Below is one of the proofs from the first maths book ever, entitled “The Elements”, and written by a mathematician (or group of mathematicians: we don’t know) named Euclid, in Alexandria, Egypt, about 300 BC.
For centuries “The Elements” was also the basic textbook for everyone who got to study maths beyond everyday arithmetic. Most people didn’t get enough years of school to do that, but for those who did, maths was Euclid.
In the 20th century, Euclid was replaced by newly-written textbooks, but those textbooks considered themselves obliged to cover most of Euclid’s arguments, and more or less in Euclid’s terms. Below the excerpt from Euclid is an excerpt from C V Durell’s A new geometry for schools, first published in 1939 and reprinted again and again at least until 1970, as a textbook for the School Certificate and then O level (rough equivalent to GCSE). This is how I was taught maths when at school.
Durell uses Euclid’s proof of Pythagoras’s theorem, although there are hundreds of other proofs, and many generally reckoned to be more vivid (click here for three).
Durell sets out the proof in a way easier to read than Euclid, with each line of reasoning under the previous line, and generous spacing between lines. In fact, Durell’s setting-out is a model of clarity which newer textbooks would do well to copy, and which working mathematicians use as a model when they write by hand.
That is not just because Durell was good at it, though he was; for most of the centuries since Euclid, paper had been very expensive, and squashing up your writing into the minimum space was an economic obligation.
You notice that Durell has proofs (a lot of them: there is a section in the book instructing students on how to write formal proofs), and that words (chosen, concise words) play an indispensable part in those proofs. He does not give long chains of algebraic manipulation with no words (not even in his Algebra textbook – see excerpt below). (Click here for a discussion of how proof dropped out of school maths).