Pons asinorum

I recently discovered a draft of a blog post I started to write nearly a year ago, shortly after my previous post on the subject of Euclidean geometry.  I’m not sure why I didn’t publish it at the time (possibly because I was planning to extend it in some now-forgotten direction).  Here it is now, with minimal editing:

Undoubtedly the most famous proposition in Euclid’s Elements is I-47 (that is, the 47th proposition in book I), better known as Pythagoras’ Theorem.  This, as you probably know (though you may not have known its number in Euclid), is the statement that the square of the hypotenuse of a right-angled triangle is the sum of the squares on the other two sides.

Perhaps the second most famous one, although certainly having nothing like the same level of recognition among non-mathematicians, is I-5, which states that the two angles at the base of an isosceles triangle (i.e. one having two sides of the same length)  are equal.  Like Pythagoras’ theorem and unlike most of Euclid’s other propositions, this one has a name.  In fact, it has been known by several names, but the most popular is pons asinorum – the bridge of asses (in Latin).

There are at least two competing explanations as to the origin of this name.   One is that this is the first proposition in the Elements that might cause some people difficulty in understanding, as well as being quite necessary for the proof of many of the later propositions, so that it functions as a kind of figurative bridge that ignorant people (considered to be donkeys) are unable to cross to allow them further into the study of geometry.  Another, rather more complimentary to donkeys, is that the diagram of the construction given in Euclid’s proof resembles a steep-sided bridge that horses would have difficulty climbing but sure-footed donkeys would have no trouble with.

Interestingly, while pons asinorum is used in the English-speaking mathematical world to refer to Euclid I-5, it is apparently used in the French-speaking world to refer to I-47, i.e. Pythagoras’ Theorem.

Elementary, my dear

About 11 years ago, while I was studying for my PhD in abstract algebra (and, coincidentally, happened to have a friend who worked at a bookshop who was able to get me occasional books at a discount rate), I decided to treat myself to a copy of Euclid’s Elements, one of the classic texts of mathematics (with no discernible connection whatever to my official research topic).  This is a collection of 13 books covering not just plane and solid geometry (Euclidean, of course!) but also quite a bit of elementary number theory (although treated in a fairly geometrical way).

I went for the 3 volume Dover edition of Heath’s annotated translation of the complete 13 books, originally dating from about 1925.  After ploughing through the entire 150-page introduction, I must confess I didn’t get very far through my planned systematic study of the books themselves; in fact, the bookmark I found in there the other day indicates I got about as far as proposition 8 in book 1 (there being 48 propositions in that book alone)!

Recently I have been working through several of my old maths books, and scaring myself with the realisation of quite how much I’ve forgotten.   Euclid is the latest one to come down from the shelf and I’ve been enjoying working through some of the proofs in book 1 and glancing through some of the later books.  I’m not yet sure whether I’ll make another attempt at systematically working through the whole lot, though I expect I probably won’t get round to it any time soon.