pumpkinstrokemarrow

The start point was a very short proof of the famous theorem of Pythagoras illustrated. The start square is ABC from which one constructs the large square. One then observes that CBED is also a square and the theorem emerges.

I objected that this depended on knowing all about the areas of triangles and squares, and that the area of a whole was the sum of the areas of the parts. These being matters which I did not think that Euclid knew about. How do you get from pencils, straight edges and a pair of compasses to area?

Eventually I get around to asking Google who takes me to a long list of proofs of the theorem, one of which avoided the use of the concept of area but did require you to know the rule about the ratios of the sides of similar triangles. This also being a matter which I did not think that Euclid knew about. Try Google again and this time find my way to the helpful site at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html where I am able to inspect Euclid's proof of the theorem.

Which is a bit longer than that illustrated above but only depends on the ideas that the area of a triangle is half that of a suitable parallelogram constructed on one of its sides, that the areas of all parallelograms constructed on a given side between given parallels are the same and that the area of an oblong is the product of the lengths of two adjacent sides. Then there is a catch in that the demonstration of the second of these requires the additional idea that the area of a whole is the sum of the area of the parts - which is obviously true in the simple case of an oblong chopped into two smaller oblongs but which is not so obviously true otherwise.

So I have still to find a proof of the theorem which does not appeal to notions about area or require limit arguments. This last being necessary, I think, to prove the ratio rule of similar triangles.

All of which made me regret that my undergraduate course on mathematical logic was not illustrated by recourse to Euclid. I think it would have made a good teaching vehicle. As things were, I do not think I did any geometry of that sort once I had acquired my O-level ticket at the tender age of 15 or so. On the other hand all this has prompted me to make use of the parallel rules which I have sentimentally retained but not used since about the same time.

PS: first brussel sprouts of the season yesterday. Not brilliant but OK.