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Euclid’s first task (Book 1, Proposition 1) was to show that, given any line, he could make an equilateral triangle, i.e. a triangle with three equal sides, with that line as one side:

Step 1: Put the compass point on one end of the given line and draw a circle that passes through the other end of the line.

Step 2: Repeat the first step with the compass on the other end of the line. You now have two intersecting circles.

Step 3: Draw a line from one of the intersections of the circle to the end points of the original line.

The Elements , Proposition 1.

He then meticulously progressed from proposition to proposition, revealing a host of properties of lines, triangles and circles. For example, Proposition 9 shows how to ‘bisect’ an angle, that is construct an angle that is exactly half of a given angle. Proposition 32 states that the internal angles of a triangle always add up to two right angles, or 180 degrees. The Elements is a magnum opus of pedantry and rigour. Nothing is ever assumed. Every line follows logically from the line before. Yet from only a few basic axioms, Euclid amassed an impressive body of compelling results.

The grand finale of the first book is Proposition 47. The commentary from a 1570 edition of the first English translation reads: ‘This most excellent and notable Theoreme was first invented of the greate philosopher Pithagoras, who for the exceeding ioy conceived of the invention thereof, offered in sacrifice an Oxe, as recorde Hierone, Proclus, Lycius, & Vitruvius. And it hath bene commonly called of barbarous writers of the latter time Dulcarnon.’ Dulcarnon means two-horned, or ‘at wit’s end’ – possibly because the diagram of the proof has two horn-like squares, or possibly because understanding it is indeed horribly difficult.

There is nothing pretty about Euclid’s proof of Pythagoras’s Theorem. It is long, meticulous and convoluted, and requires a diagram full of lines and superimposed triangles. Arthur Schopenhauer, the nineteenth-century German philosopher, said it was so unnecessarily complicated that it was a ‘brilliant piece of perversity’. To be fair to Euclid, he was not trying to be playful (as was Dudeney), or aesthetic (as was Annairizi) or intuitive (as was Baravalle). Euclid’s driving concern was the of his deductive system.

While Pythagoras saw wonder in numbers, Euclid in The Elements reveals a deeper beauty, a watertight system of mathematical truths. On page after page he demonstrates that mathematical knowledge is of a different order than any other. The propositions of The Elements are true in perpetuity. They do not become less certain, or indeed less relevant with time (which is why Euclid is still taught at school and why Greek playwrights, poets and historians are not). The Euclidean method is awe-inspiring. The seventeenth-century English polymath Thomas Hobbes is said to have glanced at a copy of The Elements that lay open in a library when he was a 40-year-old man. He read one of the propositions and exclaimed: ‘By God, this is impossible!’ So, he read the previous proposition, and then the one before that, and so on, until he was convinced it all made sense. In the process, he fell in love with geometry for the certainty it prescribed, and the deductive approach influenced his most famous works of political philosophy. Since The Elements, logical reasoning has been the gold standard of all human enquiry.

Euclid started off by carving up two-dimensional space into the family of shapes known as polygons, which are those shapes made from only straight lines. With his compass and straightedge he was able to construct not just an equilateral triangle, but also a square, a pentagon and a hexagon. Polygons for which every side has the same length and the angles between the sides are all equal are called regular polygons. Interestingly, Euclid’s method, however, is not effective for all of them. The heptagon (seven sides), for example, cannot be constructed with a compass and straightedge.