Alex's Adventures in Numberland - Alex Bellos [38]
Regiomontanus’s statue and rugby problem.
Perhaps the most stunning result in Euclidean geometry, though, is one that reveals an astonishing property about triangles. Let’s first consider where the centre of a triangle is. This is a surprisingly unclear issue. In fact, there are four ways we can define the centre of a triangle and they all represent different points (unless the triangle is equilateral, when they all coincide). The first is called the orthocentre, and is the intersection of the lines from each vertex that meet the opposing sides perpendicularly, which are called the altitudes. It is already pretty amazing to think that, for any triangle, the altitudes always meet at the same point.
Construction of the Euler line.
The second is the circumcentre, which is the intersection of the perpendiculars drawn from the halfway points of the sides. Again, it is very neat that these lines will also always meet, whatever triangle you choose.
The third is the centroid, which is the intersection of the lines that go from the vertex to the midpoints of opposing lines. They always meet too. Finally, the midcircle is a circle that passes through the midpoints of each side and also the intersections of the sides and the altitudes. Every triangle has a midcircle, and its centre is the fourth type of middle point a triangle can have.
In 1767 Leonhard Euler proved that for all triangles the orthocentre, the circumcentre, the centroid and the centre of the midcircle are always on the same line. This is mind-blowing. Whatever the shape of a triangle these four points have a dazzlingly uniform relation to each other. The harmony is truly wondrous. Pythagoras would have been most awed.
Though this may be hard to fathom these days, The Elements was a literary sensation. Until the twentieth century, it is said to have had more editions printed than any other book except the Bible. This was all the more remarkable since The Elements is no easy read. One edition, however, is worth mentioning for its unorthodox approach in making the text accessible. Oliver Byrne, whose day job was Surveyor of Her Majesty’s Settlements in the Falkland Islands, rewrote Euclid in colour. Instead of the long proofs, he drew illustrations in which angles, lines and areas were marked in geometrical blocks of red, yellow, blue or black. His Elements…in which coloured diagrams and symbols are used instead of letters for the greater ease of learning was published in 1847 and has been described as ‘one of the oddest and most beautiful books of the whole nineteenth century’. In 1851 it was one of the few British books on display at the Great Exhibition, though the public failed to see the excitement. Indeed, Byrne’s publishers went bust in 1853, with more than 75 percent of stock of The Elements unsold. Its high production costs had contributed to the bankruptcy.
Byrne’s illustrated proofs did make Euclid more intuitive, predating colour-coordinated textbooks of recent years. Aesthetically it was also ahead of its time. The gaudy primary colours, asymmetrical layout, angularity, abstract shapes and plentiful empty space anticipated the paintings of many twentieth-century artists. Byrne’s book looks like a tribute to Piet Mondrian, published 25 years before Mondrian was even born.
As masterful as the Euclidean method was, it could not solve all problems; some quite simple ones, in fact, are unsolvable using just a compass and ruler. The Greeks suffered for this. In 430 bc Athens was struck by a plague of typhoid. Its citizens consulted the oracle at Delos, who advised them to double the size of Apollo’s altar, which was cube-shaped. Relieved that such an apparey easy task would