line L is the limit of secants of C through P as the second point of intersection approaches P, i.e. for every angle containing L as a side, there is a neighborhood of P such that all secants of C through P make with L a smaller angle than the given one.
The first property, that a tangent line should meet a circle only once, is the one I was taught in high school, and it is inadequate for discussing tangents to curves more complicated than globally convex ones. The property of not cutting the curve, or of not crossing it, is adequate for describing tangents to all locally convex curves, and the property of making an angle of zero, or of being a limit of secants is of course Newton's most sophisticated definition suited for describing tangents to all curves. I am now not surprised to learn that Newton read Euclid before producing his own work. I had no idea this was in Euclid, and I have had the same eye opening experience in several other places as I read this amazing book.
Thanks to Green Lion for this lovely edition, and to Robin Hartshorne for writing his book in such a way as to force me to read Euclid.
While teaching the first 4 books of this work this summer to brilliant 8-10 year olds I noticed that proposition 35 or 36 in book III already contains the fundamental theorem of similarity, and one can finesse the beautiful but complicated presentation in book 5 i believe. Moreover using the prop in book 3 gives similarity for non archimedean geometries as well, unlike the version in the later book. the point is that the theories of area and similarity are equivalent and the fundamental result for all of them is pythagoras' theorem.
