The tears - or, more accurately the four-hour sessions of homework, ending in complete incomprehension and occasional migraine - were mine. This was maths, and therefore important, so I had to do it and couldn't. The late Bill Shankly said that football was not a matter of life and death, but something much more important. So was maths. The cleverest pupils could do in ten minutes what I couldn't do in four hours. Across Europe, I'm quite sure that millions of people are in the same position. Hundreds of thousands of FE students in the UK are set up to fail maths yet again, with no hope of passing, and in a state of despair or passive revolt.
Euclid appears in illustrations as an old man with downcast eyes, sparse hair and a straggly beard. His work is available, free, here, with explanations from Professor David E Joyce of Clark University, Worcester, Massachussetts. I am extremely grateful to Professor Joyce for enabling me to lay the ghost of a man I saw as my tormentor. Professor Joyce describes Euclid's elements as "beautiful and influential". For most of those who approach it, it does not have these qualities, because, with the best of intentions, it is mistaught.
To begin at the beginning, Euclid used a plain straight edge, compasses of some sort, and some means of writing, probably akin to a pencil. He did not own a protractor, or a marked ruler. These items are universal in secondary schools, and were my undoing. They encourage people to approach problems by measuring them, and this runs counter to Euclid's system of logic - you can't be sure that your measurement is completely accurate. You can, on the other hand, be sure of the logic that stems from Euclid's simple axioms - for Professor Joyce, Common Notions - and postulates. The only angle measurement in Euclid's work is the right angle - others are described in relation to it. The key to his logic is is that if two items are equal to a third item, they are equal to each other. Mathematicians have pointed to the limitations of Euclid's logic, as these suppositions are themselves not proved. For example, Euclid says that all right angles are equal to each other, but does not set out to prove it, perhaps because it is not possible to prove it - he would have had no way of proving that right angles drawn on two pieces of papyrus, were in fact equal to each other.
Proposition 1 shows the method to perfection, and is easy to teach. Draw a straight line, not too long for your piece of paper, but do not measure it. Set your compasses to the length of the line, and draw a circle centred on one end of it. Do the same from the other end. Both of your circles are equal to each other, because they have the same radius - the distance from the centre of a circle to its edge is always the same, or it isn't a circle. So, joining the two ends of the line, to the point at which the circles intersect, creates a triangle in which all sides have this radius, and are therefore equal. It works every time, and is a perfect introduction to Euclid's system of logic. The rest follows.
Professor Joyce has performed a most important service, in making this work freely available to those of us who have needed it - pro bono publico, of the first order.
Conclusion: Protractors and marked rulers should not be used in the teaching of Euclidian geometry. They are a worse distraction than mobile phones. A phone, on the other hand, can provide an unmarked straight edge...