Not everyone did Euclidean geometry at school. A friend of mine who went to a secondary modern school and left at 15 only learned it later, but when he did he recalls that "I was bowled over by the fact that you could prove things, clearly and beyond doubt." Had he passed the eleven-plus and gone to grammar school, he would certainly have taken a serious geometry course, but now you can go to school until you are 18, take A-levels and go to university without ever doing such a course.

What has replaced it? A bit of this and a bit of that, but the trouble with bits and pieces is that they don't always hang together well, and things get learned by rote, and applied with calculator in hand.

This is rather strange because we are happy to inveigh against rote learning in third-world countries. We think people should learn to think for themselves, and find it regrettable when school fails to inculcate reasoned and rational argument.

What we need is rational thinking in terms of abstract concepts, and this is easiest when using abstractions we all understand, such as points and lines, rather than "sets" that often confuse students. I've known students at university being confused about the difference between the empty set, and the set consisting of zero, and even more abstractly the set consisting of the empty set, which is different again. Points and lines, on the other hand, seem real enough to most people, though they are in fact abstractions from reality - after all, lines in the material world have a certain thickness, and points a certain size. But as abstractions they are not hard to understand, and the arguments are logical, and visual, which helps. Moreover, Euclidean geometry, and more generally Greek rationalism, has a glorious history. It inspired early Islam, where scholars in Baghdad translated Euclid and other Greek authors into Arabic. These Arabic works were later translated into Latin and inspired new learning in medieval Europe. Later, during the Renaissance, Greek manuscripts were translated directly into Latin, and into the living languages of Europe. This allowed Euclid's work to become the basis for teaching geometry, and learning geometry became synonymous with reading Euclid.