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mathwonk

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One of the most rewarding mathematical experiences I had in my career was finally teaching geometry from Euclid, in my 60's, guided in that endeavor by the excellent book of Hartshorne: Geometry, Euclid and beyond. I found that the reputation of Euclid for being unrigorous is highly overstated, and it in fact provides the background for the basic modern theories of mathematics, i.e. both algebra and analysis, in elementary geometric terms.

There are two deep concepts that are in fact equivalent, something I had not before realized, namely area and proportion, or similarity. Euclid begins with a fairly careful development of area, culminating in the Pythagorean theorem, and derives the concept of similarity from it, in a way that gives a glimpse of the Dedekind definition of real numbers. He then shows how conversely one could derive the Pythagorean theorem from the principle of similarity.

The opposite development, postulating similarity and deriving the theory of area, is nowadays done in the Birkhoff approach to geometry. This backwards derivation is "easier" but at the cost of assuming a much more sophisticated concept as opposed to an elementary one in my opinion, and hence very unnatural. Today's math student may prefer it since we are steeped in the theory of real numbers and proportion from early on. But to understand the origin of the idea of real numbers I greatly recommend studying Euclid.

He also presages the theory of algebraic operations by means of elementary geometry, giving simple geometric interpretations of the laws of multiplication and its properties, distributivity for instance. He even shows how to solve quadratic equations geometrically. (To see this interpretation one only has to consider the product of two segments as the rectangle with them as sides.) After relating the study of circles to the study of triangles, the construction of a regular pentagon is a tour de force in Euclid.

To approach Euclid I highly recommend beginning with Hartshorne, as I had avoided Euclid all my life because of being put off by the beginning, the vague definitions for example. Once I got into the theorems I began to see the beauty and clarity of it. I think micromass had a similarly positive experience with Euclid, according to his comments here.

After teaching from Hartshorne and Euclid to college students, I had the privilege of teaching a shorter version of the same course to a brilliant group of 10 year olds, who also enjoyed it, and it helped me to really grasp the material while trying to make it completely accessible to bright but naive students. In fact, asked to introduce similarity but without time to reach the fundamental result, Proposition VI.2, I realized the same idea was already contained in Prop. III.35 and I presented it that way. At the end of the course I understood how Archimedes had refined the treatment of area to give a foundation to the ideas of integral calculus, for example he clearly knew the so called Cavalieri principle, and used it to compute the volume of a ball.

I.e. while the fundamental theorem of calculus allows us to calculate area and volume formulas from height and area formulas, the more basic Cavalieri principle allows one to

I ended the course for the 10 year olds with this result, as well as the field theoretic sketch of the impossibility of certain ruler and compass constructions, and the calculation of the volume of a "bicylinder", a result missing from the partially recovered "palimpsest" of Archimedes, but which can be reconstructed from his ideas. The volume of a bicylinder is an example of a calculation which is often posed as a challenge problem in calc books, but which is as easy by Archimedes' method as that of a 3-ball. One can even use his methods to calculate the volume of a 4-ball and hence also its "surface area" i.e. that of the 3 dimensional sphere S^3, and I added this result in an afterword to the course. my notes from this course are here:

http://alpha.math.uga.edu/~roy/camp2011/10.pdf

There are two deep concepts that are in fact equivalent, something I had not before realized, namely area and proportion, or similarity. Euclid begins with a fairly careful development of area, culminating in the Pythagorean theorem, and derives the concept of similarity from it, in a way that gives a glimpse of the Dedekind definition of real numbers. He then shows how conversely one could derive the Pythagorean theorem from the principle of similarity.

The opposite development, postulating similarity and deriving the theory of area, is nowadays done in the Birkhoff approach to geometry. This backwards derivation is "easier" but at the cost of assuming a much more sophisticated concept as opposed to an elementary one in my opinion, and hence very unnatural. Today's math student may prefer it since we are steeped in the theory of real numbers and proportion from early on. But to understand the origin of the idea of real numbers I greatly recommend studying Euclid.

He also presages the theory of algebraic operations by means of elementary geometry, giving simple geometric interpretations of the laws of multiplication and its properties, distributivity for instance. He even shows how to solve quadratic equations geometrically. (To see this interpretation one only has to consider the product of two segments as the rectangle with them as sides.) After relating the study of circles to the study of triangles, the construction of a regular pentagon is a tour de force in Euclid.

To approach Euclid I highly recommend beginning with Hartshorne, as I had avoided Euclid all my life because of being put off by the beginning, the vague definitions for example. Once I got into the theorems I began to see the beauty and clarity of it. I think micromass had a similarly positive experience with Euclid, according to his comments here.

After teaching from Hartshorne and Euclid to college students, I had the privilege of teaching a shorter version of the same course to a brilliant group of 10 year olds, who also enjoyed it, and it helped me to really grasp the material while trying to make it completely accessible to bright but naive students. In fact, asked to introduce similarity but without time to reach the fundamental result, Proposition VI.2, I realized the same idea was already contained in Prop. III.35 and I presented it that way. At the end of the course I understood how Archimedes had refined the treatment of area to give a foundation to the ideas of integral calculus, for example he clearly knew the so called Cavalieri principle, and used it to compute the volume of a ball.

I.e. while the fundamental theorem of calculus allows us to calculate area and volume formulas from height and area formulas, the more basic Cavalieri principle allows one to

*compare*areas and volumes of different figures, and hence to deduce new such formulas from old. I.e. although Archimedes could not go directly from a formula for the slice areas to a volume formula, he knew that two figures with the*same*slice areas have the*same*volume. So even without the FTC, Archimedes deduced the volume of a 3-ball by comparing it to the difference of the volumes of a circular cylinder and a cone, which he deduced in turn from comparisons with cubes and polygonal cylinders.I ended the course for the 10 year olds with this result, as well as the field theoretic sketch of the impossibility of certain ruler and compass constructions, and the calculation of the volume of a "bicylinder", a result missing from the partially recovered "palimpsest" of Archimedes, but which can be reconstructed from his ideas. The volume of a bicylinder is an example of a calculation which is often posed as a challenge problem in calc books, but which is as easy by Archimedes' method as that of a 3-ball. One can even use his methods to calculate the volume of a 4-ball and hence also its "surface area" i.e. that of the 3 dimensional sphere S^3, and I added this result in an afterword to the course. my notes from this course are here:

http://alpha.math.uga.edu/~roy/camp2011/10.pdf

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